ACI 445.1R-12 - Report on Torsion in Structural Concrete (2013)

ACI 445.1R-12

Report on Torsion in Structural Concrete Reported by Joint ACI-ASCE Committee 445

First Printing April 2013 ®

America Ameri can n Con Concr crete ete Ins Insti titu tute te Advanc Adv ancing ing con concre crete te kno knowle wledge dge

Report on Torsion in Structural Concrete Copyright by the American Concrete Institute, Farmington Hills, MI. All rights reserved. This material may not be reproduced or copied, in whole or part, in any printed, mechanical, electronic, film, or other distribution and storage media, without the written consent of ACI. The technical committees responsible responsibl e for ACI committee reports and standards strive to avoid ambiguities, omissions, and errors in these documents. In spite of these efforts, the users of ACI documents occasionally find information or requirements that may be subject to more than one interpretation or may be incomplete or incorrect. Users who have suggestions for the improvement of ACI documents are requested to contact ACI via the errata website at www.concrete.org/committees/errata.asp.. Proper use of this document includes periodically checking for errata for www.concrete.org/committees/errata.asp the most up-to-date revisions. ACI committee documents are intended for the use of individuals who are competent to evaluate the significance and limitations of its content and recommendations and who will accept responsibility for the application of the material it contains. Individuals who use this publication i n any way assume all risk and accept total responsibility for the application and use of this information. All information in this publication is provided “as is” without warranty of any kind, either express or implied, including but not limited to, the implied warranties of merchantability, fitness for a particular purpose or non-infringement. ACI and its members disclaim liability for damages of any kind, including any special, indirect, incidental, or consequential damages, including without limitation, l ost revenues or lost profits, which may result from the use of this publication. It is the responsibility of the user of this document to establish health and safety practices appropriate to the specific circumstances involved with its use. ACI does not make any representations with regard to health and safety issues and the use of this document. The user must determine the applicability of all regulatory limitations before applying the document and must comply with all applicable laws and regulations, including but not limited to, United States Occupational Safety and Health Administration (OSHA ) health and safety standards. Participation by governmental representatives in the work of the American Concrete Institute and in the development of Institute standards does not constitute governmental endorsement of ACI or the standards that it develops. Order information: ACI documents are available in print, by download, on CD-ROM, through electronic subscription, or reprint and may be obtained by contacting ACI. Most ACI standards and committee reports are gathered together in the annually revised ACI Manual of Concrete Practice (MCP). American Concrete Institute 38800 Country Club Drive Farmington Hills, MI 48331 U.S.A.

Phone: 248-848-3700 Fax: 248-848-3701 www.concrete.org ISBN-13: 978-0-87031-810-8 ISBN: 0-87031-810-1

First Printing April 2013 ®

America Ameri can n Con Concr crete ete Ins Insti titu tute te Advanc Adv ancing ing con concre crete te kno knowle wledge dge

Report on Torsion in Structural Concrete Copyright by the American Concrete Institute, Farmington Hills, MI. All rights reserved. This material may not be reproduced or copied, in whole or part, in any printed, mechanical, electronic, film, or other distribution and storage media, without the written consent of ACI. The technical committees responsible responsibl e for ACI committee reports and standards strive to avoid ambiguities, omissions, and errors in these documents. In spite of these efforts, the users of ACI documents occasionally find information or requirements that may be subject to more than one interpretation or may be incomplete or incorrect. Users who have suggestions for the improvement of ACI documents are requested to contact ACI via the errata website at www.concrete.org/committees/errata.asp.. Proper use of this document includes periodically checking for errata for www.concrete.org/committees/errata.asp the most up-to-date revisions. ACI committee documents are intended for the use of individuals who are competent to evaluate the significance and limitations of its content and recommendations and who will accept responsibility for the application of the material it contains. Individuals who use this publication i n any way assume all risk and accept total responsibility for the application and use of this information. All information in this publication is provided “as is” without warranty of any kind, either express or implied, including but not limited to, the implied warranties of merchantability, fitness for a particular purpose or non-infringement. ACI and its members disclaim liability for damages of any kind, including any special, indirect, incidental, or consequential damages, including without limitation, l ost revenues or lost profits, which may result from the use of this publication. It is the responsibility of the user of this document to establish health and safety practices appropriate to the specific circumstances involved with its use. ACI does not make any representations with regard to health and safety issues and the use of this document. The user must determine the applicability of all regulatory limitations before applying the document and must comply with all applicable laws and regulations, including but not limited to, United States Occupational Safety and Health Administration (OSHA ) health and safety standards. Participation by governmental representatives in the work of the American Concrete Institute and in the development of Institute standards does not constitute governmental endorsement of ACI or the standards that it develops. Order information: ACI documents are available in print, by download, on CD-ROM, through electronic subscription, or reprint and may be obtained by contacting ACI. Most ACI standards and committee reports are gathered together in the annually revised ACI Manual of Concrete Practice (MCP). American Concrete Institute 38800 Country Club Drive Farmington Hills, MI 48331 U.S.A.

Phone: 248-848-3700 Fax: 248-848-3701 www.concrete.org ISBN-13: 978-0-87031-810-8 ISBN: 0-87031-810-1

ACI 445.1R-12 Report on Torsion in Structural Concrete Reported by Joint ACI-ASCE Committee 445 Daniel A. Kuchma, Chair

Perry Adebar Neal S. Anderson Robert B. Anderson Mark A. Ascheim Oguzhan Bayrak Zdenek P. Bažant Abdeldjelil Belarbi *† Evan C. Bentz John F. Bonacci

Robert W. Barnes Jr., Secretary

Hakim Bouadi Michael D. Brown Michael P. Collins David Darwin Walter H. Dilger* Marc O. Eberhard Catherine E. French Robert J. Frosch Gary G. Greene*

Neil M. Hawkins Thomas T. C. Hsu* Gary J. Klein Zhongguo John Ma Adolfo B. Matamoros Denis Mitchell Yi-Lung Mo* Lawrence C. Novak Carlos E. Ospina

Stavroula J. Pantazopoulou Maria A. Polak Julio A. Ramirez Karl-Heinz Reineck David H. Sanders* Raj Valluvan James K. Wight

*

Subcommittee members who produced this report. Subcomittee Chair. The committee would like to thank t he following individuals for their contribution t o this report: Mohammad Ali, Neal S. Anderson, Shri Bhide, Michael D. Collins, Maria Cristina Vidigal de Lima, Leonard Elfgren, Christos Karayannis, Liang-Jenq Leu, Mohammad Mansour, Basile Rabbat, Khaldoun Rahal, and Paul Zia. †

The behavior of members subjected to torsion combined with bending moment, axial load, and shear is discussed. This report deals with design issues, including compatibility torsion, spandrel beams, torsional limit design, open sections, and size effects. The final two chapters are devoted to the detailing requirements of transverse and longitudinal reinforcement in torsional members with detailed, step-by-step design examples for two beams under torsion using ACI ( ACI ( ACI 318-11), 318-11 ), European ( EC2-04 EC2-04), ), and Canadian Standards Association ( CSA-A23.3-04 ) standards. Two design examples are given to illustrate the steps involved in torsion design. Design Example Example 1 is a rectan rectangular gular reinfor reinforced ced concrete concrete beam under under pure torsion, and Design Example 2 is a prestr prestressed essed concre concrete te girder under combined torsion, shear, and flexure.

A clear understa understanding nding of the effects of torsion on concr concrete ete members is essential to the safe, economical design of reinforced and prestressed concrete members. This report begins with a brief and systematic summary of the 180-year history of torsion of structural concrete members, new and updated theories and their applications, and a historical overview outlining the development of research on torsion of structural concrete members. Historical theories and truss models include classical theories of Navier, Saint-Venant, and Bredt; the three-dimensional (3-D) space truss of Rausch; the equilibriu equilibrium m (plasticity) truss truss model of Nielson Nielson as well as Lampert and Thürlimann; the compression field theory (CFT) by Collins and Mitchell; and the softened truss model (STM) by Hsu and Mo. Mo. This report emphasizes that it is essential esse ntial to the analysis of torsion in reinforced concrete that members should: 1) satisfy the equilibrium condition (Mohr’s stress circle); 2) obey the compatibility condition (Mohr’s strain circle); and 3) establish the constitutive relationships of materials such as the “softened” stress-strain relationship of concrete and “smeared” stress-strain relationship of steel bars.

Keywords: combined action (loading); compatibility compatibility torsion; compression field theory; equilibrium torsion; interaction diagrams; prestressed concrete; reinforced concrete; shear flow zone; skew bending; softened truss model; spandrel beams; struts; torsion detailing; torsion redistribution; warping.

CONTENTS CHAPTER 1—INTRODUCTION AND SCOPE, p. 2 1.1—Introduction, 1.1—Introducti on, p. 2 1.2—Scope, p. 3

ACI Committee Reports, Guides, and Commentaries are intended for guidance in planning, designing, executing, and inspecting construction. This document is intended for the use of individuals who are competent to evaluate the significance and limitations of its content and recommendations and who will accept responsibility for the application of the material it contains. The American Concrete Institute disclaims any and all responsibility for the stated principles. The Institute shall not be liable for any loss or damage arising therefrom. Reference to this document shall not be made in contract documents. If items found in this document are desired by the Architect/Engineer to be a part of the contract documents, they shall be restated in mandatory language for incorporation by the Architect/Engineer.

CHAPTER 2—NOTATION AND DEFINITIONS, p. 3 2.1—Notation,, p. 3 2.1—Notation 2.2—Definitions, 2.2—Definition s, p. 5 ACI 445.1R-12 was adopted and published April 2013. Copyright © 2013, American Concrete Institute. All rights reserved including rights of reproduction and use in any form or by any means, including the making of copies by any photo process, or by electronic or mechanical device, printed, written, or oral, or recording for sound or visual reproduction or for use in any knowledge or retrieval system or device, unless permission in writing is obtained from the copyright proprietors.

1

2

REPORT ON TORSION IN STRUCTURAL CONCRETE (ACI 445.1R-12)

CHAPTER 3—HISTORICAL OVERVIEW OF TORSION THEORIES AND THEORETICAL MODELS, p. 5 3.1—Navier’s theory, p. 5 3.2—Thin-tube theory, p. 5 3.3—Historical development of theories for reinforced concrete members subjected to torsion, p. 6 3.4—Concluding remarks, p. 13 CHAPTER 4—BEHAVIOR OF MEMBERS SUBJECTED TO PURE TORSION, p. 13 4.1—General, p. 13 4.2—Plain concrete, p. 13 4.3—Reinforced concrete, p. 15 4.4—Prestressed concrete, p. 17 4.5—High-strength concrete, p. 18 4.6—Concluding remarks, p. 19 CHAPTER 5—ANALYTICAL MODELS FOR PURE TORSION, p. 20 5.1—General, p. 20 5.2—Equilibrium conditions, p. 20 5.3—Compatibility conditions, p. 20 5.4—Stress strain relationships, p. 22 5.5—Compression field theory, p. 23 5.6—Softened truss model, p. 25 5.7—Graphical methods, p. 26 CHAPTER 6—MEMBERS SUBJECTED TO TORSION COMBINED WITH OTHER ACTIONS, p. 28 6.1—General, p. 28 6.2—Torsion and flexure, p. 29 6.3—Torsion and shear, p. 33 6.4—Torsion and axial load, p. 36 6.5—Torsion, shear, and flexure, p. 37 CHAPTER 7—ADDITIONAL DESIGN ISSUES RELATED TO TORSION, p. 39 7.1—General, p. 39 7.2—Compatibility torsion and torsional moment redistribution, p. 39 7.3—Precast spandrel beams, p. 47 7.4—Torsion limit design, p. 48 7.5—Treatment of open sections, p. 51 7.6—Size effect on the strength of concrete beams in torsion, p. 53 CHAPTER 8—DETAILING FOR TORSIONAL MEMBERS, p. 53 8.1—General, p. 53 8.2—Transverse reinforcement, p. 55 8.3—Longitudinal reinforcement, p. 57 8.4—Detailing at supports, p. 58 CHAPTER 9—DESIGN EXAMPLES, p. 59 9.1—Torsion design philosophy, p. 59 9.2—Torsion design procedures, p. 59

9.4—Design Example 1: solid rectangular reinforced concrete beam under pure torsion, p. 67 9.5—Design Example 2: Prestressed concrete box girder under combined torsion, shear, and flexure, p. 74

CHAPTER 10—REFERENCES, p. 86 CHAPTER 1—INTRODUCTION AND SCOPE 1.1—Introduction Accounting for the effects of torsion is essential to the safe design of structural concrete members, requiring a full knowledge of the effects of torsion and a sound understanding of the analytical models that can easily be used for design. For over three decades, considerable research has been conducted on the behavior of reinforced concrete members under pure torsion and torsion combined with other loadings. Likewise, analytical models have been developed based on the truss model concept. Several of these models were developed to predict the full load history of a member, whereas others are simplified and used only to calculate torsional strength. Many models developed since the 1980s account for softening of diagonally cracked concrete. This report reviews and summarizes the evolution of torsion design provisions in ACI 318, followed with a summary of the present state of knowledge on torsion for design and analysis of structural concrete beam-type members. Despite a vast amount of research in torsion, provisions of torsion design did not appear in ACI 318 until 1971 (ACI 318-71), although ACI 318-63 included a simple clause regarding detailing for torsion. Code provisions in 1971 were based on Portland Cement Association (PCA) tests ( Hsu 1968b). These provisions were applicable only to rectangular nonprestressed concrete members. In 1995, ACI 318-95 adopted an approach based on a thin-tube, space truss model previously used in the Canadian Standards Association (CSA-A23.3-77) code and the Comité Euro-International du Béton (CEB)-FIP code (1978). This model permitted treatment of sections with arbitrary shape and prestressed concrete (Ghoneim and MacGregor 1993; MacGregor and Ghoneim 1995). The ACI 318-02 code extended the application of the (ACI 318) 1995 torsion provisions to include prestressed hollow sections. ACI 318 allows the use of alternative design methods for torsional members with a cross section aspect ratio of 3 or greater, like the procedures of pre-1995 editions of ACI 318 or the Prestressed Concrete Institute (PCI) method (Zia and Hsu 1978). This report reviews and summarizes the present state of knowledge on torsion and reviews their use as a framework for design and analysis of structural concrete beamtype members. Chapter 3 presents a historical background outlining the development of research on torsion of structural concrete members. The general behavior of reinforced and prestressed concrete members under pure torsion is discussed in Chapter 4. In Chapter 5, the compression field theory (CFT) and softened truss model (STM) are presented in detail. Chapter 5 also includes a description of two graphical methods (Rahal 2000a,b; Leu and Lee 2000). The behavior of members subjected to torsion combined with shear, flexure,


and axial load is discussed in Chapter 6. Chapter 7 introduces additional design issues related to torsion, such as precast spandrel beams, torsion limit design, size effect, open sections, and torsional moment distribution. Detailing of torsional members is described in Chapter 8. Chapter 9 covers detailed design examples of several beams subjected to torsion using ACI 318, EC2-04, and CSA-A23.3-04 design equations, and additional graphical design methods reported by researchers.

D

1.2—Scope Theories presented in this report were developed and verified for building members of typical size. For application to largescale members, size effects should be considered. They could present a serious safety issue when using the shear strength equations provided in the design standard, which cannot take into account the shear strength reduction in large-scale members caused by loss of aggregate interlock behavior. Experimental information on large-scale torsional members is lacking.

Dc =

D0 = D1 =

Db =

Ds e E c E ps

2.1—Notation a = moment arm for bending, mm (in.) ac = geometric property index ao = depth of equivalent rectangular stress block in concrete strut of torsional member, mm (in.) A = area of yield surface, mm2 (in.2) Acp = area enclosed by outside perimeter of concrete cross section, mm2 (in.2) Al = total area of longitudinal reinforcement to resist torsion, mm2 (in.2) Ao = gross area enclosed by shear flow path, mm2 (in.2) (noted as Atb in Eq. (7.2.6)) Aoh = area enclosed by centerline of outermost closed transverse torsional reinforcement, mm 2 (in.2) A ps = area of prestressing reinforcement in flexural tension zone, mm2 (in.2) As = area of nonprestressed longitudinal tension reinforcement, mm2 (in.2) As′ = area of longitudinal compression reinforcement, mm2 (in.2) At = area of one leg of a closed stirrup resisting torsion within spacing s, mm2 (in.2) (noted as Atb in Eq. (7.2.6)) b = width of compression face of member, mm (in.) bc = width of stirrups, mm (in.) B = integral of T w C = cross-sectional constant to define torsional properties of a beam d v = distance between top and bottom longitudinal reinforcement, mm (in.)

= = = =

E ps′ = E s

CHAPTER 2—NOTATION AND DEFINITIONS The material presented is a summary of research carried out worldwide and spanning more than four decades, making unification of the symbols and notations used by the various researchers and design codes a challenge. In some cases, mostly for graphs and figures, the notation is kept as originally published.

=

=

EJ w = f c′ = f c*

=

f ck = f cm = f d = f ds = f l f p l

= =

f yl

=

f p

=

f p0.1 = f pc

=

f pk = f po = f pu

=

f p,ud = f r f t f t′ f t* f tp

= = = = =

3

cross-sectional depth used in fracture mechanics calculations, mm (in.) size effect constant for computing σ N for plain concrete section normalized constant to represent characteristic structural dimensions used in fracture mechanics calculations size effect constant for computing σ N for reinforced concrete section total energy dissipated on discontinuous concrete yield surface total energy dissipated by reinforcement moment arm for torsion, mm (in.) modulus of elasticity of concrete, MPa (psi) modulus of elasticity of prestressed reinforcement in flexural tension zone, MPa (psi) tangential modulus of Ramberg-Osgood curve at zero load MPa (psi) modulus of elasticity of reinforcement and structural steel, MPa (psi) rigidity of beam under warping torque, N·m2 (lb-in.2) characteristic concrete cylinder compressive strength, MPa (psi) concrete effective (plastic) compressive stress, MPa (psi) characteristic compressive strength of concrete, MPa (psi); f ck = f cm – 8 MPa ( f ck = f cm –1200 psi) mean compressive strength of concrete, MPa (psi) diagonal concrete stress, MPa (psi) diagonal concrete stress corresponding to strain εds, MPa (psi) reinforcement stress in l direction, MPa (psi) prestressing reinforcement stress in the l direction, MPa (psi) specified yield strength of longitudinal reinforcement, MPa (psi) stress in prestressing reinforcement; f p becomes f p l or f tp when applied to longitudinal and transverse reinforcement, respectively, MPa (psi) characteristic yield strength of prestressing reinforcing strands, MPa (psi); f p0.1 = 0.9 f u compressive stress in concrete due to prestress, MPa (psi) characteristic tensile strength of prestressing reinforcing strands, MPa (psi); f pk = f pu effective prestress after losses in prestressing reinforcement, MPa (psi) specified tensile strength of prestressing reinforcement, MPa (psi) design ultimate strength of prestressing reinforcing strands, MPa (psi); f p,ud = f pk / γs (γ s = 1.15) modulus of rupture of concrete, MPa (psi) reinforcement stress in t direction, MPa (psi) uniaxial tensile strength of concrete, MPa (psi) concrete effective (plastic) tensile stress, MPa (psi) prestressing reinforcement stress in t direction, MPa (psi)

4


specified yield strength of transverse reinforcement, MPa (psi) f y = specified yield strength of reinforcement, MPa (psi) f yd = design yield strength reinforcing steel, MPa (psi); f yd = f y / γs (γ s = 1.15) f yl = yield strength of the torsional longitudinal reinforcement, MPa (psi) f yv = torsional hoop yield strength reinforcement, MPa (psi) G = shear modulus, MPa (psi) h = overall thickness or height of a member, mm (in.) H o = horizontal force in radial direction, N (lb) (Chapter 7) I p = polar moment of inertia, mm 4 (in.4) k 1 = ratio of average stress to peak stress K = value from Mohr-Coulomb yield criterion K f = flexural stiffness of floor beams, N·m2 (lb-in.2) K ts = torsional stiffness of spandrel beam, N·m/rad (in.-lb/rad) = span length of beam, mm (in.) l l f = length of flexural beam, mm (in.) = width of shear flow q along top wall (Fig. 4.2(a) l q and (b)), mm (in.) m = ratio of effective (plastic) compressive stress to effective (plastic) tensile stress of concrete M = applied flexural moment at section, N·m (in.-lb) M o = pure flexural strength of section, N·m (in.-lb) n = integer value n R = number of redundants nV = coefficient describing an under-reinforced, partially under-reinforced, or completely over-reinforced section N = applied axial load at section, N (lb) N o = pure axial strength of section, N (lb) ph = perimeter of centerline of outermost closed transverse torsional reinforcement, mm (in.) po = perimeter of outer concrete cross section, mm (in.) (sometimes noted as pcp) P = applied concentrated load, N (lb) q = shear flow, N/m (lb/in.) r = ratio of top-to-bottom yield forces of the longitudinal reinforcement r = size effect constant for computing σ N R = shape parameter used in Ramberg-Osgood s = center-to-center spacing of longitudinal and transverse reinforcements, mm (in.) sl = center-to-center spacing of longitudinal reinforcement, mm (in.) st = center-to-center spacing of transverse reinforcement, mm (in.) t = wall thickness of hollow section, mm (in.) t d = thickness of shear flow zone, mm (in.) T = applied torsional moment at section, N·m (in.-lb) T c = nominal torsional strength provided by concrete, N·m (in.-lb) T cr = torsional cracking resistance of cross section, N·m (in.-lb) T f = applied torsional moment, N·m (in.-lb) (Chapter 9) T max = maximum torsional moment, N·m (in.-lb) (Chapter 7) T n = nominal torsional moment strength, N·m (in.-lb) f ty

=

pure torsional strength of section, N·m (in.-lb) nominal torsional strength provided by reinforcement, N·m (in.-lb) T u = factored torsional moment at section, N·m (in.-lb) T w = warping torsional moment, N·m (in.-lb) T xu = factored balanced torsional strength, N·m (in.-lb) T xub = balanced torsional strength, N·m (in.-lb) nondimensional balanced torsional strength, N·m T xub = (in.-lb) v = shearing stress due to shear, MPa (psi) v* = plastic flow rate (Chapter 7) vu = ultimate shear stress, MPa (psi) V = applied shear force at section, N (lb) V c = nominal shear strength provided by concrete, N (lb) V o = pure shear strength of section, N (lb) V u = factored shear force at section, N (lb) w = ultimate distributed load on helical stair, N/m (lb/ft) (Chapter 7) W = external work, N/m (lb/ft) x = shorter overall dimension of rectangular part of cross section, mm (in.) x1 = distance section centroid and an infinitesimally small area of yield surface, mm (in.) y = longer overall dimension of rectangular part of cross section, mm (in.) z = distance along axis of beam, mm (in.) α, β = Saint-Venant’s coefficients for homogeneous torsional section * * α , β = rotational angles in beam subjected to torsion (Chapter 7) α1 = stress block factor given as ratio of f d to f c′ (Chapter 5) β = factor relating effect of longitudinal strain on shear strength of concrete (American Association of State Highway and Transportation Officials (AASHTO) LRFD (general message) β1 = factor relating depth of equivalent rectangular compressive stress block to neutral axis depth; also, block factor given as ratio of ao to t d (Fig. 4.5) γ 1 = angle along helical stair (in plan) at which maximum torsional moment is assumed to occur γ 2 = angle along helical stair (in plan) at which vertical moment is assumed to be zero γ lt = shear strain εd = strain in d direction εdec = strain in prestressing reinforcement at decompression of concrete εds = maximum strain at concrete strut surface (Fig. 4.3) εh = strain in hoop direction εl ε yl = yield strain in l direction εo = strain at peak compressive stress f c′ in concrete ε p = peak strain in concrete εr = strain in r direction εs = strain in nonprestressed reinforcement; εs becomes εl or εt when applied to longitudinal or transverse reinforcement, respectively εt = strain in t direction εty = yield strain in t direction T o T s

= =


εuk

=

ε x ζ η

= = =

ηlb

=

ηt

=

l

ηtb

=

θ

=

ξ

=

ρ ρ p ρt ρtp σ

= = = = =

σ0

=

σd

=

σ

=

l l

l

σmax = σ N = σr = σt

=

σ∞

=

τ = τmax = τ t = l

ν νc

= =

νt ϕ φ φc

= = = =

φ p

=

φs

=

characteristic total elongation of reinforcing steel at ultimate load longitudinal strain at midheight of concrete section softening coefficient of concrete strut normalized reinforcement ratio of longitudinal reinforcement balanced normalized reinforcement ratio of longitudinal reinforcement normalized reinforcement ratio of transverse steel reinforcement balanced normalized reinforcement ratio of transverse steel reinforcement angle between axis of strut, compression diagonal, or compression field and tension chord of the member; also, the angle between l -t direction/axis and d-r direction/axis, radians coefficient equal to 1 for rectangular sections and to π /4 for circular cross sections; ξ can be taken as unity for all shapes of cross sections with only negligible loss of accuracy for Ao and po reinforcement ratio in l direction prestressing reinforcement ratio in l direction reinforcement ratio in t direction prestressing reinforcement ratio in t direction compressive stress acting in combination with torsional moment, psi (MPa) nominal torsional strength according to the current code specifications based on plastic limit analysis, MPa (psi) principal stress in d direction for concrete struts, MPa (psi) normal stress in longitudinal direction for reinforced concrete, MPa (psi) maximum principal tensile stress, MPa (psi) nominal strength of structure, MPa (psi) principal stress in r direction for the concrete struts, MPa (psi) normal stress in the transverse direction for reinforced concrete, MPa (psi) strength of plain beams according to elastic analysis with maximum stress limited by material strength, MPa (psi) shearing stress due to torsion and shear, MPa (psi) maximum shear stress, MPa (psi) applied shear stress in l -t coordinate for reinforced concrete, MPa (psi) uniform plastic effectiveness factor (Chapter 7) plastic effectiveness factor for compression (Chapter 7) plastic effectiveness factor for tension (Chapter 7) friction angle strength reduction factor strength reduction factor for concrete (0.65 for cast-in-place, 0.70 for precast concrete) strength reduction factor for prestressing tendons (0.90) strength reduction factor for nonprestressed reinforcing bars (0.85)

Φ

=

Φ′

=

Φ′′

=

Ψ ω ω s ω s ω t

= = = = =

l

l

5

angle of twist in torsional beam, radians/m (radians/in.) second derivative of rotation with respect to beam’s axis z third derivative of rotation with respect to beam’s axis z bending curvature of concrete strut reinforcement index in l direction functional indicator of an index of reinforcement reinforcement ratio index reinforcement index in t direction

2.2—Definitions ACI provides a comprehensive list of definitions through an online resource, “ACI Concrete Terminology,” http:// terminology.concrete.org . CHAPTER 3—HISTORICAL OVERVIEW OF TORSION THEORIES AND THEORETICAL MODELS 3.1—Navier’s theory A theory for torsion of elastic homogeneous members was first developed by C. L. Navier (1826) for circular cross sections. His theory, which was based on equilibrium conditions, compatibility conditions, and a linear stress-strain relationship like Hooke’s Law, has guided the development of various theories about the behavior of reinforced concrete members subjected to torsion after cracking. 3.2—Thin-tube theory Navier’s torsion theory for members of circular sections was followed by Saint-Venant’s (1856) solution for rectangular sections. Saint-Venant’s torsional constants considered warping of rectangular cross sections. According to SaintVenant’s circulatory shear flow theory, the most efficient cross section to resist torsion is a thin tube. Bredt (1896) was able to derive simple equations for thin tubes. His thin-tube theory states that the shear stress multiplied by wall thickness has a constant value around the perimeter and that this shear flow is found by dividing the torsion by twice the area enclosed by the shear flow path. Bredt’s theory has served as the basis for modern theories of cracked reinforced concrete members subjected to Saint-Venant torsion. 3.2.1 Two- and three-dimensional plane truss models— The first theoretical models for shear in cracked reinforced concrete members date back to the turn of the century when Ritter (1899) and Mörsch (1902) formulated the two-dimensional (2-D) plane truss model concept, where reinforced concrete members were modeled as an assembly of two types of linear elements—concrete struts and reinforcement ties. The axis of concrete struts in the model was assumed to be inclined at 45 degrees to longitudinal members, and shear strength was assumed to be controlled by the yielding of transverse reinforcement ties. By extending the 2-D plane truss model, Rausch (1929) developed a three-dimensional (3-D) space truss model for torsion that consisted of longitudinal and hoop reinforcement-resisting tension and concrete struts-resisting compression. He also assumed that

6


the shear flow path would follow the centerline of the hoop reinforcement. 3.2.2 Skew-bending and space truss theories —In later years, research on torsion followed two theoretical tracks— skew-bending and space truss. Lessig (1959) first proposed a skew-bending theory for reinforced concrete members with two modes of failure, Mode 1 and Mode 2, as explained in 3.3.5. The skew-bending theory used only equilibrium equations and assumed that all reinforcement yielded before failure. Lessig’s research was followed by the skew-bending theory of Walsh et al. (1966) and Collins et al. (1968a,b), who proposed a third failure mode, Mode 3, and used all three modes to derive nondimensional torsion-bending moment interaction equations (Walsh et al. 1967), as described in 3.3.5. Based on three modes of failure, a nondimensional interaction surface of torsion, shear, and flexure was derived by Elfgren (1972a,b) and Elfgren et al. (1974a,b). Rausch’s space truss theory for torsion was generalized by Lampert and Thürlimann (1969, 1971), who showed how the angle of inclination of the compression diagonals at failure could be determined from equilibrium if both the hoops and longitudinal reinforcement were assumed to yield. Lampert and Collins (1972) showed that predictions of skew-bending and space truss theories were in close agreement. 3.2.3 Compression field theory (CFT)—The truss model with linear elements developed by Rausch was replaced in the 1960s by a new type of truss model with membrane elements that were subjected to in-plane normal and shearing stresses. In determining the torsional strength of members where some reinforcement does not yield, consider compatibility conditions. Such conditions were introduced by Baumann (1972) for shear and by Collins (1973) for torsion. Mitchell and Collins (1974) incorporated compatibility conditions in their CFT, which also relied on equilibrium equations and nonlinear material models for concrete and reinforcement. Unlike previous models, the CFT calculates cracked member torsional behavior up to the peak torque. A compatibility condition derived by minimizing the strain energy in the system is used to calculate the angle of inclination in the truss model struts. 3.2.4 Softened truss model (STM)—In 1985, Hsu and Mo (1985a,b,c) proposed the STM by softening the concrete stress-strain curve. All of the aforementioned models satisfy Navier’s theory. Earlier models overestimated test strengths (Hsu 1968c), whereas the CFT, which uses spalling of concrete cover, and STM, which uses softening of concrete, have been shown to predict test results accurately (McMullen and El-Degwy 1985).

3.3—Historical development of theories for reinforced concrete members subjected to torsion 3.3.1 General—Section 3.1 summarizes the historical models developed to describe reinforced concrete members subjected to torsion, covering almost two centuries of research from 1826 to the early twenty-first century (2007). Classical theories include Navier (1826), Saint-Venant (1856), Bredt (1896), and Bach (1911). This review shows that the three principles of mechanics of materials (equilibrium, compatibility conditions, and materials stress-strain

Fig. 3.3.2—ACI Committee 438—Torsion, Mexico City, October 1976: Tom Hsu, Lennart Elfgren, Phil Ferguson, Art McMullen, Emory Kemp, Gordon Fisher, Paul Zia, and Michael Collins.

relationships) have been the basis for research in torsion of reinforced concrete members. (Equation notation in this section are provided in Chapter 2.) 3.3.2 Twentieth century—In the first 60 years of the twentieth century, progress in reinforced concrete theories was made primarily on flexure. Early flexural theories for reinforced concrete assumed plane sections remained plane and stress-strain relationships of concrete and reinforcement were linear. Equilibrium conditions for longitudinal stresses were used to determine the location of the neutral axis and stresses in concrete and reinforcement caused by the moment. The contribution of concrete tensile stresses was disregarded if concrete cracking was expected. Later flexural theories accounted for the nonlinear stress-strain response of the concrete and steel reinforcement so that the complete moment-curvature relationship for a section could be predicted. In terms of shear and torsion research, a significant achievement was made with the development of truss models (Ritter 1899; Mörsch 1902; Rausch 1929). Research in torsion of reinforced concrete underwent significant advances during the last 40 years of the twentieth century. Two theories were developed—skew-bending and truss models with membrane elements. Skew-bending includes the theories of Lessig (1959), Yudin (1962), Collins et al. (1968a), Hsu (1968a), and Elfgren (1972a,b). Truss models include the theories of Nielsen (1967), Lampert and Thürlimann (1968, 1969), Collins (1973), Mitchell and Collins (1974), Elfgren et al. (1974a,b), Collins and Mitchell (1980), and Hsu and Mo (1985a,b,c). Several researchers involved in the development of theories for torsion were members of ACI Committee 438 for Torsion, which is now the Joint ACIASCE Committee 445 for Shear and Torsion ( Fig. 3.3.2). Development of these modern truss models was based on the same three principles of mechanics, which, in terms of torsion and shear, include the softened stress-strain relationship of concrete. 3.3.3 Classical torsion theory for homogeneous members— Navier (1826) derived a theory for torsion of homogeneous elastic members with circular cross sections. His theory is based on the three principles of mechanics of materials:


equilibrium, compatibility conditions, and Hooke’s Law. Navier’s work also includes the linear theory for flexure. His book is recognized as the first on the mechanics of materials The three principles of the mechanics of materials have become well known as Navier’s theory. His theory defines the torsional rigidity for circular sections as GI p. By extending the formulas for the polar moment of inertia of circular sections to square sections, Navier found that the calculated strength for specimens tested by Duleau (1820) overestimated measured values by approximately 20 percent. He acknowledged that “the formulae for square members do not depict as accurately the behavior as those for circular members.” This inconsistency was explained three decades later by Saint-Venant (1856), who recognized that Navier’s polar moment of inertia could not reflect the warping deformation of rectangular cross sections. To obtain the correct solution, Saint-Venant developed the semi-inverse method to solve all 15 differential and algebraic equations in the theory of elasticity developed by Cauchy (1828). By satisfying equilibrium, compatibility, and Hooke’s Law at each differential element of a member, Saint-Venant developed a solution that considered the warping displacements of rectangular cross sections. In Saint-Venant’s rigorous derivation, torsional rigidity is defined as GC . The torsional constant C is taken as β x3 y, where the coefficient β is a function of the x and varies between 0.141 ( y / x = 1) and 0.333 ( y / x ratio y / = ∞). The maximum shear stress τmax occurs on the outside face of the rectangular section at midpoint of each long side and is equal to T / α x2 y, where α is a coefficient that varies x = 1) and 0.333 ( y / x = ∞). The relationship between 0.208 ( y / between α and β is α = β / k , where k = 1 −

∞

8

π

2

1

∑

n =1,3,5,...

n 2 cosh

nπy

2 x

According to Saint-Venant’s circulatory shear flow pattern, the maximum shear stresses occur at the outer periphery of a cross section, and the most efficient cross section to resist torsion is a thin tube, as shown in Fig. 3.3.3. Bredt (1896) was able to derive a simple equilibrium equation for thin tubes by assuming the entire tube to be uniformly and fully stressed q

=

T

2 Ao

(3.3.3)

The area Ao is formed by sweeping the lever arm (symbol a in Fig. 3.3.3) around the axis of twist, a term later called the “lever arm area” (Hsu 1988). The torsional constant C for thin tubes with constant thickness was also simplified to C = po. Bach’s formula (1911) is a simplification of Saint4 Ao2t / Venant’s theory for thin-walled open sections, such as T, L, and I sections. Because the coefficient β for each rectangular component of such sections can be approximated as 1/3, the torsional constant for the entire section can be taken as the sum (Σ) of the components (C = 1/3Σ x3 y). The theories of

7

Fig. 3.3.3—Torsion of thin tube and lever arm area Ao ( Hsu 1993 after Bredt 1896 ).

Navier, Saint-Venant, Bredt, and Bach are applicable to reinforced concrete beams before cracking. They also laid the foundation for developing theories to calculate the behavior of cracked reinforced concrete members subjected to torsion. 3.3.4 Theories for reinforced concrete under flexure, shear, and torsion—Reinforced concrete was first developed in 1867 when Joseph Monier obtained a patent for reinforcing his concrete flowerpots with wrought iron wires. The concept of using steel reinforcement to overcome the weakness of concrete in tension was quickly adapted to buildings and bridges, making the use of reinforced concrete for construction a widely accepted application in the last quarter of the nineteenth century. Such growth in applications gave rise to the demand for theories to analyze and design reinforced concrete structures. 3.3.4.1 Flexure theory—As reported by Delhumeau (1999), the first flexure theory to emerge was the linear flexure theory developed by Hennebique’s firm near the end of the nineteenth century. This theory served as the basis for the allowable stress design method addressed in the first ACI code [National Association of Cement Users (NACU) 1910]. Theories for nonlinear flexure occupied the attention of researchers until 1963 when strength design was incorporated in ACI 318 (ACI 318-63). Both linear and nonlinear theories for flexure satisfy the three principles of the mechanics of materials: equilibrium of parallel coplanar forces, Bernoulli’s linear strain compatibility, and the constitutive laws of materials. A linear stress-strain curve of reinforcement and concrete is used for the linear flexure theory and a nonlinear stress-strain curve of concrete is used for the nonlinear flexure theory. 3.3.4.2 Torsion and shear theory—Following the adoption of flexural strength design in ACI 318-63, the attention of researchers turned to more complex problems of torsion and shear in beams. Computer analysis of structures was becoming available, making it feasible for design engineers to compute the magnitude of torsions in their buildings. In addition, concrete box-girder bridges, which were becoming a competitive bridge type, needed to be designed for torsion. Shear is essentially a two-dimensional problem, requiring an understanding of the interaction of two principal stresses and strains in a membrane element. Torsion is complicated because it is a three-dimensional problem involving

8


Fig. 3.3.4.2—Number of published papers on torsion from 1900 to 2007 (plot prepared by A. Belarbi).

the shear problem of membrane elements in a tube and the warping of tube walls that cause flexure in concrete struts. Figure 3.3.4.2 shows that the number of globally published papers on torsion began to surge around 1960 and peaked around 1970. The principles of equilibrium, compatibility conditions, and materials stress-strain relationships that were needed to solve torsion problems in reinforced concrete members were primarily developed between 1960 and 1985 (Lampert and Thürlimann 1971; Lampert and Collins 1972; Elfgren et al. 1974a,b; Collins and Mitchell 1980; Hsu and Mo 1983). Theories and tests produced before 1980 are summarized in detail by Hsu (1984). By 1985, researchers solved the basic problems of reinforced concrete design by applying Navier’s theory. Further research was necessary to refine the constitutive laws of materials for torsion and shear. The experimental work needed to generate new advancements is tedious and requires highly sophisticated testing equipment. Only two universities in North America— the University of Toronto and University of Houston—are capable of studying the behavior of reinforced concrete shell and panel elements subjected to in-plane shear and normal stresses. The study of softened concrete in shear elements, which has been the subject of extensive research worldwide in the last three decades, continues to be a major research topic. The need for larger and more complex specimens has increased ongoing work at both universities. The high cost of experimental research needed for new developments in torsion imposes a limiting constraint on new research. For example, studying the behavior of full-scale girders with open sections that involve Saint-Venant and warping torsion is expensive. The future of torsion research is largely tied to available equipment or to the combined efforts of many institutions, or both. 3.3.5 Space truss model using struts and ties —The first theory for shear design of reinforced concrete was developed at the turn of the twentieth century when Ritter (1899) and Mörsch (1902) formulated the concept of plane trusses with struts and ties. They modeled a reinforced concrete member

as a truss with two types of linear elements: struts made out of concrete and ties made out of steel reinforcement. The Ritter and Mörsch model represents the struts and ties as lines without cross-sectional dimensions, where forces satisfy equilibrium at points of intersection—a model with the advantage of conceptual clarity. Extending the 2-D plane truss model to a 3-D space truss model, Rausch (1929) developed a theory for torsion of reinforced concrete. Rausch’s space truss model, as shown in Fig. 3.3.5a, is made up of 45-degree diagonal concrete struts, longitudinal reinforcing bars, and hoop reinforcing bars connected at the joints by hinges. Torsional moment is carried by the concrete struts in axial compression (dotted lines), and by the straight reinforcing bars in axial tension (solid lines) in the longitudinal (horizontal) and lateral (hoop) directions. Equilibrium of the joints in the longitudinal, lateral, and radial directions requires that the forces in the longitudinal bars ( X ), in the hoop bars (Q), and in the inclined struts ( D) should be evenly distributed among all cells and joints. To satisfy equilibrium, the relationship between these forces should be X = Q = D / √2. As shown in Fig. 3.3.5a, the series of hoop forces Q at the joints constitute a shear flow q = Q / s. Using Bredt’s lever arm area concept, T can be related to q (or Q / s) by 2 Ao, as expressed by Bredt’s equation (Eq. (3.3.3)). The term Ao refers to the area enclosed by a series of straight lines connecting joints of the cross section. Assuming that ultimate torque is reached when the forces in the transverse reinforcement reach the yield stress, then q = Q / s = A f t ty / st and Eq. (3.3.3) becomes Tn

= 2 Aoh

At f ty st

(3.3.5)

Although the space truss model has the advantage of conceptual clarity in terms of simple assemblage of compression struts and tension ties, Rausch’s equation (Eq.


9

Fig. 3.3.5a—Rausch’s space truss model with struts and ties ( Hsu 1984 after Rausch 1929).

(3.3.5)) was found to greatly overestimate the experimentally measured torsional strengths of reinforced concrete members. As a result, empirical formulas proposed to correct limitations of Rausch’s equation were incorporated in the design codes of several countries until the 1990s. Figure 3.3.5b compares T n with Aoh A f t ty / st for a series of PCA tests (Hsu 1968b,c). The straight solid line represents calculated strengths using Eq. (3.3.5) (that is, αt = 2.0) and the dashed line represents calculated strengths using the torsion provisions of ACI 318-71. The lever arm area Ao is defined by the centerline of closed stirrups and denoted as Aoh. In view of the unconservative nature of the values calculated with Eq. (3.3.5) (Fig. 3.3.5b), in particular when T n is large, an empirical formula was proposed for ACI 318-71. This formula is simply a modification of Eq. (3.3.5), which reduced the slope of 2, shown in Fig. 3.3.5b, down to 1.2, and added a “concrete contribution T c,” which represents the vertical intercept of the experimental curve. The background of “concrete contribution T c,” which is included in early ACI Codes (1971 through 1995) and later dropped, is provided in Hsu (1997). From the experimental tests shown in Fig. 3.3.5b, Lampert and Thürlimann deduced that Ao should be less than Aoh. They recommended that Ao be taken as the area enclosed by a line joining the centers of the longitudinal bars in the corners of the hoops. With this change in the definition of Ao, the agreement between predictions and experimental results was considerably improved. This new definition of Ao was subsequently incorporated into the CEB-FIP code (1978). A different approach, to reconcile Rausch’s space truss model with test results, was achieved in 1983 with development of the STM (Hsu and Mo 1983) for torsion, using a softened stress-strain relationship of concrete developed by Vecchio and Collins (1981). A more recent STM is provided in 5.6. 3.3.6 Skew-bending theory for torsion 3.3.6.1 Beams without web reinforcement— When beams without transverse reinforcement are subjected to torsion, they fail quickly and abruptly at cracking. Their torsional behavior is similar to that of plain concrete beams and, therefore, can be predicted by Saint-Venant’s theory. The experimental torsional strength of a plain concrete beam, however, was found to be greater than the theoretical strength calcu-

Fig. 3.3.5b—Experimental Tn versus Aoh(Atf ty /st) curve compared with Rausch’s formula and 1971 ACI code ( Hsu 1984).

lated with Saint-Venant’s theory ( T n = α x2 yf t′ ), assuming the principal tensile stress reaches a value of f t′ . To find a reason for this discrepancy, Hsu (1968a) used high-speed photography to record failure process and observed that plain concrete members failed abruptly in a skew-bending mode. Based on the skew-bending failure, the torsional strength was derived as T n = (1/3) x2 y(0.85 f r ). As a result, the constant 1/3 is an approximation of Saint-Venant’s coefficient α y ratio becomes large. This coefficient was used when the x / as the calibration for the torsional strength of beams without reinforcement, and for the “concrete contribution” of the torsional strength of beams with web reinforcement (Hsu 1968b,c). Formulas based on the parameter (1/3) x2 yf t′ were incorporated into the ACI Code from 1971 to 1995. 3.3.6.2 Beams with web reinforcement —Skew-bending theory was also applied to concrete beams reinforced with both longitudinal and transverse reinforcement subjected to torsion, shear, and flexure. Such beams could fail in three modes (Fig. 3.3.6.2). The first mode has a compression zone along the top face and failure is caused by yielding of the bottom longitudinal reinforcement and transverse reinforcement along the three remaining faces. The second mode has a compression zone along a side face and failure is caused by the yielding of the longitudinal reinforcement on the opposite side face, and in the transverse reinforcement along the three faces in tension. The third mode has a compression zone along the bottom face where failure is caused by yielding along the top longitudinal reinforcement and transverse reinforcement along the three remaining faces. Lessig (1959) was the first to propose a skew-bending theory in connection with Mode 1 and Mode 2 failures. Equilibrium equations were established by taking moment equilibrium about an axis AB (Fig. 3.3.6.2) parallel to the

10


Fig. 3.3.6.2—Three failure modes in skew-bending theories ( Lessig 1959; Yudin 1962; Collins et al. 1968a; Elfgren et al. 1974a,b).

skew compression zone and force equilibrium about the axis perpendicular to the center of the compression zone. The crack angles defining the skew failure surfaces were determined by minimizing the torsional resistance. The resulting equations were complicated. Yudin (1962) took a different approach by deriving three equilibrium equations: moment equilibrium about the longitudinal and transverse axes through the compression zone center, and force equilibrium about the axis perpendicular to the compression zone center. Assuming all the crack angles to be 45 degrees, he showed that his equation for ultimate torque was the same as Rausch’s. Lessig and Yudin’s research was followed by the skew-bending theory of Walsh et al. (1966, 1967) and Collins et al. (1968a,b), who also identified Mode 3 failure. The three modes of failure—1, 2, and 3—are governed by the dominant effect of positive flexure moment, torsion shear, and negative flexure moment, respectively. For

combined torsion and flexure, Lampert and Collins (1972) showed that both the skew-bending theory and the space truss theory predict the same parabolic shape for the interaction curve. Elfgren et al. (1974a,b) then derived a nondimensional interaction surface for combined torsion, shear, and flexure based on the three modes of skew-bending failure. If the inclination angle of cracks in each wall is adjusted so that both longitudinal and transverse reinforcement can yield, the interaction surface becomes identical to that derived from the equilibrium (plasticity) truss model (Elfgren 1972a,b), as explained in 5.5. 3.3.7 Truss models with membrane elements 3.3.7.1 Equilibrium (plasticity) truss model for torsion— Rausch’s space truss model with struts and ties ( 1929) was replaced by a new and more realistic type of truss model in the 1960s. As shown in Fig. 3.3.7.1a, the new model is made up of membrane elements treated as trusses made up


11

Fig. 3.3.7.1a—Membrane element in shear ( Hsu 1993).

of struts and ties after cracking. Three equilibrium equations for the elements were first derived by Nielsen (1967) and Lampert and Thürlimann (1968).

σ = σd cos2θ + σr sin2θ + ρ f

(3.3.7.1a)

σ t = σ d sin 2 θ + σ r cos2 θ + ρt f t

(3.3.7.1b)

τ t = (–σd + σr )sinθcosθ

(3.3.7.1c)

l

l

l

l

Because Eq. (3.3.7.1a) through (3.3.7.1c) describe the transformation of stresses, they form a system of stresses that should satisfy Mohr’s stress circle, a type of equilibrium condition called Mohr’s circular stress equilibrium. The longitudinal and transverse reinforcement stresses are f yl , f t = assumed to yield at failure ( f l = f ty), and the concrete tensile stress is neglected ( σr = 0). Equations (3.3.7.1a) through (3.3.7.1c) were simplified to yield the shear stress τl t in the case of pure shear without normal stresses ( σl = σt = 0)

τ t = ρ f yρt f ty l

l

l

(3.3.7.1d)

A reinforced concrete tube subjected to torsion can be visualized as an assembly of shear elements similar to the one shown in Fig. 3.3.3. The torsional moment at yielding, T n, can then be derived q / t ) into Eq. (3.3.3) by substituting the shear stress τl t ( τl t = s t A s t and substituting ρl = A / and = / ρ l l t t t

Tu

= 2 Ao

A f y At f ty l

s

l

l

st

= A f t ty / st ), Lampert and Thürlimann’s Eq. (3.3.7.1e) looks identical to Rausch’s Eq. (3.3.5), except the term Ao is defined differently. The former is based on the centerline of longitudinal reinforcement and the latter is based on the centerline of hoop reinforcement. Equation (3.3.7.1e) was first used in the European CEB-FIP code in 1978 and ACI 318-95. By combining Eq. (3.3.7.1e) with equations for shear and flexure, Elfgren (1972a,b) and Elfgren et al. (1974a) established a nondimensional interaction surface as shown in Fig. 3.3.7.1b, based on the equilibrium (plasticity) truss model as explained in 6.5. Elfgren also showed the interaction surface derived from the truss model is identical to the skew-bending theory. Actual testing has shown, however, that Elfgren’s interaction surface overestimates the experimental results in the region adjacent to torsion and shear because softening concrete was not considered. 3.3.7.2 Compatibility truss model for torsion 3.3.7.2.1 General—According to Navier, the strain compatibility (or geometric) condition should be considered if the angle of twist is to be related to the torsional moment. Assuming the membrane element to behave as a truss as shown in Fig. 3.3.7.1a(b), three strain compatibility equations have been derived by Baumann (1972) and Mitchell and Collins (1974).

ε = ε d cos2 θ + εr sin 2 θ

(3.7.2.1a)

ε t = ε d sin 2 θ + εr cos2 θ

(3.7.2.1b)

l

γ t l

(3.3.7.1e)

Equation (3.3.7.1e) was first derived by Lampert and Thürlimann (1968, 1971). In the case of equal yield force sl in both longitudinal and transverse reinforcement ( A f l y l /

2

= (−ε d + εr )sin θ cos θ

(3.7.2.1c)

Because Eq. (3.7.2.1a) through (3.7.2.1c) describe the transformation and distribution of strains, they should satisfy Mohr’s strain circle. This type of compatibility condition is therefore called Mohr’s circular strain compatibility.

12


Fig. 3.3.7.1b—Nondimensional interaction surface for torsion, shear, and flexure ( Hsu 1984). (The numbering of the failure modes is different than in Fig. 3.3.6.2.)

In addition, there are two compatibility equations for a reinforced concrete tube. One relates the angle of twist, Φ, to the shear strain γ lt of the membrane element

Φ=

po

2 Ao

γ t l

(3.7.2.1d)

The other describes the flexure of diagonal concrete struts. The curvature of concrete struts Ψ at an angle θ is related to the angle of twist Φ by

Ψ = Φsin2θ

(3.7.2.1e)

3.3.7.2.2 Compression field theory (CFT)—In a major development of the truss model for torsion, Mitchell and Collins (1974) established the CFT to calculate the complete torque-twist response of reinforced concrete members, irrespective of sectional shape or amount of reinforcement. The CFT combines four equilibrium equations [ Eq. (3.3.3), (3.3.7.1a), (3.3.7.1b), and (3.3.7.1c)], five compatibility equations [Eq. (3.7.2.1a) through (3.7.2.1e)], and constitutive equations of concrete and reinforcement. Concrete diagonals thickness t d and the area enclosed by the shear flow, Ao, are determined as described in 5.5. The assumption that the diagonal compressive strains are reduced linearly with depth below the surface was experimentally verified (Collins 1973). Because the concrete cover is assumed to have spalled before the section reaches the maximum strength, this theory is also known as the “spalling model.” Details of the CFT are provided in 5.5. 3.3.7.2.3 Softened truss model—A reinforced concrete membrane element subjected to shear is a two-dimensional problem; the shear stress can be resolved into a principal tension stress and a principal compressive stress in the 45-degree direction. Robinson and Demorieux (1972) found that when a membrane element is subjected to a biaxial stress condition, the principal compressive stress is reduced (or “softened”) by the principal tensile stress in the perpendicular direction. Performing biaxial tests of shear elements,

Collins and Chockalingam (1979) and Vecchio and Collins (1981, 1982, 1986) showed that the concrete softening coefficient in the principal compressive direction is related to the principal tensile strain, not the principal tensile stress. Softening was further quantified experimentally by researchers at the University of Houston (Belarbi and Hsu 1994, 1995; Pang and Hsu 1995; Hsu et al. 1995a,b; Zhang and Hsu 1998), who confirmed the importance of principal tensile strain and identified concrete strength as an additional essential variable. An STM (Hsu and Mo 1985a,b,c; Hsu 1993) was developed based on a softened concrete stressstrain curve. Details of this theory are provided in 4.6. In the STM (Hsu and Mo 1985a,b,c), the softened concrete stressstrain curve is used in the flexural compression zone of the concrete struts to determine the neutral axis position. The distance from the neutral axis to the extreme compression fiber is known as thickness of shear flow zone, t d; the lower the softening coefficient, the greater the thickness t d. Because the softening coefficient varies from approximately 1/4 to 1/2, the thickness t d, obtained from the softened stress-strain curve, is expected to be approximately two to four times the thickness from the non-softened stress-strain curve. Based on Vecchio and Collins’s softened stress-strain curve (1981), Hsu and Mo (1985a,b,c) studied the torsional behavior of cracked reinforced concrete members. They found that Rausch’s Eq. (3.3.5) is valid if the thickness t d and the lever arm area Ao are calculated by a truss model that incorporates a softened stress-strain curve. Hsu and Mo’s STM satisfies the Mohr’s stress circle, the Mohr’s strain circle, and the softened stress-strain curve of concrete. It was further shown to provide accurate estimates of experimental results available at the time (Hsu and Mo 1985a,b,c), including the PCA tests in Fig. 3.3.5b. When accounting for the modifications proposed by Hsu and Mo, the equation by Rausch becomes a curve that starts at the origin, closely tracing the PCA test points. This occurs because Ao decreases with increasing T n and an increasing percentage of reinforcement. The depen Ao on T n is expressed by simple equations dency of t d and (Hsu 1990, 1993) t d =

Ao

= Acp −

t d po

2

≈ Acp −

t d pcp

2

4T n Acp f c′

= Acp −

(3.3.7.2.3a)

2Tn pcp Acp f c′

(3.3.7.2.3b)

As indicated by Eq. (3.3.7.2.3b), the lever arm area Ao is decreased by increases in the thickness t d. Because the torsional strength of a reinforced concrete member is directly proportional to Ao, the strong effect of the softening coefficient on the torsional strength is evident, explaining why Rausch’s equation for T n (Eq. (3.3.5)) overestimated the test results, as shown in Fig. 3.3.5b. Because the shear flow zone thickness t d occupies such a crucial position in the STM, torsion research is now focused on a better understanding of the shear flow zone (Alkhrdaji and Belarbi 2003).


3.3.8 Derivation of ACI 318: torsion design equations—As discussed previously, ACI equations for designing members under torsion are based on the space truss model theory. There are two basic design equations: one for torsional hoop reinforcement, and another for torsional longitudinal reinforcement. From the truss model (Fig. 3.3.7.1a(b)), equilibrium of the top face of the element relates the shear flow q to the force in the torsional hoop reinforcement per unit length ( A f t / t s) through the equation q =

At f t s

cotθ

(3.3.8a)

Substituting Eq. (3.3.8a) into Bredt’s Eq. (3.3.3) and assuming yielding of the hoop reinforcement renders the ACI 318 requirement for transverse reinforcement for torsion. At s

=

T u

φ2 Ao f yv cot θ

(3.3.8b)

Similarly, from the equilibrium of the left face of the element, shown in Fig. 3.3.7.1a(b), the shear flow q can be related to the force in the torsional longitudinal reinforcement per unit length ( A f l / l sl ) through the equation q

=

A f l

s

l

tan θ

(3.3.8c)

l

The term Al is defined in ACI 318 as the total area of torsional longitudinal reinforcement in the cross section, assuming that po = ph, where ph is the perimeter of the centerline of the outermost hoop bars. Equation (3.3.8c) then becomes q

=

A f l

ph

l

tan θ

(3.3.8d)

An expression for the amount of torsional longitudinal reinforcement in ACI 318 is obtained by equating the shear flow q of Eq. (3.3.8d) and (3.3.8a) A

l

 A  f =  t  ph yv cot 2 θ  s  f y

(3.3.8e)

l

3.4—Concluding remarks This historical review identifies research studies that have provided the foundation for the current understanding of torsion in reinforced concrete members. It took almost the entire twentieth century for researchers to successfully apply Navier’s theory to cracked reinforced concrete. The application of Navier’s theory to the flexure theory of reinforced concrete was relatively easy because of the one-dimensional condition of equilibrium and compatibility, combined with the non-softened constitutive laws of materials. Shear and

13

torsion proved to be much more difficult because they were, respectively, two-dimensional and three-dimensional problems. From a historical perspective, the development of the current torsional theories spanned over 40 years. Equilibrium equations were developed in the 1960s, compatibility conditions were added in the 1970s, and the formulation and expressions of softening coefficients were introduced in the 1980s and 90s. Since 1960, the bulk of work has focused on beams subjected to monotonic torsional loading. Early tests on members subjected to reverse cyclic loading were conducted by Collins and Chockalingam (1979) on reinforced concrete beams subjected to pure torsion and by Stevens et al. (1991) on membrane elements subjected to reversed cyclic shear up to reinforcement yielding. Significant recent advances have applied rational models such as the STM and the modified CFT to shear elements subjected to cyclic loading. Vecchio (1999) and Palermo and Vecchio (2003, 2004) applied the MCFT for cyclic loading up to yielding of the reinforcement. Mansour et al. (2001), Zhu et al. (2001), Zhu and Hsu (2002), and Mansour and Hsu (2005a,b) studied cyclic torsion and shear, including the unloading and reloading branches of the softened stress-strain curves (hysteretic loop) and the effect of Poisson’s ratio in a membrane element. The latter was crucial in determining the descending branch of the softened stress-strain curve. Belarbi and Greene tested several full-scale box girders under cyclic torsion and combined torsion and shear (Belarbi and Greene 2004; Greene and Belarbi 2008). The cyclic shear tests of membrane elements ( Mansour and Hsu 2005a,b) permitted the establishment of cyclic stressstrain curves of concrete and reinforcement. These cyclic constitutive laws laid the foundation for the Cyclic Softened Membrane Model (Mansour and Hsu 2005a,b), which could be used for the design of seismic-resistant reinforced concrete structures (Hsu et al. 2006).

CHAPTER 4—BEHAVIOR OF MEMBERS SUBJECTED TO PURE TORSION 4.1—General The torsional strength of concrete beams depends on many factors, such as the amount and distribution of transverse and longitudinal reinforcement, concrete strength, cross section shape, and, for rectangular cross sections, aspect ratio (depth-to-width ratio). Concrete strength and amount of reinforcement can change the failure characteristics. The failure mechanism varies from ductile to sudden and brittle, depending on the longitudinal and transverse reinforcement volumetric ratios. Chapter 4 details the behavior of plain concrete, reinforced concrete, and prestressed concrete under pure torsion. The behavior of high-strength concrete members is also discussed. The effects of other stress resultants, such as shear and flexure, are discussed in Chapter 6. 4.2—Plain concrete Saint-Venant’s elastic theory accurately describes the torsional behavior of plain concrete members at low torque before cracking. This theory assumes that the cross section

14


Fig. 4.2—Torsion stress factors for solid rectangular section (Collins 1973).

shape remains unchanged after twisting, the angle of twist per unit length is constant along the element length, warping deformation perpendicular to the cross section is identical throughout the member length, and failure occurs when the maximum principal stress equals the direct concrete tensile strength. Member shear stresses resist applied torsional moment. For solid circular shapes, the shear stresses are zero at the member center, increasing linearly to a maximum at the outside surface. In solid rectangular shapes, the shear stresses increase from zero at the center to a maximum at the center of the longer faces, but they decrease along the edge and are zero again at the corners. When considering a small element in the member subjected to pure torsion, only shear stresses develop and, therefore, the principal tension and compression stresses equal the shear stress. Applying Saint-Venant’s theory to concrete, the member would fail in a brittle manner once the maximum shear stress equals the concrete tensile cracking strength. After cracking, the elastic theory does not accurately describe the concrete behavior because various nonlinearities exist, such as the stress-strain behavior in compression, post-peak softening in tension, and tensile cracking. When the elastic theory is used, it considerably underestimates the torque at failure of a plain concrete member because concrete exhibits a complex structural response (Hsu 1984; Karayannis 2000). Other theories, such as the plastic theory and skew-bending theory, were developed to estimate the failure strength of plain concrete members. The plastic theory is not theoretically satisfactory because it assumes that concrete develops plasticity. As in the elastic theory, failure is assumed to occur when the maximum principal tensile stress of concrete reaches the tensile strength. Concrete, however, does not develop significant plastic behavior in tension and the torsional failure of plain concrete is brittle ( Hsu 1984). Unlike the elastic theory, the plastic theory tends to overestimate the failure strength. The skew-bending theory assumes that plain concrete members subjected to torsion fail by bending about an axis parallel to the wider cross-sectional face and inclined at an angle of 45 degrees with respect to the longitudinal axis. Failure occurs when the tensile stress on

the inclined face due to flexure reaches a reduced modulus of rupture of concrete (Hsu 1968b). The formula derived is T u = (1/3) x2 y(0.85 f r ). The torsional stress factor is a constant 1/3, which lies between those of the elastic theory and the plastic theory, as shown in Fig. 4.2. The skew-bending theory accurately describes the failure of concrete elements with rectangular cross sections, but it should be considered as an assembly of component rectangles when applied to flanged sections (Hsu 1968b, 1984). For thin-walled tubes, the relationship between torsional shear stress τ (= q / t d) and the torsional moment T can be found from Bredt’s Eq. (4.2). Collins (1973) suggested that if a solid section was replaced by a tube with the same exterior dimensions but with a wall po, the cracking torques could still be thickness of 0.75 Ao / determined from Eq. (4.2). Figure 4.2 compares the torsional shear stress τ for solid rectangular sections predicted by elastic theory, plastic theory, skew-bending, and thin-walled tube model. By further assuming that the wall thickness t d = pcp and that Ao = (2/3) Acp, the torsional stress τ can 0.75 Acp / pcp) be related to T by the parameter ( Acp2 /

τ=

T

2 Ao t d

=

T 2 cp

( A / pcp )

(4.2)

Equation (4.2) was embraced by CSA-A23.3-77 and later adopted by ACI 318-95. For example, the ACI code uses it to address compatibility torsion of spandrel beams. When the nominal torsional stress τn needed to form a torsional plastic hinge was found to be 0.33 √ f c′ MPa (4√ f c′ psi) (Hsu and Burton 1974; Hsu and Hwang 1977), the design torsional pcp) moment T n of the spandrel beam was 0.33 f c′ ( Acp2 / pcp) psi). MPa (equivalent to 4 f c′ ( Acp2 / In addition to the aforementioned theories, there are approaches to the torsional behavior of concrete elements that are based on truss models. Because these models are developed on modeling the flow of forces in a cracked reinforced concrete element, as stated by Karayannis (1995), these approaches are more suitable for the description of the post-elastic behavior of reinforced concrete elements, with rectangular cross sections reinforced in the longitudinal and transverse directions. Karayannis and Chalioris (2000) confirmed that a plain concrete member subjected to pure torsion fails in tension due to shear. The behavior of concrete elements in torsion is therefore primarily governed by the material tensile response, particularly its tensile cracking characteristics. Tension cracking makes the plain concrete behavior in torsion nonlinear. Before the peak torque is reached, some microcracking and softening regions are developed. The cracking is a localized phenomenon, and its boundary is the strain softening region called the fracture process zone. Bažant and Oh (1983) regarded this tension-softening region as being caused not only by microcracking but also by bond rupture; therefore, the fracture process zone has to be assumed wider than the region of visible microcracks. The zone width can be considered as a material property that, except for the influence of the stress distribution, depends


15

on the nature and size of the aggregate. The role of the size effect on the fracture process zone width should also be considered. Between the microcracks of this zone, there are less damaged or even elastic parts; this can be considered a result of the strong tendency for eventual localization of the damage and the simultaneous recovery of the undamaged or less-damaged parts between the final cracks.

4.3—Reinforced concrete Before the concrete cracks, a member under pure torsion is assumed to behave elastically with a nearly linear torquetwist curve. After cracking, the resistance mechanism changes and stirrups and longitudinal reinforcement carry most of the tensile load whereas the concrete carries the compression. The longitudinal reinforcement tensile strains cause the member to lengthen. Increased torque causes a nearly linear increase in the beam length. A longitudinal restraint acts as a compressive prestress force and increases the torsional strength. Members reinforced for torsion typically have closed stirrups and at least one longitudinal bar in each stirrup corner, as originally required in ACI 318-63. Members with only longitudinal reinforcement behave similarly to a plain concrete member in regards to torsional rigidity and torsional strength, and they fail after cracking in a brittle manner. An uncracked beam loaded in pure torsion has only shear stresses on a plane perpendicular to the longitudinal axis. The associated principal tensile stress causes inclined cracks on the beam surface in the principal compressive stress direction. These cracks spiral around the beam. The tensile cracks are due to strain in the principal tensile stress direction, and they affect the concrete acting in compression. The tensile strains have the effect of weakening the concrete’s ability to withstand compression in the perpendicular direction. This is known as “softening” of the concrete compressive strength. Tests performed by Vecchio and Collins (1982, 1986) identified the principal tensile strains as the primary variable influencing tension softening and quantified them. Continued work by Belarbi and Hsu (1994, 1995), Pang and Hsu (1995), Hsu and Zhang (1997), Hsu and Zhu (2002), and Hsu and Mo (2010) has improved the understanding of compression softening. Other studies, such as those of Vecchio and Collins (1993) and Koutchoukali and Belarbi (2001), have also been reported based on further calibration of their original models. After cracking, the longitudinal and transverse reinforcement are in tension, and the concrete struts between the diagonal cracks are in compression, as shown in Fig. 4.3a. The faces of the beam are warped as the beam is twisted, causing bending stresses, in addition to compressive stresses, in the concrete struts (Mitchell and Collins 1974). The concrete between inclined cracks is capable of acting in tension and will increase the overall beam torsional stiffness. This effect, known as tension stiffening, is most predominant immediately after the first cracks appear and decreases with increased torsion. The volumetric ratios ρl and ρt have a significant effect on the behavior of reinforced concrete beams subject to pure torsion. The ratios affect the angle of the inclined cracks on

Fig. 4.3a—Diagonal cracks in members under pure torsion ( Mitchell and Collins 1974).

the surface, crack width, beam strength and stiffness, and failure mode. The reinforcement ratios have a strong influence on the inclined crack angles and the size of the cracks. The crack width is related to the principal tensile strain. When the volumetric ratios ρl and ρt are equal, θ is approximately 45 degrees and the cracks have the smallest width because the strains in the longitudinal and transverse directions are also equal. For beams with ρt larger than ρl , the angle is greater than 45 degrees relative to the horizontal and the strain larger in the longitudinal direction, which causes wider cracks. Similarly, in beams with ρt less than ρl , the angle is less than 45 degrees relative to the horizontal, the transverse strain is larger, and the cracks are also wider than when θ is 45 degrees (Mitchell and Collins 1974; Hsu 1993). The volumetric ratios of longitudinal and transverse reinforcement to concrete affect the failure mechanism. Large ρl and ρt result in reinforcement stresses less than yield when the concrete reaches its ultimate compressive strain at the surface. In this condition, the concrete is crushed and the beam experiences sudden and brittle failure. Beams of this type are called over-reinforced. Figure 4.3b shows an example of beam failure boundaries (Leu and Lee 2000). The condition known as balanced failure point is met when concrete reaches its ultimate strain at the point where both transverse and longitudinal reinforcement begin to yield simultaneously. Beams with ρl and ρt less than the reinforcement ratios at the balanced condition are called underreinforced (Region I). Under-reinforced beams are capable of continued twist as the reinforcement yields, producing a ductile failure. Over-reinforced beams are in Region IV where the failure is brittle. For lower ratios of either ρl or ρt , the longitudinal or transverse reinforcement yields before the concrete is crushed. Beams of this type are called partially under-reinforced (Regions II and III). The volumetric ratios of longitudinal and transverse reinforcement have a strong influence on the beam’s postcracking torsional stiffness and strength as shown in Fig. 4.3c. For under-reinforced beams with equal volumetric ratios ρl and ρt , an increase in the total reinforcement increases the torsional rigidity, which is the slope of the torque-twist curve

16


Fig. 4.3d—Effects of concrete cover on the torque-twist behavior ( Mitchell and Collins 1974). (Note: 1 in.-kip = 0.113 kN·m.) Fig. 4.3b—Failure boundaries for f c′ = 4.0 ksi (27.5 MPa) and f y = 45.2 ksi (311.9) MPa ( Leu and Lee 2000).

Fig. 4.3c—Torque-twist curves of beams with various percentages of reinforcement ( Hsu 1968a). (Note: 1 in.-kip = 0.113 kN·m; 1 in. = 0.0254 m.)

after cracking. It also increases the torsional yield strength, which is the amount of torque the beam can sustain at the onset of yielding. As the total amount of reinforcement increases, however, the post-yield twist decreases. Therefore, an increase in the total amount of reinforcement results in a stiffer and stronger beam, but at the cost of decreased ductility. The minimum amount of reinforcing steel is the amount causing the reinforcement to yield at the same torque that causes cracking. Where this occurs, the total reinforcement ratio, including both longitudinal and transverse reinforcement, is approximately 0.01 for normal-strength concrete (Hsu 1984). A beam with less reinforcement than the minimum fails in a brittle manner after cracking. The behavior of solid members compared with hollow ones

shows that the inner core of solid members is not effective after cracking. The cracking strength of hollow members is less than that of solid members because, before cracking, the shearing stresses are acting over the entire section including the inner core. After cracking, however, the outer perimeter of the cross section resists the larger part of the torque with minimum contribution from the inside core; torsional strength and general behavior are similar in solid and hollow sections after cracking as long as wall thickness is large enough to accommodate the shear flow zone (Hsu 1968a). Leonhardt and Schelling (1974) confirmed these findings when they performed an extensive test series on beams in pure torsion where they systematically varied the cross section type, including rectangular with different aspect ratios, circular, and hollow box, T-shaped, and inverted L-shaped beams. They made two observations: 1) the importance of detailing by properly anchoring the stirrups and using small stirrup spacings; and 2) the importance of softening concrete in the inclined struts. High concrete shearing stresses outside of the stirrups create potential for concrete spalling under high torques. To satisfy equilibrium, the concrete compressive stress that resists torsion after cracking changes directions at the outside corners of the beam. The resultant tensile stress in the concrete causes the concrete cover to spall off when the tensile stress is high enough. Typically the spall occurs at the stirrups. The resulting section is smaller and has a reduced torsional strength (Mitchell and Collins 1974). Mitchell and Collins (1974) tested two beams, PT5 and PT6, to investigate the effects of spalling. The beams had similar reinforcing cages and concrete strength, but PT5 had a cover of 1.5 mm (1/16 in.) and PT6 had a cover of 40 mm (1-9/16 in.). The results of the tests showed that, although the concrete outside the hoop reinforcement had a significant effect on the cracking torque, it had little effect on the peak torque after significant spalling occurred (Fig. 4.3d). In tests performed by Rahal and Collins (1995b), specimens with small covers did not experience spalling until after the torsional strength had been reached. Beams with larger covers spalled before reaching the torsional strength.


17

Fig. 4.4a—Cowan’s failure criteria ( Hsu 1984).

In these tests, the beams had stirrup dimensions of 245 x 545 mm (9.65 x 21.5 in.), and cover thicknesses of 22.5 and 42.5 mm (0.89 and 1.637 in.) for the thin and thick concrete covers, respectively. The concrete cover thickness is the critical parameter for spalling. Experiments have shown that spalling occurs if the cover thickness is greater than 30 percent of the ratio of area to the perimeter of the cross section. For this calculation, the sectional area and perimeter do not consider the area removed by voids in hollow sections (Rahal and Collins 1996).

4.4—Prestressed concrete Adding longitudinal prestressing to a concrete member subjected to torsion will increase its cracking torque and, to a lesser extent, its failure torque. The longitudinal prestressing force creates a compression-shear biaxial state of stress that delays cracking of concrete. This effect is illustrated by Cowan’s failure criteria, shown in Fig. 4.4a. The Mohr-Coulomb failure criterion has been simplified into two straight lines in Fig. 4.4a(a). If the Mohr circle of stresses touches the straight line BD described by t = c – tanϕ, where ϕ is the friction angle (a typical value for concrete is 37 degrees), failure is by sliding. If the Mohr circle touches the tension cutoff line DE described by σ = f t′ , failure is by separation. In Fig. 4.4a(b), σmax = OE = FE – FO = FP – FO. The variable σmax can then be expressed as 2

 f  σ max =  pc  + τ 2 −  2  τ = f t ′ 1 +

f pc f t ′

f pc

2

(4.4a)

(4.4b)

Cowan’s failure envelope using Mohr’s circle is a simple and concise model for explaining the compression-shear state of stress. Equation (4.4b) for shear stress is derived from the geometry of Cowan’s failure envelope, and it is a function of

Fig. 4.4b—Ultimate torque-versus-reinforcement factor for prestressed and reinforced beams under pure torsion ( Hsu 1984). (Note: 1 in.-kip = 0.113 kN·m.)

only the prestress and the concrete tensile strength. In effect, the longitudinal prestress increases the cracking strength, as shown in Fig. 4.4b. Expressions for prestressed member torsional strength are similar to those of reinforced concrete members with a factor that increases the strength due to the prestress. Pure torsion tests have shown that the prestressed beams appear to require approximately the same minimum torsional reinforcement as the nonprestressed beams (Hsu 1984). Similar to beams without prestressing, prestressed beams without web reinforcement fail shortly after cracking. Testing has shown that as the amount of longitudinal reinforcement and prestress increase, the torque at cracking,

18


Fig. 4.4c—Effect of longitudinal reinforcement ( Mitchell and Collins 1978 ). (Note: 1 in.-kip = 0.113 kN·m; 1 in. = 0.0254 m.)

hoop yield, and ultimate increase, whereas the twist at failure decreases. This effect is illustrated by the experimental work shown in Fig. 4.4c (Mitchell and Collins 1978). Prestressing does not influence ultimate torsional strength if the longitudinal reinforcement does not yield before failure, as shown in Specimen P5 of Fig. 4.4c. Prestressed beams under sustained, long-term torque exhibit increased rotation due to concrete creep. Beams tested to failure after being subjected to sustained torque showed no reduction in strength or stiffness. After a sustained torque, the ratio of beam rotation to the instantaneous rotation increases with larger values of torque and decreases with increased amounts of transverse reinforcement (Allos and Rashid 1989).

4.5—High-strength concrete High-strength concrete (HSC) is more brittle than normalstrength concrete (NSC). In some ways, NSC can be treated as a perfectly plastic material, whereas HSC very much resembles a perfect elastic and brittle material ( Rasmussen and Baker 1995; Iravani 1996; ACI Committee 363 1992). The ascending portion of the stress-strain curve is steep and approximately linear, whereas the descending curve almost vanishes and shows no strain softening. High-strength concrete has a peak load strain that is higher than the typical 0.0020 of NSC and an ultimate load strain less than the typical 0.0035 (Wafa et al. 1995). In the case of pure torsion, the use of HSC allows a higher service stress in the reinforcing bars and produces a higher stiffness in the structure, enhancing both the strength and stiffness of a member. For a given amount of reinforcement, the initial crack width is larger for HSC because larger reinforcement strains are induced at cracking, but the number

of cracks is fewer in HSC. Wider cracks in HSC could also be the result of reduced aggregate interlock. Shortly after the longitudinal reinforcement yield, the width of the largest crack increases rapidly. After this point, all further deformation occurs at this crack. This behavior is exaggerated in high-strength beams due to the smooth-faced cracks that are less effective in transmitting shear ( Koutchoukali and Belarbi 2001). Rasmussen and Baker (1995) observed diagonal cracks in both NSC and HSC beams on the larger faces of the cross section because the shear stresses are largest on these faces. This is similar to the assumptions of the elastic theory. After first cracking, the number of cracks parallel to the initial cracks increases as the applied torque increases. In this way, characteristic spiral cracks develop around the beam. Cracking of an HSC beam is more brittle than that of an NSC beam because in HSC, not only the matrix but also the aggregate splits (the paste in HSC is strengthened relative to the aggregate), producing a more brittle response. That is, cracks in NSC matrix propagate around the aggregate, whereas cracks in HSC also pass through the aggregate. Before cracking, surface compressive principal strains for HSC beams are smaller compared with those of NSC beams because of the higher elastic modulus for HSC. Whereas HSC delays beam cracking, after cracking the shape and angle of inclination of cracks are the same for all concrete strengths (Rasmussen and Baker 1995). An HSC beam has a higher torsional stiffness than an NSC beam for a given torque and a given cross section because of the higher modulus of elasticity for HSC. Figure 4.5 shows the torquetwist diagrams for an NSC (36.3 MPa [5.1 ksi]) beam and an HSC (109.8 MPa [15.9 ksi]) beam under pure torsion, reported by Rasmussen and Baker (1995). They showed that the torque-twist curves are linear in the uncracked range and the HSC beam has greater torsional stiffness. After cracking, the torsional stiffness is reduced significantly in both beams. The post-cracking stiffness of an HSC beam remains almost linear up to failure, whereas the NSC beam curve is nonlinear. The surface compression strain at failure for an HSC beam is slightly less than that for an NSC beam, but because of the higher modulus of rupture and consequent higher torsional stiffness for HSC, the twist at peak torque for both NSC and HSC beams is approximately the same (Rasmussen and Baker 1995). The increase in both longitudinal and transverse reinforcement stresses immediately after cracking is larger for an HSC beam than an NSC beam, resulting in brittle cracking of HSC. For the same torque, the reinforcement stresses in an HSC beam are less than in an NSC beam. Therefore, for a given torque, the required amount of beam reinforcement is less using HSC. A general comparison of the ratio of transverse reinforcement stresses and longitudinal reinforcement stresses at failure shows that this ratio increases when concrete strength increases. For HSC, the bond between concrete and reinforcement is better due to denser concrete. Therefore, the vertical component of diagonal concrete compression is easily transferred to the transverse reinforcement so that it attracts a relatively larger amount of torque using HSC. The lower bond between the


stirrup and concrete using NSC results in a portion of the vertical component of the diagonal concrete compression being carried by longitudinal reinforcement bending and shear (Rasmussen and Baker 1995). Beams elongate because of increased cracking state under increasing torque. The longitudinal strains at failure increase as concrete strength increases. For the beams tested by Rasmussen and Baker (1995), the longitudinal elongation at failure for the HSC beam (B110.1) was approximately 35 percent larger than for the NSC beam (B30.3). For a given torque and given cross section, however, the longitudinal elongation of an HSC beam was approximately 50 percent lower than that of an NSC beam. Accounting for the spalling effect is essential for a correct evaluation of the ultimate torque and failure mechanism of over-reinforced HSC beams. In the case of some over-reinforced NSC concrete beams, the phenomenon of spalling does not take place, and the torsional strength is evaluated considering the entire cross section. In the case of over-reinforced HSC beams, the cover is reasonably neglected. In fact, due to the higher load-carrying strength of HSC with respect to NSC, spalling of the concrete cover from the stirrups appears to occur more frequently (Cerioni et al. 1998). This is because, even though HSC is stronger, the tensile strength does not increase in the same proportion as the compressive strength. Based on the results of 14 HSC prestressed beams tested under pure torsion, Wafa et al. (1995) concluded that increasing the concrete strength and the prestressing force increases the cracking strength, ultimate torsional strength, and torsional stiffness. Wafa et al. reported that among the several models used—the space truss model with softening of concrete (Hsu and Mo 1985a), space truss model with spalling of concrete cover (Mitchell and Collins 1974), and skew-bending theory (Hsu 1984)— the space truss theory with softening of concrete gave the best estimate of the test beams’ torsional strength. Wafa et al. (1995) observed that the tensile strength increase is not in direct proportion to the compressive strength. The ratios of the modulus of rupture to the compressive strength were 9 percent and 11.4 percent, respectively, for the HSC and NSC. The post-cracking torsional stiffness, as represented by the slope of the torque-twist curve, is influenced by the concrete strength, the amount of torsional reinforcement, and the aspect ratio. The higher the concrete compressive strength, the higher the stiffness is, irrespective of the aspect ratio and the prestressing force. Torsional strength increases with an increase in concrete strength and prestressing level and with a decrease in aspect ratio. All tested beams exhibited compression zones located on the longer sides of the beams. Tension cracks initially formed on one longer face and progressed to form spirals. Crushing on the other longer face characterized the failure pattern.

4.6—Concluding remarks Saint-Venant’s elastic theory accurately describes plain concrete members’ torsional behavior at low torque and before cracking. Applying Saint-Venant’s theory to concrete,

19

Fig. 4.5—Torque-twist relation for NSC and HSC beams ( Rasmussen and Baker 1995). (Note: 1 m = 39.4 in., 1 kN·m= 8.85 in.-kip; 1 MPa = 0.145 ksi.)

the plain concrete member fails in a brittle manner once the maximum shear stress equals the concrete tensile cracking strength. For reinforced concrete members, because the elastic theory does not accurately describe the member behavior after cracking, other theories such as truss model and skew-bending were developed to estimate the failure strength. In members with mild reinforcement, the longitudinal and transverse reinforcement are in tension after cracking and the concrete struts between diagonal cracks are in compression. The beam faces are warped as the beam is twisted, causing bending and compressive stresses in the concrete strut. The total amount and the relative amount of longitudinal and transverse steel reinforcement have a significant effect on the behavior of reinforced beams subject to pure torsion. The behavior of solid members compared with hollow ones shows the inner core of solid members is ineffective after cracking, making the cracking strength of hollow members less than solid ones. Torsional strength and general behavior, however, are similar in solid and hollow sections. The most significant effect of adding longitudinal prestressing to a concrete member subjected to torsion is the increase in its cracking strength. Testing has shown that as the amount of longitudinal reinforcement and the prestress force increase, the torque at cracking, hoop yield, and ultimate strength all increase whereas the twist at failure decreases. The use of HSC in torsional members allows a higher service stress in the reinforcing bars and produces a higher sectional stiffness in the structure. Cracking of an HSC beam is more brittle than that of an NSC beam; in HSC, both the matrix and aggregate split, producing a more brittle response. An increase in twist immediately after cracking is more significant for HSC beams because of the higher energy released, with more brittle crushing as concrete strength increases. The post-cracking stiffness of an HSC beam remains almost linear up to failure, whereas the curve for an NSC beam is nonlinear.

20


CHAPTER 5—ANALYTICAL MODELS FOR PURE TORSION 5.1—General In 1929, using space truss concepts, Rausch developed an analytical model to predict the torsional strength of reinforced concrete members. Unfortunately, Rausch’s equation was unconservative by more than 30 percent for underreinforced beams (Hsu 1968a,b). Although Rausch’s model assumed the lever arm area Ao to be the area within the hoop reinforcement centerline, Aoh, the actual Ao would be much smaller in members with a high percentage of reinforcement when concrete softening is considered. The correct determination of the lever arm area Ao and the centerline of shear flow depend on the means used to determine the thickness of the shear flow zone, t d. Since the late 1960s, the truss model theory has undergone four major developments. First, Lampert and Thürlimann (1968, 1969) introduced the variable angle truss model and discovered the bending phenomenon in diagonal concrete struts. Second, Collins (1973) derived compatibility equations to determine the angle of the diagonal concrete struts. It should be noted that Mitchell and Collins (1974) developed a space truss model with concrete cover spalling to determine the shear flow zone thickness. Third, the softening phenomenon in the concrete struts, discovered by Robinson and Demorieux (1972), was quantified by Vecchio and Collins (1981) using a softened coefficient. Fourth, combining the equilibrium, compatibility, and softened stress-strain relationships, Hsu and Mo (1985a) developed a softened truss model (STM) theory to determine the shear and torsional behavior of reinforced concrete members throughout the post-crack loading history up to the peak strength. Using the STM theory, shear flow zone thickness t d can be determined accurately to calculate the torsional strength of reinforced concrete members. The test data for solid and hollow beams suggests that, once cracking has occurred, concrete in the center member has little effect on torsional strength of the cross section and is ignored (Hsu 1968b; Lampert and Thürlimann 1968; Leonhardt and Schelling 1974). Beams are considered equivalent tubular members. This observation is the basis of the torsion design procedures introduced in ACI 318-95 (MacGregor and Ghoneim 1995). 5.2—Equilibrium conditions 5.2.1 Shear elements in shear flow zone —A prismatic reinforced concrete member subjected to an external torque T is shown in Fig. 5.2.1(a). This external torque is resisted by an internal torque formed by the circulatory shear flow q along the cross section periphery. This shear flow q occupies a zone, called the shear flow zone, with a thickness denoted by t d. The thickness is a variable determined from equilibrium and compatibility conditions; it is different than the given wall thickness t of a hollow member. An element A in the shear flow zone (Fig. 5.2.1(a)) is , as shown in Fig. 5.2.1(b). subjected to a shear stress τl t = q / td The in-plane equilibrium of this element should satisfy Eq. (5.2.1a) through (5.2.1c) as originally derived by Nielsen (1967), Lampert and Thürlimann (1968, 1969), and Mitchell

Fig. 5.2.1—Hollow box subjected to torsion ( Hsu 1993).

and Collins (1974), but given herein in the most general form (Hsu 1993)

σ = σ d cos2 θ + σ r sin2 θ + ρ f + ρ p f p

(5.2.1a)

σ t = σ d sin 2 θ + σ r cos2 θ + ρt ft + ρtp f tp

(5.2.1b)

τ t = ( −σ d + σ r ) sin θ cos θ

(5.2.1c)

l

l

l

l

l

l

In Eq. (5.2.1a) through (5.2.1c) the reinforcement ratios ρl , ρt , ρ pl , and ρtp, should be taken with respect to the shear flow zone thickness t d as described in 5.3. These three equa tions are expressed in the form of stress transformation to show that they satisfy Mohr’s stress circle. Because Eq. (5.2.1a) and (5.2.1b) involve the reinforcement ratios ρl , ρt , ρ pl , and ρtp, they are coupled with the compatibility equations through the variable t d. The thickness t d is a geometric variable that is determined by equilibrium conditions as well as compatibility conditions. The determination of t d is similar to that of the flexural neutral axis, which requires plane section compatibility conditions. 5.2.2 Bredt’s equilibrium equation—Equations (5.2.1a) through (5.2.1c) were derived from the equilibrium of a member element in the shear flow zone. To maintain equilibrium of the whole cross section, a fourth equation derived by Bredt (1896) should be satisfied. For a shear flow zone thickness t d, the shear stress τl t is

τ t = l

T

2 Ao t d

(5.2.2)

This also introduces an additional variable, the torque T . The shear flow zone thickness t d is incorporated in Eq. (5.2.2) explicitly, as well as implicitly through Ao, which is a function of t d.

5.3—Compatibility conditions 5.3.1 Shear elements—As shown in Fig. 5.2.1(a), the element A in the shear flow zone is subjected to a shear


21

stress. The in-plane deformation of this element should satisfy three compatibility equations (Eq. (5.3.1a) through (5.3.1c)) as originally derived by Baumann (1972) and Mitchell and Collins (1974). The equations are in the form of strain transformation to show they satisfy Mohr’s strain circle (Hsu 1993)

ε = ε d cos2 θ + εr sin2 θ

(5.3.1a)

ε t = ε d sin 2 θ + εr cos2 θ

(5.3.1b)

l

γ t l

2

= ( −ε d + εr ) sin θ cos θ

(5.3.1c)

Element A is subjected to in-plane shear as well as out-ofplane deformation as demonstrated (5.3.2). 5.3.2 Shear strain due to twisting—When a tube is subjected to torsion, the relationship between the shear strain γ lt in the tube wall and the member angle of twist, Φ, can be derived from the compatibility condition of warping deformation (Bredt 1896).

Φ=

po

2 Ao

γ t l

(5.3.2)

It is clear from Eq. (5.3.2) that the angle of twist, Φ, will produce a shear strain γ lt in the shear flow zone elements. This shear strain γ lt will induce the reinforcement strains εl and εt in the d -r direction. The relationships between the strains in the l -t direction ( εl , εt , and γ lt ) and the strains in the d -r direction (εd and εr ) are described by the three transformation equations, Eq. (5.3.1a) through (5.3.1c). 5.3.3 Bending of diagonal concrete struts—In a torsional member, the angle of twist, Φ, also produces warping in member walls that in turn causes bending in the concrete struts. The concrete struts are therefore subjected to compression due to circulatory shear and bending due to warping of walls. The relationship between the angle of twist, Φ, and the bending curvature of concrete struts, Ψ, is given in Eq. (5.3.3). Figure 5.3.2(a) shows a box member with four walls of thickness t subjected to torsional moment T . Each wall contains a shear flow zone with a thickness of t d. The perimeter of the centerline of shear flow q has a width of l q along the top wall. The length of the member in the longitudinal direction is taken to be l qcosθ, so that the diagonal line in the center plane of the shear flow in the top wall OABC has an angle of inclination θ with respect to the longitudinal axis. When this member is subjected to an angle of twist, Φ, the center plane OABC becomes a hyperbolic paraboloid surface OADC, as shown in Fig. 5.3.2(b). The plane edge CB rotates to the position CD through an angle Φl qcosθ. The curve OD has a curvature of Ψ that is related to Φ and θ by

Ψ = Φsin2θ

(5.3.3)

Fig. 5.3.2—Bending of a concrete strut in the wall of a box section subjected to torsion ( Mitchell and Collins 1974; Hsu and Mo 1985a).

Equation (5.3.3) was originally derived by Mitchell and Collins (1974) using Mohr’s circle, as shown in Fig. 5.3.2(c). The illustration in Fig. 5.3.2 and the derivation of bending curvature are based on Hsu (1993). Although the imposed curvature is illustrated by a rectangular box section, this equation is applicable for any arbitrary, bulky section with multiple walls. 5.3.4 Strain distribution in concrete struts—The curvature Ψ derived in Eq. (5.3.3) produces a nonuniform strain distribution in the concrete struts. Figure 5.3.4(a) shows a concrete strut unit width in a hollow section with a wall thickness of t . The tension area in the cross section inner portion is disregarded. The area in the outer portion, which is in compression, is considered effective in resisting the shear flow. The compression zone depth from the neutral axis to the extreme compression fiber is defined as the shear flow zone thickness t d. Within this thickness t d, the strain distribution is assumed to be linear, as shown in Fig. 5.3.4(b), based on Bernoulli’s

22


Fig. 5.3.4—Strains and stresses in concrete struts ( Hsu and Mo 1985a).

plane section hypothesis used in bending theory ( Mitchell and Collins 1974). The thickness t d can therefore be related to the curvature Ψ and the maximum surface strain εds by the simple relationship t d =

ε ds Ψ

(5.3.4a)

The average strain εd can be related to the maximum strain εds, as shown in Fig. 5.3.4(d), by

ε d =

ε ds

(5.3.4b)

2

Equations (5.3.2), (5.3.3), (5.3.4a), and (5.3.4b) are the four additional compatibility equations for torsion. They introduce four additional variables, Φ, Ψ, t d, and εds.

5.4—Stress strain relationships 5.4.1 Concrete struts—The stress-strain relationship of concrete in compression, as shown in Fig. 5.3.4(e), can be expressed analytically by two branches of parabolic curves (Hsu and Mo 1985a) Ascending branch:

  ε d   ε d  2  σ d = ζ f c′ 2   −      ζε o   ζε o  

ε d <1 ζε o

(5.4.1a(a))

Descending branch:

  ε d / ζε o − 1 2  σ d = ζ f c′ 1 −       2 / ζ − 1 

ε d >1 ζε o

(5.4.1a(b))

where ζ is a softening coefficient. Section 5.6.1 discusses the softening coefficient as presented in Eq. (5.6.2d) with values ranging between 0.2 and 0.6. The stress-strain relationship of concrete in tension is irrelevant, if the concrete tensile stress σr is assumed to be zero in the equilibrium Eq. (5.2.1a) through (5.2.1c). 5.4.2 Mild reinforcement —When the tensile strength of the concrete is neglected, the stress-strain relationship of mild reinforcement is taken as the elastic perfectly plastic relationship, expressed as f l = E sεl

ε < ε y l

(5.4.2a(a))

f l = f yl

ε ≥ ε y

(5.4.2a(b))

f t = E sεt

εt < εty

(5.4.2b(a))

f t = f ty

εt ≥ εty

(5.4.2b(b))

l

l

l

5.4.3 Prestressing reinforcement —In the case of prestressed concrete, the nonlinear stress-strain relationship of prestressed strands is given (Mattock 1979) f p = E ps(εdec + εs)

f p ≤ 0.7 f pu

(5.4.3a(a))


f p

=

E ps ′ (ε dec

23

+ εs )

 1 + ( E ′ (ε + ε )) / f R  pu }  { ps dec s 

1 / R

f p > 0.7 f pu

(5.4.3a(b)) In structures where prestress is applied in both directions, such as nuclear containment vessels, Eq. (5.4.3a(a)) and (5.4.3a(b)) are applicable to prestressing reinforcement in both the longitudinal and transverse directions. When applied to longitudinal reinforcement, f p is equal to f lp, and εs is equal to εl. When applied to transverse reinforcement, f p is equal to f tp, and εs is equal to εt . The equilibrium equations, compatibility conditions, and stress-strain relationships described are used by two main theories for torsional analysis, namely, the compression field theory (CFT) (Mitchell and Collins 1974) and the STM (Hsu and Mo 1985a). The primary difference between these two theories lies in the stress-strain relationship of concrete in compression. The STM uses a softened stress-strain curve as shown in Eq. (5.4.1a(a)) and (5.4.1a(b)), where ζ is a softening coefficient that varies from approximately 0.2 to 0.6. In CFT, the stress-strain curve is not softened, and the softening coefficients ζ in Eq. (5.4.1a(a)) and (5.4.1a(b)) are taken as unity. To compensate for the unconservative nature of this non-softening assumption, CFT also assumes spalling concrete cover before reaching the peak torque. As a result, CFT is also called the “spalling theory.” The CFT and the STM are described in detail in 5.5 and 5.6, respectively, by combining the stress-strain relationships of concrete and reinforcement given in Eq. (5.4.1a(a)) and (5.4.1a(b)), and through (5.4.3a(b)) with the equilibrium Eq. (5.2.1a) through (5.2.1c) and (5.2.2) and compatibility Eq. (5.3.1a) through (5.3.4b). The solution algorithms of the two theories are also different.

5.5—Compression field theory Collins and Mitchell (1980) and Mitchell and Collins (1974, 1978) proposed the CFT for structural concrete in torsion. This section provides a basic summary of the theory. Before the truss analogy equilibrium equations can be used to design a member for torsion, the diagonal struts’ inclination should be determined from the compatibility equations. Wagner (1929) dealt with an analogous problem in studying the post-buckling shear resistance of thin-webbed metal girders. He assumed that after buckling, the thin webs would not resist compression and the shear would be carried by a field of diagonal tension. To determine the angle of inclination of the diagonal tension, Wagner considered the systems’ deformations. He assumed that the angle of inclination of the diagonal tensile stress would coincide with the angle of inclination of the principal tensile strain. This approach became known as Tension Field Theory. Applying Wagner’s approach to reinforced concrete, the concrete is assumed to be unable to carry tension, and the shear is assumed to be carried by a diagonal compression field. The inclination θ of the diagonal compression represents the principal strain

Fig. 5.5a—Spalling of the concrete cover due to torsion ( Mitchell and Collins 1974).

direction; therefore, it can be derived from compatibility Eq. (5.3.1a) and (5.3.1b), by combining them to form tan2 θ =

ε + ε d εt + ε d l

(5.5a)

Equation (5.5a) shows the angle θ can be expressed as a function of the strain in the concrete diagonals ( εd ), the strain in the longitudinal reinforcement ( εl ), and the strain in the transverse reinforcement ( εt ). The full behavioral response of reinforced concrete members subjected to torsion can be predicted by using compatibility (Eq. (5.5)), equilibrium ( Eq. (5.2.1a) through (5.2.1c)), the stress-strain relationships of concrete [Eq. (5.4.1a(a)) and (5.4.1a(b))], and those for mild reinforcement [Eq. (5.4.2a(a)) through (5.4.2b(b))]. Besides these fundamental equations, additional torsional aspects were considered in the CFT. In resisting the torsion, not all of the concrete is effective in providing diagonal compressive stresses. Estimating the equilibrium of a corner element for a beam in torsion (Fig. 5.5a) reveals that the compression in the concrete tends to push off the corner whereas the tension in the hoops holds it in place. Because concrete is weak in tension, the concrete outside of the hoops spalls off at higher torsions. Because of concrete cover spalling, the effective outer surface of the concrete is assumed to coincide with the hoop centerline. Examining the deformed shape of the twisted beam in Fig. 5.5b reveals that the diagonal concrete stresses vary in magnitude over the thickness of the effective concrete tube, from zero at the inside to a value f ds corresponding to the strain εds at the effective outer surface. As in the case of

24


given in 5.2 and 5.3. The following solution algorithm is the approach used by Rahal (2000a,b). From the equilibrium of an element in the shear flow zone, shown in Fig. 5.2.1(b), two equations can be derived

∆ N At f t

q=

po

s

(5.5d)

and tan θ =

At f t po s

∆ N

(5.5e)

Substituting Eq. (5.5d) into Bredt’s Eq. (5.2.2), the torsional moment T can be related to the longitudinal and transverse reinforcement forces, ∆ N and A f t t, as T

Fig. 5.5b—Effective wall thickness of a twisted beam (Collins and Mitchell 1980).

flexure, this actual stress distribution can be replaced by a ao, uniform stress of α1 f c′ = f d acting over a depth of β1t d = where the stress block factors α1 and β1 depend on the shape of the concrete stress-strain curve and the value of surface compression strain εds. The centerline dimensions of the resulting tube of uniformly stressed concrete of thickness ao are assumed to define the shear flow path q. This path lies ao /2 inside the centerline of the hoop, as shown in Fig. 5.5b. Given the shear flow path, the terms Ao (the area enclosed by the shear flow) and po (the perimeter of the shear flow path) can be determined using Eq. (5.5b) and (5.5c), which assume a thin tube Ao

= Aoh −

ao

2

ph

po = ph – 4ao

(5.5b)

(5.5c)

∆ N At f t po

s

(5.5f)

Like flexure, the compression zone depth is a function of reinforcement tensile forces, and is derived from equilibrium conditions. In a shear element, however, the reinforcement includes both longitudinal and transverse reinforcement forces, ∆ N and A f t t . Eliminating θ from Eq. (5.2.1a) through (5.2.1c) results in ao

=

∆ N At f t + α fc′po αl 1 fc′s

(5.5g)

l

Once the compression block depth ao is known, the terms Ao and po in Eq. (5.5b) and (5.5c) can be calculated. To determine the longitudinal and transverse beam strains, εl and εt , which correspond to the chosen value of εds, the basic compatibility Eq. (5.3.1a) through (5.3.4a) were manipulated to develop two expressions

 α β f ′A s  εt =  1 1 c oh − 1 ε ds  2 ph At f t 

(5.5h)

 α β f ′A p  ε =  1 1 c oh o − 1 ε ds   2 ph ∆N

(5.5i)

l

Based on the equilibrium of an element in the shear flow zone, shown in Fig. 5.2.1(b), the shear flow, q, in a box section can be expressed in terms of the longitudinal reinforcement force, ∆ N , and the transverse reinforcement force, A f t t. The longitudinal reinforcement force ∆ N is assumed to be distributed uniformly along the shear flow path po. Equilibrium Eq. (5.2.1a) through (5.2.1b) are simplified in three respects: 1) for pure torsion, σl = σt = 0; 2) concrete tensile stress is neglected, σr = 0; and 3) when no prestressing reinforcement is used, ρ p l = ρtp = 0. In the CFT, several approximate algorithms have been developed to plot the torque-twist curve using the equilibrium and compatibility relationships

= 2 Ao

A trial-and-error process can be used to solve the eight variables (θ, ao, εt , εl , Ao, po, ∆ N , and A f t t) using Eq. (5.5a) through (5.5c), (5.5e), and (5.5h) through (5.5i). First, select a value of εds and assume a pair of forces, A f t t and ∆ N , from which the four variables θ, ao, εt , and εl , can be calculated from Eq. (5.5e), (5.5g), (5.5h), and (5.5i), respectively. Given the value of ao, the variables Ao and po can be calculated from Eq. (5.5b) and (5.5c). Based on the values of εt , εl , and εd (related to εds by geometry), another θ can be calculated from Eq. (5.5a). If the θ from Eq. (5.5a) is not equal


to the θ from Eq. (5.5e), assume another pair of A f ∆ N , t t and and repeat the cycle until Eq. (5.5a) and Eq. (5.5e) produce the same θ. Once these eight variables are solved, the torque T can be calculated from Eq. (5.5f) and the angle of twist, Φ, from Eq. (5.3.3). In Eq. (5.3.3), the curvature of the concrete struts, Ψ, can be obtained from Eq. (5.3.4a), where the neutral axis depth t d is equal to ao / β1, and εds is the chosen value. The torque-twist response (T -versus-Φ curve) is then determined by repeating these calculations for a series of εds values.

5.6—Softened truss model 5.6.1 General—The softened truss model is built on satisfying the equilibrium Eq. (5.2.1a) through (5.2.2), the compatibility Eq. (5.3.1a) through (5.3.4b), the stress-strain materials relationships in Eq. (5.4.1a(a)) and (5.4.1a(b)) for concrete, and Eq. (5.4.2a(a)) through (5.4.2b(b)) for reinforcement. The properties of concrete stress block and shear flow zone are derived from the softened stress-strain relationships of concrete. 5.6.2 Concrete stress block —As shown in Fig. 5.3.2, the diagonal concrete struts in the shear flow zone are subjected to axial stress and bending. The compressive strains in the concrete struts are assumed to vary linearly from εds at the surface to zero at the neutral axis, as shown in Fig. 5.3.4(b). The compression zone depth t d is defined as the shear flow zone. The distribution of compressive stresses within the thickness t d is shown by the solid curve in Fig. 5.3.4(c), based on the softened stress-strain relationship shown in Fig. 5.3.4(e). The peak stress is ζ f c′ and the average compressive stress is σd . The average stress σd of the concrete stress block in Fig. 5.3.4(c) can be expressed as follows

σd = ζ f c′

(5.6.2a)

where the coefficient k 1 is the ratio of the average stress to the peak stress. By integrating the stress-strain curve in Eq. (5.4.1a(a)) and (5.4.1a(b)), the coefficient k 1 can be expressed k 1

k 1

=

ε ds  1 ε ds  1− ζε o  3 ζε o  

ε ds ≤1 ε p

(5.6.2b(a))

 ζ2   1 ε ds  ζ 2 ε ds  1 ε d s  = 1 − + 1−  1− 2  ( 2 − ζ)2 ζε o  3 ζε o    ( 2 − ζ)   3 ζε o 

ε ds >1 ε p

(5.6.2b(b))

The coefficient k 1 is tabulated as a function of εds and ζ in the work by Hsu (1993), and equals approximately 0.8. The softening coefficient was first proposed by Zhang and Hsu (1998) as a function of tensile strain εr and concrete strength f c′

ζ=

5.8 f c′(MPa )

1 1 + 400ε r and

5.8 f c′(MPa )

≤ 0.9

(5.6.2c)

25

This equation is valid for HSC up to 100 MPa (14.5 ksi). As proposed by Belarbi and Hsu (1995), for NSC up to 42 MPa (6.0 ksi), ζ becomes

ζ=

0.9 1 + 400ε r

(5.6.2d)

The softening coefficient ζ usually varies in the range of 0.2 (0.045) to 0.6 (0.003). 5.6.3 Centerline of shear flow zone and formulas for A o and po—As shown in Fig. 5.3.4(c), the resultant force of the compression stress block C is located at a distance k 2t d from the surface, where the coefficient k 2 defines the location of the resultant force C. By integrating the concrete stress-strain curve given in Eq. (5.4.1a(a)) and (5.4.1a(b)), the coefficient k 2 is found to vary in the range of 0.40 to 0.45, depending on concrete strength. To simplify, the centerline of the shear flow is assumed to coincide with the centerline of the shear flow zone, located at a distance 0.5 t d from the extreme compression fiber. This assumption also results in a compatible agreement between theory and tests. Based on this simple centerline assumption of the shear flow, formulas for calculating the lever arm area Ao and the perimeter po for a thick tube, whose thickness is usually quite large with respect to the overall dimension of the cross section and different from the thin tube assumption in CFT when the softening of concrete is considered, are Ao

= Ac −

1 pc t d 2

+ t d2

po = pc – 4t d

(5.6.3a)

(5.6.3b)

When the softening coefficient ζ of concrete is considered, the shear flow zone thickness t d is usually large with respect to the overall cross section dimensions. The formula for calculating Ao, therefore, is expressed in Eq. (5.6.3a) and should include the third term t d2 . The thin tube formula used in the CFT (Eq. (5.5b)) is not used because it neglects the ao2 term. 5.6.4 Solution algorithm—The 18 governing equations for a torsional member are: four equilibrium, (Eq. (5.2.1a) through (5.2.2); seven compatibility, (Eq. (5.3.1a) through (5.3.4b)); seven constitutive (Eq. (5.4.2a(a)) through (5.4.3a(b)) and (5.6.2a) through (5.6.2d)). For a member subjected to pure torsion, the normal stresses σl and σt acting on an element in the shear flow zone are equal to zero ( σl = σt = 0). When the concrete tensile strength is neglected, σr = 0. Disregarding these three variables (σl , σt , and σr ) in the analysis, the 18 equations contain 19 unknown variables as follows: seven stress or force variables, including τl t, σd , f l, f t, f p tp, and T ; 10 strain or geoml , f etry variables, including εl , εt , γ lt , εd , εr , θ, Φ, Ψ, t d, and εds; and two material coefficients, including ζ and k 1. If one of the unknown variables is selected—for example, εd —then the remaining 18 unknown variables can be solved using the 18 equations.

26


Next, select the concrete strain εd as the independent variable because it varies monotonically from zero to maximum. The solutions for a series of εd values make it possible to trace the loading history. A solution algorithm is given by Hsu (1993) and is briefly described. This method requires the derivations of Eq. (5.6.5) through (5.6.8b). 5.6.5 Thickness t d as a function of strain—The thickness of the shear flow zone t d can be expressed in terms of strain using the seven compatibility equations: Eq. (5.3.1a) through (5.3.4b). The substitutions and manipulations result in t d =

Ao po

 ( − ε d ) ( εr − ε d )   ε −ε ε −ε  d )( t d )  (

(5.6.5)

l

The variable t d is expressed in terms of strains in all d , r , l , and t directions (εd , εr , εl , εt ). The compressive strain is defined as negative and tensile as positive. The variable t d is also involved in equilibrium equations through the terms Ao, po, ρl , and ρt . Therefore, the variable t d should first be assumed and then checked by Eq. (5.6.5). 5.6.6 Longitudinal strain εl as a function of longitudinal stresses f l, f lp —The strain εl can be related to the reinforce f p ment stresses f l and l by eliminating the angle α from the equilibrium Eq. (5.2.1a) using compatibility equations (Eq. (5.3.1c) to (5.3.4b)). These manipulations result in

ε = ε d + l

Ao ( − ε d ) ( −σ d )

( A f + A l

l

l p

f p l

)

(5.6.6)

The three unknown variables εl , f l, and f p l can be solved using Eq. (5.6.6) and the stress-strain relationships, Eq. (5.4.2a), (5.4.2b), (5.4.3a), and (5.4.3b), for longitudinal reinforcement. 5.6.7 Transverse strain εt as a function of transverse stresses f t, f tp—Similarly, the strain εt can be related to the f tp by eliminating the angle θ reinforcement stresses f t and from equilibrium Eq. (5.2.1b) using the compatibility equations (Eq. (5.3.1b) through (5.3.4b)).

ε t = ε d +

Ao s ( − ε d ) ( − σ d )

(

po At ft

+ Atp f tp )

(5.6.7)

The three unknown variables εt , f t, and f tp can be solved using Eq. (5.6.7) and the two stress-strain relationships, Eq. (5.4.2c) and (5.4.2d) and (5.4.3a) and (5.4.3b), for transverse reinforcement. 5.6.8 Useful compatibility equations—By combining compatibility Eq. (5.3.1a) and (5.3.1b), the two variables εr and θ can be expressed directly in terms of strains εl , εt , and εd as

εr = ε + εt − ε d

(5.6.8a)

ε + ε d ε t + ε d

(5.6.8b)

l

tan2 θ =

l

Equations (5.6.8a) and (5.6.8b) will facilitate solution procedures (5.6.9). 5.6.9 Solution procedure and flow chart —The solution procedure is illustrated by a flow chart shown in Fig. 5.6.9. This solution procedure uses 12 equations, Eq. (5.4.2a) through (5.4.3b) and (5.6.2a) through (5.6.8b), to solve 12 unknown variables (ζ, k 1, σd , f l, f t , f p tp, εl , εt , εr , εds, and l , f t d) for each selected value of εd . From these stresses and strains, the remaining six variables ( θ, τl t , T , γ lt , Φ, Ψ) can be calculated using the appropriate equilibrium or compatibility equations. Selecting a series of εd values produces the entire loading history of the 18 variables. The efficiency of this solution procedure arises from the fact that the angle θ is eliminated in the expressions of Eq. (5.6.5) through (5.6.7) and is, therefore, not involved in the iteration process of t d. In addition, the derivation of Eq. (5.6.6) and (5.6.7) allows the iteration to include nonlinear stress-strain relationships (Eq. (5.4.3a) and (5.4.3b)) of prestressing strands, therefore making this solution algorithm applicable to prestressed concrete beams. A hand-calculation example illustrating this solution procedure is given by Hsu (1993). The STM theory was used by Hsu and Mo (1983, 1985a) to calculate the strength and behavior of 108 torsional beams. For the 61 under-reinforced beams with stirrup spacings within ACI code limits, the experiment-to-calculated torsional strengths had a mean value of 1.014 and a standard deviation of 0.051. The post-cracking deformations (angle of twist and strains in reinforcement and concrete) were also correctly evaluated. McMullen and El-Degwy (1985) compared the STM (softened theory) and the CFT (spalling theory) using their extensive torsion tests. They concluded that “the space truss model with softening of concrete gives a better prediction of maximum torque for the beams tested in this investigation (average T u,exp / Tu ,th = 1.03 and standard deviation = 0.091).” The space truss theory with spalling of concrete cover gave an average T u,exp / Tu ,th of 1.10 and a standard deviation of 0.129. McMullen and El-Degwy also observed that concrete cover spalled only after maximum torque was reached.

5.7—Graphical methods 5.7.1 Rahal’s method —Rahal developed a simplified method for design and analysis of reinforced concrete panels subjected to pure shear (Rahal 2000a) and extended the method to reinforced concrete beams subjected to torsional moments (Rahal 2000b). The original method for calculating f c′ of shear relates the normalized ultimate shear strength vu / reinforced concrete panels to the reinforcement indexes ω t and ω L in the transverse and longitudinal directions, respectively. The reinforcement indexes are given by

ω t =

ρt f ty f c′

;

ω L =

ρ f y f c′ l

l

(5.7.1a)

Figure 5.7.1 gives the relationship between the indices and the normalized shear strength obtained using the results of the modified CFT (Collins and Mitchell 1991). Each curve


f c′ and ω L in the figure represents the relationship between vu / ω at a given t . At relatively low ω t values, the strains in the transverse rein forcement exceed the yield strains before the ultimate strength is reached. Beyond a specific level of reinforcement, the concrete is crushed before the reinforcement yields. Figure 5.7.1 shows a curve passing through those points beyond which concrete is crushed before the transverse reinforcement yields (over-reinforced case). The figure also shows a similar curve for the over-reinforced case in the longitudinal direction. The two “yield” curves divide Fig. 5.7.1 into four regions. The relative position of a point of coordinates ( ω t; ω L), with respect to these curves or regions, indicates the expected failure mode of an element with these reinforcement ratios. Four failure modes are possible: 1) Partially over-reinforced; only longitudinal reinforcement yields (Zone III) 2) Only transverse reinforcement yields (Zone II) 3) Completely over-reinforced; concrete crushing before reinforcement yielding (Zone IV) or 4) Completely under-reinforced; longitudinal and transverse reinforcement yield (Zone I). f c′, ω t, and ω L are dimensionless, they • Because vu / provide values that can be applied to any system of units. The results of this method have been compared with experimental results from 46 reinforced concrete panels and were found to give accurate calculations of the ultimate shear strength and failure mode. The average ratio of experimental shear strength to predicted strength was 1.01 with a coefficienct of variation (COV) of 0.125 (Rahal 2000a). This method is applied to beams subjected to torsion by idealizing the section as a hollow tube and adopting simplified assumptions regarding the hollow tube thickness and shear flow zone size. Tube walls are treated as the shear panels to which Fig. 5.7.1 applies. Based on assumptions of the method, the reinforcing indexes are given by

ω t =

ω L =

At fty po

0.42sAcp f c′

+ Aps f py 0.375sAcp f c′

A f y l

l

27

Fig. 5.6.9—Flow chart for torsion analysis using STM.

(5.7.1b)

(5.7.1c)

The ultimate torsional moment is also related to ultimate shear strength in walls 2

T u

= 0.67

Acp pcp

vu

(5.7.1d)

In contrast to the STM and CFT, Rahal’s method is used to calculate approximate strength only and not the entire torque-twist relationship. Equations (5.7.1b), (5.7.1c), and (5.7.1d) depend on the gross area and outer cross section

Fig. 5.7.1—Normalized shear strength curve for reinforced concrete members ( Rahal 2000a).

perimeter, and therefore can be applied to closed sections of various shapes. 5.7.2 Leu and Lee’s Method —Leu and Lee (2000) proposed a graphical solution to the STM. Their strength

28


ηb = l

76 ( f in MPa) or 200 + f y y

11, 000 ( f in psi) 30, 000 + f y y

ηb = l

(5.7.2c)

ηtb =

76 ( f in MPa) or 100 + f y y

ηtb =

11, 000 ( f in psi) 15, 000 + f y y (5.7.2d)

3. Calculate the reinforcement indexes ω L and ω t

ω L =

η ηb

(5.7.2e)

ηt ηtb

(5.7.2f)

l

l

ω t = Fig. 5.7.2—Strength contour (ω s) diagram ( Leu and Lee 2000).

contour diagram, shown in Fig. 5.7.2, is constructed to give the torsional strength of a rectangular member when their longitudinal and transverse reinforcement indexes, ω L and ω t, are calculated. Because this graph is built on the failure mode diagram of Hsu (1993), it can also predict the four failure modes: 1) Under-reinforced 2) Partially under-reinforced in the transverse direction 3) Partially under-reinforced in the longitudinal direction 4) Over-reinforced. The diagram was constructed based on STM and, therefore, its accuracy is as good as STM prediction. In Fig. 5.7.2, a = 0.4 and a = 0.6 correspond to reinforcement grades of 40 and 60 ksi (280 and 420 MPa), respectively. The diagrams are almost identical for different reinforcement grades, except in the over-reinforced region of failure mode of Region IV. Fortunately, an over-reinforced design is usually not permitted. 1. Calculate the normalized reinforcement ratios ηl and ηt

 A  η =   f c′  Acp  f y

l

l

ηt =

f y  At pcp 



f c′  Acp s

 

(5.7.2a)

(5.7.2b)

where f y = ( f yl + f ty)/2 is the average yield strength of longitudinal and transverse reinforcement bars, f yl and f ty. 2. Calculate the balanced normalized reinforcement ratios when both longitudinal and transverse reinforcements yield simultaneously with crushing of diagonal struts ηl b and ηtb

4. Calculate the nondimensional balanced torsional strength that corresponds to the case when both longitudinal and transverse reinforcements yield simultaneously with crushing of diagonal struts, T xub T xub

=

140 ( f in MPa) or T xub 300 + f y y

=

20, 000 ( f in psi) 42, 000 + f y y (5.7.2g)

5. Calculate the balanced torsional strength T xub T xub fc′Acp 2

T xub

=

pcp

(5.7.2h)

6. Use ω L and ω t to determine the value of ω s from Fig. 5.7.2. The coordinate (ω L, ω t ) in the four failure mode regions also determines the member failure mode; and 7. Determine the torsional strength. T xu = ω sT xub

(5.7.2i)

The graphical solution using Fig. 5.7.2 is not exact. Its accuracy is affected by the aspect ratio of the cross section and the difference in the yield stresses, f yl and f ty, when calculating f y = ( f yl + f ty)/2. Thirty-eight specimens from Hsu (1968a) were used to evaluate the accuracy of the design charts. The mean and maximum errors of torsional strength were only 1.5 percent and 4 percent, respectively, when compared with the strengths of those obtained using the STM.

CHAPTER 6—MEMBERS SUBJECTED TO TORSION COMBINED WITH OTHER ACTIONS 6.1—General Torsion rarely occurs in concrete structures unaccompanied by other stress resultants. In the general case of loading,


a beam cross section can be subjected to a maximum of six stress resultants, in the orthogonal system of coordinates, as shown in Fig. 6.1. Three forces and three moments are possible: one torsional moment, two shearing forces along the major and minor axes, two bending moments along the major and minor axes, and an axial force (tension or compression). Early torsion research concerned with pure torsion was later extended to torsion combined with flexure (Hsu 1968d). This combination has attracted more attention than all other combinations of stress-resultants combined. The torsion and shear combination attracted less attention regardless of its practical importance in design. Torsion and compression is the least studied regardless of its occurrence in design, like in the case of prestressed members or columns subjected to earthquake loading. Early experimental work, which focused on the ultimate strength and stiffness of concrete beams, mostly recommended interaction curves between torsion and flexure and, in some cases, with shear. Although early theoretical work focused on the ultimate strength and shape of interaction curves, most research was applicable only to under-reinforced members in which all the reinforcement yields at ultimate state. The first model to consider strain compatibility and that was capable of obtaining the full response of concrete members subjected to pure torsion was the CFT by Mitchell and Collins (1974). This model was further developed by Onsongo and Collins (Onsongo 1978) to calculate the full response under combined torsion, flexure, and axial load. These two models apply to both under-reinforced and over-reinforced sections, a feature unavailable in previous models concerned only with ultimate strength and not strain compatibility. Various models are capable of calculating the full response of sections subjected to some combination of the six stress resultants ( Fig 6.1). One such model was developed in the 1970s by Rabbat and Collins (1977, 1978) in parallel with the development of CFT for torsion. Another model developed by Rahal and Collins (1995a), considers information on torsion and shear published in the 1980s and 1990s, like tension stiffening and concrete softening. Both models provide a unique perspective on the interaction between torsion and various stress resultants. Ewida and McMullen (1981) also developed a model to calculate the post-cracking response of sections subjected to combined torsion, shear, and flexure. Their model explained other phenomena like shear lag and effects of reinforcement on the interaction curve. Chapter 6 summarizes work on the interaction between torsion and other stress resultants. Although members reinforced only in the longitudinal direction are uncommon in practice, their behavior is discussed because in some cases they behave similarly to members with transverse and longitudinal reinforcement.

6.2—Torsion and flexure 6.2.1 General—The combination of torsional ( T ) and flexural ( M ) moments can be critical in design. Aside from the case of pure torsion, this combination has prompted most of the torsion research combined with other stress-resultants for

29

Fig. 6.1—General case of loading a beam cross section.

Fig. 6.2.1—Test setup for combined torsion and flexure.

two reasons: first, researchers understood that such studies would be prudent. Second, the beams subjected to combined T and M are easier than other combinations in testing. In the absence of shearing forces, a four-point loading arrangement is easily modified to apply constant torsional moment along with the bending moment in the central portion of a beam specimen. Figure 6.2.1 shows that the central loads in a four-point load test setup can be applied at an eccentricity e, subjecting the test region of the specimen to a uniform combination of torsion and flexure. Torsional moment to bending moment ratio is controlled by distances e and a, as shown in Fig. 6.2.1. The most significant factors affecting the behavior of members subjected to combined torsion and flexure are the presence of transverse reinforcement, the M ), and the amount and distributorque-to-bending ratio ( T / tion of the longitudinal reinforcement. Other factors include the cross section shape and concrete strength. The presence of transverse reinforcement causes a considerable change in the behavior of members subjected to torsion. Sections 6.2.1 and 6.2.2 discuss the T - M interaction for members with and without this reinforcement.

30


Fig. 6.2.3a—Torque-twist relationships at various T / M ratios for sections with symmetrical longitudinal reinforcement ( McMullen and Warwaruk 1967 ). (Note: 1 in.-kip = 0.113 kN·m; 1 in. = 0.0254 m.) Fig. 6.2.2—Normalized T-M interaction curves for members without transverse reinforcement.

6.2.2 Members without transverse reinforcement —The T - M interaction in members containing only longitudinal

reinforcement has limited practical importance, attracting a significant amount of experimental research, mainly due to its importance in understanding the behavior of members with transverse reinforcement. Behavior and ultimate strength of these members is significantly influenced by the shape of the beam cross section, the amount and distribution of the longitudinal reinforcement, and the T - M ratio (Zia 1970). In prestressed members, the amount and eccentricity of the prestressing also affect the behavior because it causes a strain variation similar to that due to a compressive force or a flexural moment ( Hsu 1968e). Research on the T - M interaction in members containing only longitudinal reinforcement is mainly experimental. Figure 6.2.2 shows examples of the interaction curves recommended by various researchers. For square and rectangular sections, Kemp et al. (1961) and Hsu (1968a) independently suggested the trilinear interaction. Victor and Ferguson (1968) suggested a similar trilinear interaction for L-sections, and a square interaction for T-beams. Lim and Mirza (1968) also proposed a square interaction curve for T-beams. Based on evaluation of numerous experimental results, Zia (1970) observed that significant reduction in the torsion strength M o between 0.5 and 1.0, and a circular interacoccurs at M / tion curve serves as a lower bound for the larger portion of the experimental results. The scatter in the results, however, is relatively large. Some test results have suggested a linear interaction (McMullen and Woodhead 1973). The interaction between T and M in prestressed members depends largely on the eccentricity of the prestressing force.

The interaction for concentrically prestressed members is similar to that of nonprestressed members. For eccentrically prestressed beams, the induced flexural moment increases torsional strength when the eccentric prestressing force counteracts tension caused by the bending moment. ACI 318 gives no direct information on the T - M interaction in members without transverse reinforcement. The minimum amounts of longitudinal reinforcement for torsion and flexure and transverse reinforcement for torsion required for the combination are usually calculated independently for each stress resultant, and the sole interaction is from superposition of the amount of longitudinal reinforcement required for each stress-resultant. 6.2.3 Members with transverse reinforcement —Members reinforced in the longitudinal and transverse directions exhibit significant post-cracking strength and ductility compared with members reinforced in the longitudinal direction only. Behavior of these members also depends significantly on the distribution and amount of longitudinal reinforcement. Cases of symmetrically and unsymmetrically reinforced beams are discussed later. Figure 6.2.3a shows the torque-twist relationship of four 152 x 305 mm (6 x 12 in.) under-reinforced solid reinforced concrete specimens with symmetrical longitudinal reinforcement tested by McMullen M ratios ranging from and Warwaruk (1967) at variable T / M = 0.25. As shown in Fig. 6.2.3a, the addipure torsion to T / tion of flexural moment significantly reduced the torsional strength, ductility, and post-cracking stiffness. Figure 6.2.3b shows the experimentally observed normalized T - M interaction from two series of symmetrically reinforced specimens. Five under-reinforced beams, each 152 x 305 mm (6 in. x 12 in.), were tested by McMullen and Warwaruk (1967). The torque-twist results of four of those beams are shown in Fig. 6.2.3b. In addition, five symmetrically prestressed beams, each 305 x 432 mm (12 x 17 in.), were tested by Mardukhi and Collins (Mardukhi 1974). This


Fig. 6.2.3b—Normalized T-M interaction curves in symmetrically longitudinally reinforced members with transverse reinforcement.

study showed that a flexural moment equal to 60 percent of the ultimate flexural strength caused only about a 10 percent reduction in the torsional strength. Similarly, a torsional moment equal to 40 percent of the ultimate pure torsion strength caused an approximately 20 percent reduction in the flexural strength. The behavior of unsymmetrically reinforced beams differs significantly from the previous case. Figure 6.2.3c shows the torque-twist relationship of three under-reinforced hollow reinforced concrete specimens tested by Onsongo M ratios from and Collins (Onsongo 1978) at variable T / 0.63 to 4.27. Figure 6.2.3d shows the observed interaction from four series of unsymmetrically reinforced beams, one series tested by McMullen and Warwaruk (1967), two by Onsongo and Collins (Onsongo 1978), and one by the Zurich group (Lampert and Thürlimann 1968, 1969). The ratio of compression to tension longitudinal reinforcement As f y) ranged from 0.1 to 0.27. yield force r = ( As′ f y / Figures 6.2.3c and 6.2.3d show that the addition of a small flexural moment can significantly increase the torsional strength and post-cracking stiffness of unsymmetrically reinforced beams. In pure torsion, the additional bottom longitudinal reinforcement available in unsymmetrically reinforced sections does not increase the ultimate strength because the weaker top reinforcement is critical. The addition of a flexural moment introduces compression in the weaker top reinforcement and increases its resistance to torsional shear stresses. Based on the same experimental study (Onsongo 1978), it appears that a small axial compressive force substantially increases post-cracking stiffness of a torsionally cracked member with strength also enhanced. This is because the compressive axial load counteracts the widening of spiral cracks. This point may be particularly relevant in the case of structures under seismically-induced torsion.

31

Fig. 6.2.3c—Torque-twist relationships at various T / M ratios for sections with unsymmetrical longitudinal reinforcement (Onsongo 1978 ). (Note: 1 kN·m= 8.85 in-kip; 1 m = 39.4 in.)

The peak in the interaction curve depends on the ratio r and if the beam is under-reinforced or over-reinforced in the transverse direction. An increase of up to 30 percent in torsional strength was observed with the addition of flexural moment equal to 40 percent of the pure flexural strength in the under-reinforced tests by McMullen and Warwaruk (1967). Onsongo (1978), however, observed a 25 percent and 6 percent increase in under-reinforced and over-reinforced beams, respectively. The interaction curve seems to be less affected by the ratio r in over-reinforced members, where Onsongo reported only a 6 percent increase in torsional strength in spite of the significantly low value of r (approximately 0.1). Zararis and Penelis (1986), who tested unsymmetrically reinforced T-beams under combined flexure and torsion, observed a trend similar to that shown in Fig. 6.2.3d. Due to weakness in the smaller top reinforcement, the presence of a flexural moment increased torsional strength by approximately 18 percent. Presence of a flexural moment reduces the torsional ductility of a member with symmetrical or unsymmetrical longitudinal reinforcement, as shown in Fig. 6.2.3a and 6.2.3c. The torque-twist diagrams show that the ultimate torsional strength was reached at a smaller twist. Similarly, the pres M = 4.27 in Fig. ence of relatively small flexural moment, T / 6.2.3c, also affected the curvature of unsymmetrically reinforced members subjected to torsion. Figure 6.2.3e shows the moment-curvature diagrams of two unsymmetrically reinforced hollow specimens tested by Onsongo and Collins (Onsongo 1978). The flexural moment was predominant in M = 0.63), whereas torsion was predomione specimen (T / M = 4.27). Torsional shearing stresses nant in the other ( T / were equilibrated by tensile stresses in the longitudinal direction. Upon cracking, the weaker top reinforcement elongates more than the stronger bottom reinforcement, causing greater reverse curvature than that caused by lesspredominant flexural moment in the torsion predominant M = 4.27). Therefore, the section is subjected to an beam (T /

32


Fig. 6.2.3e—Observed flexural moment-curvature relationships at various T / M ratios (Onsongo 1978 ). (Note: 1 kN·m= 8.85 in.-kip; 1 m = 39.4 in.) 2

Fig. 6.2.3d—Normalized T-M interaction curves in unsymmetrically longitudinally reinforced members with transverse reinforcement.

overall curvature opposite in direction to the relatively small flexural moment. M ratio affects the diagonal compression angle and The T / the beam crack pattern. The presence of flexural moment introduces tensile and compressive strains in the bottom and top faces of the section, respectively. Compression in the top face delays cracking, in some cases until ultimate strength is reached (Johnston and Zia 1975). In the case of pure torsion, the crack angle on the bottom face is diagonal, and in the case of pure flexure, the crack angle becomes normal to the beam longitudinal axis. The angle on the sides is steep near the bottom of the section due to tensile strains and becomes flatter near the top from the compressive strains caused by flexural moment. Numerous theories (McMullen and Warwaruk 1967; Walsh et al. 1967; Lampert and Collins 1972; Elfgren et al. 1974a) recognize the phenomenon shown in Fig. 6.2.3d. Interaction curves recommended by Lampert and Collins (1972) have been simplified. They suggested that in under-reinforced sections, an increase in torsional strength is related to the ratio of the yield force of the compression As f y). The reinforcement to tension reinforcement, r = ( As′ f y / flexural moment creates a tensile force M / dv in the bottom stringer and an equal compressive force in the top stringer. The torsion induces a total tensile force of Tpo /2 Aocotθ in the longitudinal reinforcement, where d v is the distance between the top and bottom longitudinal reinforcement, po and Ao are the shear flow perimeter and area, respectively, and θ is the diagonal crack angle. The longitudinal reinforcement forces are additive, as shown in Fig. 6.2.3f . Two simple equations (Eq. (6.2.3a) and (6.2.3b)) are derived based on this theory. Equation (6.2.3a) is for the case where the bottom longitudinal reinforcement yields along with the stirrups

 T  M r   + =1  T o  M o

(6.2.3a)

Equation (6.2.3b) applies when the weaker top longitudinal reinforcement yields along with the stirrups 2

 T  1 M − = 1  T  r M o o 

(6.2.3b)

Figure 6.2.3g shows the interaction curves for members with r values of 0.3, 0.5, and 1. The increase in torsional strength calculated using both equations is larger than the experimentally-observed increase shown in Fig. 6.2.3d. Both equations give acceptable results when pure torsion strength T o is taken as the conservatively calculated value, as demonstrated by Lampert and Collins (1972). Since 1971, ACI code procedures for the design of members subjected to flexure and torsion are based on the superposition of reinforcement required to resist torsion (longitudinal and transverse) and flexural moment (longitudinal). Experience has proved that these procedures are conservative and suitable for design. Calculating cross section strength using the code equation is not a simple task, especially in common cases where shear is present and the longitudinal reinforcement is not symmetrically placed in the top and bottom sides of the cross section. Research by Greene and Belarbi (Greene and Belarbi 2006a; Greene 2006) expands the STM described in Chapter 5 to members under torsion combined with shear and flexure. The combined-action STM can be used to predict the loaddeformation response for hollow or solid reinforced concrete members and also to generate torque-flexure interaction curves for members under torsion combined with shear and flexure. The model considers a softened stress-strain relationship for concrete and the tension-stiffening effect of concrete surrounding the reinforcement to predict service-level deformation (Greene and Belarbi 2006b). Figure 6.2.3h shows the interaction curves created using the combined-action STM


33

Fig. 6.2.3f—Superposition of forces due to torsion and bending ( Hsu 1993).

Fig. 6.3.1—Shearing stress due to shear, torsion, and combined shear and torsion.

action curve. ACI code equations were used to calculate T o and M o for use in Elfgren’s model. Figure 6.2.3h(a) shows that for this test series, ACI 318-11 is unconservative in its calculation of T o, which affects the calculation of the interaction curves. Fig. 6.2.3g—Normalized T-M interaction curves for members with unsymmetrical longitudinal reinforcement ( Lampert and Collins 1972).

Fig. 6.2.3h—Interaction curves predicted by combinedaction STM and Elfgren Model for McMullen and Warwaruk (1967). Series 1 and 2 (Greene 2006 ). (Note: 1 kN·m= 8.85 in.-kip.)

for two series of beams under combined torsion and flexure tested by McMullen and Warwaruk (1967). Members of Series 1 in Fig. 6.2.3h(a) were symmetrically reinforced and members of Series 2 in Fig. 6.2.3h(b) had more reinforcement on the flexural tension side. For comparison, the model by Elfgren et al. (1974a) was also used to predict the inter-

6.3—Torsion and shear 6.3.1 General—Torsion and shear cause shear stresses across a section, as shown in Fig. 6.3.1. The shear stresses τ due to the torsional moment T circulate around the section, whereas the vertical shear force V induces shear stress v. The side of the section where the stresses are additive ( τ + v) is critical in design due to the great intensity of the shearing stresses. Intense stresses increase tensile strains in transverse and longitudinal steel reinforcement and compressive and tensile strains in the concrete. In spite of its practicality, work on T-V interaction is relatively limited mainly because shear cannot be induced without causing a flexural moment along the beam’s test region. Therefore, the torsion-shear condition is achieved only at the point of inflection. Otherwise, an experimentally observed T -V interaction curve is only a projection of a part of the T -V - M curve on the T -V coordinate plane. Additionally, the pure shear strength of a beam cannot be determined experimentally. Typically, this maximum should be defined from a calculated value. 6.3.2 Members without transverse reinforcement — Nylander (1945) reported that tests on members containing only longitudinal reinforcement showed considerable scatter, with most tests falling between a linear and circular interaction curve. Other tests by Birkland (1965), Hamilton (1966), and Ersoy and Ferguson (1968) have shown that a circular interaction is more accurate. Nylander’s model (1945) and

34


Fig. 6.3.3b—Various normalized T-V interaction curves in the literature. Fig. 6.3.3a—Normalized T-V linear and circular interaction curves for members with transverse reinforcement.

those proposed by other researchers conservatively recommend a linear interaction. Mirza and McCutcheon (1968) tested quarter-scale specimens and found that the longitudinal reinforcement has a significant effect on the interaction curve. The result’s scatter was considerable and a lower-bound linear interaction was recommended. McMullen and Woodhead (1973) tested eccentrically prestressed beams under various combinations of torsion, shear, and flexure and found that Eq. (6.3.2) adequately fit the results 2

 T  +   = 1 V o  T o  V

(6.3.2)

When torsion design provisions were introduced in ACI 318-71, a circular interaction between V c and T c was assumed. This approach slightly increased the design time in members with transverse reinforcement. Abandoned in ACI 318-95, concrete contribution to torsion, T c, was disregarded to simplify design. Considerable scatter produced by test specimens without transverse reinforcement confirms the sound principle that members subjected to such combined actions should contain a minimum amount of transverse and longitudinal reinforcement. 6.3.3 Members with transverse reinforcement— Development of models for torsion and shear has made it possible to calculate shear strength with accuracy similar to the flexure theory’s. Early work on T -V interaction was largely experimental. Figure 6.3.3a shows some of the experimental results and Fig. 6.3.3b presents some proposed interaction curves. Experimental results were scattered, with most strength values falling between linear and circular curves (Fig.

6.3.3a). Because the pure shear strength V o is significantly affected by the accompanying bending moment, it cannot be accurately represented and a calculated value is used. Different researchers use varying definitions of V o, making it difficult to compare the suggested curves in Fig. 6.3.3b. Figure 6.3.3a shows that the presence of torsional moment reduces shear strength, especially if the torque is more than 25 percent of pure torsional strength. The amount of transverse reinforcement is considered the main factor affecting the shape of the curve. Based on an experimental testing program, Klus (1968) suggested a bilinear interaction curve as shown in Fig. 6.3.3b. The theoretical model and the experimental results of Ewida and McMullen (1981) showed the amount of reinforcement has a considerable effect on the interaction curve. They suggested a simplified equation nV

 V  +   = 1 T o  V o  T

where nV =

(6.3.3a)

1.2 for under-reinforced sections in which all reinforcement (both longitudinal and transverse) yields when the section reaches ultimate strength = 1.75 for partially under-reinforced sections in which only stirrups yield or only the longitudinal reinforcement yields when the section reaches ultimate strength = 3.0 for completely over-reinforced sections in which concrete crushes before yielding in any of the reinforcement. All three curves are shown in Fig. 6.3.3b. For underreinforced sections (nV = 1.2), the curve is almost linear, whereas for completely over-reinforced sections, the interaction is closer to a circular curve. Klus’s (1968) bilinear interaction curve lies between the linear and circular curves.


Elfgren et al. (1974b) developed a theoretical model for the ultimate strength of members subjected to combined torsion, shear, and flexure as discussed in 6.5. The T-V interaction curve is circular when Failure Modes 1 and 2 govern, but it is close to a straight line when Failure Mode 3 governs. These three failure modes were introduced in 3.3.5. Ewida and McMullen (1981) observed that the addition of a relatively small shear force (20 percent of the shear strength calculated using ACI equations) caused a slight torsional strength increase of 3 percent, a phenomenon they called “shear lag.” Rahal and Collins (1995a) confirmed the significant effect the amount of reinforcement has on the interaction curve. They also showed that in completely over-reinforced sections, the addition of shear causes redistribution in the shear stresses and a subsequent slight increase in the torsional strength. ACI 318 adopts a superposition approach in which the reinforcement required to resist T is added to that required to resist V . To control the diagonal cracks and ensure that the stirrups yield before crushing occurs in the concrete, ACI 318 sets an upper limit on the shear stresses due to torsion and shear. Figure 6.3.3c (Rahal and Collins 2003b) shows a comparison between the calculations of ACI 318-02 and the 1998 AASHTO LRFD Bridge Design Specifications with test results from a series of large-scale beams subjected to a combination of T , V , and a relatively small M (Rahal and Collins 1995b). The test setup for this study ensured failure near the point of inflection to accurately represent the interaction of torsion and shear. Loads were introduced to the test region through wing beams instead of by typical concentrated loads applied at the beam top. This loading minimized disturbance in the test region, representing the actual situation better than the typical setup. ACI design provisions give conservative results if the recommended value of θ = 45 degrees is used for the angle of inclination of the diagonal struts. Using the lowest allowable value of θ = 30 degrees gives less consistent results, overestimating failure loads at high torsion-to-shear ratios. Nonetheless, the AASHTO (1998) provisions resulted in a calculated value for θ of approximately 36 degrees for the series of beams studied. This value compared well with the inclinations of cracks observed in the tests, producing better correlation with failure loads. Calculations from ACI and AASHTO resulted in a nearly linear interaction curve, with a cut-off horizontal plateau defined by the limit set on torsional strength to avoid concrete crushing before the reinforcement yields. Variation in the intensity and direction of shear stresses (Fig. 6.3.1) affects the concrete crack patterns and reinforcement strains. Figure 6.3.3d shows the angle of diagonal compressive stresses and crack orientation at variable T / V ratios calculated by Rabbat and Collins (1977, 1978) using a Variable Angle Space Truss Model. In pure shear, the side cracks are parallel, where in pure torsion, they are nearly perpendicular. Crack orientation on the side where shear stresses are subtractive ( τ – v) depends on the relative intensity of stresses τ and v. This trend was confirmed in tests by Rahal and Collins (1995b), Greene (2006), and Greene and Belarbi (2008).

35

Fig. 6.3.3c—Comparison between observed and calculated T-V interaction curves ( Rahal and Collins 2003b). (Note: 1 kN·m= 8.85 in.-kip; 1 MPa = 0.145 ksi.)

Fig. 6.3.3d—Variation of angle of diagonal compressive stresses depending on T / V ( Rabbat and Collins 1978 ).

Stresses in the transverse and longitudinal reinforcement are also significantly larger on the side where the shear stresses are additive (τ + v) rather than subtractive ( τ – v). The difference in longitudinal stress magnitude on vertical sides of the section shown in Fig. 6.3.1 causes a lateral curvature, as shown in Fig. 6.3.3e, tested by Rahal and Collins (1995b). They observed lateral curvatures in two of the specimens tested at a T / V of 76 mm (3.0 in.) (Specimen RC2-4) and 156 mm (6.14 in.) (Specimen RC2-1). Calculations of the model (Rahal and Collins 1995a) plotted with

36


Fig. 6.3.3e—Observed and calculated lateral curvature in sections subjected to combined torsion and shear ( Rahal and Collins 1995a).

Fig. 6.4a—Torsion-axial compression interaction in rein forced concrete (tests from Bishara and Peir [1968]). (Note: 1 in.-kip = 0.113 kN·m.)

dashed lines show that available theoretical models capture this phenomenon.

the descending branch plotted in Fig. 6.4a demonstrate suitability to capturing the behavior of Bishara’s tests. The results provided (Bishara and Peir 1968), however, were less conservative than experimental results from prestressed beams without stirrups tested by Humphreys (1957) and Zia (1961). The square root factor in Eq. (6.4a) was labeled the prestressed factor by Hsu (1968b). He showed that this prestressed factor can be derived theoretically from an element subjected to shear and compression, assuming the principal tensile stress reached the tensile strength of concrete. The prestressed factor can also be derived from the skew-bending theory of a beam in pure torsion. To ensure best fit with the test results, the constant 12 in the prestressed factor was reduced to 10, resulting in Eq. (6.4b).

6.4—Torsion and axial load Elements subjected to combined torsion and compressive forces, N , include bridge piers under gravity and horizontal loads, and prestressed concrete beams subjected to torsion. Cracked spandrel beams subjected to torsion tend to elongate. Columns and walls restraining the elongation induce beam compression acting in combination with the compatibility torsion and other stress-resultants. Though the combination of torsional moment ( T ) and axial force ( N ) can be critical in design for these cases, it has not been adequately studied, particularly in experimental research on nonprestressed elements and combined torsion and axial tension. Lack of adequate studies is due to the relative rarity of this combination compared with others discussed. Torsion tests on concrete beams subjected to various prestressing levels simulate conditions similar to those of reinforced concrete beams subjected to combined torsion and axial compression. Experimental results on members without stirrups by Humphreys (1957) and Zia (1961) showed trends similar to those observed by Bishara and Peir (1968), which are given in Fig. 6.4a. A drop in torsional strength beyond the “transformation point” was clear only in Zia’s tests on prestressed concrete beams subjected to torsion. Humphreys reported a 170 percent increase in torsional strength and Zia a 190 percent increase in torsional strength due to prestressing. Based on their experimental findings, Bishara and Peir (1968) recommended Eq. (6.4) to calculate an increase in torsional strength in the presence of an axial compressive load T

= T o

1 + 12

σ f c′

(6.4a)

The descending part is taken as a line joining the transformation point (defined as the point at σ = 0.65 f c′ and T u = 3T o) and pure compression strength. Results from Eq. (6.4) and

1 + 10 ( σ f c′)

(6.4b)

The torsional strength based on a 45-degree angle of the compression diagonals and spalled dimensions of the section described previously is 2.1 kN·m (18.6 in.-kip)—38 percent of experimental strength. This strength is increased to 4.62 kN·m (40.9 in.-kip)—84 percent of experimental strength—if unspalled dimensions are used. For prestressed members, ACI 318 allows a 37.5 degrees value for the angle of the compression diagonals, leading to a strength of 6.05 kN·m (53.3 in.-kip). This value, which is plotted in Fig. 6.4a, remains considerably smaller than the experimentally observed maximum torque of 17.2 kN·m (151.9 in.-kip) acting in combination with a compressive stress of approximately 0.65 f c′. The general procedure for the torsion and shear design method in AASHTO (1998) specifications accounts for the effect of compression and prestressing on torsional strength. Figure 6.4a shows calculations for the AASHTO LRFD method (general method) based on unspalled dimensions, which are more likely for small concrete cover (19 mm [0.75 in.]) used in Bishara and Peir’s (1968) tests. Although this method underestimates an increase in torsional strength at


Fig. 6.4b—Torque-twist relationships of reinforced concrete columns under different compressive stresses ( Bishara and Peir 1968 ). (Note: 1 in.-kip = 0.113 kN·m; 1 in. = 0.0254 m.)

higher compressive loads, it produces slightly better predictions than ACI 318. Figure 6.4a also shows the results of a detailed analysis based on the modified compression field theory (CFT) (Rahal and Collins 1995a, 2003a). Similar to the AASHTO LRFD method (general method), this model underestimates the torsional strength increase at high compressive loads while providing generally conservative results. Figure 6.4b shows the torque-twist relationships for some tests by Bishara and Peir (1968). Compressive stress increased torsional strength and torque at peak strength, but decreased torsional ductility in members with compressive stresses greater than approximately 0.4 f c′. Combined torsion and axial tension in beams and columns is uncommon. The rarity of this combination and difficulties testing reinforced concrete sections under this forced combination account for little experimental data on the subject. Figure 6.4c shows the torsion-axial tension interaction calculated using diagonal CFT (Rahal and Collins 1995a, 2003a) for an under-reinforced section. Axial tension reduced torsional strength most significantly after tensile forces reach 85 percent of pure tensile strength.

6.5—Torsion, shear, and flexure In most cases of practical importance, torsion acts in combination with shear and flexure. Research on pure torsion, torsion combined with flexure, and torsion combined with shear in members with or without reinforcement is aimed at understanding the general behavior of the three stress-resultants T , M , and V . Similar to work on the combinations discussed above, early research (Hsu 1968a; Johnston 1971; McGee and Zia 1973; Elfgren et al. 1974a) focused on studying ultimate strength and sought to produce three-dimensional interaction surfaces. Further work created the ability to obtain the full response of reinforced and prestressed concrete beams subjected to various stress combinations (Rabbat and Collins 1978; Rahal and Collins 1995a, 2003a). Elfgren et al.’s (1974a) theory assumes that a rectangular box section subjected to torsion, shear, and flexure could

37

Fig. 6.4c—Torque-axial tension calculated using diagonal compression field theory ( Rahal and Collins 1995a).

Fig. 6.5a—Interaction surface for torsion, shear, and flexure ( Hsu 1993).

fail in one of three modes. In first mode failure, the bottom longitudinal reinforcement and transverse reinforcement yield on the side where shear and torsional stresses are additive. In second mode failure, the top longitudinal reinforcement and transverse reinforcement yield on the additive side. Third mode failure occurs when longitudinal and transverse reinforcement on the additive side yield. Nondimensionalized interaction relationships for M , V , and T for the three failure modes are given by Eq. (6.5a) through (6.5c). Figure 6.5a illustrates the three interaction relationships for the case As f y). A systematic derivation of of r = 1/3, where r = ( As′ f y / Elfgren’s nondimensional interaction surface is found in Hsu’s (1993) book.

Mode 1:

Mode 2:

M M o

2

2

 V   T  + r   + r   = 1  V o   T o 

1  M 

2

 V  −   +   r  M o   V o 

(6.5a)

2

 T  +   = 1  T o 

(6.5b)

38

REPORT ON TORSION IN STRUCTURAL CONCRETE (ACI 445.1R-12) 2

Mode 3:

 V   V  o 

2

 T   VT  +   + 2   T o   VoT o  

2d v po

1 + r = 2r

(6.5c)

Researchers in Zürich carried out work in this area. Their results are reported in Lüchinger (1977), Müller (1976, 1978), and Thürlimann (1978). Based on their work, Elfgren (1979) provided a kinematics model that gives the same equations, Eq. (6.5a) through (6.5c), derived earlier with equilibrium methods (Elfgren 1972a,b; Elfgren et al. 1974a,b). The kinematics model is illustrated in Fig. 6.5b and Fig. 6.5c. Beam deformations under combined torsion, shear, and flexure were studied by Petersson (1972), Karlsson (1973),

Fig. 6.5b—Kinematics failure model: (a) General view; (b) Model seen from above; (c) Deformations in bottom; (d) Bending moment diagram ( Elfgren 1979).

and Gabrielsson (1999). Both theoretical and experimental work has been carried out by Teutsch (1980) and Kordina and Teutsch (1983, 1985). The beam shown in Fig. 6.5c was loaded at midspan with an eccentric point load acting downward. The beam was rectangular, with dimensions 100 x 200 x 3300 mm (3.94 x 7.87 x 130 in.) (width x height x length), and the stirrup strength was 0.236 MN/m (13.5 kip/in.). The relationship among torsional moment T , vertical shear force V , and flexural moment M in the failure section was M :T :V = 0.1:0.5:0.2 (Elfgren 1979). The numerals shown along the cracks in Fig. 6.5c refer to the applied load when this part of the crack became visible (units in multiples of 10 kN [2.25 kip]). In the left beam, two failure cracks, ABC and FED, are indicated as well as a rotation hinge AD (compare with Fig. 6.5b). The combined-action STM (Greene and Belarbi 2006a, 2009a,b) can also generate torsional-flexural moment interaction curves for members under torsion combined with shear and flexure. Figure 6.5d shows the interaction curves for two series of beams under combined torsion, shear, and flexure tested by McMullen and Warwaruk (1967). Members of Series 5 and 6 were unsymmetrically reinforced with more longitudinal reinforcement in the bottom side. For comparison, the model by Elfgren et al. (1974a) was also used to predict the interaction curve. ACI 318 code equations were used to calculate T o, M o, and V o for use in Elfgren’s model. Figure 6.5d(a) shows that for this test series, ACI 318-11 is unconservative in its calculation of T o, which affects the calculation of the interaction curves, similar to the observation related to Fig. 6.2.3h(a). Leonhardt et al. (1968) performed large-scale tests on two prestressed concrete box beams representing hollow box bridges loaded in torsion, shear, and flexure. Figure 6.5e shows crack patterns after failure. The beams had a depth of 0.78 m (2.6 ft) and a span of 6.00 m (19.7 ft), and web, top, and bottom flange widths of only 80 mm (3.1 in.). The beams were eccentrically loaded at a midspan diaphragm, and the bottom slab reinforcement was varied between an orthogonal mesh and mesh inclined at 45 degrees. In both beams, the web near the eccentrically applied load failed in

Fig. 6.5c—Crack pattern and failure mechanism for a beam loaded in combined torsion, shear, and flexure ( Elfgren 1979).


39

inclined compression (Fig. 6.5e(a)), confirming the importance of determining the strength of struts considering the web strain and cracking conditions.

Fig. 6.5d—Interaction curves predicted by combined-action TM and the Elfgren model for McMullen and Warwaruk (1967) Series 5 and 6 (torsion, shear, and flexure) and Specimen 1-1 (pure torsion) (Greene 2006 ). (Note: 1 kN·m= 8.85 in.-kip; 1 m = 39.4 in.)

CHAPTER 7—ADDITIONAL DESIGN ISSUES RELATED TO TORSION 7.1—General Chapter 7 addresses additional issues encountered in the design of structural concrete members subjected to torsion. When considering torsion, it is necessary to distinguish between when the torsional moment cannot be reduced by redistribution of internal forces (equilibrium torsion) and when the torsional moment can be reduced by redistribution of internal forces (compatibility torsion) after cracking. To allow torsional moment redistribution, a limit analysis method similar to moment redistribution is addressed. This method requires the formation of torsional plastic hinges at critical sections. Also addressed are precast spandrel beams, torsion limit design, treatment of open concrete sections subjected to torsion, and the size effect on torsional members. 7.2—Compatibility torsion and torsional moment redistribution

Fig. 6.5e—Crack pattern and failure of prestressed hollow box-beam BM 2 with high shear and torsion loaded eccentrically ( Leonhardt et al. 1968 ).

7.2.1 Basic concepts of torsional moment redistribution 7.2.1.1 General—Torsion and flexural moment redistribution are similar in their dependency on adequate ductility in plastic hinge regions. No fundamental difference, therefore, exists between flexural analysis for the case of pure flexure and of torsion. Three standard cases of torsion redistribution have been reported: 1) A spandrel beam loaded by a floor beam 2) An L-beam 3) A spandrel beam in a frame subjected to lateral load. 7.2.1.2 Spandrel beam loaded by floor beam (Collins and Lampert 1973; Hsu and Burton 1974; Hsu and Hwang 1977 ; Abul Mansur and Rangan 1978 )—Figure 7.2.1.2 shows a portion of a three-dimensional structural frame, including columns, spandrel, and floor beams. A load P, applied to the floor beam, produces end rotation, which in turn produces a torsional moment in the spandrel beam. The interaction of floor beam and spandrel beam can be studied using the shaded portion in the shape of a T-specimen. This T-specimen is cut at the flexural inflection points, which can be simulated by hinges. When the T-specimen is loaded, a concentrated torsional moment will occur at the spandrel beam midspan. Whereas cracking reduces flexural stiffness by a factor of approximately 2, it reduces torsional stiffness by a factor of approximately 10. Therefore, a significant redistribution of internal actions begins to occur at cracking. When the spandrel beam develops torsional hinges at its ends, additional load will result in higher flexural moment in the floor beam. This response is equivalent to moment redistribution from the spandrel to floor beam. Designing the spandrel beam based on this redistribution could be economical. 7.2.1.3 L-beam ( Bishara and Londot 1979)—Figure 7.2.1.3 shows a floor system with an L-shaped spandrel beam. Bishara and Londot (1979) studied the response of L-beams under combined torsion and flexure. Due to torsional stiff-

40


Fig. 7.2.1.2—Spandrel beam under concentrated torsion (Hsu and Burton 1974).

Fig. 7.2.1.4a—Spandrel beam under lateral sway (P (Pantazopoulou and Moehle M oehle 1990). 1990 ).

Fig. 7.2.1.3—Floor system with spandrel beams.

ness of the spandrel beam, the slab cannot be assumed to be hinged at the spandrel beam, and vertical loading on the slab causes distributed compatibility torsion in the spandrel beam. When torsion redistribution occurs, a plastic hinge forms at a yield line that defines the slab-beam connection. This could potentially result in a torsional moment that affects the spandrel beam and that could be lower than that computed by elastic analysis. These tests also showed that before yielding, torsional cracking caused a reduction in both torsional and flexural stiffnesses, leading to redistribution of internal forces. Reductions in flexural stiffness range from 40 to 60 percent to as much as 80 to 90 percent for torsional stiffness in beams with transverse reinforcement. 7.2.1.4 Spandrel beam in a frame subjected to lateral load (Pantazopoulou (Pantazopoulou and Moehle 1990) 1990)—When a typical building frame is subjected to lateral sway, beam flexural action results in contraflexure along the span length, as shown in Fig. 7.2.1.4a. 7.2.1.4a. Therefore, in each bay, there is a region where the slab is in tension and another where the slab is in compression. The beam is likely to respond asymmetrically. asymmetrically. The surface of the floor beam or main beam elongates as the beam is subjected to negativ negativee moment. The same elongation develops in the slab at the slab-beam interface. Elongation distorts the slab in shear to satisfy deformation compatibility and the slab reinforcement develops tensile stresses that are transferred to the spandrel (transverse) beams. These actions are illustrated by an equivalent structure shown in Fig. 7.2.1.4b. 7.2.1.4b. Slab action is modeled as truss elements posi-

tioned eccentric to the centroidal axes of the beam elements and connected to the beams through short rigid segments. For any curvature distribution along the longitudinal or main beams as identified in the figure, this configuration creates a corresponding elongation at mid-depth of the slab. Truss forces are transferred to the transverse beam by modeled truss elements in the slab, creating a torsional lever arm in the transverse (spandrel) beam. The result of these stresses on the spandrel beam (the transverse beam in Fig. 7.2.1.4b), 7.2.1.4b), is twofold: 1) flexural moments in the slab plane develop in the spandrel beam; and 2) torsional moments develop in the spandrel beam as a result of the eccentric action of the slab relative to the shear center of the spandrel beam cross section. Torsion Torsion and flexure in the spandrel beam are maximum at the column, and decrease with distance from the joint. This combination of torsion and flexure in the spandrel beams of exterior connections is commonly responsible for the diagonal cracking often observed on the exterior and interior faces of such beams. These actions can also be tested as shown in Fig. 7.2.1.4c. 7.2.1.4c. Two separate actions can cause moment redistribution in this case: 1. Plasticity could occur in the slab under axial load, a condition often idealized as plastic deformation in the truss members modeling the slab of Fig. 7.2.1.4b. 7.2.1.4b. Alternatively, the slab could reach its maximum load strength and redistribution could occur before reinforcement in the spandrel beam yields (Collins (Collins and Lampert 1973; 1973 ; Pantazopoulou and Moehle 1990). 1990 ). This redistribution could result in less torsion on the spandrel beam than that computed from elastic analysis; and 2. A torsional plastic hinge could form at a location across the spandrel beam axis, resulting in redistribution of torsion along the spandrel beam axis. For example, formation of the plastic hinge at the connection between the spandrel beam and


41

Fig. 7.2.1.4b—Idealization of spandrel beam under lateral sway (Pantazopoulou (Pantazopoulou and Moehle 1990). 1990).

Fig. 7.2.1.4c—Testing 7.2.1.4c—Testing of spandrel beams under lateral sway (P ( Pantazopoulou and Moehle M oehle 1990). 1990 ).

the column (location of maximum torsion in the elastic analysis) would reduce the torsion affecting the spandrel beam. 7.2.2 Design benefits from torsion redistribution—Plastic hinges and moment redistribution permit full strength use of the strength of the flexural member’ member’ss cross section. With torsion, elastic analysis typically results in high torsional moments, which in turn require both stirrups and longitudinal reinforcement beyond minimum requirements. Torsion Torsion redistribution redistribut ion can reduce design torsional moments, allowing for more economical designs. Disregarding garding structural torsion—Although one 7.2.3 Disre common practice is to ignore spandrel beams’ torsional stiffness and similar members under gravity loads, effectively assuming full torsional redistribution, it is also common to disregard spandrel beams’ torsional moments resulting from wind or earthquake actions. Sections 7.2.4 through 7.2.6 address codes, experimental evidence, and discussions of their limitations.

7.2.4 Existing code status—In ACI ACI 318, a flexural moment mo ment redistribution formula has been maintained since 1971. In 1977, ACI 318 introduced the torsional limit design, allowing the torsional plastic hinge to have a torsional shear

stress of 0.33 f c′ MPa (4 f c′ psi). Alternatively, Section 8.6.1 of ACI 318-77 318-77 specifies that any set of reasonable assumptions can be used for computing relative torsional stiffnesses of structural elements. Commentary indicates that member stiffnesses should reflect the degree of cracking and inelastic action occurring along the member length before yielding. The stress limit of 0.33 f c′ MPa (4 f c′ psi), however, however, has been extended to prestressed and nonprestressed members subjected to an axial tensile or compressive force in the subsequent editions, including ACI including ACI 318-11. 318-11. In commentary Section 8.6.1 of ACI 318-77, 318-77, in the case of compatibility torsion (torsion not required to satisfy equilibrium), torsional stiffness can be disregarded. Section 8.6.1 commentary effectively effectively permits up to 100 percent torsional

42


moment redistribution from the elastic analysis without relating it to specific ductility requirements to satisfy a high redistribution redistribut ion value. The gap between flexural and torsional redistribution redistribut ion provisions is due primarily to limited research in literature for torsional moment redistribution. There are three areas where the gap is manifested: 1. Testing relating torsional redistribution to member ductility is limited 2. Factors affecting torsional ductility and, therefore, the permissible torsional redistribution, are not understood 3. The concept of permissible torsional redistribution not affecting service load deflections has not been experimentally verified. In the following sections, literature available available in these three areas is reviewed and specific recommendations for needed research are given. 7.2.5 Torsion redistribution experiments 7.2.5.1 Early tests—Collins and Lampert (1973) tested (1973) tested six spandrel beam specimens with a concentrated load applied at midspan of the floor beam. They found that specimens designed assuming the members had zero torsional stiffness behaved as satisfactorily as specimens designed assuming uncracked stiffness values. They also showed that the ratio of torsional-to-flexural stiffness will drop at cracking, causing redistribution of the torsion and flexural moments. They concluded that for compatibility torsion, design should be for a twist and not a torque, and the main function of torsional reinforcement is to distribute the cracks caused by twist. The compatibility torsion magnitude is overestimated if gross stiffness is used. If zero torsional stiffness is assumed, design procedures become simple, and only minimum torsional reinforcement is needed to ensure ductility and limit crack width. Hsu and Burton (1974) focused (1974) focused on addressing three questions for spandrel beams subjected to torsion: 1. Would Would torsional strength be reached if a spandrel beam were designed by the ACI ACI philosophy of neglecting torsional stiffness and assuming 100 percent redistributi redistribution? on? 2. Can other values of torsion redistribution be used effectively? 3. How serious is service load cracking? To address these questions, Hsu and Burton (1974) tested (1974) tested two series (A and B) of spandrel beams: 1. Series A: Concentrated load at midspan of the floor beam. 2. Series B: Uniform loads on the floor beam simulated by four concentrated loads. Based on elastic stiffness analysis, the torsional moment distribution to spandrel beam at its joint with the floor beam is calculated according to Eq. (7.2.5.1a) 3 P l f M = 16 3 K f 1+ 4 K u

(7.2.5.1a)

Assuming that spandrel beam is torsionally fixed at both ends, the spandrel beam torsional moment magnitude is

T =

M

(7.2.5.1b)

2

In each series, the spandrel beam torsional moment is calculated assuming varying ratios between the spandrel beam torsional stiffness and the floor beam flexural stiffness. stiffness. These ratios varied from zero to 100 percent torsional redistribution. The latter condition is where torsional moment does not affect the spandrel beam, but is redistributed to positive flexure in the floor beam. Figure 7.2.5.1a shows 7.2.5.1a shows the results for Series A tests and Fig. 7.2.5.1b for 7.2.5.1b for Series B. Three pairs of companion specimens from this testing program are examined: A1 and B1, A3 and B2, and A5 and B3. Specimens A1 (Fig. ( Fig. 7.2.5.1a) 7.2.5.1a) and B1 (Fig. ( Fig. 7.2.5.1b) 7.2.5.1b) were designed assuming uncracked stiffness of both floor and spandrel beams. Specimens A3 and B2 were designed according to the limit design theory, assuming the joint moment to be 48 percent of that calculated from the elastic moment distribution using the uncracked stiffnesses of the floor and spandrel beams (and therefore 52 percent torsional moment redistribution). Specimens A5 and B3 were designed by assuming 100 percent redistribution, meaning the spandrel beam in these two specimens was not designed to sustain any torsion. 7.2.5.2 Strength and adequate values of torsion redistribution—When comparing torsional ductility and redistribution among the three specimen sets, review the experimental load-torque relationships. These curves are presented in Sets A (Fig. ( Fig. 7.2.5.1a) 7.2.5.1a) and B (Fig. ( Fig. 7.2.5.1b). 7.2.5.1b). Curves for Specimens A1 and B1 (no torsional redistribution) redistribution) can be divided into three stages: 1. The first stage represents elastic behavior before cracking. 2. The second stage is where crack development caused the torsional stiffness to decrease and the torsional moment remained essentially constant whereas the load increased. When the load reached a magnitude at which the floor beam bottom longitudinal reinforcement yielded, as indicated by in the figures, the third stage began. 3. In the third stage, a torsional plastic hinge developed under the load and accelerated the twist deformation of the spandrel beam. A new load-carrying mechanism evolved in which the load was primarily sustained by the spandrel beam torsional strength. The torsional moment again increased, primarily resulting in a slight increase in the load-carrying strength of the assembly. Spandrel beam stirrups and longitudinal bars then yielded before reaching assembly failure. Specimens A3 and B2 (52 percent redistribution) exhibited behavior similar to that of Specimens A1 and B1, despite the assumed higher value of torsion redistribution. In both Specimens A3 and B2, the load-torque curve had the same three stages exhibited in Specimens A1 and B1. Both specimens also exhibited the anticipated level of torsion redistribution. redistributi on. Although A3 and B2 failed at almost the same ultimate failure loads as Specimens A1 and B1, the failure load corresponded to about 50 percent less torsion at the joint between the floor beam and the spandrel beam. These results indicate the torsional redistribution assumptions for

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43

Fig. 7.2.5.1a—Series A ( Hsu and Burton 1974). (Note: 1 in.-kip = 0.113 kNm; 1 kip = 4.45 kN.)

Fig. 7.2.5.1b—Series B ( Hsu and Burton 1974). (Note: 1 in.-kip = 0.113 kNm; 1 kip = 4.45 kN.)

these two specimens closely matched the design assumption of 52 percent torsional redistribution. Specimens A5 and B3 were designed for 100 percent or complete torsion redistribution, as implied in ACI 318 commentary. As shown in Fig. 7.2.5.1a and 7.2.5.1b, torsion resisted at the joint stayed constant after cracking occurred. After cracking, the

load increased significantly, almost doubling, at the same torsion value until failure. Failure occurred at a load far below design strength, indicating that the design assumption for torsion redistribution was invalid. Figures 7.2.5.2a and 7.2.5.2b show the torque-versus-measured twist deformation for Series A and B, respectively. Both figures support conclu-

44


Fig. 7.2.5.2a—Series A torque-twist relationship ( Hsu and Burton 1974). (Note: 1 in.-kip = 0.113 kN·m; 1 in. = 0.0254 m.)

Table 7.2.5.3—Maximum crack width at service load Specimen

Design torsion redistribution, %

Maximum crack width, in. (mm)

A1

0

0.004 (0.10)

A3

52

0.005 (0.13)

A5

100

—

B1

0

0.005 (0.13)

B2

52

0.005 (0.13)

B3

100

—

sions similar to those previously mentioned. Specimens A1 and B1 exhibited high ductility, as indicated by the torquetwist relationship. Specimens A3 and B2 exhibited a twist angle 50 percent greater at failure than that of Specimens A1 and B1, which roughly corresponds to the designed torsion redistribution. Specimens A5 and B3, designed for 100 percent torsion redistribution, exhibited far lower twisting ductility. In fact, both specimens failed at 0.3 × 10 –3 degrees/ mm (8 × 10 –3 degrees/in.), far less than the angles exhibited by Specimens A1 and B1. 7.2.5.3 Cracking service load —Hsu and Burton (1974) defined service load as one-half the ultimate design load or actual ultimate test load, whichever was larger. Table 7.2.5.3 shows the maximum crack widths measured at the service load for specimen Series A and B. No crack width readings were recorded while testing Specimens A5 and B3 because these two specimens failed prematurely. A comparison of Specimens A1 and A3 shows that maximum crack widths increased only slightly for the 52 percent torsional redistribution assumption. This slight

increase may be within the measurement tolerance. Specimens B1 and B2 had the same crack widths. These tests indicate that significant torsion redistribution can be performed without affecting serviceability. Nevertheless, the results were not sufficient to support firm conclusions. Following Hsu and Burton’s (1974) approach, Hsu and Hwang (1977) tested T-shaped specimens with short spandrel beams to study the moment distributions between floor beam and spandrel in the case of high shear-to-torsion ratio. They found that the torsional limit design method can be applied to spandrel beam end regions, adjacent to the columns, where high shear stresses are combined with torsional moments. As a result, the torsional limit design was added to ACI 318-77 code. Torsional hinge moment was calculated based on a torsional shear stress of 0.33 f c′ in MPa (4 f c′ in psi). Conclusions reached by Hsu and Burton (1974) were validated by Abul Mansur and Rangan (1978), who conducted experiments on seven beams with configurations shown in Fig. 7.2.5.3. In a related set of experiments, Abul Mansur and Rangan (1978) studied the effect of applying a concentrated load directly on the joint. This work showed that applying a concentrated load in this fashion did not affect torsion redistribution results, which were generally in agreement with Hsu and Burton’s work (1974). 7.2.6 Factors affecting torsion redistribution—Section or member ductility is a primary factor affecting permissible redistribution. With flexure, ACI 318-95 relates ductility to the amount of longitudinal reinforcement present in the section. A similar torsional ductility reinforcement ratio would be appropriate, accounting for transverse and longitudinal reinforcement areas coupled with torsion support


45

Fig. 7.2.5.2b—Series B torque-twist relationship ( Hsu and Burton 1974). (Note: 1 in.-kip = 0.113 kN·m; 1 in. = 0.0254 m.)

conditions. Before 1995, ACI 318 adopted the skew-bending theory for idealizing member behavior under torsion. The approach was complex and produced several inconsistencies, which led to the introduction of space truss analogy to ACI 318-95. Before this change in philosophy, when the skewbending theory was prevalent, Chakraborty (1977) proposed a balanced torsional reinforcement concept. He used a spacetruss analogy for developing the balanced reinforcement concept, referring to early work on the space-truss analogy by Mitchell and Collins (1974). Chakraborty’s work was the subject of commentary in several professional journal discussions, most notably by Collins and Mitchell (1978). Discussions focused primarily on the difficulty of using Chakraborty’s approach and an apparent difficulty in deriving similar conclusions from the specimens tested. These difficulties stemmed primarily from the fact that Chakraborty’s (1977) approach considered both the truss-analogy from the modified compression field theory (CFT) of Mitchell and Collins, and compatibility between the concrete and reinforcement. This approach resulted in a complex equation (Eq. (7.2.6)) for the balanced stirrup reinforcement area for one leg. The equation proposed by Chakraborty is

Atb ph Ao s

=

0.85 f c′β1 2

0.003 E s

f ty f ty

+

 β1  p  2   o

0.003 Es  1 −

(7.2.6)

ph

The above approach of combining both compatibility and the space-truss analogy to compute torsional strength, or

Fig. 7.2.5.3—Test configuration ( Abul Mansur and Rangan 1978 ). (Note: 1 mm = 0.0394 in.)

balanced reinforcement, was not adopted by ACI 318 when the committee changed to the space-truss analogy in 1995. Numerous articles have reviewed ductility and serviceability, and their relationship to the angle θ. The crack width model of Thürlimann et al. (1983) is shown in Fig. 7.2.6a. This model minimizes strains in the entire concrete medium. Thürlimann also suggested that within these limits of strut inclination, the maximum diagonal compressive stress is independent of the angle of inclination. Further, the minimum crack strain corresponds to an inclination angle of 45 degrees that in turn corresponds to the simultaneous yielding of both the longitudinal (main) and transverse (stirrups) reinforcement (Fig. 7.2.6b). Their analysis was based on shallow beams to which Bernoulli’s hypothesis of plane strain is applicable. In such beams, the elastic stress distribution implies that maximum diagonal compression stress is inclined at 45 degrees to the beam axis. Another interpretation of angle limits suggested by Thürlimann et al. (1983) is that as the truss-model stress field deviates from the elastic stress field that the crack strains,

46


Fig. 7.2.6a—Crack width model (Thürlimann et al. 1983).

Fig. 7.2.6b—Cracking strain and inclination angle θ (Thürlimann et al. 1983).

thereby increasing crack widths. Ali and White (1997) showed that for disturbed (D) regions in general, where the Bernoulli hypothesis is not applicable, large deviations from the elastic stress distribution result in loss of ductility and serviceability due to increased concrete strains. This deviation corresponds to large deviations from the 45-degree angle between diagonal compression and longitudinal reinforcement for shallow beams. The interaction between torsion and flexure has been shown to affect flexural ductility of members subjected to both torsion and flexure. An extensive testing program of L-beams conducted by Bishara and Londot (1979) showed that increased torsional loads resulted in reduced flexural rotations. This reduced flexural rotation is shown in Fig. 7.2.6c, where the flexural rotation is plotted with respect to a normalized torsional stress. 7.2.7 Needed research—As in the case of flexure, a preferred design procedure would specify the relationship between permissible moment redistribution and measure of ductility. To allow for complete yielding of reinforcement, ACI 318 limits the angle θ in the range of 30 to 60 degrees. The angle θ is generally taken as 45 degrees for reinforced concrete beams and 37.5 degrees for prestressed beams. A comprehensive study of prestressed and nonprestressed

Fig. 7.2.6c—Effect of torsion-flexure interaction on ductility ( Bishara and Londot 1979).

Fig. 7.2.7—Torsion in spandrel beam due to lateral load (Pantazopoulou and Moehle 1990). (Note: 1 in.-kip = 0.113 kN·m; 1 in. = 0.0254 m.)

beams, however, is needed to understand the relationship between permissible moment redistribution and θ. Understanding the direct relationship between θ and ductility is also essential. It is important to include the interaction among torsion, shear, and flexure and its effect on ductility. The beam configuration in Fig. 7.2.1.2 (described in more detail in Hsu and Burton (1974)) is ideally suited for such a study. Curves similar to those in Fig. 7.2.6c, once developed, could be readily used in code provisions. Simple provisions for calculating the slab-action torsion developed in a spandrel beam under lateral loads (Pantazopoulou and Moehle 1990) would be equally beneficial. Tests by Pantazopoulou and Moehle (1990) showed that, due to slab action, torsional moments can develop in the spandrel beam. Figure 7.2.7 shows flexural and torsional moment distribution and the corresponding deflection and twist along the longitudinal axis of a spandrel beam due to slab action under lateral loads. The left side shows measured flexure with moment at the top and deflection at the bottom, and the right side shows torque with torque and twist at the bottom. The supported end is to the right of each graph whereas the free end is to the left.


47

Fig. 7.3.1a—Typical L-shaped spandrel beam ( Lucier et al. 2007 ). (Note: 1 in. = 0.0254 m.)

7.3—Precast spandrel beams 7.3.1 General—To this point, the discussion has focused on monolithic, cast-in-place concrete structures in which torsional moment redistribution could occur when members subjected to torsion are designed with adequate ductility. The situation is different in precast structures where precast members, either reinforced or more commonly prestressed, are usually simply supported and, without continuity, there is no torsional moment redistribution. These members are designed for the full torsional moment to maintain equilibrium. Precast L-shaped spandrel beams, which are common to their type, are characterized by a plate- or wall-like web with a continuous ledge running along the bottom of one web side. This ledge provides deck beam support, as shown in Fig. 7.3.1a. A variation on the L-shaped spandrel beam is the corbel spandrel beam on which the continuous ledge is replaced by a series of discrete corbels that support deck beams. Another variation is the pocketed spandrel beam in which the discrete corbels are replaced by a series of pockets (recesses) cast in the spandrel that support deck beams. All these spandrel beam types are subjected to significant torsion caused by the series of large, concentrated, eccentric loads along the span. Typically, L-shaped spandrel beams are 1.5 to 2.1 m (5 to 7 ft) deep with spans ranging from 9.1 to 15.2 m (30 to 50 ft) in length. They are usually simply supported at the columns. Beam ends are connected to columns to prevent torsional rotation. Deck beams are often connected to the spandrel web to provide lateral restraint along the span. These end and intermediate connections greatly affect the L-shaped spandrel beam torsional behavior. Eccentrically applied loads on unsymmetrical L-shaped spandrel beams cause vertical displacement as well as significant lateral displacement and rotation. The torsional and shear effects are largest near the spandrel end. This complex structural behavior, coupled with heavy loadings, often results in heavy reinforcement in the end regions. Similar significant torsion effects may occur in inverted T-beams with severely unbalanced loads, as shown in Fig. 7.3.1b. 7.3.2 Torsion design—Design for torsion in precast concrete members received much attention in the late 1950s and early 1960s when precast prestressed concrete became

Fig. 7.3.1b—Equilibrium torsion in T-beam with unbalanced load.

widely accepted as an alternative to cast-in-place concrete construction. Before publication of a paper by Zia and McGee (1974), the primary guidance on torsion design for engineers were recommendations developed by former ACI Committee 438 in 1969. The ACI 438 report formed the basis for the first comprehensive torsion design provisions to be included in the ACI code (ACI 318-71). The Zia-McGee paper extended the ACI code torsion provisions to prestressed concrete. In 1978, Zia and Hsu updated the Zia and McGee (1974) paper to conform to ACI 318-77 code format and also introduced refinements. Papers by Zia and Hsu (1978, 2004) were widely used by the precast prestressed concrete industry for nearly 30 years for its reliability and ease of use. The ACI 318 code and the Zia-Hsu approach present a unified method for the torsion and shear design of prestressed and nonprestressed concrete flexural members following similar design procedures. The ACI method was developed with a primary focus on compact closed sections generally found in cast-in-place reinforced concrete structures and was based on a thin-walled tube, space-truss analogy. The Zia-Hsu method was developed with a primary focus on noncompact flanged sections more common in precast structural systems and the skewed-bending concept. In cast-in-place structures, the torsional effect is often minimized due to stress redis-

48


tribution in an indeterminate system, whereas in precast structures, the torsional effect is more severe and connection details can significantly affect the torsional behavior of structural members (Klein 1986; Raths 1984). The ACI torsion design procedure produces design results comparable to those offered by the Zia-Hsu method (Prestressed Concrete Industry (PCI) 1999), which was updated based on the Zia-McGee method (PCI 1978) developed from the tests of small laboratory specimens. The aspect ratio of their component rectangles rarely exceeded 1:3, in contrast to large precast members such as L-shaped spandrel beams. Large spandrel beams may develop plate-bending action in addition to torsion, and therefore behave differently from the small laboratory specimens. The Zia-Hsu method has been calibrated by two tests (Klein 1986) of full-size members with support conditions similar to those used in practice. Torsion behavior of slender spandrel beams in these tests was much different from compact beams. Spandrel beams showed no signs of cover spalling or stirrup debonding, for which closed stirrups are required. End region behavior of slender precast spandrel beams is dictated by connection details. Upper horizontal reaction prevents torsional rotation and causes out-of-plane web bending. Vertical shear force from the reaction is carried across the same diagonal crack. Out-of-plane bending caused by horizontal forces is essentially the same as bending produced by a concentrated load or reaction in the corner of a structural slab. Tests conducted by Logan (2007) and by researchers at North Carolina State University (Lucier et al. 2007; Hassan et al. 2007) confirmed that out-of-plane bending dominates torsional response in slender spandrel beams. Lucier et al. (2007) evaluated the need for closed ties in slender spandrel beams using fullscale tests. Twelve precast spandrel beams, each nearly 14 m (15.3 yd) long, were tested to failure in the laboratory. End regions of the beams were designed for plate-bending and shear, with and without closed ties. Several beams failed in flexure or ledge punching. The ledge punching failures were primarily due to interaction with global tension and shear forces in the ledge. Beams with reinforcement to prevent such flexure and ledge failures failed due to combined torsion and shear. All torsion/shear failures occurred just inside the bearing reaction along a failure plane inclined at an angle of approximately 45 degrees. Although the beams with closed ties performed somewhat better, all of the beams sustained test loads well in excess of their calculated nominal strengths. These test results suggest that slender spandrel beams can be designed based on the following four principles: 1. Torsion acting on the 45-degree failure plane can be divided into plate-bending and twist components, each equal to T u / √2. 2. Plate-bending component of torsion requires equal amounts of longitudinal and transverse reinforcement evenly distributed over the height of the beam. 3. The twist component of torsion is resisted by out-ofplane shear stresses, which are greatest near the section’s top and bottom, as predicted by Saint-Venant’s equations and the “soap bubble” analogy. Closed ties are required only where the concrete section cannot safely resist the twist component.

4. Reinforcement requirements on the inside face and outside face may be considered separately; shear increases the required reinforcement on the inside face and decreases it on the outside face.

7.4—Torsion limit design 7.4.1 Basic concepts of limit design—There are two fundamental limit design theories (Gvozdev 1938, 1960; Melan 1938; Horne 1949; Lubliner 1990): 1. Lower-bound approach 2. Upper-bound approach The lower-bound approach requires a stress field that satisfies equilibrium everywhere and that does not violate appropriate yield conditions or material failure criterion at any location. The classic example of lower-bound approach is the truss model in which a stress field is assumed and the ultimate strength is calculated accordingly. For members made of plastic materials (those that satisfy the basic theory of plasticity assumptions), the resulting ultimate strength is easily proved to provide a conservative, lower-bound solution to the true ultimate strength of the structural member under consideration. In the upper-bound approach, a strain field or failure mechanism is developed that does not violate strain compatibility conditions or appropriate yield conditions and that does not exceed material failure criterion at any location. A classic example of the upper-bound approach is the yield line theory of slabs in which a failure mechanism is assumed and the ultimate strength is calculated accordingly. For members made of materials satisfying basic theory of plasticity assumptions, the resulting ultimate strength can easily be proved to provide an upper-bound, or unconservative, solution to the true ultimate strength of the structural member under consideration. Exact solutions are obtained if the results of the upper-bound and lower-bound solutions are identical. The advantage of applying either the lower- or upperbound approach lies in their simplicity. Relaxation of either equilibrium or strain compatibility conditions in solving the governing equations significantly simplifies the solution procedure. There are, however, disadvantages: 1. The potential exists for large over- or under-estimation of true ultimate strength; and 2. No information is provided for important quantities, such as deflections and rotations at service or failure load limit states; these are quantities that require classical deflection analysis considering all conditions of equilibrium, compatibility, and constitutive relations governing the material behavior. Research in limit design faces the following challenges: 1. The appropriate stress or strain field needs to be derived and validated from ultimate strength calculations that are neither too conservative nor too unconservative for the cases of lower and upper bounds, respectively. 2. Parameters should be developed for the stress or strain field such that optimization techniques can be used to achieve realistic lower and upper ultimate strength estimations.


3. Both the lower- and upper-bound approaches should be recognized as valid for plastic materials; any approach is valid if it satisfies the basic assumptions of the theory of plasticity of either perfectly rigid plastic or linear-elastic, perfectly-plastic materials. In both cases, the material should ideally exhibit unlimited strains under the same yield stress. Reinforcement—in particular, mild reinforcement—exhibits behavior closely satisfying these requirements. Certain assumptions and limitations, however, should be applied to achieve proper application of the plasticity theory to structural concrete members. The first two challenges are explored in 7.4.2 and 7.4.3 because they pertain to reinforced concrete members subjected to torsion. The third challenge is far-reaching, as it relates to the general applicability of the theory of plasticity to plain concrete and reinforced concrete members. A thorough discussion of this challenge is provided in Chen (1982). 7.4.2 Lower-bound approach 7.4.2.1 General—The lower-bound method requires a stress field that satisfies equilibrium everywhere and does not violate the appropriate yield conditions at any point. From a design perspective, this approach means that if every region of a structure is proportioned and detailed for a stressresultant field that satisfies all equilibrium conditions, then the structure should safely carry the design ultimate load. A well-known example of this approach is the Hillerborg’s (1960) strip method for reinforced concrete slab design. Rangan et al. (1978) proposed a closed-form, lower-bound method for the torsion design of reinforced concrete structures with the following steps: 1. Set up all equilibrium equations in a statically indeterminate structure in terms of the redundancy. Let the number of redundancies be represented as n. 2. Assign the maximum torsional moment, which is normally near a torsionally-restrained support, equal to the limiting value T o defined as

= 0.33 fc′( x 2 y / 3) (N-mm) 2 To = 4 fc′( x y / 3) (in.-lb) To

(7.4.2.1)

Here, x and y are the shorter and longer overall rectangular cross section dimensions, respectively, and f c′ is the characteristic concrete cylinder compressive strength. As an alternative, the value of T o can be set by the designer. This condition will result in an equation to solve the redundancies. 3. To obtain the other ( n R – 1) equations, introduce ( n R – 1) known quantities in the stress-resultant field. For this purpose, the position of inflection points and the maximum values of any stress resultants other than the torsional moment can be assumed. All or some of these can then be used to obtain the required (n R – 1) equations. 4. The above n R equations can be solved simultaneously, and the resultant stress field is completely formulated. 7.4.2.2 Helical stair example—Rangan et al. (1978) applied the four steps (7.4.2.1) to the design of a helical stair with fixed support, subject to uniformly distributed

49

Fig. 7.4.2.2—Elevation of helical stair ( Rangan et al. 1978 ).

loading. Elastic theory indicated that a helical stair is indeterminate to the sixth degrees. When the loading is symmetrical, however, the resultant-stress field is fully described in terms of only two redundancies: M o and H o. The term M o is the flexural moment acting in a tangential plane, and H o is the horizontal force in the radial direction, both acting at midspan, as shown in Fig. 7.4.2.2. For this problem, six equilibrium equations were first formulated in terms of M o and H o, along with reactions at the two supports. In their paper, Rangan et al. (1978) assumed the maximum torsional moment to occur at an angle Φ = γ 1 on the helical stair. The maximum T o is therefore expressed as (T f )Φ = γ 1

= To = 0.33

(T f )Φ = γ 1

= To = 4

f c′

x 2 y

3

( N-mm )

2

f c′

x y

3

(7.4.2.2a)

(in.-llb)

Development of another equilibrium equation required solving for the redundancies. The moment M rf was assumed to be zero at Φ = γ 2 (inflection point). The resulting equations were in the form w = f (γ 1,γ 2)

(7.4.2.2b)

For any selection of angle γ 1 or γ 2, the resulting ultimate load provided a lower-bound solution to the true ultimate strength of the stair. The most realistic estimate of ultimate load is obtained when the ultimate load in Eq. (7.4.2.2b) is maximized, or when

∂w ∂w = 0 and =0 ∂γ 1 ∂γ 2

(7.4.2.2c)

Substituting Eq. (7.4.2.1) and (7.4.2.2a) through (7.4.2.2c) into the equilibrium equations for the helical stairs and proceeding with additional mathematical derivations (Rangan et al. 1978), an equation for the ultimate helical stair strength was provided as a function of the stair dimensions and the characteristic concrete cylinder compressive strength f c′.

50


Fig. 7.4.2.3—Curved guideway structure.

7.4.2.3 Curved girder example— Collapse load cannot be determined using a closed-form solution for all torsional cases, except for possibly the simplest cases, such as the helical stair case solved by Rangan. A trial-and-error procedure, therefore, is usually employed to solve for more complex members subjected to torsion, such as the case of curved girders. Geometrical techniques have been presented for the trial-and-error procedure (Boulton and Boonsukha 1959; Jordaan et al. 1974), but most published studies use computer analyses (Yoo and Heins 1972) to obtain the collapse load. Representative solutions from the literature provide various iterative procedures. For example, Yoo and Heins (1972) determined the sequence of hinge formation leading to collapse using an elastoplastic analysis and finitedifference techniques. Badawy and Jordaan (1977) used a methodical technique to set up a system of nonlinear, simultaneous equations for each failure mode. The equations satisfied both equilibrium and yield conditions, and provide a lower-bound estimate of ultimate strength. Badawy and Jordaan (1977) indicated that: “In some cases, the number of equations that result from a lower bound analysis is less than the number of unknowns, and the required solution should correspond to the maximum load. The problem is generally reduced to one requiring an optimization type procedure, in which the load is maximized under the conditions or constraints of equilibrium and yield criterion.” A typical optimization technique normally employed to satisfy those constraints simultaneously, and therefore solve for the ultimate carrying strength, is the Lagrange multiplier method as described in Badawy and Jordaan (1977). The curved girder (Fig. 7.4.2.3) and associated torsion solution problem represent a condition examined by several researchers because of its frequent occurrence in design. Separate studies by Boulton and Boonsukha (1959), Jordaan et al. (1974),Yoo and Heins (1972), and Badawy and Jordaan (1977) all shared a similar general lower-bound framework as applied to a curved girder subjected to torsion. 7.4.3 Upper-bound approach —The use of upper-bound approaches in the design of reinforced concrete structures is common, particularly in slab design using the yield line method. Studies of the general upper-bound approach appli-

Fig. 7.4.3—Square prismatic reinforced concrete torsional member (Wang and Hsu 1997 ).

cability to reinforced concrete structures are numerous. For example, Wang (1995) and Wang et al. (1993) have discussed its applicability to reinforced concrete deep beams. Wang and Hsu (1997) extended that work to reinforced concrete beams subjected to pure torsion. Both deep beams and torsional members can be viewed as special cases of shear failure. In the case of pure torsion, shear stresses develop in the member to resist the applied torque. The application of plasticity theory may then successfully lead to analytical solutions of ultimate torsional strength of prismatic elements made of homogeneous and isotropic materials. One analytical solution method is the “sand-heap analogy” developed by Sadowsky (1949), which is based on the theory of plasticity. This theory has proven effective for estimating the ultimate torsional strength of metal elements. The same approach for structures made of reinforced concrete, however, is complex and not as straightforward. Wang and Hsu (1997) used the modified Coulomb-Mohr failure criterion as the constitutive law of concrete and assumed that the plastic flow of concrete after yielding is associated with this failure surface. They employed a generalized energy dissipation rate formulation along a yield line in accordance with the failure criterion, as well as the upperbound approach, to calculate the ultimate strength of a beam subjected to pure torsion. They compared their approach to the results of experiments and found substantial agreement with them. Expanding on this work, Wang and Hsu (1997) considered a square prismatic reinforced concrete beam subjected to pure torsion, as shown in Fig. 7.4.3. By introducing a permissible failure mechanism, as illustrated in Fig. 7.4.3, the work equation along the yield surface is formulated as v*

= Φ x1 sec(α* − β* )

(7.4.3a)

Refer to Fig 7.4.3 for identification of α* – β*. Therefore, on an infinitesimally small area dA of the yield surface, Eq. (7.4.3b) applies


dA = 2 x1 sec β* dx1

Ds

= 2bc2 Φ ρl fly tan(α* − β* ) + ρt f ty tan β* 

(7.4.3c)

The energy dissipated on the discontinuous concrete yield surface can also be found by integration, resulting in the equation Dc

=

Φ 3

b3 sec(α *

∂T = 0

(7.4.3b)

By counting all reinforcement bars on the relevant section, the total energy dissipated by the reinforcement is represented by

− β* ) ft * (sin α* + K cot α* cos α* ) (7.4.3d)

K is obtained from the Mohr-Coulomb yield criterion and

takes the form

51

(7.4.3i)

The solution of Eq. (7.4.3i) results in an expression of the form T

=

f (m, ft , ω s , ac ) *

Where the first two functional terms represent concrete properties, ω s is a reinforcement index, and the last term represents geometric properties. Wang and Hsu (1997) obtained a closed form solution for Eq. (7.4.3j), which they compared with experimental results. Thirty-nine test results from previous empirical studies conducted at the University of Houston were analyzed and compared with the proposed theoretical solution. Researchers found that the best fit was obtained when a uniform plastic effectiveness factor for concrete under torsion was used, as given by

ν = 0.75 + 0.125ω

s l

K

1 = m + 2(1 − 4

m + 1) 



(7.4.3e)

where m=

f c* f t *

(7.4.3f)

The quantities f c* and f t* are related to the concrete compressive and tensile strength, f c′ and f t′ , using plastic effectiveness factors νc and νt *

m=

f c

* t

f

=

νc f c′ νt f t ′

(7.4.3g)

Although a discussion on plastic effectiveness factors is generic to concrete structures and outside the scope of this report, it is noteworthy that plastic effectiveness factors are direct functions of concrete ductility. The higher the strains withstood by the concrete medium, the higher the plastic effectiveness factors. Many formulas give the plastic effectiveness factors as inversely proportional to concrete strengths because ductility decreases as concrete strength increases. Another factor affecting plastic effectiveness is the degrees of concrete confinement. Higher confinement levels, due either to applied triaxial pressure or to the presence of confining reinforcement, increases concrete ductility and increases the plastic effectiveness factors. For a complete discussion of the plastic effectiveness factors, refer to Chen (1982). Equations (7.4.3c) through (7.4.3g) are related to applied torsion by the equation of externally applied work to the dissipated work, where the external work is given as W = T Φ

(7.4.3h)

The lowest upper-bound solution can then be found by

(7.4.3j)

(7.4.3k)

This equation indicates that an increase in reinforcement ratio provides better confinement to the concrete medium, allowing higher ductility and, therefore, higher plastic effectiveness. The measured-versus-predicted ultimate torsional strength of the 39 tested beams averaged 0.98 with a standard deviation of 0.09, which is a good indicator of the applicability of the upper-limit approach derived by Wang and Hsu (1997). 7.4.4 Approach comparison— Marti and Kong (1987) provided a similar yield line approach to predict the overall load-deformation response of orthogonally reinforced concrete slabs subjected to pure torsional moments in the reinforcement direction. In a companion paper, Marti et al. (1987) presented the results of torsion tests for nine orthogonally reinforced concrete slabs, comparing the results to various theoretical predictions, including the yield line approach provided by Marti and Kong (1987) . They found the measured ultimate resistances to be 5 to 46 percent below yield line theory predictions. They also found that a close prediction was achieved by a lower-bound limit analysis approach with an assumed effective plastic concrete strength equal to 45 percent of the measured uniaxial compressive strength. No similar research comparing lower- and upperbound approaches to each other or to existing codes for beams subjected to torsion, either pure or combined with flexure, has been published. Such a study would have shed more light on the validity of upper- and lower-bound assumptions, as well as the degrees of conservatism or unconservatism in the lower- and upper-bound approaches, respectively.

7.5—Treatment of open sections 7.5.1 Types of torsional resistance— When subjected to torsion, a member develops two types of torsional resistance—Saint-Venant and warping. For solid and hollow members with a relatively bulky cross section, Saint-Venant (1856) torsion normally predominates with warping torsion generally neglected. In contrast, for members with a thin-

52


Hwang and Hsu (1983) used the Fourier Series method to solve the warping torsion problem. Under torsional stresses, the angle of rotation was assumed to be a continuous function between the supports, expressed as ∞

Φ( z) = Bo + ∑ Bn sin n =1

Fig. 7.5.2—Typical B-Φ′′ curve ( Hwang and Hsu 1983).

walled open section, warping torsional resistance predominates and the Saint-Venant torsional resistance does not occur. Past research has usually ignored secondary warping phenomenon (Vlasov 1961; Zbirohowski-Koscia 1968), mainly due to the mathematical difficulties encountered in formulating this type of torsional resistance and the very narrow thickness of metal structures. The wall thickness of reinforced concrete open sections, however, is often substantial compared with overall cross-sectional dimensions. Therefore, this secondary warping may play a less significant role in reinforced concrete structures. The torsional behavior of a homogeneous elastic member with a thin-walled open section is predicted by Vlasov’s theory. The theory was later generalized by Hwang (1978) to predict the torsional behavior of inelastic reinforced concrete members with open sections in the post-cracking stage. The mathematical formulation proposed by Hwang (1978) and Hwang and Hsu (1983) is given Eq. (7.5.2a). 7.5.2 Consideration of warping torsion-formulation— Applying the concept of sectorial area, Vlasov (1961) derived the basic equation for warping torsion, which relates the warping torque T w to the third derivative of the rotation with respect to z, Φ′′ T w = – EJ wΦ′′

(7.5.2a)

The quantity EJ w is used to characterize rigidity of the beam subjected to warping torsion. The term J w is called the sectional moment of inertia and is calculated by applying procedures summarized by the original work of Vlasov. Integrating Eq. (7.5.2a) results in the equation B = – EJ wΦ′′

(7.5.2b)

where B is the integral of T w. A typical B-Φ′′ curve is shown as a solid line in Fig. 7.5.2. This B-Φ′′ curve can be constructed by a method proposed by Hwang (1978) and Hwang and Hsu (1983) based on a bimaterial model. A point on the curve signifies a certain load level. At the specific load level, the slope of a straight line connecting this point with the origin is defined as the postcracking warping rigidity. The presence of an end diaphragm provides a warping restraint, which is considered in their model by incorporating an applied bidirectional moment.

∞ nπ nπ z + ∑ Bn cos z (7.5.2c) n =1 L L

where Bo, Bn, and Bn′ are constant in the Fourier series expansion of a beam’s curvature under warping. Both the Saint-Venant torsion (1856) and warping torsion expressed the basic governing equation for mixed torsion as T ( z ) = GC( z ) Φ(′z)

− EJ w Φ (′′′z)

(7.5.2d)

The solution for Eq. (7.5.2d) was calculated using matrix notation, which is discussed in Hwang (1978). Before cracking, the GC and EJ w terms are constants for any given section and the solution is easily identified. After cracking, however, both GC and EJ w vary as functions of the load level, making it essential to employ a trial-and-error procedure to obtain the solution for mixed torsion (Hwang 1978). Krpan and Collins (1981a) also developed an analytical procedure using Vlasov’s theory and incorporating CFT to predict the elastic pre- and post-cracking response of thin-walled reinforced concrete beams. The post-cracking response was categorized as either a cracked elastic response or an inelastic warping torsion response. The authors recommended using the post-cracking section properties rather than the uncracked elastic when analyzing the overall structure values to achieve a more accurate assessment of the actual force distributions in the structure. 7.5.3 Experimental verification— To verify the proposed solution, Hwang and Hsu (1983) designed two reinforced concrete beam specimens with open sections and tested their torsional behavior. The results were used to evaluate the Fourier Series method proposed for mixed torsion analysis. A comparison of experimental and predicted data revealed that the Saint-Venant torsional resistance had to be modified by certain empirical factors, as addressed in Hwang (1978). Incorporating these modification factors into the SaintVenant torsional portion, they found that a mixed torsion analysis using the Fourier Series method predicted with reasonable accuracy both the pre- and post-cracking stages of the T -Φ′ curves of the two tested specimens. Figure 7.5.3 shows test results and analytical predictions for a thin-walled reinforced concrete specimen tested by Krpan and Collins (1981b). That work indicated that to accurately predict the strain in transverse reinforcement, it was necessary to account for the interaction between Saint-Venant’s torsion and warping torsion. Test results obtained by the authors, shown as numbered points in Fig. 7.5.3, were compared with analytical predictions made according to the method proposed by Krpan and Collins (1981a). Figure 7.5.3 shows these results and confirms the ability of the developed analytical method to predict measured torque and twist. This method also accurately predicted the elastic pre- and post-cracking


53

response of thin-walled reinforced concrete beams. The postcracking response for the cracked elastic response, or inelastic warping torsion response, was also well predicted.

7.6—Size effect on the strength of concrete beams in torsion Concrete size effect (Bažant 1984; Bažant and Planas 1998; RILEM Committee QFS 2004) is explained by energy release due to stress redistribution caused by a large crack or cracking zone formed before the point of maximum load. This size effect is transitional between the small-size asymptotic case of no size effect corresponding to plastic limit analysis, and the large-size asymptotic size effect corresponding to linear elastic fracture mechanics (LEFM), in which the fracture process is assumed to take place at one point only at the tip of a sharp crack. In small concrete structures, typical of normal laboratory tests, the size effect is normally almost undetectable, whereas in extrapolations to large structures in the field, the size effect may be strong. Size effect is expected for all the brittle failures of concrete structures, which include torsion. For reinforced concrete beams with little or no shear reinforcement, size effect for shear has been observed (Bažant and Yu 2005a,b; Bentz 2005). Torsional failure is a similar phenomenon, making size effect expected in that case as well. There is significantly less experimental data on size effect in torsion than for that of shear failure, although some experimental data do exist (Humphreys 1957; Hsu 1968a; McMullen and Daniel 1975). These were collected and evaluated by Bažant and Sener (1987) and reproduced in Fig. 7.6a (top and middle) D1. Term D is the characas plots of log σ N / σ1 versus log D / teristic structural dimension, chosen here as the cross section size, whereas σ1 and D1 are normalizing constants that are different for each plot. The term σ N is the nominal strength of the structure; for beams it is geometrically similar in three D3, where T max is the dimensions. It is defined as σ N = T max / maximum torsional moment, and D is the cross section depth defined in Bažant et al. (1988). The size range of these data, which does not exceed 1:2.7, is too limited comparing to the scatter width. If the plastic limit analysis approach to torsion was valid, the size effect would have to be absent and the trends in Fig. 7.6a would be horizontal. Clearly this is not the case. Although these data are too scattered to confirm any particular formula, they nevertheless indicate the downward trend of size effect. Bažant et al. (1988) obtained further test data, with strict geometrical scaling in three dimensions and a greater size range (1:4). They used square beams made of microconcrete with a maximum aggregate size of 4.8 mm (0.18 in.) and cross-sectional side dimensions of 25.4, 50.8, and 101.6 mm (1, 2, and 4 in.) ( Fig. 7.6b). They used standard reduced-scale deformed reinforcement taken from PCA tests (Hsu 1968a) and tested both plain concrete beams and those with longitudinal reinforcement without stirrups (except at beam ends). Results are shown in the bottom of Fig. 7.6a. There was much less scatter observed in this study than the data in the top and middle of Fig. 7.6a. Decreasing strength with the increase of size is clearly confirmed due to size effect. The straight lines of slope (–1/2) in these plots

Fig. 7.5.3—Comparison of predicted and observed torquetwist relationship (Krpan and Collins 1981b).

represent the size effect for similar fractures according to LEFM, which is the strongest possible size effect. The data trend closely approaches this slope, which means that the size effect is strong for plain concrete beams and beams with longitudinal reinforcement without stirrups. Based on the works of Bažant and Planas (1998) and Bažant (2002), theoretically-justified formulas were developed to account for the size effect in beams subjected to torsion. The equation for reinforced concrete beams, which fail after significant fracture growth, is given in simplified form as

σ N = σ0(1 + D / D0)–1/2

(7.6a)

and for plain concrete beams, which fail at fracture initiation, is given in simplified form as

σ N = σ∞(1 + rDb / D) 1/ r

(7.6b)

where σ0 is the nominal torsional strength according to the current code specifications based on plastic limit analysis; σ∞ is the plain beams strength according to elastic analysis with the maximum stress limited by material strength; and D0, Db, and r are constants. Simple prediction of these three constants and their dependence on the longitudinal reinforcement ratio, stirrup ratio, stirrup spacing, longitudinal and transverse prestress (if any), and some other geometrical ratios is lacking at present. Experiments should be conducted to verify and calibrate the theory to predict these constants. For very large beams, the Weibull statistical size effect should be supplanted to Eq. (7.6b), as shown in general in Bažant (2002).

CHAPTER 8—DETAILING FOR TORSIONAL MEMBERS 8.1—General Torsional moment in a reinforced concrete member is resisted by a circulatory shear flow in a tube along the cross section periphery. The tube can be idealized as a space truss made up of reinforcement ties and concrete struts, as shown in Fig. 3.3.5a. The shear flow induces tensile forces in both the hoop reinforcement and longitudinal reinforcement.

54


Fig. 7.6a—Nominal torsional failure strengths of beams with rectangular cross section and various sizes. Left: plain concrete beams. Right: longitudinally reinforced concrete beams without stirrups ( Bažant and Sener 1987 ). Top: test data of Humphreys (1957). Middle: data of Hsu (1968a) and of McMullen and Daniel (1975). Bottom: Data on microconcrete beams tested by Bažant et al. (1988).

Good reinforcement detailing is required to ensure that the hoop and longitudinal reinforcement can develop their yield strength to resist circulatory shear flow. Good detailing demands consideration of the interaction between the member longitudinal and transverse reinforcement. Although each member type brings about different detailing conditions, the designer should be mindful of this overall force interaction in the member. Transverse reinforcement, oriented either horizontally or vertically, should contain

a longitudinal bar at the corners. Enclosure of the longitudinal reinforcement by the transverse reinforcement provides the necessary equilibrium at the joint in the three principal directions, where the three-dimensional force flow is equilibrated.

8.2—Transverse reinforcement 8.2.1 General— Once proportioned for torsion and shear, the transverse reinforcement is laid out at a specific longitudinal spacing along the member span. The objective of transverse


55

Fig. 7.6b—Plain and longitudinally reinforced microconcrete beams tested by Bažant et al. (1988) and the typical mode of torsional failure.

reinforcement for torsion and shear is to provide the reinforcement around the perimeter to enclose the member core. Typically, this reinforcement has a smaller diameter than the longitudinal reinforcement due to spacing, placement, bending, and proportioning needs. The transverse reinforcement should enclose the perimeter as closely as possible while maintaining clear cover requirements. A closed stirrup is imperative for torsional detailing. In the simplest case—a basic rectangular

member cross section—stirrups are provided in a closed rectangular shape to encase the rectangular member core. The hooks of the closed stirrup are developed into the core with 135-degree bends. These bends ensure the hooks are wellanchored to the member core and prevent hook pullout under high torsional loads. Figure 8.2.1a provides an example of a simple rectangular closed stirrup.

56


Other common examples of cast-in-place member types subject to torsional loads are shown in Fig. 8.2.1b. The key to providing transverse reinforcement in a member subject to torsion is to start with the largest rectangular cross section and provide a rectangular closed stirrup in that section. Alternately, multiple-leg configurations can also be used for this purpose with single or multi-leg pieces or bar layouts to reinforce the cross section. Any protrusions, apertures, ledges, corbels, or other geometric outcroppings are provided with

Fig. 8.2.1a—A typical closed stirrup used in a simple rectangular cross section.

supplemental ties or stirrups developed back into the rectangular core of the individual member. As shown in Fig. 8.2.1c, good detailing usually dictates that additional ties or stirrups reinforce any protrusions. These ties also have a semi-closed detail with 135-degree or greater hooks developed into the core, which is the region enclosed by the closed stirrup shape. In addition to the closed stirrups and longitudinal bars shown in Fig. 8.2.1b and 8.2.1c, local reinforcement in the disturbed regions or D-regions should accommodate specific load concentrations. To be effective in any size member subjected to torsion, spacing between the closed ties should not exceed about one-half of the smallest dimension of the member, except for slender precast spandrel beams, such as those used in parking structures. In these members, torsional forces cause out-of-plane bending in the web. As described in 7.3, limited testing of such members has not shown signs of spalling or stirrup debonding for which closed stirrups are required. In load tests, slender precast spandrel beams have performed exceptionally well without closed ties. The current state of practice on spandrel beam behavior is contained in a recent study at North Carolina State University (Lucier et al. 2010). 8.2.2 Hooks and development considerations— Stirrups or ties are best terminated with 135- or 180-degree bends. Hooks should be developed into the main core of the member, where greater confinement is present. This detail is important in isolated members, where hook confinement is only provided by the member core, and no other external geometric conditions provide confinement. Practical considerations might dictate the use of simpler stirrup geometry, usually employing 90-degree hooks. When 90-degree hooks are used, confinement should be provided at locations where a slab frames into the beam side or elsewhere as needed.

Fig. 8.2.1b—Examples of transverse torsional detailing in cast-i n-place concrete members.


57

Fig. 8.2.1c—Examples of transverse torsional detailing in precast concrete members.

Fig. 8.2.2a—Recommended two-piece closed single and multiple U-stirrups for members subjected to torsion ( ACI Committee 315 2004).

In Fig. 8.2.2a, examples from the ACI Detailing Manual (ACI Committee 315 2004) suggest using 90-degrees hooks under various side confinement conditions. Examples of poor detailing are reproduced in Fig. 8.2.2b. These recommendations are adopted from the research of Mitchell and Collins (1976).

8.3—Longitudinal reinforcement Longitudinal reinforcement is also proportioned according to torsional requirements and provided around the member

cross section perimeter. In beam regions or B-regions, special details need not be provided aside from equal spacing or proportioning of the reinforcement around the perimeter. Splices are proportioned in accordance with ACI requirements. At the end of a cast-in-place member, the perimeter longitudinal reinforcement may have to be developed into a column or other type of rigid vertical member providing torsional restraint. CSA-A23.3-04 also includes the requirement that “A longitudinal reinforcing bar or bonded prestressing tendon

58


Fig. 8.2.2b—Ineffective closed stirrup types for members subjected to torsion ( ACI Committee 315 2004).

shall be placed in each corner of closed transverse steel reinforcement required for torsion. The nominal diameter of the bar or tendon shall not be less than s /16.” The corner bars help to support the outward thrusts in the zones between the hoops (Mitchell and Collins 1976). Precast concrete members usually have semi-rigid connections consisting of field-welded angles and plates. Although this is not a full torsionally restrained connection, the end region should be examined along with subsequent development of the longitudinal, perimeter reinforcement at the member ends. One common way of developing reinforcement is to use conventional hooks at the member end. In some cases, the hooks might not fit in typically thin precast concrete members. Another common detail is the use of U-bars placed horizontally at the member end and lapped with the longitudinal reinforcement. The U-bars provide additional end confinement to a given precast member. Likewise, such reinforcement can provide sufficient confinement around the connection plate studs or tail bar reinforcement.

Fig. 8.4—Support detailing requirements in a precast spandrel member, dependent on the support connection locations ( Raths 1984).

8.4—Detailing at supports Precast concrete members require other special detailing considerations due to their horizontal support conditions. Torsional forces in precast members are often equilibrated by out-of-plane, horizontal, or sometimes vertical, reactions at discrete locations along the member depth. The member end conditions and subsequent details are highly dependent on the support configuration. Figure 8.4 illustrates a common precast spandrel beam and the horizontal force couple that is typically developed at the end. Additional reinforcement is thereby required at the member end near the top to accommodate a potential 45-degree crack that typically develops at the end location due to the couple resisting torsion, as shown in Fig. 8.4(a)


and 8.4(b). A possible reinforcement scheme to address this condition is shown in Fig. 8.4(c) and 8.4(d). Similar conditions often exist in other precast members due to their specific support and horizontal restraint conditions. Although these are D-region locations, they are complicated by the threedimensional or out-of-plane nature of the problem.

CHAPTER 9—DESIGN EXAMPLES 9.1—Torsion design philosophy Design philosophy for torsion in the ACI 318-11 building code is based on a thin-walled tube, space truss analogy in which compression diagonals wrap around the tube and the tensile contribution of concrete is neglected. Both solid and hollow members are considered tubes in accordance with Saint-Venant’s circulatory shear flow pattern both before and after cracking. The outer part of the cross section centered along the stirrups is assumed to provide torsional resistance. The contribution of core concrete cross section is neglected. Once a reinforced concrete beam has cracked in torsion, the torsional resistance is provided primarily by closed stirrups and longitudinal bars located near the member’s surface and diagonal compression struts. The inclined angle of the diagonal compression struts is permitted to be taken as 45 degrees for nonprestressed and lightly prestressed members, and 37.5 degrees for most prestressed members. Accordingly, ACI 318-11 makes the specific assumptions in torsion design that: a) Concrete tensile strength in torsion is neglected b) Torsion has no effect on the shear strength of concrete c) Torsion stress determination is based on the closed thinwalled tube with uniform stress distribution and specific thickness, called shear flow d) The torsional, flexural, and shear strength are accounted for by adding longitudinal reinforcement calculated for torsion and flexure e) The longitudinal reinforcement are calculated for torsion and shear The design of torsional resistance in Section 6.3 of EC2-04 is also based on a truss model using the thin-walled closed section theory with an effective wall thickness. The angle between the concrete compression strut and the member’s longitudinal axis, θ, may be taken between 22 and 45 degrees. Both the solid and hollow cross section can be modeled by an equivalent hollow section neglecting the core concrete contribution to calculate the torsional resistance, which is limited by the strength of the concrete struts. The longitudinal and transverse reinforcement contributions to torsional resistance are accounted for after the thin-wall cracks. Effects of combined torsion and shear may be superimposed assuming the same value for the strut inclination angle. The required longitudinal and transverse reinforcement for torsion should be added to the existing longitudinal reinforcement for bending and transverse reinforcement for shear, respectively. The Canadian code (CSA-A23.3-04) provides a General Design Method for torsion derived from the modified compression field theory (CFT) and represents solid cross sections by an equivalent thin-walled tube with a wall thickness determined by cross section dimensions. The space truss

59

analogy provides the basic concept for torsion design assuming that the tension contribution in concrete is neglected and the diagonal compression struts spiraling around the member with variable inclined angle that depend on the loading condition and reinforcement ratio. The bending moment and longitudinal forces due to torsion and shear are considered resistant to four chords, one in each corner of the space truss and the shears by the shear flows in the walls. Dimensions of a cross section are limited to prevent crushing of the diagonal compression struts. In addition to this AASHTO LRFD method (general method), CSA-23.3-04 provides a simplified method for a restricted group of structural members, which states that the inclined angle of diagonal concrete compression strut is fixed at 35 degrees.

9.2—Torsion design procedures 9.2.1 Torsion design in ACI 318-11 — According to SaintVenant’s circulatory shear flow pattern, the most efficient cross section to resist torsion is tube-shaped. Therefore, torsion of a reinforced concrete member is a three-dimensional (3-D) problem because it involves the shear in a reinforced concrete two-dimensional (2-D) wall element of a hollow tube and the out-of-wall bending of the concrete struts. In ACI 318-11, two simplifications are made. First, the concrete strut bending is neglected and the amount of hoop steel required in the tube determined from Bredt’s (1896) equilibrium equation of a cross section q y = T u /2 Ao

(9.2.1a)

where the symbol q y is the shear flow at yield (N/mm [lb/ in.]); T u is the torsional moment (N·mm [in.-lb]); and Ao [mm2 (in.2)] is the lever arm area enclosed by the centerline of the shear flow. Second, the hoop and longitudinal steel are assumed to yield at ultimate strength. To design steel reinforcement in a 2-D shear element, it is possible to use only three equilibrium equations (Hsu 1993). Combining the three equations creates a simple equation for yield shear flow q y

=

( At f y / st )( A f y / s ) l

l

(9.2.1b)

where f y is yield stress of hoop steel and longitudinal steel (MPa [psi]); At , Al are area of hoop steel and longitudinal steel (mm2 [in.2]), respectively; and st , sl are spacing of hoop steel and longitudinal steel (mm [in.]), respectively. Substituting the shear flow q y into Bredt’s (1896) equation gives Tu

= 2 A0

( At f y / s t )( Al f y / s l )

(9.2.1c)

which is the essence of the ACI code provision. The lever arm area Ao (mm2 [in.2]) is formed by sweeping the lever arm of shear flow one full circle around the axis of twist. The centerline of shear flow was taken by Rausch (1929) to be the centerline of the hoop steel bar, and the corresponding lever arm area is denoted as Aoh (mm2 [in.2]).

60


However, this definition of area Aoh was found to overestimate the torsional strength by as much as 30 percent. Therefore, the ACI code provides a simple, approximate formula for calculating the lever arm area as Ao = 0.85 Aoh

(9.2.1d)

To provide a more accurate formula for the ultimate torsional strength, consider the softening of concrete struts in the reinforced concrete 2-D wall elements of a tube. Under a biaxial tension-compression stress condition, the compressive stress-strain curve of the 2-D elements should be multiplied by a softening coefficient. This softening coefficient is a function of the principal tensile strain ( Zhang and Hsu 1998) and varies from approximately 0.25 to 0.50. Applying this softened stress-strain curve of concrete to the study of reinforced concrete tubes under torsion (Hsu 1990, 1993), the thickness t d (mm [in.]) of the shear flow zone and lever arm area can be determined as

= 4Tu / Acp fc′ Ao = Acp − (2Tu pcp / Acp f c′) td

(9.2.1e)

where Acp is the area enclosed by the outer boundary of cross section (mm2 [in.2]); and pcp is the periphery of the outer boundary (mm [in.]). These formulas are given in the ACI code commentary, and the background was given in a paper by Hsu (1997). 9.2.2 Torsion design in EC2-04 — Section 6.3 of EC2-04 requires a full design procedure for a reinforced concrete member under torsion covering both ultimate and serviceability limit states in cases where the static equilibrium of the structure depends on torsional resistance of the elements. In conventional statically indeterminate reinforced concrete structures, torsion arises from consideration of compatibility and it is normally unnecessary to consider torsion at the ultimate limit state. However, even if torsion arises from consideration of compatibility only, it may lead to excessive cracking in the serviceability limit state. Therefore, a minimum reinforcement of stirrups and longitudinal bars should be provided to prevent excessive cracking, as indicated in EC2-04 for cracking control (Section 7.3) and detailing beams (Section 9.2). In normal slab-and-beam or framed structures, specific calculations for torsion are usually unnecessary when torsional cracking is being adequately controlled by shear and minimum flexural reinforcement. Where torsion is essential for equilibrium of the structure, EC2-04 should be consulted. One example of this is when structure arrangement is such that loads are imposed mainly on one face of a beam without corresponding rotational restraints provided. The design of torsional resistance moment is based on a truss model using a thin-walled closed section theory with inclined angle θ between the concrete compression strut and the beam axis. The angle θ should be limited and recommended limits are: 1 ≤ cotθ ≤ 2.5 (45 degrees ≥ θ ≥ 22 degrees). With a solid section, the section can be modeled

by an equivalent hollow section from which the torsional resistance is calculated. Complex shapes, such as T-sections, can be divided into a series of subsections modeled as an equivalent thin-walled section, and the total torsional resistance taken as the sum of the capacities of each individual element. The effects of combined torsion and shear for both hollow and solid members can be superimposed assuming the same value for the strut inclination angle θ. A common value for angle θ is 45 degrees. Eurocode 8 (EN 1998-1:2004) determines that: “In the critical regions of primary seismic beams, the strut inclination θ in the truss model shall be 45 degrees” (Paragraph 5.5.3.1.2(2) of EC2-08). However, in members not designated to resist seismic actions, a reduced value of angle θ could be considered to decrease the required transverse reinforcement and required longitudinal reinforcement. This way, fewer stirrups and more longitudinal bars could be provided. Required torsional reinforcement is added to the required stirrups and bars calculated from the shear and flexural design, respectively. Strength of materials used in EC2-04 is based on characteristic values and depend on whether the value is used for strength or stiffness. The characteristic value used to calculate strength corresponds to the 95 percent fractile of strength from material tests. The characteristic value for stiffness corresponds to mean strength from material tests. Design values are based on multiplying the characteristic value for resistance by the safety factors α and β. Design procedure in accordance with EC2-04: - Step 1: Calculation of the equivalent thin-walled section characteristics such as t ef , Ak , and uk (also refer to Fig. 6.11 of EC2-04 for notation) t ef = A / u ≥ 2c and, in the case of a hollow section, t ef < t real

where t ef =

effective wall thickness of the equivalent thinwalled section [mm (in.)] A = total area of the cross section within the outer circumference, including inner hollow areas (for example, A = bh in a rectangular cross section with width and height equal to b and h, respectively) [mm2 (in.2)] u = outer circumference of the cross section (for example, u = 2(b + h) in a rectangular cross section) c = distance between edge and center of the longitudinal reinforcement (centroid cover) [mm (in.)] t real = real thickness of a hollow section [mm (in.)] Ak = area enclosed by the centerlines of connecting walls, including inner hollow areas (for example, Ak = (b – t ef )(h – t ef ) in a rectangular cross section) [mm2 (in.2)] uk = perimeter of the area Ak (for example, uk = 2(b + h – 2t ef ) in a rectangular cross section) - Step 2: Assume the value of angle of compression struts, θ, based on the expression: 1 ≤ cotθ ≤ 2.5 (45 degrees ≥ θ ≥ 22 degrees). For combined shear and torsion, the same value of θ should be assumed and the common value is 45 degrees.


- Step 3: Check the maximum resistance of the member subjected to torsion and shear. This is limited by the strength of the concrete struts. If the following relationship is not satisfied, the member cross section dimensions, the concrete compressive strength, or both, should be increased T Ed T Rd.max

+

V Ed V Rd.max

≤1

(9.2.2a)

where T Ed =

design torsional moment [N·m (in.-lb)] T Rd.max = design torsional resistance moment according to the following relationship [N·m (in.-lb)] T Rd,max = 2vαcw f cd Ak t ef sinθcosθ

(9.2.2b)

V Ed = design shear force [N (lb)] V Rd.max = maximum design shear resistance according to the

following relationship [N·m (in.-lb)] V Rd , max =

bw = z =

v

=

α c bwzvfcd ( cot θ + tan θ)

width of the web of the cross section [mm (in.)] inner lever arm, for a member with constant depth, corresponding to the bending moment in the element under consideration. In the shear analysis of reinforced concrete without axial force, the approximate value z = 0.9d may normally be used ( d is the effective depth of the cross section) [mm (in.)] strength reduction factor for concrete cracked in shear, recommended values (values for use in a country may be found in its National Annex): v = 0.6(1 − fck / 250) [ f ck in MPa] v = 0.6(1 − fck / 36. 26) [ f y in ksi]]

αcw

=

f ck = f cd =

(9.2.2d)

coefficient taking into account the state of compressive stress

1   1 + σ / f cp cd α cw =  1.25  2.5(1 − σ cp / fcd ) 

σcp =

(9.2.2c)

 0 < σ cp ≤ 0.25 f cd    0.25 f cd < σ cp ≤ 0.5 f cd  0.5 fcd < σ cp ≤ f cd   non-prestressed

(9.2.2e)

mean compressive concrete stress due to design axial force (measured positive) [MPa (psi)] characteristic compressive concrete strength [MPa (psi)] design compressive concrete strength (= f ck / γc , where γ c is the partial factor for concrete equal to 1.5 for ultimate limit state and persistent and transient design situations) [MPa (psi)]

61

- Step 4: Calculation of the required cross-sectional area of the longitudinal reinforcement for torsion, Σ Asl :

∑ As =

T Ed uk cot θ

l

2 Ak

f y d

(9.2.2f)

l

where f yl d is the design yield stress of the longitudinal rein forcement [MPa (psi)]. Notes: The longitudinal reinforcement for torsion should be added to the required longitudinal reinforcement for flexure. The longitudinal reinforcement should generally be distributed over the length of side, zi, ( zi is the side length of wall i defined by the distance between intersection points with the adjacent walls [refer to Fig. 6.11 of EC2-04]), but for smaller sections it may be concentrated at the ends of this length. According to EC2-04 provisions (Section 9.2.3(4)), longitudinal bars for torsion should be arranged such that there is at least one bar at each corner, with the others being distributed uniformly around the inner periphery of the links, with a spacing not greater than 350 mm (14 in.). - Step 5: Calculation of the required cross-sectional area of the transversal reinforcement for torsion Asw s

=

T Ed

2 Ak f ywd cot θ

(9.2.2g)

where Asw =

cross-sectional area of the transversal reinforcement (stirrups) [mm2 (in.2)] s = spacing of the stirrups [mm (in.)] f ywd = design yield stress of transversal reinforcement [MPa (psi)] Notes: The transversal reinforcement for torsion should be added to the existing transverse reinforcement for shear. The torsion links (stirrups) should be closed and anchored by means of laps or hooked ends and form an angle of 90 degrees with the axis of the structural element. Refer to Fig. 9.6 of EC2-04 for recommended shapes. According to provisions of EC2-04 (Section 9.2.3(3)), longitudinal spacing of the torsion stirrups should not exceed u /8, or the requirements about the maximum longitudinal spacing between shear assemblies (Section 9.2.2(6) of EC2-04) or the lesser dimension of the beam cross section. - Step 6: Check the value of the angle of compression struts, θ, based on the calculated and provided longitudinal and transversal reinforcement from Steps 4 and 5

θcalc = tan −

1

Asw uk f ywd

s ∑ Asl f y d

(9.2.2h)

l

Note: In case of a significant difference between the calculated and the initially assumed angle of compression struts, reassume the angle θ (Step 2) and recalculate Step 3 through 6. 9.2.3 Torsion design in CSA-A23.3-04—The Canadian code stipulates that the effect of torsion should be considered

62


in design only if the torsion due to factored loads, T f [N·m (in.-lb)], exceeds 0.25T cr . The cracking torque T cr [N·m (in.lb)] is assumed to be reached when the principal tensile stress f l [MPa (psi)] (equivalent to the shear stress v in pure torsion) equals the factored tensile strength of the concrete, f cr [MPa (psi)]. For the calculation of T cr , the following assumptions are made: - Solid cross sections are represented by an equivalent thin-walled tube with a wall thickness [mm (in.)]

The longitudinal strain ε x is affected by the bending moment: shear, torsion, and if present, by axial load and prestressing in the member. In the presence of bending moment, shear, and torsion, the strain at mid-depth of the section, ε x, is computed from the expression

ε x = ( M f

/ d + V f2

+ [0.9 phT f

/ 2 Ao ]2 ) / ( 2 Es As )

(9.2.3f)

where t c = 0.75 Ac / pc

(9.2.3a)

- Bredt’s classical equation for tubular section applies [MPa (psi)] v = T f /(2 Aot c)

(9.2.3b)

- Area enclosed by shear flow path [mm 2 (in.2)] Ao = 2/3 Ac

(9.2.3c)

M f = V f = ph =

moment due to factored loads [N·m (in.-lb)] shear force due to factored loads [N (lb)] perimeter of the centerline of the closed transverse reinforcement [mm (in.)] E s = modulus of elasticity of the reinforcement [MPa (psi)] As = area of flexural reinforcement on the flexural tension side of the member [mm 2 (in.2)] θ = 29 + 7000ε x (degrees) For a given angle θ, the transverse reinforcement to resist the factored torque T f is derived from equilibrium and given by

- Factored design tensile strength for normal concrete [MPa (psi)] fcr

= 0.38φc

fcr

= 0.38φc 12

fc′ [ f c′ in MPa]

(9.2.3d)

fc′ [ f c′ in psi]

For non-prestressed concrete members, this results in the following expression Tcr

= ( Ac / pc )0.38φc

Tcr

= ( Ac / pc )0. 38φc 12

fc′ [ f c′ in MPa ] fc′ [ f f c′ in psi]

(9.2.3e)

The symbols not defined in the above equations are as follows: Ac = area enclosed by outside perimeter pc of concrete section [mm2 (in.2)] f c′ = specified compressive strength of concrete φc = resistance factor for concrete (= 0.65) If torsion is not negligible (T f > T cr ), torsion reinforcement should be provided. The General Design Method (CSAA23.3-04) for torsion was derived from the MCFT, which represents a holistic approach for both shear and torsion design. For torsion, the basic concept is the space truss analogy, originally envisioned by Rausch (1929), assuming a 45-degree angle for the compression struts. This AASHTO LRFD (general method), originally developed for shear, requires a longitudinal strain indicator ε x and the level of f c′ to estimate θ and β. In the case normalized shear stress vu / of a member subjected to pure torsion, it is not necessary to consider β. For torsion, the General Design Method ( CSAA23.3-04) assumes that: a) Concrete in the cracked member carries no tension b) The angle θ of the diagonal compression struts spiraling around the member is variable and depends on longitudinal strain at mid-depth of section, ε x

At =

T f s

2 Ao φs f y cot θ

(9.2.3g)

where s =

spacing of transverse reinforcement measured parallel to the axis of the member [mm (in.)] φs = resistance factor for non-prestressed reinforcing bars f y = specified yield strength of transverse reinforcement [MPa (psi)] Additional longitudinal reinforcement is required to resist the longitudinal forces generated by torsion. As usual, the transverse reinforcement due to torsion should be added to the shear reinforcement. Dimensions of the cross section of the member have to be such that crushing of the diagonal compression struts is prevented. This is achieved if the combined stress due to shear and torsion does not exceed 25 percent of the factored compressive strength of the concrete. This is expressed by Eq. (9.2.3h) V f bw d v

+

T f ph 2

1.7 Aoh

≤ 0.25φc f c′

(9.2.3h)

9.2.4 A comparison of torsion design procedures for ACI, EC2, and CSA— The design philosophy and procedures for pure torsion and combined loads including bending, shear, and torsion are discussed previously according to ACI 318-11, EC2-04, and CSA-A23.3-04. All the design procedure and equation citations from these codes are summarized and compared in Table 9.2.4.

9.3—Introduction to design examples Two examples were selected to illustrate the steps involved in torsion design: 1) a solid reinforced concrete rectangular beam under pure torsion; and 2) A prestressed box girder under combined loading including torsion. Although this


′ f c

2 . , 0 4 . ≤ 0 1  h . 9  3 . 1 - p h f Ao 1 1 . T 7 1 1 .   1 n q + o i t E    c p e V d v S − w

[

′ f c

2

2 1

2

φ c λ 8 3 . 0

2

   A p   

2

2

c

c

f

=

V

b

r

o

f p p A

] i s p n i ′

f

5 . 0

f

c

+

[

ε x   0 2

vy

b f ′ f c w

 ) 

−

N

0 0 7

9 . 2

0 +  

+

y + T f s f

φ o

s

A s E [

A 2

2

) p

2

= θ

=

o c

A p T o h p A E

9 2

2 7 . 0

θ t

] p

f

≥

V

−

s

t

A

f

V (

n i m

   A s      

 

T c

T

7 1 1 1 4 . d f c , 6 3 . n a [ . 3 0 . , 3 w vy 1 . 1 f 1 1 3 - b . 1 ′ c 1 d f n 1 1 a , n , 2 6 o 1 . 1 i - 0 t 8 . c . 1 1 0 e 3 . , S 1 1 = 1 n i m 1 1   .  q v s E A

′ f c c φ 5

] i s p n i

]

a 2 - P 1 M 1 i n . ′ 4 q f c [ 0 - E ′ 3 . , f c . 3 1 2 9 . φ c A . λ - 2 8 1 . A 1 3 0 S n   C o c pc i A t  c  e = S r c

] a P M n i ′

63

=

+

v

f

x a

v

M d

s m

= ε x

z f t t e

τ = d E

R T

,

=

< ) 0 3 4 - T ( 6 2 . d →  3 . n 6 a   n 9 1 o 2 i - ≤ t c 6 . e q V V S E

k d ) 7 E 2 A 1 - T ( 6 2 2 . d = 3 . n 6 a t f e n 6 τ r o 2 i t  c 6 .  c e q t S E p c / 2

4 0 2 C E

d E

n o i s r o t e r u p

+

A

s e d o c A S C d n a , ] 2 a P C M E ] n i i s , , ′ p I f 1 n . [ 1 i C 1 ′ 5 . f   A 8 1 [ 1 A p r 1 n       o 3 o    A p I i f ′ C t  f 2  ′ s A c 1 e f e S    r λ λ u φ φ < < d e T T c o r p n e g i l b a n s n a e o c i d s t s r o c n t e f o f d i d e e r e e s o n d r r t o u r i c a o a s t d r g e f e o f c e t r s h r o o t f i i d p e n n n d n i g o i a m s s r t i e n e r d t e e m a n D o o p i m s m r o o t C e r u — P 4 . 2 . 1 9 e l b a T

 x 

o c

θ n i s f

t k e A f d c c

α ν 2 =

x a m , d R

T

n

i ′ f c

    

[

)

α

t ) o c h ,

8 + w / b f k y N ( . u 1 i n / 6 9 d ′ . 5 m f c q 7 . (

)

E  0    8 d n  0 n . a i 0 m

(

= t ρ

=

x a

s m

= f

. q E , ] a 1 . P 1 M . n 2 i . ′ , 1 f 1 [ . 1 3 . n    5 o ) ′ f . i 3 3 1 t 1 c e 1 2 + n S 1 d o d ( i w V b t c n a  e )   φ S 8 ≤ 1 p A 1 . T 7 1 1 ( . q E c

c

p c

p c

c

c

c

u

T T

a 3 P . 2 M . n 9 i ′ d f c n [ a ) N f k y 2 5 . / ( 9   2 . ′ c  . q f 2 3 . 1 6 E 8 n 0 . o i 0 t c    e = S t ρ

t e

c

p 2 c

x a m , d R

d E

a m

c

p 2 c

x a m , d R

d E

]

θ s

x a m , d

V

h h 2 o u

u

] a P M n i

′ f c [

′ f c

] i s p n i ′

f c [

  ′ f c 8

+

2 V c d w

1 b λ    2 φ ≤ =

d V w b c

t n e l a v i u q e n o e i h t t c f e o s d s e e l i t r l e a p w o n r i p h t e t a l u c l a C

s s o r c e h t e f t o a s u n q e o d i s a n e e r m a i d n o i f i t e c s n e i m r e t e D

2

3

2

h o

p h A . T u 7 1

, 6 . o 3 . t n 5 . i 1 d θ , 1 e t 5 . n m o o r 3 . i o u c o t f 5 . c s 1 e n T A t 1 S a f y r n d t 2 o n φ i a ) t c ) 1 e 0 2 - t ≥ S 2 s - 1 ( A 1 1 . 1 q ( . q E E

] i s p n i ′

f c [

′ f c λ 2 =

d V w b c

s p u r r n i o t s i s f r o o t t n e u r o u p m r a f o e h d t e r e i t u a l q u e c r l a C

e b n a ] c a ) P 3 M 2 n i 1 ′ f c 1 ( y [ . q b b w f vy E d , e s ′ f c 2 2 1 . s e 5 . r 5 p 7 5 . x . 2 1 e 0 = 1 n i n o    m i t t A s c e    S d n a s l i a t e d ’ s n p o u t r i r c i t s s e e s s h t o k r c c e e h h c t r d f o n a t c e l e S

4

. n i 2 6 . v d 3 2 7 . ( 0 m m 0 0  6  n i m

i s p n i ′ c f [ ]

v

w y b f

′ f c

5 7 . 2 0

=

n i m

   A s    t

64


] a P M n i ′

4    1 f 1 o T h o 1 t p . i 5 A n 2 4 q . 0 E d  , e  2 + . m 9 . r o ) p f 3 V . − 1 s s 1 n a V r 5 d t . 0 n e a b − f 6 . n V ( 0 a c 1 . 1 θ f 3 t T . 2 h o o p 1 1 c 5 A 1 2 + 4 . f n d 0 v o n d θ i t a M = t o c c e t F = S 2

C x e n n A

t y

, f 7 h . p 3 . t 5 . 1 A 1 = n l o A i t c e S

f

4 0 2 C E

6 . 7 . 7 . 1 . 2 . 1 . 3 3 3 d d d n n a a n a 2 . 4 . 6 . : 2 1 s 3 . . 2 . 1 . 1 . d 3 3 . 3 4 n . , . a 3 4 n 2 . r . 2 t o 1 4 i d s t . n n o g n c 3 i o e t n a i c 6 S n . t c o e i c 3 : i e r S o . S h t t c : 3 f g e s n , : n S i e i 4 s e . r : t e r h r 2 r e t . v s t e d 4 p e . o e g c o v n r s e s 2 e i t p r n s e r i s t o e s l r t e e s t c r l e t e n p i e c e s o r m s S g P C o n e i n c t e t e c r t e o e f r c r c n n n i o o e R C C

s r a b n l o a i s n r i o d t u t i e r g u n p o l r o e f h t d e e i r t a u l u q c e r l a C

5

5 . 1

. 1

g

A p c 5

(

=

r

3 . 1 . 5 d n a 3 . 4 . 2 n o i t c e S

d n a 3 . . 8 1 . n 2 o 9 i t c e S

1 1 8 1 3 I C A

] i s p n i ′

′ f c

[ 1 . 5 .    p 1 2 p c c 1 A p n    o i t   c ′  e f c 2 S 1

f c [

   p 2 c

A

   T

,

l

f c

λ φ < T u

u

A

p c

p

 ′  

λ φ <

s r e b m e m e t ) e r s n o c g i n n s o i r c d t o d a o d e l s s d n e e a , r n r t s i a e b e r h s p m , r o c g o f r i n e e d r n d n u e d u B e ( c o r p n g i s e D

=

T c

] a P M n i

l

r a b l a n n o i t d i u c t i e g s n s o s o l e r c h t e h t k c r e o h c f s d l i a n t a e t d c e l e S

)

r c

y

,

3 . 0

2

g

T

l

l

φ λ 8

A p c

[

A

2 1 c

c

(

f c

[

t y

1

′ f c

c

0

, 3 t l . f y f y 5 . f f h 5 ) . h p 1 4 p s   1 2 -  t    n 1 t A s o 1 A s i (   .    t c q − e − : S E c p p ) c A 2 ′ ′ A f y d 2 c f c f l y - f 2 n a 1 1 5 1 5 ( . = = q n n i E i m m

l y

3 . 0

+

2

] i s p n i ′

′ f c

p φ φλ 8

c

p

)

l

t o c

2 f c p 1 c

, f φ 1 . φ 8λ 9 3 . 2 . 2 . - + 0 1 1 1 1 1 . ′ n q o E f i t c φ λ e 8 S 3 . p c

4 0 3 . 3 2 A A S C

d t b

θ

′ f c

f

1 m k . θ t d y f t c f y 2 f . o k 9 c 6 k A 2 d u . n u 2 0 a T = ) ≥ 3 n i ( l m , 2 . A 3 A . ∑ 6 N 8 n 1 2 o . i 9 t 6 c . . e q q S E E d n a

2

[

′ c

l

a P M n i

f c

f c

[

2

]

] i s p n i ′

s s o r c d s n i a e r l t a e i r p e o r t a p m l a e n h i o t e t c n i e s m r e t e D

d d n n a , a e s u d q a r o t o t l , n e d r e a m r e o h o t c s m a d g f n e e r i h o d t t n e c e a b n f i e t m r a e l t u e l c D a c

e b n a c s t c e f f d e e n d o r a i s r g e o r s t i f d i e n i m r e t e D

1

2

3


′ f c c φ 5

v

w

c

− d w b

f

c

1

≤ x a m ,

d E

θ

d R s V V o 0 c 3 +

) 6 d 4 d E ( n T 2 . a 3 . , 6 9 2 - d n 6 c o i , f t v z w c 4 e 1 - b S 6 c . q α E

x a m , d R

θ

)

n i s

T t f k e A θ f d c n c

)

a t

α ν + 2 θ t = o c

(

=

(

x a m , d R

T

x a m , d R

V

, ) 9 - ) 1 8 ] 1 1 ( 1 a ] i . 1 P s q ( p M . , E n , q n 1 i . 2 i ′ c . E ′ c 1 f . 3 , f . 1 [ 2 [ . 1 . ′ 1 1 3 . ′ c 1 n 5 f c f . n i o 1 2 o t 1 i λ 1 t 2 c c e n λ 2 e S o = S , i = ) t c c d 3 e c d V b w w 1 S V b 1 d ( n . q a E

] a P M n i ′

] i s p n i ′

[

[

f c

f c

d w b    d u u V M ) 9 8 6 0 0 . 0 ( 0 0 7

′ f c

0 0 7

+ ′ f c λ 6

2 1

. 0

  

] a P M n i

′ c

f d w [ b       ′ d u u f c 3 V M

+

λ 6

o , f p p 4 o : . t t A 3 n i n u 3 . − r o 1 1 t i f t d s 1 ] 1 c e N a i s n 1 n e m 5 e P . z p o d o S r 0 i s i t s M i : f n 0 + t o + c s n a 2 0 e 1 i n ′ e s c θ ] p 3 0 n r 2  θ f e S 1 t   ′ [ c p 1 f 1 0 m n t - f a A : o r 0 o p 1 - T e c t m 1 o h t 1 e 1 [ c e s c v o 1 p A E f f y d e r r d v c . 9 2 + T × V b s . d o b s l f 0 φ o c n n f y s a q  a d w ′ A n n f c s ε x i a o 6 n E  E b ′ A c 0 c c , e φ o + [ - f 0 0 r 7 2 f 1 g 4 2 2 4 . 2 ) r 5 . a = o 1 p i 6 1 0 1 a 1 ≥ . . β β e q V d 3 e 1 v s t + h c . − f 1 λ c λ . A A s n E 1 s q f o 1 a φ φ V t d e E ( = s l n e i , = = β r 1 g o s i i n . c e + u 5 r a t V V q c f v . r e e e M d 3 a . h S R 1 e T h = 1 S ε x d e n c a r n i o 3 e f r 1 ; n i ) r θ 6 e 5 t r ( a . e o 2 r , q d . h E c a t s e 2 e a E d V . ; e w h y 9 h f s u ) z n t t q 4 ( f o ) u e 3 i . ≥ t o 3 o d ( 2 a h . 3 w t θ c e a . e s 6 i t s r d S a 2 A n c . w o n f n : a 6 l t o v r f i a z w t n e I t n ⇒ + n o b ) . c b c θ e o i d e 4 i t E m t t m e ( S V e 1 1 α c c o : e c 2 e r c m . t ≥ 6 s o - S : a 2 f . n e θ t s n s n f 6 m = i t o o n e o e e x r a c r c m o c e i m , r d c t r d w o a e e R y n c f c f e i v r a e n V z t h o s S i s t f c e i w : s e s t r s f A e m r n f r e a u e r m e = m m a i s e h s , n u d h i R d s m e V M i r n i x g a u i s q M e e D R ] i ] 7 . , s a n 4 P p . . o n 2 i ] ] M 1 3 n o . t i i 1 i a n c t 1 , s i ′ ′ c P e n c 1 c f p ] f e o S ] [ a [ i M i t n S n : P s i t d c o n : i d w ′ p e i n 3 ′ e t b w M e t f c S c n 1    b f c e : c i n [ e t m r S d u u  ′ i [ c 1   ′ e V c ) n v f c V M d e φ f y c 1 u n d 9 f ) [ r w ty t . u 9 o n [ d f f o q − φ 8 c a 1 V M m ′ b w f y b e c 6 u ′ c E 1 f ) f ′ ′ n 0 c 0 r c ( c i V f , f 2 0 o 3 . f . 0 o e 0 f - q 7 r 3 e 1 . 1 ( 2 = n λ c 1 E 0 r 6 i 1 + 5 . 0 n ( v 2 e λ 5 ′ 7 r A s a 7 a . e 4 c . 2 . 7 t f r + = h . 1 s 0 q a ′ s 1 i 0 c = f e s E 2 d c = h e = , m λ s V b w λ 1 r 1 c d n u 6 w i n r . i . d V 6 b m m a . e m , 0 , . r i v v 0 e 1 i    2 n    h . A A u i S 1 = q = e M 1 c c V V R ]

, 2 4 . . 0 0 1 ≤ . 8 3 t h . 1 1 1 f o 1 1 A . T 7 n q . o 1 i t E c + e p S V v

V

65

. 0

 

=

c

V

=

c

V

2

+

d V b w c

] i s p n i ′

f c [

  ′  f c 8

+

c

V

d w b

      φ φ ≤

h h 2 o

p A . T u 7 1

≤

h h 2 o

p A . T u 7 1

s s o r c e h t e f t o a s u n q e o d i s a n e e r m a i d n o i f i t e c s n e i m r e t e D

e t n n i e m r m e e t c e c e r r o f d f o o n r t i e n r a e g e h i s s s e r r o d e v f r a s n e a h t r S

4

5

66

REPORT ON TORSION IN STRUCTURAL CONCRETE (ACI 445.1R-12) : n o s i s r n a o t r t r e o f b t n n e a c θ t m 7 o e 1 o c c t r 1 n f f y o i 1 T s f . n q d φ o i e e A r E m , r 2 e 3 o ≥ s . f r 0 e 1 t . v A s s 3 . n a r 1 1 t d n o e r i t i u c e q e S R r o f n n i o o t i t c c e e S s : d n e i l l ) o a ( 1 s r o w - 2 . t n 3 i . r o θ h t 6 f t t n t o n n i c o e d e t m ) E d l w 2 a c T f k y e e ( v S c i 2 . u : r A o 3 q n f . 2 e g n 6 i ≥ e i s e h e r t d w s e f A s o n s r o e s i s v m s r r o n e t a t r e t h d t e r e i t u a l q u e c l R a C s e d o e c r n t r o a i n p f n 6 m i . d o e . e c r 3 l e 5 . m r a s 1 θ o n r t 1 e o v n f o c g s o s n a v a u i y i n t t r f T d a c o r e ) f t A 1 o d S 2 2 : e e φ l r n 1 g i o 1 = i n u s ( a q . t e r o q A s e R t h t t . r E , o 6 e u f . r t m s t 5 . n 1 u e s n s o m 1 A i s e s r e v s n n a o r i t r e s o n i t r m o r f e t t e n d e o m t e n c g r i o s f e i n d e n r o i s r o T

6

: t ] n a e P m 1 M e c n r 1 i o 1 c f . ′ f n q [ i E e r , 2 e . b w f y s r 8 . ′ e 2 f c . v s 1 n 1 6 a n 0 r . t o 0 i m t c = u e m S i n i m , s n v i A M

] i s p n i

′ f c [

w b ′ f c

2 7 . 0

=

n i m , v

A

e s r n e i v s    e r n 1 e . ) a s . r 8 t . n r e i f 3 . o 1 2 v s b e g 1 v 6 . n a n r w i n d 3 t 2 e c i o 7 . ( n n t 0 f y a p o o m i s c e n l S r i m s a : o t t 0 n t i n 0 d n d e 6 n e u m t    a m i e r e c n g c i a r n r o e f o o m h l f s = n i s m e x u e a h r s m t m i d x d a A M

. q : e n n E d o o : i r 5 s n a r i 7 e . o t t 0 n h s r e r o = f o m f s e x a c s p  r     m r p u o s u f r r θ i n r t t ) i ) i s t → e o r 2 ( s ° e ) c h  0 , 4 e 2 h 3 8 + b e 9 . t s 3 t h 0 = ( / ( r f 1  . 3 α e 6 f 2 n o . u i v d . s n o C  g 2 m 5 g E 9 n n i o i a 7 n n t . c n i r i t c c α a o i e a N t p n S   0    s t c o l o : p s 6 n i e . i c s a l 9 r b n S m + i o e a n t d w i = 1 u d d t x u n t i a d a i g s m 5 g n r a n o 7 e l . l o h 0 m s m e = u u h m t x m a i x d i m a x d s M a A M ] ] I i a I n 8 s i P . : C f C p e t A f r 3 M . n n o A o : 5 i e : n . e n ′ i r s c g g ′ 1 ) o r f c n a n i e 1 m 3 f e e i i [ s [ v 2 r 6 c h 5 c c . . s n r s o 5 a a ty o t 4 1 n t o t w i f 1 w f y b f p r p . . a s o 1 s r 1 b r ( n c o 1 ′ t e i . ′ c l f 1 l f c a a t f e f n S r q 2 n n n n n t 1 i o : e E n o i o i d e i e i 5 , s s 5 t d t 5 m u u r b e r 7 c t t m . e 7 e o w 5 3 i c e i e c 3 e t . v g r S g c . . 0 s 1 0 d e n o , n r S , n n f o n o o 1 = a = l n 1 l f 1 a o t r n 1 1 n i n i n - m i r i i n e 8 o a u r  m m r 8 t e t m i   m   1 u e 1 e v c  v u h 3 n e 3 m r s m s s e m e A s A s i i r S e x e e m i 1 2 1 2 x a v a v n h s t e + + c i s M M n r d o M n t t a d f r a A s A s t r t A     e c n r i n o n e o f r i o n i s e i r t s o e r r c e e t s e v s s d s n r s n a e a o v r r r s t a n c l e a r a h t r o t f o s t r s k o l e f c i e t h a t t h e c d e n e n m d t i n m e a n e c r t e r c m t o e e f l e D S )

(

)

(

7

- 4   e 1 f c r 1 T h 1 o p o f . 5 A n q 2 4 i . e 0 r E , o  t  l 2 . n + a 9 i n . i ) p d d 3 V u . 1 e t − i 1 1 - m r s g 6 2 o V n . 1 f 5 . o 0 1 s 0 l n 1 l . d − a r a 3 f . n a t n 1 V ( e o i 1 b s r n n θ t o i o a o t c c d t c + e e f r v i S M d u t : q n = e t R e F m 2

2

r p

F f f p t l

= p

A

l

n o i t c e S : n o i s r o t r o f t n e m ) e 4 c r ( o 3 f . n 2 i . e 9 r l a n i d u t i g n o l f o g n i c a p S

n o i t c e S : t n e m e c r o f θ d n 8 i 2 - t e w o f y r 6 l . c k k a q u A n E d 2 i E , T d ) u t i 3 ( ≥ g 2 . s n 3 o . A l 6 Σ l a n o i s r o t d e r i u q e R l

e c ) r o 2 f n 2 i 1 θ e r 1 2 ( t l . a q o n c i E d ,    u v l t 7 . y i y f 3 f g . n 5   o .  l h l 1 p a 1 n n o t o i i t A s s r c e o S = t l : d t A e r n i e u m q e R t n e m e c r o f n i e r l n o a i n r s i d t u o t i r o g f n o l e n i m r e t e D

] ] r i a s o ) f P p t 4 n M i n 2 ′ n e 1 ′ i f c c m 1 [ f e ( [ . c q   r l v  y  o E   y f f y f l v y , f n 3 f i   .    e h r 5 h p . l 5 p a .      t n 1  t i A s d 1 A s   u n  t i o   − g i t g − g n c A l ′ A o e ′ c f y f c l y l S f 2 f 1 m : 5 5 u n o m i = = s i n n i n r i o m i m , t , A M A e c r o f n i e r l a n n i o s i d l i t u c a t e i t e s g d s n t o l n s o e r k c m c r e o h f c d n a t c e l e S l

8

l

r o f . t 3 n 6 . e 5 . m 1 e 1 c r d o n f n a i 2 e r . . l 6 a 5 . n 1 i d 1 u t i n o g i n t o c e l f S o : n g i n o i r c s o a t p S


67

Fig. 9.3—Design process flow chart for combined shear and torsion effects.

report focuses on recent torsion developments and theories, the design examples are solved by three major building codes: ACI 318-11, EC2-04, and CSA-A23.3-04. Example 1 was also solved by the two graphical methods proposed by Rahal (2000b) and Leu and Lee (2000). These graphical methods are not suitable to treat prestressed concrete sections under combined loads and, therefore, were not used for Example 2. Following each design example, a comparison table summarizes results of the various codes and methods. Design equations and expressions may vary slightly from one code to another, but the general design procedure is the same. In the case of torsion combined with shear, design procedures for all three codes follow the flow chart given in Fig. 9.3. Hsu (1997) provides background on ACI 318-95 that is still applicable to the updated ACI 318-11. His paper includes a detailed design example of a prestressed hollow box girder subjected to torsion, shear, and flexure that constitutes Example 2.

9.4—Design Example 1: solid rectangular reinforced concrete beam under pure torsion 9.4.1 Design problem statement— As shown in Fig. 9.4.1, the cross-sectional dimensions of beam are bw = 300 mm (12 in.) and h = 500 mm (20 in.). The characteristic concrete cylinder compressive strength is f c′ = 20 MPa (2900 psi) [Class C20/25], and the characteristic steel yield strength is f y = 420 MPa (60,000 psi). The applied torsional moment is T u = 30 kN·m (266 in.-kip). The mean cylinder strength of the concrete: f cm = f c′ + 8 MPa ⇒ f cm = 28 MPa (4000 psi). Assume 40 mm (1.5 in.) from exterior face to stirrup centerline typical. 9.4.2 Solution according to ACI 318-11 1. Determine if torsion effects may be disregarded (ACI 318-11, Section 11.5.1). The torsion effects can be disregarded if the following expression is valid

68


ph = 2( xo + yo) = 1280 mm (50.4 in.) xo, yo = horizontal and vertical dimension of the centerline

of outermost closed transverse torsional reinforcement, respectively xo = (300 – 2 × 40) = 220 mm (9 in.) yo = (500 – 2 × 40) = 420 mm (17 in.) Aoh = area enclosed by centerline of outermost closed transverse torsional reinforcement, mm (in.) Aoh = xo yo = 92,400 mm2 (153 in.2) Therefore Tu ph 2 oh

1.7 A

Fig. 9.4.1—Rectangular reinforced concrete cross section subjected to pure torsion.

φ

where φ = λ =

strength reduction factor for torsion and shear = 0.75 modification factor of lightweight concrete (normalweight concrete, λ = 1.0) pcp = perimeter of outer concrete cross section, mm (in.) = 2 × (bw + h) = 1600 mm (64.0 in.) Acp = total area enclosed by the outside perimeter of concrete cross section, mm (in.) = bw × h = 150,000 mm2 (240 in.2) Therefore,

 f ′   A   20   150, 000  φλ  c    = 0.7 5(1.0)   N·mm  12    1600   12   pcp  2 cp

2

= 3.93 kN·m (36.3 in.-kip) T u = 30.0 kN·m (266 in.-kip). Torsion effects must be considered. 2. Determine if dimensions of the cross section are adequate (ACI 318-11, Section 11.5.3.1). Dimensions of the cross section are adequate if 2  V c  Tu ph  + ≤ φ  b d + 8  1.7 Aoh2    w

f c′ 

12

 

In case of pure torsion, the expression becomes Tu ph 2 1.7 Aoh

where ph =

≤φ

5 f c′ 6

, taking

V c bw d

=

× 1280

1.7 × 92, 4002

= 2.65 MPa (347.6 psi)

2λ f c′ 12

perimeter of centerline of outermost closed transverse torsional reinforcement, mm (in.)

5 f c′ 6

5 20 6

= 0.75

= 2.80 MPa (403 psi)

Because the expression in Eq. (11-18) (ACI 318-11) is valid, the cross section dimensions are adequate. 3. Calculate the amount of stirrups required for pure torsion (ACI 318-11, Section 11.5.3.3). To calculate transverse reinforcement for torsion, Eq. (11-20) and (11-21) in ACI 318 can be transformed into At

≥

s

T u

φ2 f yt Ao cot θ

where At = s = f yt =

area of one leg of a closed stirrup resisting torsion spacing of the stirrups design strength of torsion transverse reinforcement = 420 MPa (60,000 psi) Ao = gross area enclosed by shear flow path = 0.85 Aoh θ = angle of compression diagonals in truss analogy for torsion = 45 degrees Therefore, At

2

=

whereas

 f c′   Acp2  T u < φλ      12   pcp 

 V u   b d  w 

30 × 106

s

≥

T u

φ2 f

yt

Ao cot θ

=

30 × 10

6

0.75 × 2 × 420 × 0.85 × 92 , 400 × 1

= 0.61 mm2 /mm (0.0227 in.2 /in.) The minimum transverse reinforcement is computed by Eq. (11-23) in ACI 318. In the case of pure torsion, this minimum is expressed by 0.75  At  =  s   min 2

f c′ bw

12 f yv

=

0.75 × 20 × 300 24 × 420

= 0.1 mm2 /mm (0.0040 in.2 /in.) but shall not be less than


A = total area of the outside perimeter of the concrete cross

0.35 bw 0.175 × 300  At  = =  s   min 2 f yv 420 = 0.125 mm2 /mm (0.005 in.2 /in.) The maximum spacing of the stirrups is given by (ACI 318-11, Section 11.5.6.1)

 ph / 8   = 160 mm (6.50 in.) 300 mm 

smax = min 

For stirrups ∅8 ( d s = 8.0 mm [0.315 in.]), At = 50.3 mm (0.078 in.2) and s ≤ 81 mm (3.20 in.). Select: s = 80 mm (3.15 in.). For stirrups No. 3 ( d s = 9.5 mm (0.375 in.)), At = 71.0 mm2 (0.11 in.2) and s ≤ 115 mm (4.55 in.). Select: s = 110 mm (4.50 in.). 4. Calculate the longitudinal bars required for pure torsion (ACI 318-11, Section 11.5.3.7). Total longitudinal reinforcement for torsion ( Al ) (Eq. (11-22) in ACI 318) is calculated by 2

A

l

=

At s

ph

f yt f y

cot 2 θ = 0. 610 × 1280 × 1 × 1

l

section = bwh = 300 × 500 = 150,000 mm2 (240 in.2) pc = outside perimeter of the concrete cross section = 2(bw + h) = 2(300 + 500) = 1600 mm (64.0 in.) cl = distance between edge of member and center of the longitudinal reinforcement = 50 mm (2.0 in.) The area enclosed by the centerlines of the connecting thin-walls is calculated by Ak = (bw – t ef )(h – t ef ) = 200 × 400 = 80,000 mm 2 (124.0 in.2)

The perimeter of the area enclosed by the centerlines of the connecting thin-walls is determined by uk

2. Determine if dimensions of the cross section are adequate (evaluate the strength of concrete struts) (EC2-04, Section 6.3.2(4)). Dimensions of the cross section are adequate if (Eq. (6-29) in EC2-04) T Ed T Rd , max

Minimum longitudinal reinforcement (Eq. (11-24) in ACI 318) is calculated by

l

=

5 fc ′Acp 12 f y

l

 A  p  s  

−

t

h

f yt f y

l

=

× 150, 000 − 0..610 × 1280 × 1 12 ⋅ 420

5 20

< 0 mm2 Therefore, Al = 780.8 mm2 (1.18 in.2) At least one longitudinal bar is required in each stirrup’s corner and bar spacing distributed around the perimeter of the closed stirrups is 305 mm (12 in.). Therefore, the number of longitudinal bars is at least six and each bar requires780.8/6 = 130 mm2 (0.20 in. 2). For six longitudinal bars ∅14 (d s = 14.0 mm [0.551 in.]), Al = 6 × 153.9 = 923 mm2 (1.43 in.2). For six longitudinal bars No. 5 (d s = 15.9 mm [0.625 in.]), Al = 6 × 200 = 1200 mm2 (1.86 in.2). 9.4.3 Solution according to EC2-04 code 1. Calculate the terms of the equivalent thin-walled section (EC2-04, Section 6.3.2(1)). Effective wall thickness is calculated by t ef

 A / pc  150,000/1600 = 94 mm  = max  = max     2 × 50 = 100 mm   2c  ⇒ t ef = 100 mm (4.0 in.) l

where

= 2 (bw − t ef ) + (h − t ef ) = 2(200 + 400) = 1200 mm (48.0 in.)

= 780.8 mm2 (1.18 in.2)

A , min

69

+

V Ed V Rd,max

pure torsion ≤ 1   → T Ed <= T Rd ,max

where T Ed = V Ed =

design torsional moment = T u design shear force = V u T Rd, max = design torsional resistance moment V Rd, max = design shear resistance force The design torsional resistance moment is given as (Eq. (6-30) in EC2-04) T Rd ,max

= 2vαc fcd Ak tef sin θ cos θ

where v =

strength reduction factor for concrete cracked in shear = 0.6(1 – f c ′/250) = 0.6(1 – 20/250) = 0.552 ( f c′ in MPa) αc = 1 (for non-prestressed elements) θ = angle of compression diagonals in truss analogy for torsion 1 ≤ cotθ ≤ 2.5; assume θ = 35 degrees f cd = design compressive strength of concrete = f ck / γc = 20/1.5 = 13.3 MPa (1933 psi) Therefore, T Rd , max = 2 vα c fcd Ak t ef sin θ cos θ

= 2 × 0. 552 × 1 × 13. 3 × 80, 000 × 100 × sin 35° cos 35°(10 −6 ) = 55.2 kN•m (513.4 in.-kip) ⇒ T Rd , max = 55.2 kN•m (513.4 in.-kip) > T u = 30 kN•m (266 in.-kip)

70


Because the above expression is valid, dimensions of the cross section are adequate. 3. Calculate the amount of stirrups required for pure torsion (EC2-04, Section 6.3.2 (2)). The amount of stirrups required is calculated using the equation

4. Calculate the longitudinal bars required for pure torsion (EC2-04, Section 6.3.2(3)). The total longitudinal reinforcement needed for torsion (Σ Al ) is given as (Eq. (6-28) in EC2-04)

∑ A ≥ l

At s

≥

T u

Tu uk cot θ

2 Ak f yd

=

30 × 106 × 1200 × cot 35° 2 × 80,000 × 365. 2

⇒ ∑ A

l

≥ 880 mm2 (1.37 in.2)

2 Ak f ywd cot θ

where At = area of one leg of a closed stirrup resisting torsion s = spacing of the stirrups f ywd = design strength of transverse reinforcement = f ty / γs = 420/1.15 = 365.2 MPa (52,174 psi) Therefore,

Minimum longitudinal reinforcement is determined by (Eq. (9.1N) in EC2-04) A , min = 0.26 l

f ctm f yk

bt d

≥ 0.0013bt d

where 30 × 10 2 × 80, 000 × 365.2 cot 35° 6

T u

2 Ak f ywd cot θ

=

= 0.359 mm2 /mm (0.014 in.2 /in.)

⇒

At s

≥ 0.359 mm2 /mm (0.014 in.2 /in.)

The ratio of the required transverse reinforcement is given by

ρt =

At sbw sin α

α = 90   → ρt =

At sbw

=

0.359 300

= 0.0012

where α is angle between the stirrups and longitudinal axis. The ratio of the minimum stirrups is (Eq. (9.5N) in EC2-04)

ρt = (0.08

)

fc′ / f yk

= 0.08

20 / 420 = 0. 00085

The maximum longitudinal spacing of the stirrups is given as (Eq. (9.6N) and Section 9.2.3(3) in EC2-04). u / 8     smax = min 0.75d (1 + cot α )  ⇒  min(b , h)   w   1, 600 / 8 = 200    smax = min 0.75 ⋅ 450 = 338 = 200 mm (8.0 in.)   300  

For stirrups ∅8 ( d s = 8.0 mm [0.315 in.]), At = 50.3 mm2 (0.078 in.2) and s ≤ 137 mm (5.40 in.). Select: s = 125 mm (5.0 in.). For stirrups No. 3 ( d s = 9.5 mm [0.375 in.]), At = 71 mm2 (0.110 in.2) and s ≤ 194 mm (7.66 in.). Select: s = 180 mm (7.1 in.).

f ctm =

mean tension strength of concrete = 0.30 f ck 2/3 = 0.3(202/3) = 2.21 MPa (319 psi) bw = mean width of tension zone = 300 mm (12 in.) Therefore,

A

l

0.26 ( 2.21 / 420) 300 × 450 = 184. 7 mm2  = max   2 0.0013 × 300 × 450 = 175 mm  = 185 mm2 (0.287 in.2)

Longitudinal bars are arranged with at least one bar at each corner of the stirrups and the others distributed uniformly around the inner periphery of the torsion links (closed stirrups) with a maximum spacing of 350 mm (13.8 in.). Therefore, the number of longitudinal bars is at least six and each bar requires 880/6 = 147 mm 2 (0.23 in.2). For six longitudinal bars ∅14 ( d s = 14 mm [0.55 in.]), Al = 6 × 154 = 924 mm 2 (1.43 in.2). For six longitudinal bars No. 5 ( d s = 15.8 mm [0.625 in.]), Al = 6 × 198 = 762 mm2 (1.88 in.2). 9.4.4 Solution according to CSA-A23.3-04 code 1. Determine if torsion effects may be disregarded (CSAA23.3-04, Section 11.2.9.1). If the magnitude of the torsion, T f , satisfies the following expressions (Eq. (11-2) in CSA-A23.3-04), torsional effects need not be considered T f < 0.25T cr

 Ac 2  0.38λφc  pc  

T cr = 

f c′ 1 +

 A 2  =  c  0.38λφ λ c 0.38λφ c f c′  pc  f p f cp

 150, 0002  = 0.38 × 1 × 0.65  1600  

20 N·mm

= 15.5 kN·m (144 in.-kip) where = Acp = 150,000 mm2 (240 in.2)

Ac

f c′


pcp = outside perimeter of concrete cross section =

where

1600 mm (64.0 in.) factor to account for low-density concrete 1.0φc = resistance factor for concrete = 0.65 ′ f c specified compressive strength of concrete, which is the same as the characteristic concrete cylinder compressive strength in previous definition = 20 MPa (2.9 ksi) Therefore,

At = s = Ao =

pc

λ

= = = =

area of one leg of a closed stirrup resisting torsion spacing of the stirrups gross area enclosed by shear flow path = 0.85 Aoh θ = angle of inclination of compression stresses to the longitudinal member axis 4. Calculate angle of inclination of compression strut (CSA-A23.3-04, Section 11.3.6.4). The angle of inclination of the diagonal compression strut is given by the expression (Eq. (11-12) in CSA-A23.3-04)

T f (= 30.0 kN·m) > 0.25 T cr (= 0.25 × 15.5 = 3.9 kN·m)

Torsion effect, therefore, must be considered. 2. Determine if dimensions of the cross section are adequate (CSA-A23.3-04, Section 11.3.10.4). Dimensions of the cross section are adequate if the equation below (Eq. (11-19) in CSA-A23.3-04) is satisfied 2

 V f − V p   b d  w v 

2

 T p  +  f h2  ≤ 0.25φc f c′  1.7 Aoh 

In case of pure torsion, the expression is simplified to T f ph 2 oh

1.7 A

θ = 29 + 7000 ε x The longitudinal strain indicator (11-13) in CSA-A23.3-04) M f

ε x =

d v

2

+

0.9 ph T f

Therefore,

2 oh

1.7 A

30 × 106

× 1232

1.7 × 84, 860

2

= 3.02 MPa (394 psi)

Because 3.02 MPa < 3.25 MPa, dimensions of the cross section are adequate. 3. Calculate the stirrups required for pure torsion ( CSAA23.3-04, Section 11.3.10.3); The equation of nominal torsional strength T n is same in all codes except for differences in strength reduction factors. For design, the equation for T n is rearranged to express the required area of transverse reinforcement per unit length (Eq. (11-7) in CSA-A23.3-04)

s

≥

2 Es As

=

0.9 ph T f 4 Es As Ao

0. 9 × 1232 × 30 × 106 4 × 200, 000 × 413 × 72, 130

= 0.00140

Therefore

θ = 29 + 7000 × 0.00140 = 38.8 degrees

0.25 f c f c′ = 0.25 × 0.65 × 20 = 3.25 MPa (471 psi)

At

2 Ao

with Ao = 0.85 Aoh = 0.85 × 84,860 = 72,130 mm 2 (120 in.2) and the longitudinal reinforcement As = 413 mm2 (0.64 in.2) (established in Section 6 of this example), the following is obtained

ε x = =

(V f

For non-prestressed sections subjected to pure torsion, the expression for ε x is simplified to

ph = 2[(300 – 2 × 46) + (500 – 2 × 46)] = 1232 mm (50.0 in. 2)

Tu ph

defined by (Eq.

 0.9 phT f  − V p ) +  + 0.5 N f − Ap f po  2 Ao   2[ Es As + E p Ap ]

ε x =

Aoh = (300 – 2 × 46)(500 – 2 × 46) = 84,864 mm 2 (141.0 in.2)

ε x is

2

≤ 0.25φc f c ′

With a concrete cover of 40 mm (1-1/2 in.) and 10M (No. 3) stirrups (diameter 12 mm [1/2 in.]), the following calculations/values apply

71

T f

2 Ao φs f y cot θ

5. Calculate stirrups required for pure torsion. The required stirrup area per unit length is calculated using the equation (Eq. (11-17) in CSA-A23.3-04) At s

≥

T f

2 Ao φ s f y cot θ

=

30 × 106 2 × 72,130 × 0. 85 × 420 × cot 38. 8°

= 0.468 mm2 /mm (0.0175 in.2 /in.) The minimum transverse reinforcement is (Eq. (11-1) in CSA-A23.3-04)

 Av  = 0.06  s   min

f c′

bw f yv

= 0.06 ×

20 ×

300 420

= 0.192 mm2 /mm (0.00775 in.2 /in.)

72


The maximum spacing of the stirrups is (CSA-A23.3-04, Section 11.3.8.1) smax

 0.7d v  = min   = 277 mm (11.2 in.) 600 mm 

1. Determine if torsion effects may be disregarded. The torsion effects can be disregarded if this expression is valid: T u ≤ 0.25φT cr

= 0.328

Tcr

where d v =

M f

l

d v

2

+ cot θ

(V f

− 0.5Vs − V p )

2

0.45 ph T f  0.45 ph T f  + = cot θ  2 Ao  2 Ao 

where F lt = required tension force in longitudinal reinforcement =

φ A f y l

Therefore A

l

≥

Ac2 pc

= 0.328

20

150, 0002 N•mm 1600

= 20. 6 kN·m (193.87 in.-kip)

effective shear depth = max(0.9d , 0.72h) = max(396 mm, 360 mm) [(16.0 in., 14.4 in.)] For stirrups 10M ( d s = 11.3 mm [0.444 in.]), At = 100 mm2 (0.155 in.2) → s ≤ 210 mm (8.32 in.). Select s = 200 mm (8.0 in.). For stirrups No. 3 ( d s = 9.5 mm [0.375 in.]), 71.0 mm 2 ( At = 0.110 in.2) → s ≤ 150 mm (5.89 in.). Select: s = 150 mm (5. 9 in.). 6. Calculate the amount of longitudinal bars required for pure torsion (CSA-A23.3-04, Section 11.3.9.2 and 11.3.10.6). Total longitudinal reinforcement is calculated using the following equation (Eq. (11-21) and (11-14) in CSA-A23.3-04) F t =

f c′

0.45 ph T f 2φ s Ao f y

cot θ =

0.45 × 1232 × 30 × 106 2 × 0. 85 × 72, 130 × 420

cot 38.8°

The torque is disregarded if T u = 30 kN·m (266 in.-kip) is smaller than 0.25φT cr = 0.25 × 0.75 × 20.6 = 3.9 kN·m (35.8 in.-kip). Torsion effects, therefore, must be considered. 2. Calculate normalized shear stress and determine if size of cross section is adequate. Shear stress in the walls of the cross section is calculated by T u v f c′

φ

0.67 Ac2 f c′

=

30 × 106 × 1600 0.75 0.67(150, 000) 2 20

= 0.212

The normalized shear stress fits well within Region I in Fig. 5.7.1. Therefore, the section can be designed as underreinforced, and the section dimensions are adequate. 3. Calculate the required reinforcement. The most straightforward design of an under-reinforced section uses equal amounts of longitudinal and transverse f c′). Therefore, ω l = 0.212 reinforcement indexes ( ω l = ω t = v / and ω t = 0.212. From Eq. (5.7.1b) and (5.7.1c), the longitu dinal and transverse reinforcement are At s

= 402 mm2 (0.61 in.2) One longitudinal reinforcing bar should be placed in each corner of the closed transverse reinforcement required for torsion. The nominal diameter of the corner bars should be no less than s /16. If 10M stirrups are used, the bar nominal diameter can be no less than 200/16 = 12.5 mm (8/16 = 0.5 in.), and for No. 3: 150/16 = 9.38 mm (6/16 = 0.375 in.). Select: four longitudinal bars 15M (d s = 16.0 mm [0.628 in.]) at the corners and two 10M bars at mid-depth: Al = 4 × 200 + 2 × 100 = 1000 mm2 (1.55 in.2). Or, six longitudinal bars No. 3 ( d s = 9.5 mm [0.375 in.]), Al = 6 × 71 = 426 mm2 (0.66 in.2) Or, six longitudinal bars No. 4 ( d s = 12.7 mm [0.50 in.]), Al = 6 × 127 = 762 mm2 (1.18 in.2). 9.4.5 Rahal’s graphical method —Rahal’s graphical method uses the ACI general requirements, such as an upper limit on spacing of transverse and longitudinal reinforcement, an upper limit on yield strength of reinforcing rebars, and a minimum of four corner longitudinal bars. Based on ACI requirements, it also disregards the torque effect if it is smaller than 25 percent of the cracking torque. The cross section outer perimeter and area enclosed within this perimeter were calculated in the ACI design example 9.4.1 as: Ac = 150,000 mm2 (240 in.2) and pc = 1600 mm (64.0 in.).

=

pc

=

0.42 Ac f c′ fty pc

ω t =

(0.42)(150, 000)( 20) 0.212 (420)(1600)

= 0.40 mm2 /mm (0.016 in.2 /in.) A

l

=

0.375 Ac f c′ f y l

ω l =

(0.375)(150, 000)( 20) 0.212 420

= 568 mm2 (0.92 in.2) The stirrups’ maximum spacing is taken as 160.0 mm (6.30 in.), which is the smaller of ph /8 = 2(220 + 420)/8 = 160.0 mm (6.50 in.) and 300.0 mm (12.0 in.), as per ACI code. For stirrups ∅8 (d s = 8.0 mm [0.315 in.]), At = 50.3 mm2 (0.08 in.2), the maximum spacing is calculated as s ≤ 124.5 mm (4.9 in.), and it can be used: s = 120 mm (4.50 in.). For stirrups No. 3 ( d s = 9.5 mm [0.375 in.]), At = 71.0 mm2 (0.11 in.2), the maximum spacing is calculated as s ≤ 175.7 mm (6.9 in.), and it can be used: s = 160 mm (6.50 in.). To provide four longitudinal corner bars and limit the spacing to 300 mm (12 in.), six bars are needed (ACI design in 8.2.2). The minimum bar area is of 578/6 = 96 mm 2 (0.15 in.2). For six longitudinal bars ∅12 (d s = 12.0 mm [0.472 in.]), Al = 6 × 113 = 678 mm2 (1.05 in.2). For six longitudinal bars No. 4 ( d s = 12.7 mm [0.50 in.]), Al = 6 × 129 = 774 mm2 (1.20 in.2). 9.4.6 Leu and Lee’s graphical method 1. Calculate the required torsional strength.


73

Table 9.4.6— Summary of design solution of Example 1 using all five solution methods Code

Transverse reinforcement

ACI 318

EC2-04

0.61 mm /mm (0.0227 in. /in.)

780.8 mm2 (1.18 in. 2)

minimum:

0.125 mm 2 /mm (0.0042 in.2 /in.)

0 mm2 (0 in.2)

provided:

∅8/80 mm (No. 3 at 4.50 in.) 0.625 mm2 /mm (0.0244 in. 2 /in.)

6∅14 (6 No. 5) 923 mm2 (1.86 in. 2)

required:

0.359 mm2 /mm (0.014 in.2 /in.)

880 mm2 (1.37 in. 2)

minimum:

0.261 mm 2 /mm (0.0103 in.2 /in.)

185 mm2 (0.287 in. 2)

∅8/125 mm (No. 3 at 7.10 in.) 0.402 mm /mm (0.0155 in. /in.)

required:

0.468 mm2 /mm (0.0175 in. 2 /in.)

402 mm2 (0.61 in. 2)

minimum:

0.192 mm 2 /mm (0.00775 in.2 /in.)

—

provided:

10M/200.0 mm (No. 3 at 5.90 in.) 0.500 mm2 /mm (0.0186 in. 2 /in.)

Four 15M + two 10M (6 No. 3 or 6 No. 4) 1000 mm2 (0.66 or 1.18 in. 2)

required:

0.40 mm2 /mm (0.0160 in.2 /in.)

568.1 mm2 (0.92 in. 2)

minimum:

—

—

∅8/120.0 mm (No. 3 at 6.50 in.) 0.417 mm2 /mm (0.0183 in. 2 /in.)

6∅12 (6 No.4) 678 mm2 (1.18 in. 2)

required:

0.470 mm2 /mm (0.019 in.2 /in.)

615 mm2 (1.0 in. 2)

minimum:

—

—

∅8/100.0 mm (No. 3 at 5.50 in.)

6∅12 (6 No. 4) 678 mm2 (1.18 in. 2)

provided:

provided:

Nondimensional balanced torsional strength is calculated using Eq. (5.7.2g) T xub

=

140 140 ( f y in MPa) = 300 + f y 300 + 420

= 0.196

2

=

T xub fc′Ac2 pc

=

2

0.503 mm2 /mm (0.0200 in. 2 /in.)

Required reinforcement indexes (Fig. 5.7.2): ω l and ω t ω t . For convenience, assume ω l = ω t ≥ 0.70. Referring to (Fig. 5.7.2), ω s ≥ 0.73 → ω l = Balanced normalized reinforcement ratios ηl b and ηtb are calculated by Eq. (5.7.2c) and (5.7.2d)

ηb =

Balanced torsional strength T xub is determined by Eq. (5.7.2h) T xub

2

6∅14 (6 No. 5) 924 mm2 (1.88 in. 2)

CSA A23.3-04

Leu and Lee (2000)

Longitudinal reinforcement

required:

provided:

Rahal (2000b)

2

l

0.196 × 20 × 150, 0002 N·mm 1600

ηtb =

76

( f y in MPa)

=

76 200 + 420

= 0.125

76 ( f in MPa) 100 + f y y

=

76 100 + 420

= 0.135

200 + f y

= 55.1 kN·m (512 in.-kip)

η and ηt are

Required normalized reinforcement ratios determined using Eq. (5.7.2e) and (5.7.2f)

where Ac =

total area of the outside perimeter of the concrete cross section = bwh = 300 × 500 = 150,000 mm2 (240 in.2) pc = outside perimeter of the concrete cross section = 2(bw + h) = 2(300 + 500) = 1600 mm (64.0 in.) The torsional strength indicated by Eq. (5.7.2h) is greater than required

ω sT xub ≥

T u f

→ ω s × 55.1 ≥

30 0.75

→ f s ≥ 0.73ω

l

η = ω η b ≥ 0.70 × 0.125 → η ≥ 0.0861 l

l

l

l

ηt = ω t η tb ≥ 0.70 × 0.135 → ηt ≥ 0.105 Required longitudinal bars are calculated with Eq. (5.7.2a) f A

y 420 A η ≥ 0.0868 → f ′ A ≥ 0.0868 → ≥ 0.0861 20 150, 000 c cp → A ≥ 615 mm2 (1.0 in.2) l

l

l

l

where T u = φ =

applied torsional moment = 30.0 kN·m (266 in.-kip) strength reduction factor for shear and torsion; which is assumed to be the same as that in ACI 318 = 0.75ω s = strength contour value (Fig. 5.7.2) 2. Calculate the longitudinal bars and stirrups required for pure torsion.

At least one longitudinal bar is placed at each corner of the stirrups with the others distributed uniformly around the inner periphery of the closed stirrups with a maximum spacing of 300 mm (12 in.), as indicated in ACI 318. Therefore, the number of longitudinal bars is at least six and each bar requires 615/6 = 102.5 mm2 (0.17 in.2).

74


For six longitudinal bars ∅12 (d s = 12.0 mm [0.472 in.]), Al = 6 × 113 = 678 mm2 (1.05 in.2). For six longitudinal bars No. 4 (d s = 12.7 mm [0.500 in.]), Al = 6 × 127 = 762 mm2 (1.18 in.2). Required stirrups are calculated using Eq. (5.7.2b) f A p

y t cp ηt ≥ 0.1036 → f ′ A s ≥ 0.1036 c cp

→ →

At s

420 1600 At 20 150, 000s

≥ 0.1036

≥ 0.470 mm2 /mm (0.019 in.2 /in.)

Using ACI 318, the maximum stirrups’ spacing is calculated to be 160 mm (6.3 in.). For stirrups ∅8 (d s = 8.0 mm [0.315 in.]) → At = 50.3 mm2 (0.08 in.2) and s ≤ 107 mm (4.2 in.) and it can be used: s = 100 mm (4.0 in.). For stirrups No. 3 ( d s = 9.5 mm [0.375 in.]) → At = 71 mm2 (0.11 in.2) and s ≤ 151 mm (5.9 in.) and it can be used: s = 150 mm (5.5 in.).

9.5—Design Example 2: Prestressed concrete box girder under combined torsion, shear, and flexure 9.5.1 Design problem statement 9.5.1.1 Design problem description— Design the shear and torsional reinforcement of a box girder. A 3658 mm (12 ft) wide and 1270 mm (4 ft 2 in.) deep box girder with overhanging flanges (Fig. 9.5.1.1(a)) was designed as an alternative to the double-tee girder in Dade County, FL ( Hsu 1997). The standard prestressed box girder is simply supported, 24.00 m (79.00 ft) long, and prestressed with 64 strands at 1860 MPa (270 ksi), 13.0 mm (1/2 in.), seven-wire strands as shown in Fig. 9.5.1.1(b). Total prestress force is 6076 kN (1366 kips) after prestress loss. The design of flexural reinforcement is omitted for simplicity. The concrete cover is 40 mm (1.5 in.), and material strengths are normalweight concrete: f c′ = 48.0 MPa (7000 psi) and f y = 420 MPa (60,000 psi). 9.5.1.2 Sectional properties L = 24.00 m (79.00 ft) h = 1270 mm (50.00 in.) d = 1016 mm (40.00 in.) at 0.3 L from support t = 251 mm (9.88 in.) (average of stem width) bw = 502 mm (20 in.) A = 1.523 × 106 mm2 (2361.4 in.2) I = 319.8 × 109 mm2 (768,336 in.4) yt = 516 mm (20.34 in.) yb = 753 mm (29.66 in.) λ = modification factor of lightweight concrete (λ = 1.0) 9.5.1.3 Loading criteria— The standard girders are designed to carry a train of cars, each 22.86 m (75 ft 0 in.) long. Each car has two trucks with a center-to-center distance of 16.46 m (54 ft 0 in.). Each truck consists of two axles 1981 mm (6 ft 6 in.) apart. The crush live load of each car is 513.8 kN (115.5 kip). The maximum web reinforcement amount was obtained at section 0.3 L from the support under a derailment

load, which consists of two truckloads located symmetrically at a distance 3200 mm (10 ft 6 in.) from midspan. Each axle load is taken as 24 percent of the crush live load (513.8 kN/4) with 100 percent impact and a maximum side shift of 914 mm (36.0 in.). The self-weight of the girder is 34.4 kN/m (2.36 kip/ft). The girder is also subjected to a superimposed dead load caused by the track rails’ weight, rail plinth pads, power rail, guard rail, cableway, acoustic barrier, and other permanent loads. At derailment, this superimposed dead load is assumed to produce a uniform vertical load of 12.8 kN/m (0.88 kip/ft) and a uniformly distributed torque of 3.16 kN-m/m (0.71 ft-kip/ft). This torque is neglected in the calculation because the magnitude of the distributed torque is small, and the torque is acting in a direction opposite to the derailment torque. 9.5.2 Solution according to ACI 318-11 1. Determine the factored forces for ( ACI 318-11, Section 9.2.1). Factored dead and live loads—

The load factor for live loads is taken as 1.6. The derailment load per axle is calculated by Pu,L = 16 ×

513.8 × 2 = 411 kN/axle (92.4 kip/axle) 4

The derailment torque per axle is calculated by T u,L = 16 × 513.8 × 2 × 0. 914

4

= 375.7 kN·m/axle (277 ft-kip/axle) The load factor for dead loads is taken as 1.2. The girder weight is calculated by wu,g = 1.2(34.4) = 41.3 kN/m (2.83 kip/ft)

The superimposed dead weight is calculated by wu,s = 1.2(12.8) = 15.4 kN/m (1.05 kip/ft) Factored shear, torque, and bending moment— The V u, T u, and M u at 0.3 L from the support are V u = (wu,g + wu,s)(0.2 L) + 2Pu,L = (41.3 + 15.3) × 0.2 ×

24.00 + 2 × 411 = 1094 kN (246 kips) T u = 2T u,L = 2 × 375.7 = 752 kN·m (554 ft-kip) M u = 0.5(wu,g + wu,s)( L – 0.3 L)(0.3 L) + 2Pu,L(0.3 L) = 0.5(41.3 + 15.4)(24.00 – 7.2) × 7.2 + 2 × 411 × 7.2 = 9347 kN·m (6922 ft-kip) 2. Determine if torsion effects may be disregarded. Check outstanding flanges— pcp is As indicated by Fig. 9.5.1.1(a), the parameter Acp2 / determined by (disregard overhanging flanges) Acp = 1854 × 203 + 0.5(1854 + 1791) × 1067 = 2.32 × 106 mm2 (3597 in.2) pcp = 1854 + 1791 + 2 × 1270 = 6185 mm (243.5 in.) Acp2 / pcp =(2.32 × 10 6)2 /6185 = 8.70 × 108 mm3 (53,090 in.3) Check threshold torque ( ACI 318-11 , Section 11.5.1)—


75

Fig. 9.5.1.1—Cross section and elevation of box girder. Ag =

2.32 × 106 – (1270 – 419)(1320) = 1.20 × 10 6 mm2 (1855 in.2)( fT c /4) r

=

φ(0.083)λ

f c′

 f pc  1+   = φ(0.083)λ pcp  0.33 f c′ 

Ag2

f c′

 6076 / A  1 +  pcp  0.33 f c′ 

Ag2

6 2

=0.75(0.083)(1.0)

(1.20 × 10 )  6076 / 1523 − 48.0 1 +  10

6

6185



0.33 48.0



= 276 kN·m (202 ft-kip) < 752 kN·m (554 ft-kip) Factored torsional moment should be considered in design. 3. Determine if dimensions of the cross section are adequate (ACI 318-11, Section 11.5.3.1). Check cross section—

Assume a clear concrete cover of 40 mm (1.5 in.) and 13 mm No. 4 bars for web reinforcement Aoh = 0.5[(1854 – 93) + (1791 – 93)](1270 – 93) = 2.04 × 106 mm2 (3177 in.2) ph = (1854 – 93) + (1791 – 93) + 2(1270 – 93) = 5813 mm (229.5 in.) =

(753 – 127) –

9754 − 7224

508 9754 = 496 mm (19.47 in.) at 0.3 L from support d = yt + e = 516 + 496 = 1012 mm (39.81 in.) at 0.3 L from support d = 0.8h = 0.8 × 1270 = 1016 mm (40.00 in.) governs bw = 2t = 2 × 251 = 502 mm (20.0 in.) bwd = 502 × 1016 = 510 × 103 mm2 (790 in.2) The interaction equation for hollow box sections is (Eq. (11-18) in ACI 318-11) e

76


 V   T   V + ≤ φ + 8 × 0. 083  b d    1.7 A t    b d u

u

w

c

oh

w

 f c′ when t < 

ph

Ao = Acp –

= 2.15 + 0.86 = 3.01 MPa (437 psi)

Vu d M u

Ats

 Vu d  bwd ,  0.6 × 0.083λ f c′ + 700 × 0. 0689 M  u 

=

M u

φ fc′Acp

s

=

φ2 Ao f yv cot θ

2

=

752 (1000)

2

0.75 × 2 (2.21 × 106 ) × 420 × 1. 303

0.414 mm2 /mm (0.0166 in.2 /in.) smax ph /8 = 5813/8 = 727 mm (28.7 in.) > 305 mm (12 in.) (ACI 318-11, Section 11.5.6); These calculations indicate that 305 mm (12 in.) spacing governs.

≤1

(Eq. (11-9) in ACI 318-11)

Transverse reinforcement for vertical walls ( ACI 318-11 , Section 11.5.3.8)—

OK

Transverse reinforcement in the vertical walls is contributed by both torsion and shear: 1 Av = 0.414 + 0. 5 × 2. 043 s 2 s = 1.44 mm2 /mm (0.0575 in.2 /in.)

 510 × 103  = 469 kN (105.4 kip)  1000  

 

510 × 10 1000

3

= 2 × 0.083(1.0)  48.0

  

2 ( 254)

 

 588 × 1000 + 8 × 0. 083  510 × 103

 

48. 0

= 0.75(1.15 + 4.61) = 4.31 MPa (627 psi) > 3.01 MPa (437 psi) OK 4. Calculate number of transverse bars required Design of shear reinforcement ( ACI 318-11 , Section 11.4.7)— V c = V c,min = 587 kN (132.2 kip)

− φV c (1094 − 0.75 × 587)1000 = φdf yv 0.75 × 1016 × 420

12 f yt

Select two layers of ∅18 mm bars (No. 6 bars) in each vertical wall at 305 mm (12 in.) spacing

= 587 kN (132.13 kip) governs

f c′ = 0.75 

f c′ bw

= 0.259 mm2 /mm (0.01 in.2 /in.) < 1.44 mm2 /mm (0.0575 in.2 /in.) OK (Eq. (11-23) in ACI 318-11)

(Eq. (11-3) in ACI 318-11)

 V φ  c + 8 × 0.083  bw d

+

 At 1 Av  = 0.375  s + 2 s   min

V c,min = 2 × 0.083 λ f c′ bwd

s

2 × 752 × 6185 (1000)

= =

1094 × 1016 = 0.119 < 1 9347 × 1000

Vu

= 0.51 mm 2 /mm (0.021 in.2 /in.)

= 2.32 × 106 –

T u

At

=

12 f yt

2Tu pcp

V c = (0.6 × 0.083(1.0) 48.0 + 700 × 0.00689 × 0.119)

Av

f c′ bw

0.75 × 48 × 2. 32 × 106 = 2.21 × 106 mm2 (3582.7 in.2) Assume θ = 37.5 degrees, as recommended by the code provision for prestressed members:

 V u   T u  1094 × 1000 751.4 × 106 +  = +  b d  1.7 Aoh t   510 × 103 1.7 (2.04 × 106 ) 251 w  

Vu d

s

= 0.75

smax = 305.0 mm (12.00 in.) for torsion governs Design of torsional hoop reinforcement ( ACI 318-11 , Section 11.5.3.6)—

6 = 2. 04 × 10 = 351 mm (13.85 in.) > t = 251 in. ph 5829

where

Av, min

>

Aoh

V c =

2.043 mm2 /mm (0.0818 in.2 /in.)

=

Aoh

(305)

= 1.666 mm2 /mm (0.0656 in.2 /in.)

> 1.440 mm2 /mm (0.0575 in.2 /in.)

OK

Transverse reinforcement for horizontal walls—

Transverse reinforcement in horizontal walls is contributed by torsion only: At s

= 0.414 mm2 /mm (0.0166 in.2 /in.)

Select two layers of ∅10 mm bars (No. 3 bars) in each horizontal wall at 305 mm (12 in.) spacing:


- Required average compressive strength (psi)

2 × 78.5 = 0.515 mm2 /mm (0.0203 in.2 /in.) 305

f cr ′ = f c′ + 1400 (for f c′ > 5000 psi)

> 0.259 mm2 /mm (0.01 in.2 /in.) The transverse reinforcement in the top wall should be added to the flexural reinforcement required in the top flange acting as a transverse continuous slab. 5. Calculate number of longitudinal bars required.

f cr ′ = 7000 + 1400 = 8400 psi (57.9 MPa)

- Mean compressive strength (based on f cr ′) f cm = 57.9 MPa (8400 psi)

Design of torsional longitudinal reinforcement (Eq. (11-22) in ACI 318-11)—

=

A

l

At s

- Characteristic compressive strength (MPa) (EC2-04, Section 3.1.2(5))

 f yv  2  cot θ = 0.414 × 5813 × 1 × 1.3032  f y 

ph 

f c′ = f cm – 8 ⇒ f ck = 49.9 MPa (7240 psi)

l

2

77

2

= 4085.9 mm (6.46 in. ) Check minimum limitation for A / t s and Al,min (Eq. (11-23) in ACI 318-11) At s

= 0.375

f c′

bw f yt

- Strength class for concrete according to Section 3.1.2 (Table 3.1) in EC2-04 corresponding to mean compressive strength Strength Class C50, therefore: f c′ = 50 MPa (7250 psi)

2

2

= 0.259 mm /mm (0.01 in. /in.) - Design compressive strength (EC2-04 , Section 3.1.6(1)) 2

2

< 0.414 mm /mm (0.0166 in. /in.) f cd A ,min = l

5 fc′Ag

f yl

 A   f  −  t  ph  yv   s   f yl  (Eq. (11-24) in ACI 318-11)

f c′

50  = 1   = 33.3 MPa (4833 psi) γ c  1.5 

= α cc

- Mean compressive strength according to EC2-04 (Section 3.1.6(5)) and the strength class of concrete f c′ = f cm + 8 ⇒ f cm = 58.0 MPa (8410 psi)

=

5 × 0.083 48.0 (1. 2 × 106 )

Concrete tensile strength ( EC2-04, Section 3.1.6 (2)—

– 0.414 × 5813 × 1 420 = 8215 – 2407 = 5808 mm2 (9.17 in.2) governs

- Mean tensile strength f t′ = 0.3 f c′2/3 = 4.07 MPa (590 psi)

Select 36 ∅16 bars (No. 5 bars) longitudinal bars 2

2

2

- Design tensile strength 2

Al = 36 × 201 = 7236 mm (11.22 in. ) > 5808 mm (9.17 in. ) f ctd

Arrangement of reinforcing bars—

The arrangement of the reinforcing bars for torsion and shear is summarized in Table 9.4.6. This reinforcement arrangement could be conservatively used throughout the girder length. 9.5.3 EC2-04 code 9.5.3.1 Material properties 9.5.3.1.1 Concrete Concrete compressive cylinder strength ( EC2-04 , Section 3.1.2 and 3.1.6)—

- Specified compressive strength (The characteristic concrete cylinder compressive strength) f c′ = 48.0 MPa (7000 psi)

= α ct

f ctk ,0.05

γ c

= 1×

0.7 f t ′ 1.5

= 1.89 MPa (275 psi)

- Modulus of elasticity 0.3

E m

 f  = 22  cm   10 

= 37.28 GPa (5405 ksi) (EC2-04, Section

3.1.3, Table 3.1) Parabola-rectangle diagram for concrete under compression, as shown in Fig. 9.5.3.1.1 ( EC2-04 , Section 3.1.7)

78


Area As = 99 mm2 (0.153 in.2) Characteristic tensile strength f pk = f pu = 1860 MPa (270 ksi) Design ultimate strength (EC2-04, Section 2.4.2.4) f p,ud =

f pk

γ s

=

1860 = 1617 MPa (234.78 ksi) 1.15

Characteristic yield strength (EC2-04, Section 3.3.3(1)) Fig. 9.5.3.1.1—Parabola-rectangle diagram for concrete under compression.

f p0.1k = 0.9 f pu = 1674 MPa (243 ksi)

Design yield strength (EC2-04, Section 3.3.6(6)) f pd =

  ε   σ c = f cd 1 −  1 − c     ε c2   for n

f p 0.1k

γ s

=

1674 = 1456 MPa (211 ksi) 1.15

Modulus of elasticity (EC2-04, Section 3.3.6(3))

εc < εc2 σc = f cd for εc ≥ εc2

E p = 196 GPa (28,420 ksi)

where 4

 90 − f c′  100  

n = 1.4 + 23.4 

Strain at design yield strength

= 2.00

ε c2

= (2 + 0.085 ( f c′− 50)

0.53

) / 100

= 0.0020

  90 − f c′  / 100 ε cu2 =  2.6 + 35   100     

ε p, yd =

f pd E p

= 0.0074

4

= 0.0035

9.5.3.1.2 Reinforcing steel Type: Ribbed reinforcing bars (high bond)— Diameter for longitudinal bars = 12 mm [ ∅12] (0.472 in.) Diameter for closed stirrups = 10 mm [ ∅10] (0.394 in.) Characteristic yield strength f y = 420 MPa (60 ksi) Design yield strength (EC2-04, Section 2.4.2.4)— f yd =

f y

γ s

Ultimate design strain (EC2-04, Section 3.3.6(7))

ε p,ud = 0.0200 9.5.3.2 Sectional areas and concrete cover 9.5.3.2.1 Gross concrete area - Area of the gross cross section including overhanging flanges

= 2.65 × 106 mm2 (4107 in.2)

=

420 = 365 MPa (52 ksi) 1.15

Modulus of elasticity (EC2-04, Section 3.2.7(4)) E s = 200 GPa (29,000 ksi)

Strain at design yield strength

ε y =

f yd E s

- Area of the gross cross section (disregarding overhanging flanges) = 2.32 × 106 mm2 (3596 in.2) - Area of the concrete cross section (disregarding overhanging flanges) Ac = 1.20 × 106 mm2 (1860 in.2)

= 0.0018

Ultimate design strain (EC2-04, Section 3.2.7(2))

εud = 0.9εuk = 0.0200 9.5.3.1.3 Prestressing reinforcing strands Type: Low relaxation, 270K, seven-wire strands— Diameter = 13.0 mm (0.50 in.)

9.5.3.2.2 Concrete covers ( EC2-04 , Section 4.4.1.2); - Minimum cover with regard to bond cmin,b = diameter of bar = 12 mm (0.47 in.) (reinforcing steel)

= 13 mm (0.51 in.) (prestressing steel) - Exposure class related to environmental conditions Exposure Class XD3 (Cyclic wet and dry. Parts of bridges exposed to spray containing chlorides.)


- Structural class (XD3) = 4 + 2 (service life of 100 years) – 1 (concrete class ≥ C45) = 5

79

9.5.3.4 Prestressed tendons requirements Minimum longitudinal reinforcement (EC2-04, Section 9.2.1.1)

- Minimum cover with regard to durability A

cmin,dur = 50 mm (1.97 in.) (reinforcing steel)

l

= 0.26

= 60 mm (2.36 in.) (prestressing steel) - Minimum cover cmin = max(cmin,b; cmin,dur ; 10 mm [0.39 in.])

= 50 mm (1.97 in.) for reinforcing steel = 60 mm (2.36 in.) for prestressing steel

f t ′ f yk

bt d

≥ 0.0013bt d

where bt = 470 mm (18.5 in.) mean width of tension zone, and d = 1172 mm (46.1 in.) distance from extreme top fiber to the centroid of the reinforcement. Therefore, A ,min = 716 mm 2 (1.1 in.2) l

- Nominal cover cnom = cmin + 10 mm (0.39 in.)

The area of the prestressed tendons is

= 60 mm (2.36 in.) for reinforcing steel = 70 mm (2.76 in.) for prestressing steel - Distance from center of longitudinal reinforcing bars to extreme concrete fiber (cover from bars centroid) = 66 mm (2.56 in.) > cmin + 12 mm/2 (0.47 in./2) - Distance from center of prestressed tendons to extreme bottom fiber (cover from tendons centroid) = 102 mm (3.94 in.) > cmin + 13 mm/2 (0.51 in./2) 9.5.3.3 Factored shear, torque, bending moment, and prestress Factored shear force V u = 1.4 × (34.4 + 12.8) × 0.2 × 24.0 + 1.5 × (513.8/2) × 2

= 1089 kN (245 kip) Factored torsional moment T u = 1.5 × (513.8/2)2 × 0.914 = 705 kN·m (520 ft-kip)

Factored bending moment

AP = 64 × 99 = 6336 mm2 (9.82 in.2) ≥ As ,min l

9.5.3.5 Shear design 9.5.3.5.1 Design shear resistance of a member without shear reinforcement ( EC2-04 , Section 6.2.2(1)) V Rd ,c

= CRd ,c k (100ρ

where d = bw = C Rd,c = k =

ρ = l

A p bw d

σ cp =

Total prestress losses 20 percent or ω = P / t Po = 0.80 Therefore, prestress force at time t = 0 Po = P / t ω = 7595 kN (1708 kip)

fck )

+ k1σ cp  bw d ≥ (vmin + k1σ cp ) bw d

≤ 0.02 ⇒ ρ = 0.0133

N Ed Ac

l

=

Pt Ac

= 5.1 MPa (734.4 psi) (compressive)

k 1 = vmin =

0.15 0.035k 3/2 f ck 1/2 ⇒ vmin = 0.428 Therefore V Rd,c = 698 kN (156.2 kip)

+ 1.5(513.8/2)(2 × 7.2) = 9561 kN·m (7074 ft-kip)

Pt = 6076 kN (1366 kip)

1 / 3

l

517 + 494 = 1011 mm (39.8 in.) at 0.3 L = 7.2 m (23.7 ft) (9.5.1.1(b)) 470 mm (18.50 in.) 0.18/ γc = 0.18/1.5 = 0.12 1+ 200 d ≤ 2 ⇒ k = 1.44

M u = 0.5 × 1.4 × (34.4 + 12.8) × 0.7 × 0.3× (24.00) 2

Prestress force at time t = ∞

OK

Because V Rd,c < V Ed = 1089 kN (245 kip), shear reinforcement must be provided

9.5.3.5.2 Minimum shear reinforcement ( EC2-04 , Section 9.2.2(5)) The ratio of the minimum stirrups is

ρw,min = (0.08 Therefore

fck

)

f yk

= 0.08

49. 9 / (1. 2 × 420) = 0.00113

80


 Asw   A  ≥ 0.00113 →  sw  ≥ 0.5311 mm 2 /mm (0.0209 in.2 /in.)  sb   s  min w  min 9.5.3.5.3 Maximum effective cross-sectional area of the shear reinforcement ( EC2-04 , Section 6.2.3(3))

V Rd ,s

=

Asw s

zf yd cot θ ≥ V Ed ⇒

sbw

α vb f 1  A  ≤ α c vf cd →  sw  ≤ c w cd  s  max 2 2 f yd

s

where v = 0.6(1 – f ck /250) = 0.48 αc = 1 + σcp / f cd = 1.154 (because σcp ≤ 0.25 f cd ) Therefore

Asw s

(Eq. (6.14) in EC2-04)

With the above data and

s

1. 154 × 470 × 910 × 0. 48 × 33. 3 ( cot θ + tan θ)

For θ = 45 degrees V Rd,max = 3945 kN (886 kips) > V Ed ,

θ = 35 degrees V Rd,max = 3707 kN (832 kips) > V Ed , and

θ = 22 degrees

Asw s

≥

1089 × 10 3 910 × 365 × cot θ

≥ 3.28 mm2 /mm (0.131 in.2 /in.),

≥ 1.32 mm2 /mm (0.053 in.2 /in.).

Check

 Asw  Asw <  s   min s

 A  <  sw   s  max

OK

9.5.3.6 Torsion design 9.5.3.6.1 Terms of the equivalent thin-walled section ( EC2-04 , Section 6.3.2(1));

t ef

 A  375 mm  = max  u  ≤ t real = max   ≤ 178 mm 134 mm   c 2   ⇒ t ef = 178 mm (7 in.) l

f cd = 33.3 MPa (4830 psi)

V Rd , max =

zf yd cot θ

⇒

≥ 2.30 mm2 /mm (0.092 in.2 /in.), and

Asw

z = 0.9d ⇒ z = 910 mm (35.8 in.)

1 ≤ cotθ ≤ 2.5 ⇒ 22 degrees ≤ θ ≤ 45 degrees (mean value: θ = 35 degrees)

θ = 22 degrees

 V Rd,s  V Rd = min   ≥ V Ed V ,  Rd max  α c bwzvfcd V Rd , max = ( cot θ + tan θ)

s

V Ed

θ = 35 degrees

 Asw  2 2 ≤  s   max 11.86 mm /mm (0.47 in. /in.) 9.5.3.5.4 Required shear reinforcement ( EC2-04 , Section 6.2.3(4))

≥

θ = 45 degrees Asw

Asw.max f yd

Asw

where A =

2.32 × 106 mm2 (3597 in.2) total area of the cross section within the outer circumference, including inner hollow areas (disregarding overhanging flanges) u = 6185 mm (243.5 in.) outer circumference of the cross section (disregarding overhanging flanges) cl = 66 mm (2.56 in.) distance from edge to center of the longitudinal reinforcement t real = 178 mm (7 in.) minimum wall thickness of the real concrete thin-walled section. The continuous area and perimeter enclosed by centerlines of the connecting thin-walls, as shown in Fig. 9.5.3.6.1, are: Ak = 1.8 × 106 mm2 (2794 in.2) and uk = 5486 mm (216 in.). 9.5.3.6.2 Determine if cross-sectional dimensions (for example, strength of concrete struts under torsion) are adequate ( EC2-04 , Section 6.3.2(4))

Dimensions of the cross section are adequate if (Eq. (6.29) in EC2-04)

V Rd,max = 2740 kN (615 kips) > V Ed . T Ed

Thus for (Eq. (6.13) in EC2-04) V Rd = V Rd,s ≥ V Ed

T Rd , max

+

V Ed V Rd , max

≤1


81

Fig. 9.5.3.6.1—Assumed effective thin-wall section. T Ed

The design torsional resistance moment is calculated as (Eq. (6.30) in EC2-04) T Rd,max = 2vαc f cd Ak t ef sinθcosθ

where v = 0.6(1 – f ck /250) = 0.48 and αc = 1 +

σcp / f cd = 1.154

and for θ = 45 degrees

V Ed

+

T Rd ,max

V Rd , max

=

705 1089 + 4085 2740

= 0.57 ≤ 1

OK

Dimensions of the cross section, therefore, are adequate for every case (every value of θ). 9.5.3.6.3 Calculate the required stirrup area for torsion. ( EC2-04 , Section 6.3.2(2)) The required stirrup area per unit length is calculated by Asw s

≥

T Ed

2 Ak f yd cot θ

⇒

Asw s

≥

705 × 106 2 × 1.8 × 106 × 365 × cot θ

T Rd,max = 5915 kN·m (4356 ft-kip),

For

θ = 35 degrees

θ = 45 degrees

T Rd,max = 5559 kN·m (4093 ft-kip)

Asw

θ = 22 degrees

s

θ = 35 degrees

T Rd,max = 4109 kN·m (3026 ft-kip)

Design shear resistance force is determined by (Eq. (6.14) in EC2-04) V Rd , max =

α c bwzvfcd ( cot θ + tan θ)

Asw s

≥ 0.38 mm2 /mm (0.015 in.2 /in.), and

θ = 22 degrees Asw

Thus for θ = 45 degrees T Ed T Rd , max

+

V Ed V Rd , max

s

=

705 1089 + = 0.39 5910 3945

≤ 1

OK

θ = 35 degrees T Ed T Rd , max

+

V Ed V Rd , max

θ = 22 degrees

705 1089 + 5554 3707

= 0.42 ≤ 1

≥ 0.22 mm2 /mm (0.009 in.2 /in.)

where Asw and s are the area of one leg of a closed stirrup resisting torsion and the spacing of the stirrups, respectively. 9.5.3.6.4 Add stirrup areas for torsion and shear, and select the stirrups ( EC2-04 , Section 6.3.2(2)) Asw, S + T

=

≥ 0.54 mm2 /mm (0.021 in.2 /in.),

s

=

1 Asw, S 2 s

OK For

θ = 45 degrees

+

Asw,T s

(using two single-legged stirrups)

82


Asw, S + T s

≥ 2.18 mm2 /mm (0.087 in.2 /in.),

θ = 45 degrees Σ As ≥ 2943 mm (4.62 in.2), l

θ = 35 degrees Asw, S + T s

θ = 35 degrees

≥ 1.53 mm2 /mm (0.061 in.2 /in.), and

Σ As ≥ 4204 mm2 (6.60 in.2), and l

θ = 22 degrees

θ = 22 degrees

Σ As ≥ 7285 mm2 (11.50 in.2). l

Asw, S + T s

≥ 0.88 mm2 /mm (0.035 in.2 /in.).

Maximum longitudinal spacing of the stirrups for shear (Eq. (9.6N) in EC2-04) smax

α = 90° = 0.75d (1 + cot α )   → smax = 0.75d

= 758 mm (29.8 in.) where α is the inclination of the stirrups. Maximum longitudinal spacing of the stirrups for torsion (EC2-04, Section 9.2.3(3)) u / 8   773 mm      smax = min 0.75d (1 + cot θ)  = min 758 mm   min(b, h)  235 mm     

= 235 mm (9.25 in.) For two single-legged from shear and torsion For θ = 45 degrees

∅10 stirrups, spacing is calculated

s ≤ 102 mm (4.0 in.) < smax OK

θ = 22 degrees s ≤ 178 mm (7.0 in.) < smax OK

Total selection ( θ = 35 degrees): two single-legged stirrups of ∅10/100 mm (diameter 0.39 in. at 4.0 in.). 9.5.3.6.5 Longitudinal reinforcement required for torsion ( EC2-04 , Section 6.3.2(3))

Total longitudinal reinforcement for torsion (Eq. (6-28) in EC2-04)

l

2 Ak f yd

θ = 35 degrees 38∅12 → Σ Asl = 4298 mm2 (6.66 in.2) > 4204 mm2 (6.61 in.2), and

θ = 22 degrees 38∅16 → Σ Asl = 7640 mm2 (11.84 in.2) > 7285 mm2 (11.58 in.2). Final selection of longitudinal bars ( θ = 35 degrees) l

θ = 35 degrees

Σ As ≥

38∅10 → Σ Asl = 2985 mm (4.63 in. 2) > 2943 mm (4.63 in. 2),

∅12 (0.47 in.) → As = 113 mm2 (0.175 in.2).

s ≤ 72 mm (2.8 in.) < smax OK

T Ed uk cot θ

Longitudinal bars shall be arranged so that at least one bar is placed at each corner of the stirrups and the others are distributed uniformly around the inner periphery of the torsion links (closed stirrups) with a maximum spacing of 350 mm (13.8 in.) (EC2-04, Section 9.2.3(4)). Therefore, the number of the longitudinal bars is at least 36. For θ = 45 degrees

⇒ Σ As ≥ l

705 × 106

× 5486 × cot θ 2 × 1.8 × 106 × 365

Due to uniform distribution and symmetry 40∅12 → Σ Asl = 4524 mm2 (7.0 in.2). The torsional longitudinal reinforcement is in addition to the prestressing tendons. Low values of angle θ ( θ = 22 degrees) lead to a design with lower area requirements of transverse reinforcement (∅10/175 mm) and higher area requirements of longitudinal reinforcement (40 ∅16). High values of angle θ (θ = 45 degrees) lead to higher area requirements of stirrups ( ∅10/70 mm) and lower area requirements of longitudinal bars (40 ∅10). The selection of θ = 45 degrees maximizes the concrete strength components, such as V Rd,max = 3945 kN and T Rd,max = 5910 kN·m. This value of angle θ could be used when checking the adequacy of the cross section dimensions. The following relationship between the strength of concrete struts under torsion is satisfied T Ed T Rd , max

For

+

V Ed V Rd , max

≤1


83

Fig. 9.5.3.6.6—Reinforcement configuration ( EC2-04).

9.5.3.6.6 Arrangement of reinforcing bars— Total reinforcement of the member with a hollow cross section under combined loading of prestressing, torsion, shear, flexure, and axial force is depicted in detail in Fig. 9.5.3.6.6. 9.5.4 Design solution using CSA-A23.3-04 code— Load factors of the CSA-A23.3-04 code are used here to establish the forces due to factored loads. These factors are 1.25 for dead loads and 1.50 for live loads. The cross-sectional area of the prestressing strands ( f pu = 1860 MPa [270 ksi]) is 6336 mm2 (9.82 in.2), the effective prestressing force is 6076 kN (1366 kip), and the average prestress is 3.99 MPa (589 psi). The concrete strength is 48.0 MPa (7000 psi), and the yield strength of the non-prestressed reinforcement is 420 MPa (60,000 psi). 1. Determine the factored forces (CSA-A23.3-04, Annex C). Factored dead and live loads— The derailment load per axle is calculated by

 513.8  Pu,L = 1.5   4   2 = 385 kN/axle (86.6 ft/axle)

The superimposed dead weight (with a load factor of 1.25) is calculated by wu,s = 1.25 × 12.8 = 16.0 kN/mm (1.10 kip/ft) Factored shear, torque, and bending moment— At distance 0.3 L from support, the following values are

obtained V u = (43.0 + 16.0)(0.2 × 24.00) + 2 × 385 = 1053 kN (237 kip) T u = 2 × 352.3 = 705 kN·m (520 ft-kip) M u = 0.5(43.0 + 16.0)(24.00 – 7.2)7.2 + 2 × 385 × 7.2

= 9112 kN·m (6759 ft-kip) 2. Determine if torsion effects can be disregarded ( CSAA23.3-04, Section 11.2.9.1). Threshold torque—

For a hollow section with a wall thickness of less than pc (= 281 mm > 178 mm), torsion must be considered 0.75 Ac / if the torque due to factored loads, T f , exceeds 0.25T cr

The derailment torque per axle is determined by

 513.8  T u,L = 1.5   4   (2 × 0.914) = 352 kN·m (259.8 ft-kip) The girder weight (with a load factor of 1.25) is determined by wu,g = 1.25 × 34.3 = 43.0 kN/mm (2.95 kip/ft)

T cr

(1.5 A ) = g

pc

2

0.38λφc f c′ 1 +

φ p f cp 0.38λφc

f c′

where = Acp = 2.32 × 106 mm2 (3597 in.2) = pcp = outside perimeter of concrete cross section = 6185 mm (243.5 in.) Ag = gross area of section (without flanges) = 1.20 × 10 6 mm2 (1855 in.2) λ = factor to account for low-density concrete = 1.0 Ac pc

84

φc


=

resistance factor for concrete = 0.70 for precast concrete φ p = resistance factor for prestressing reinforcement = 0.90 ′ f c = specified compressive strength of concrete = 48.0 MPa (7.0 ksi) f cp = compression stress in concrete due to effective prestress = 3.99 MPa (0.6 ksi) With this information, the following calculation can be made T cr =

The angle of inclination of the diagonal compression strut is given by the expression (Eq. (11-12) in CSA-A23.3-04)

θ = 29 + 7000 ε x In the absence of an axial load normal to the cross section, the strain at mid-depth of the section is defined by (Eq. (11-13) in CSA-A23.3-04) M f

(1.5 × 1 .20 × 10 6 )2 0.9 × 3.99 0. 38 × 0.70 48 .0 1 + 6185 0. 38 × 0.70 48 .0

d v

ε x =

2

+

(V f

= 1658 kN·m (1216 ft-kip) Because 0.25T cr = 415 kN·m (304 ft-kip) < 705 kN·m (520 ft-kip), torsion must be considered in the design. 3. Determine if dimensions of the cross section are adequate. Check cross-sectional dimensions (CSA-A23.3-04 , Section 11.3.10.4)— ph, For box sections with a wall thickness of less than Aoh / the cross-sectional dimensions must satisfy the following criterion (Eq. (11-19) in CSA-A23.3-04) V f

− V p

bw d v

where V f =

+

T f

1.7 Aoh t

≤ 0.25φc f c′

V u = shear force due to factored loads = 1054 kN

(237.2 kip) V p = shear force due to prestressing factored by φ p = 0.9(6076)508/9754 = 284 kN (64.0 kip) Aoh = [(1854 + 1791)/2 – 2 × 48)](1270 – 2 × 48) = 2.03 × 106 mm2 (3142 in.2) ph = (1854 – 96) + (1791 – 96) + 2(1270 – 96) = 5801 mm (228.4 in.) Aoh / ph = 350 mm (13.8 in) > 178 mm (7.0 in.) for bottom flange > 235 mm (9.25 in.) for web > 203 mm (8 in.) for top flange d v = larger of 0.9d = 0.9 × 1016 = 914 mm and 0.72 h = 0.72 × 1270 = 914 mm (36 in.) t = minimum wall thickness = 178 mm (7.0 in.) T u = torque due to factored loads = 705 kN·m (520 ft-kip) Therefore, (1054 − 284) × 103 235 × 2 × 914

+

The terms not defined above are M f = M u = moment due to factored loads = 9166 kN·m (6761 ft-kip) Ao = 0.85 Aoh = 0.85 × 2.04 × 10 6 mm2 = 1.74 × 106 mm2 (2697 in.2) A p = area of prestressing reinforcement = 6336 mm 2 (9.8 in. 2) f po = stress in prestressing tendons (may be taken as 0.7 f pu = 1302 MPa [189.0 ksi]) E s = 200,000 MPa (29,000 ksi) As = area of non-prestressed reinforcement in tension zone (assume fourteen 15M bars with 2800 mm 2 (4.34 in.2) E p = 190,000 MPa (28,000 ksi) With this information, the following calculation can be made

ε x =

9112 × 106 914

2

Because the web governs design, the above equation was the web thickness t = 235 mm, not the bottom flange thickness. 4. Calculate θ and β (CSA-A23.3-04, Section 11.3.6.4). Angle of diagonal compression strut and shear resistance of concrete—

 0.9 × 5801× 705 × 106  + − 130 2× 633 6   2 × 1.74 × 106  = 0.00086 2(200, 000 × 2800 + 190, 000× 6336)

+

[(1054 − 284) × 103 ] 2

Therefore

θ = 29 + 7000 × 0.00086 = 35.0° The shear force resisted by the concrete is (Eq. (11-6) in CSA-A23.3-04) Vc

= φc λβ

fc′bw dv

where (Eq. (11-11) in CSA-A23.3-04)

705 × 106

1.7 × 2.03 × 106 × 235 = 2.66 MPa (386 psi) < 0.25 × 0.70 × 48.0 = 8.40 MPa (1223 psi) OK

 0.9 phT f  − V p ) +  − A f  2 Ao   p po 2[ Es As + E p Ap ] 2

β=

0.40 1 + 1500ε x

×

1300 1000 + sze

Because minimum transverse reinforcement is provided, s ze = 300 mm (12 in.) Therefore

β=

0.40 1 + 1500(0.00086)

×

1300 1000 + 300

= 0.175


and

Design of longitudinal reinforcement for torsion (CSA A23.3-04 , Section 11.3.10.6 and 11.3.9.2)— V c = 0.70 × 1.0 × 0.175(

48.0 )235 × 2 × 914 N = 365 kN (80.7 kip)

5. Calculate transverse reinforcement. Design of transverse reinforcement for shear (CSA-A23.304 , Section 11.3.5.1)— Shear with V s = V f – V c – V p = 1054 – 365 – 284 = 405 kN

Longitudinal reinforcement on the tension side of the girder shall have dimensions such that its factored resistance shall not be less than that given by the following expression. In the absence of an axial load normal to the cross section, the required longitudinal force due to torsion, shear, and flexure is

F t =

(92.6 kip) Av s

=

l

V s

φs fy d v cot θ

=

405 × 103 0.85 × 420 × 914 × cot 35. 0°

= 0.87 mm2 /mm (0.0356 in.2 /in.) Design of transverse reinforcement for torsion (CSA A23.3-04 , Section 11.3.10.3)—

The required transverse reinforcement for torsion is given by (Eq. 11-17 in CSA-A23.3-04) At s

=

T u

2 Ao φ s f y cot θ

=

=

9112 × 10 6 914

2 × 1.74 × 10

s

Av

2s

= 0.40 + 0.87/2 = 0.835 mm 2 /mm (0.0338 in.2 /in.)

The selection of two 10M bars ( Abar = 100 mm2) per web yields the following bar spacing s = 2(100)/0.835 = 240 mm; select s = 225 mm (9.17 in.) reinforcement (CSA-A23.3-04 ,

Minimum transverse Section 11.2.8.2)—

The minimum transverse reinforcement is established with the maximum web thickness at the top of the web, bw = 534 mm (21.0 in.) min Av

s

= 0.06

f c′

bw f y

= 0.06

48.0

 534   420  

= 0.528 mm2 /mm (0.0212 in.2 /in.) < 0.835 mm2 /mm (0.0338 in.2 /in.) For T f > 0.25T cr , max s = 0.35d v = 0.35(914) = 320 mm (12.6 in.) > 225 mm. (9.17 in.) OK 6. Calculate the longitudinal reinforcement.

(V f

− 0.5Vs − V p ) 2

l

2

= φ p Ap f pr = 0.9 × 6336 × 1786 × 10−3 = 10,184 kN (2290 kips)

The non-prestressed reinforcement required in tension zone is calculated by As =

+

+ cot θ

 0.45 ph T f  +  2 Ao  

 405  (10 3 )  +  0.45 × 5801 × 705 × 10 6  + cot 35.0°   1054 − 2 − 284   2 × 1.74 × 10 6     

F tr

× 0.85 × 420 × cot 35. 0°

This amount of transverse reinforcement must be provided in the top and bottom slabs. Total transverse reinforcement in one web is calculated as

d v

2

2

Factored tension resistance provided by the prestressing reinforcement with f pr = 0.96 f pu = 1786 MPa (259 ksi) (CSAA23.3-04, Section 18.6.2)

705 × 106 6

M f

= 9969 + 1108 = 11,077 kN (2503 kip)

= 0.40 mm2 /mm (0.0160 in.2 /in.)

At

85

(11, 077 − 10,184)(103 ) = 2501 mm 2 (4.26 in.2) 0.85 × 420

These calculations indicate that fourteen 15M bars are required, providing As = 2800 mm2. This reinforcement is to be distributed in the tension zone. Because the compression due to moment is much greater than the tension due to torsion an d shear, there is no need to check the tension reinforcement required in the compression zone. 9.5.5 Comparison of design solutions according to ACI 318 , CSA-A23.3-04 , and EC2-04 — The required longitudinal and transverse reinforcement calculated using ACI 318, CSA-A23.3-04, and EC2-04 is shown in Table 9.5.5. The longitudinal reinforcement (“L” Bars) given is the total amount required for the entire cross section. The required transverse reinforcement is given for the top wall (“T-1” Bars), side walls (“T-2” Bars), and bottom walls (“T-3” Bars). The transverse reinforcement in the top wall should be added to the flexural reinforcement required in the top wall acting as a transverse continuous slab. Arrangement of reinforcing bars—

Arrangement of the reinforcing bars for torsion and shear is shown in Fig. 9.5.5 and summarized in Table 9.5.5. The general location of longitudinal and transverse reinforcement in the section, which is shown in Fig. 9.5.5, is based on all three design codes.

86


Canadian Standards Association

Fig. 9.5.5—Reinforcement locations for box girder under combined torsion, shear, and flexure.

Table 9.5.5—Summary of design solution of Example 2 using design codes ACI 318-11, EC2-04, and CSA-A23.3-04 Design code Reinforcing bars

ACI 318-11 2

“L” bars

“T-1” and “T-3” bars (torsion)

5808 mm (9.2 in.2) 2

0.414 mm /mm (0.0166 in. 2 /in.)

“T-2” bars (torsion + shear) 1.44 mm2 /mm (0.0575 in.2 /in.)

EC2-04 (45, 35, and 22 degrees) 2

Proceedings of the International Conference on the Performance of Construction Materials in the New Millennium, CSA-A23.3-04

4204 mm (6.61 in. 2)

2503 mm2 (4.26 in. 2)

2.18, 1.53, 0.88 mm2 /mm (0.087, 0.061, 0.035 in.2 /in.)

0.40 mm2 /mm (0.0160 in. 2 /in.)

2.18, 1.53, 0.88 mm2 /mm (0.087, 0.061, 0.035 in.2 /in.)

0.835 mm2 /mm (0.0338 in. 2 /in.)

CHAPTER 10—REFERENCES Committee documents are listed first by document number and year of publication followed by authored documents listed alphabetically. American Concrete Institute

318-63—Building Code Requirements Concrete 318-71—Building Code Requirements Concrete 318-77—Building Code Requirements Concrete 318-95—Building Code Requirements Concrete 318-02—Building Code Requirements Concrete 318-08—Building Code Requirements Concrete (ACI 318-08) and Commentary 318-11—Building Code Requirements Concrete (ACI 318-11) and Commentary

CSA-23.3-77—Design of Concrete Structures CSA-23.3-77—Design CSA-23.3-04—Design CSA-23.3-04—D esign of Concrete Structures Abul Mansur, M., and Rangan, B., 1978, “Torsion in Spandrel Beams,” Journal of the Structural Division, V. V. 104, 10 4, pp. 1061-1075. ACI Committee 315, 2004, ACI Detailing Manual, SP-66, American Concrete Institute, Farmington Hills, MI, 166 pp. ACI Committee 363, 1992, “Report on High-Strength Concrete (ACI 363R-92),” American Concrete Institute, Farmington Hills, MI, 55 pp. ACI Committee 438, 1969, “Tentative Recommendations for the Design of Reinforced Concrete Members to Resist Torsion,” ACI Journal, V. 66, No. 1, Jan., pp. 1-8. Ali, M. A., and White, R. N., 1997, “On Extending ACI 318 to High-Strength Concrete,” Proceedings of the First International Conference on High Strength Concrete, Kona, HI, July 13-18, pp. 554-567. Alkhrdaji, T., and Belarbi, A., 2003, “Shear Flow Zone in Rectangular RC Members Subjected to Pure Torsion,”

for Reinforced for Reinforced for Reinforced for Structural for Structural for Structural for Structural

European Committee for Standardization

Eurocode 2-04—Design of Concrete Structures—Part 1-1 Eurocode 8-04—Design of Structures for Earthquake Resistance—Part 1

Cairo, Egypt, Feb. 17-20, pp. 555-564. Allos, A. E., and Rashid, A. H., 1989, “Prestressed Concrete Rectangular Beams Subjected to Sustained Torque,” ACI Structural Journal, V. 86, No. 4, July-Aug., pp. 469-472. American Association Association of State Highway and Transpor Transportatation Officials, 1998, AASHTO LRFD Bridge Design Speci fications and Commentary, SI units, second edition, Washington DC, 1091 pp. Festigkeit stigkeit , sixth edition. Bach, B., 1911, Elastizität und Fe (in German) Badawy, H., and Jordaan, I., 1977, “Plastic Analysis of Curved Beams under Point Loads,” Journal of the Structural Division, V. 103, pp. 1429-1455. Baumann, T., 1972, “Zur Frage der Netzbewehrung von Flachentragwerken (On the Problem of Net Reinforcement of Surface Structures),” Der Bauingenieur , V. 47, No. 10, pp. 367-377. Bažant, Z. P., 1984, “Size Effect in Blunt Fracture: Concrete, Rock, Metal,” Journal of Engineering Mechanics, V. 110, No. 4, pp. 518-535. Bažant, Z. P., 2002, Scaling of Structural Strength, Hermes Penton Science, Ltd. (Kogan Page Science), London, UK, 336 pp. Bažant, Z. P., and Oh, B. H., 1983, “Crack Band Theory for Fracture of Concrete,” Matériauxet Constructions, V. V. 16, 1 6, May-June, pp. 155-177. Bažant, Z. P., and Planas, J., 1998, Fracture and Size Effect in Concrete and Other Quasibrittle Materials, CRC Press, Boca Raton, FL and London, UK, 640 pp. Bažant, Z. P., and Sener, S., 1987, “Size Effect in i n Torsional Torsional Failure of Concrete Beams,” Journal of Structural Engineering, V. 113, No. 10, pp. 2125-2136. Bažant, Z. P.; Sener, S.; and Prat, P. C., 1988, “Size Effect Tests of Torsional Failure of Plain and Reinforced Concrete Beams,” Materials and Structures, V. 21, pp. 425-430.


Bažant, Z. P., and Yu, Q., 2005a, “Designing Against Size Effect on Shear Strength of Reinforced Concrete Beams Without Stirrups: I. Formulations,” Journal of Structural Engineering, V. 131, No. 12, pp. 1877-1885. Bažant, Z. P., and Yu, Q., 2005b, “Designing Against Size Effect on Shear Strength of Reinforced Concrete Beams Without Stirrups: II. Verification and Calibration,” Journal of Structural Engineering, V. 131, No. 12, pp. 1886-1897. Belarbi, A., and Greene, G. G., 2004, “Reinforced Concrete Box Girders Under Cyclic Torsion,” Proceedings of the 13th World Conference on Earthquake Engineering , Vancouver, BC, Canada, Aug. 1-6, 2004, 15 pp. Belarbi, A., and Hsu, T. T. C., 1994, “Constitutive Laws of Concrete in Tension and Reinforcing Bars Stiffened by Concrete,” ACI Structural Journal, V. 91, No. 4, July-Aug., pp. 465-474. Belarbi, A., and Hsu, T. T. C., 1995, “Constitutive Laws of Softened Concrete in Biaxial Tension-Compression,” ACI Structural Journal, V. 92, No. 5, Sept.-Oct., pp. 562-573. Bentz, E. C., 2005, “Empirical Modeling of Reinforced Concrete Shear Strength Size Effect for Members without Stirrups,” ACI Structural Journal, V. 102, No. 2, Mar.-Apr., pp. 232-241. Birkland, C. J., 1965, “Strength of Rectangular Beams with Longitudinal Reinforcement in Combined Torsion, Bending, and Shear,” Shear,” Master’s thesis, Department of Civil Engineering, University of Washington at Seattle, Seattle, WA. Bishara, A., and Londot, L., 1979, “Flexural Rotational Capacity of Spandrel Beams,” Journal of the Structural Division, V. 105, No. 1, pp. 147-161. Bishara, A., and Peir, J. C., 1968, “Reinforced Concrete Rectangular Columns in Torsion,” Journal of the Structural Division, V. 94, Dec, pp. 2913-2934. Boulton, N., and Boonsukha, B., 1959, “Plastic Collapse Loads for Circular Arc Girders,” Proceedings of the Institution of Civil Engineers, June, London, UK. Bredt, R., 1896, Kritische Bemerkungen zur Drehungselastizität, Zeitschrift des Vereines Deutscher Ingenieure , Band 40, No. 28, July 11, pp. 785-790; No. 29, July 18, pp. 813-817. (in German) Cauchy, A., 1828, “Sur les Equations qui Experimentent les Conditions d’Equilibre ou les Lois de Mouvement Interieur d’un Corps Solide,” Exercices de Mathematique, Paris, France. (in French) CEB-FIP, 1978, Model Code for Concrete Structures, CEB-FIP International Recommendation, third edition, Comite Euro-International Euro-International du Beton (CEB), 348 pp. Cerioni, R.; Ferreti, D.; and Iori, I., 1998, “Torsional Behavior of Prestressed High-Strength Concrete Beams,” Computational Modeling of Concrete Structures, pp. 797-806. Chakraborty, M., 1977, “Torsional-Balanced Steel in Concrete Beams,” Journal of the Structural Division, V. V. 103, 10 3, pp. 2181-2191. Chen, W., 1982, Plasticity in Reinforced Concrete, McGraw-Hill Book, New York, 474 pp. Collins, M. P., 1973, “Torque-Twist Characteristics of Reinforced Concrete Beams,” Inelasticity and Non-Linearity

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