Module 10

CIE 525 Reinforced Concrete Structures

Instructor: Andrew Whittaker

10. SEISMIC ANALYSIS AND DESIGN OF FRAMED BUILDINGS 10.1

Seismic Analysis and Design of Framed Buildings

Sections 10.1 through 10.8 present information on the analysis and design of reinforced framed buildings incorporating slab- and beam-column frames. Consideration is given to both gravity and lateral load analysis. Procedures for the design and detailing of special moment frames using precast concrete were introduced in ACI-318-02 (Section 21.6). Such frames are not discussed in this module. The detailing provisions are intended to produce frames that respond to design (maximum) displacements essentially like cast in-situ (monolithic) special moment frames. A list of recommended reading is presented below. Other references are provided in the body of the text. 1. ACI, 2002, Building Code Requirements for Structural Concrete, ACI 318-02, American Concrete Institute, Farmington Hills, MII 2. ATC, 2000, Seismic Evaluation and Retrofit of Concrete Buildings, Report ATC-40, Applied Technology Council, Redwood City, California. 3. FEMA, 2000, Prestandard and Commentary for the Seismic Rehabilitation of Buildings, Report FEMA 356, Federal Emergency Management Agency, Washington, D.C 4. French, C. W. and Moehle, J. P., 1991, “Effect of Floor Slab on Behavior of Slab-Beam-Column Connections”, ACI Special Publication SP-123, American Concrete Institute, pp. 225-258 5. Kitayama, K., Otani, S., and Aoyama, H., 1991, “Development of Design Criteria for RC Interior Beam-Column Connections”, ACI Special Publication SP-123, American Concrete Institute, pp. 97-123 6. Paulay, T. and Priestley, M. J. N., 1992, Seismic Design of Reinforced Concrete and Masonry Buildings, Wiley 7. Moehle, J. P., 1984, “Strong Motion Drift Estimates for R/C Structures”, Journal of Structural Engineering , Vol. 110, No. 9, ASCE, Reston, VA 8. Moehle, J. P., 1984, “Seismic Response of Vertically Irregular Structures”, Journal of Structural Engineering , Vol. 110, No. 9, ASCE, Reston, VA 9. Wong, P. C., Priestley, M. J. N., and Park, R, 1990, “Seismic Resistance of Frames with Vertically Distributed Longitudinal Reinforcement in Beams”, ACI Structural Journal , American Concrete Institute, July-August.


10.2


Modeling Elements of Reinforced Concrete Building Frames

10.2.1 Introduction Chapter 9 of ATC-40 has been uploaded to the CIE 525 web site for detailed information on modeling of reinforced concrete framed buildings and components of framed buildings. See the following URL for information: http://overlord.en http://overlord.eng.buffalo.edu/ClassHom g.buffalo.edu/ClassHomePages/cie525/Lectures/A ePages/cie525/Lectures/ATC40Ch9.pdf TC40Ch9.pdf .. Consider the sample building below from page 9-6 of ATC-40. One of the challenges associated with the analysis and design of a new building or evaluation of existing construction is the development of a mathematical model of that building.

Structural Elements frames walls diaphragms Non-structural Elements elements that influence structural behavior elements whose damage affects performance

Foundation Elements soil components structural components

What should be included in a mathematical model of the building shown above?

•

Any component, structural or non-structural, that will substantially affect the response of the building.  

What response quantities should we consider? Displacement, Acceleration, Others?


10.2


Modeling Elements of Reinforced Concrete Building Frames

10.2.1 Introduction Chapter 9 of ATC-40 has been uploaded to the CIE 525 web site for detailed information on modeling of reinforced concrete framed buildings and components of framed buildings. See the following URL for information: http://overlord.en http://overlord.eng.buffalo.edu/ClassHom g.buffalo.edu/ClassHomePages/cie525/Lectures/A ePages/cie525/Lectures/ATC40Ch9.pdf TC40Ch9.pdf .. Consider the sample building below from page 9-6 of ATC-40. One of the challenges associated with the analysis and design of a new building or evaluation of existing construction is the development of a mathematical model of that building.

Structural Elements frames walls diaphragms Non-structural Elements elements that influence structural behavior elements whose damage affects performance

Foundation Elements soil components structural components

What should be included in a mathematical model of the building shown above?

•

Any component, structural or non-structural, that will substantially affect the response of the building.  

What response quantities should we consider? Displacement, Acceleration, Others?


•


Should the foundation elements be considered? 

What is typical practice?



If foundation elements are to be considered, how?

Except for the very simplest of building frames, analysts will normally rely on specialized computer programs such as SAP2000. Some of these programs, including SAP2000 can directly represent the nonlinear load-deformation behavior of individual components. Others are only suitable for elastic analysis. Nonlinear methods of analysis are beyond the scope of CIE 525 and are not referenced by ACI 318, which is constructed around traditional elastic methods of analysis and design. For information, nonlinear methods of analysis are discussed in CIE 619, Earthquake Engineering and Structural Dynamics: a class that is generally offered in the Spring semester. Reinforced concrete framed buildings can be composed of beam-column frames, slab-column frames, or a combination of the two. Slab column frames may include wide band beams, drop panels, waffle-slab systems with solid panels between columns, and column capitals. Combinations of beam-column frames and slab-column frames are common.

•

Framing system of page 1 with perimeter spandrel beams and a flat-slab interior 

Depending on the geometry of the building, the slab-column system may be stiffer than the beam-column system

10.2.2 Beam-Column Frames The objective of modeling such frames is to capture the strength, stiffness, and deformation capacity of beams, columns, beam-column joints, and other critical components. Frames are generally considered as 2-D systems in a 3-D model. In ATC-40 and other documents, beam-column frames are defined as elements of a building. Components such as beams and columns comprise the element. The assemblage of elements comprises the building. See the figure from ATC-40 below for details.

Frame Element

Frame Components • Colum olumn n • Beam • Beam-C Beam-Colum olumn n Joint Joint



Beams and columns in a frame should be modeled considering their flexural and shear rigidities although shear rigidities can be ignored in many cases. Rigid beam-column connections are often assumed except for cases where the joint is not sufficiently strong to permit the development of component capacities at the joint.

•

What does this mean?

•

Generally using line elements with properties (stiffness) concentrated at component centerlines

•

What about cases where beam and column centerlines do not coincide? 

Reduced effective stiffnesses and strengths



Direct modeling of the eccentricity

Should the slab be modeled in beam-column frames? How could a slab influence the response of a beamcolumn frame?

•

As a diaphragm to link adjacent frames together at floor levels

•

Act compositely as a beam flange in tension and compression

F1

F2

F1

F2

Diaphragm action of a slab

Slab serving as a beam flange

The information presented above could be used to model the beam-column frame for the purpose of linear analysis. Nowadays, nonlinear analysis is possible, where such analysis can trace the post-yielding response of components and frames.



Models of beams, columns, and joints for nonlinear analysis should be capable of representing behaviors from zero deformation to maximum deformation. Information on the strength, stiffness, and deformation limits for such components is presented below. For elastic analysis of beam-column frames, components were modeled as line elements spanning between nodes at the beam-column connections. For nonlinear analysis, additional work is required. Section 9.4.2.1.2 of ATC-40 provides supplemental information on this subject. Nonlinear analytical models should (ideally) represent all likely modes of inelastic response (flexure, shear, splices, axial load). All possible locations for inelastic action in beams and columns should be identified and appropriate models prepared. For example, see the beam below that is resisting gravity and lateral loads. For elastic analysis, two nodes would typically be used to characterize the beam: the nodes at each end of the beam. For nonlinear analysis, plastic hinges (zones of inelastic response) could form at the ends of the beam and elsewhere in the span. The location of elsewhere must be identified and included in the mathematical model.

E (a) Beam span and loading

E

G Beam Column

E

G = Gravity load E = Earthquake load

E

+ M p = Positive plastic

moment strength

(b) Initial assumption

(c) Revised plastic hinging

M p = Negative plastic

moment strength

Marks assumed plastic hinge location

10.2.3 Slab-Column Frames More emphasis is placed on slab-column frames than on beam-column frames in CIE 525 because the force-deformation response of slab-column systems is often poorly understood by many design professionals. Design practice for many years in regions of high seismicity has been to ignore the contributions of slabcolumn frames to the lateral strength and lateral stiffness of building frames. 

Conservative for force-based design?

Such an approach is inappropriate for both seismic evaluation and performance-based design because





Slab-column frames can contribute substantial strength and stiffness



Damage to slab-column frames must be considered in performance evaluation

Consider the two-story slab-column frame below. How is this frame partitioned for the purpose of analysis? Does diaphragm stiffness play a role in the distribution of forces between the frames? 

Flexible diaphragms: if the vertical elements have substantial lateral stiffness and the horizontal elements are relatively flexible (timber, metal deck, thin concrete topping slabs, or large spans), displacement compatibility is not enforced. Lateral loads are distributed according to the tributary width to each frame.



Rigid diaphragms: generally the case for cast-in-place concrete slab systems, displacement compatibility is enforced between frames.



If the diaphragms are rigid, the frames could be modeled as shown below. Such a model could also be used for beam-column frames.

Three approaches can be adopted for modeling slab-column frames as shown in the sketch below. 

Equivalent beam width model : columns and slabs are represented by frame elements that are rigidly connected



: columns and slabs are represented by frame elements that are Equivalent frame model interconnected by connection springs



Finite element model : columns are represented by frame elements and the slab by plate-bending elements (must pay special attention to slaving degrees of freedom at the connections).

(a) Actual slab-column frame Column Beam

(b) Effective beam width model Connection spring Column Beam

(c) Equivalent frame model

(d) Finite element model



Information on modeling of slab-column frames is available in the literature. Sample references are 1. Allen, F. and Darvall, P., 1977, Lateral Load Equivalent Frame , ACI Structural Journal, Vol. 74, No. 7, pp 294-299 2. ACI, 1988, Recommendations for Design of Slab-Column Connections in Monolithic Reinforced Concrete Structures, ACI-ASCE Committee 352, ACI Structural Journal, Vol. 85, No. 6, pp 675696 3. Darvall, P. and Allen, F., 1984, Lateral Load Effective Width of Flat Plates with Drop Panels, ACI Structural Journal, Vol. 81, November-December, pp 613-617 4. Hwang, S.-J. and Moehle, J. P., 1993, An Experimental Study of Flat-Plate Structures Under Vertical and Horizontal Loads , Report No. UCB/EERC-93/03, University of California, Earthquake Engineering Research Center, Berkeley, CA 5. Pan, A. and Moehle, J. P., 1988, Reinforced Concrete Flat Plates Under Lateral Load , Report No. UCB/EERC-88/16, University of California, Earthquake Engineering Research Center, Berkeley, CA 6. Pecknold, D. A., 1975, Slab Effective Width for Equivalent Frame Analysis, ACI Structural Journal, Vol. 72, No. 4, pp 135-137 7. Vanderbilt, M. D. and Corley, G. W., 1983, Frame Analysis of Concrete Buildings, Concrete International, December, pp 33-43 Many of the early studies on the stiffness of slab-column systems focused on service level lateral forces (generally wind loads) that produced elastic response only. Also, most of the studies ignored the effect of gravity loads on the lateral stiffness of the slab system. Nonetheless, the presentations of Allen, Darvall, Pecknold, and Vanderbilt provide a starting point for the discussion. 

Data of Pecknold 



Slab effective width based on the use of zero-end offsets.

Data of Darvall and Allen 

Effective width coefficients based on use of zero-end offsets; if the model assumes zero-size joints, the effective width factors should be increased by (1 − c1 / l 1 )3 .

Before extending the presentation to seismic (large drift) applications, the effect of cracking on the stiffness of a slab in a slab-column system must be characterized. Vanderbilt and Corley defined a parameter β to account for cracking, namely,



β=

I eff I g

and recommended a default value of 0.33 for use with the equivalent frame method. Moehle and Diebold recommended a value of between 0.33 and 0.50 for the effective beam width model. Hwang and Moehle also studied the effect of cracking on the stiffness of slab-column systems. Analysis of an interior plate connection supported on a circular column led to an equation for β ; for moments at the column face substantially greater than the cracking moment, β ≈ 1/ 3 . 

What value for β should be used in the absence of lateral loads? 

Cracking will always be present due to construction loads, shrinkage etc, so some reduction is always warranted.

So, using the data of Darvall and Allen, and for l = l , the effective width of a cracked flat plate (no drop 1

2

panels) is

l 1 beff = (αl )(β) = (0.5l )( ) = 2 2 2 3 6 Pan, Hwang, and Moehle extended the above work for seismic applications. Consider first the experimental work of Hwang and Moehle. 

Pseudo-static testing of a 0.4-scale model of a flat-plate floor



Designed for wind-load effects only



Lateral stiffness with low gravity load and low drifts (1/800)



Lateral stiffness with high gravity load and low drifts (1/800)



Lateral stiffness with high gravity load and larger drift (1/200)



Lateral stiffness with high gravity loads and large drift (1/20)



Strain profile in reinforcement near edge of slab



Crack patterns in slab at large drifts



Lateral stiffness of test slab at different drift levels and comparison with models

Hwang and Moehle made some key recommendations regarding the lateral stiffness and strength of flat plate structures:



1. The elastic (uncracked) moment-rotation stiffness of a slab-column connection is essentially a linear function of c and l for connections having an aspect ratio c / c between 0.5 and 2, and a slab 1 1 2 1 aspect ratio l / l greater than 0.67, and is given by 2

1

beff = 5c1 +

beff = 3c1 +

l

1

4

l 1 8

for an interior frame

for an exterior (edge) frame

2. The effect of cracking can be characterized as follows for connections with f c′ = 4000 psi, f y = 60 ksi, and a minimum connection geometry that meets ACI 318 (i.e., punching shear, moment transfer etc) c

1

l

3

β=4 ≥

3. The strength of flat-plate connections can be estimated conservatively using ACI 318, wherein strength is limited by a.

Shear stresses on the critical perimeter around the column

b. Flexural strength of reinforcement placed within 1.5 h either side of the column (for a total width of c + 3h . 2

But this strength may only be achieved at drift levels in excess of 1% of the story height. 4. Flat plate slab systems can sustain large drifts (of the order of 4%) if the gravity loads are low. 5. Bottom slab reinforcement should be placed directly over the columns of flat plates to prevent progressive collapse in case a connection fails in punching shear. Consider now the work of Pan and Moehle 

Testing of a 4 scale models of a flat-plate floor



Summarizes effect of gravity shear on drift capacity and ductility ratio



Compare stiffness under low gravity load (Test 3) and high gravity load (Test 1): see Table 7.2



Effect of gravity load on ductility



Effect of gravity load on drift



Modeling Components of Reinforced Concrete Building Frames

10.3

To develop a mathematical model, the engineer must identify the stiffness of the components of the building frame? Advise on the choice of values for component stiffness, for the elastic analysis of a building frame, is given in ATC-40. The pertinent table from ATC-40 is given below.

Table 9-3. Component Initial Stiffnesses Flexural Rigidity

Shear Rigidity 2

Axial Rigidity

0.5E cIg

0.4E c Aw

E c Ag

E cIg

0.4E c Aw

E c Ag

Columns in compression

0.7E cIg

0.4E c Aw

E c Ag

Columns in tension

0.5E cIg

0.4E c Aw

E s As

Walls, uncracked

0.8E cIg

0.4E c Aw

E c Ag

Walls, cracked

0.5E cIg

0.4E c Aw

E c Ag

Flat slabs, non-prestressed

See discussion

0.4E c Aw

E c Ag

Flat slabs, prestressed

in Section 9.5.3

0.4E c Aw

E c Ag

Component Beam, non-prestressed1 Beam, prestressed1

I g for T-beams may be taken twice the I g of the web alone, or may be based on the effective section as defined in Section 9.5.4.2. 1

2

For shear stiffness, the quantity 0.4 E c has been used to represent the shear modulus, G .

3

For shear-dominated components, see the discussion and commentary in Section 9.5.3.

Once the stiffnesses of all of the components have been calculated, how is the mathematical model developed? 

Centerline dimensions?



Zero-end offsets at beam(slab)-column connections?



2D or 3D models?

10.4

Practice of Seismic Analysis and Design Per ACI 318

The key steps in the IBC force-based seismic analysis of reinforced concrete framed buildings are enumerated below: 1. Develop trial sizes of beams and columns using hand calculations; estimate the reactive weight at each floor level 2. Develop a mathematical model of the building frame using the sizes of step 1 3. Analyze the model of step 2 for frequencies and mode shapes 4. Calculate the design base shear V using the first mode period from step 3 (as modified by the IBC) as



V =

PSa (T , ξ)W R

5. Distribute the base shear force over the height of the building using

C vx =

w x hxk n

k

∑ wi hi

i =1

where k is equal to 1 for periods less than or equal to 0.5 second and 2 for periods greater than 2.5 seconds, and

Fx = CvxV where F is the lateral force at level x. Is this reasonable? x

+

=

6. Apply the lateral forces F i in conjunction with the gravity loads and (after using load factors) calculate design actions (forces) in all components.

7. Check displacements and drifts in frame by multiplying calculated elastic displacements, ∆ s , by the displacement amplification factor, C d . If the frame is too flexible, revise member sizes and return to step 2. 8. Design components for the actions of step 6 and apply prescriptive details of ACI 318.



Values for R and C d from the 2000 IBC for reinforced concrete frames are presented below. In CIE 525, attention is focused on the special reinforced concrete moment frame.

R 10.5

Ω0

C d

Chapter 21 of ACI 318

Chapter 21 of ACI 318 provides guidance for the design and detailing of components (beams, columns, joints, and walls) in seismic lateral-force-resisting systems. Design forces are calculated elsewhere in such documents as 

2000 International Building Code



2000 NEHRP Recommended Provisions

ACI 318 writes rules regarding acceptable concrete strengths and rebar types for earthquake-resisting construction, namely, 

f ′ ≥ 3000 psi c

Rebar shall conform with ASTM A706 unless tight quality control on ASTM A615 such that: 

f yact ≤ f ynom + 18000 psi; and



f act ≤ 1.25 f nom



Type 2 splices in regions of high (yield) rebar stress



u

u

No welded splices in the critical regions (twice member depth from column or beam face)

CIE 525 Reinforced Concrete Structures 10.6


Design and Detailing of SMRF Beams

10.6.1 Rules of ACI 318 ACI 318 writes rules for beams in SMRFS. What are beams? 

Flexure dominated components



Low axial loads: Pu ≤ 0.1 Ag fc′



ln ≥ 4d for shear; if span is less:



b / d ≥ 0.3 for stability



b ≥ 10′′; ≤ b

col

3

+ 2( d b ) for good moment transfer to the column 4

Prescriptive detailing requirements are used in lieu of requiring calculations for determining deformation capacity in critical regions. Limits are set on the amount of longitudinal reinforcement, namely, 

200

f y

≤ ρ ≤ 0.025

Ductility is provided through compression rebar and confinement. The limit of 0.025 is to avoid rebar congestion and recognizes that balanced failure cannot be defined once members undergo inelastic load reversals. Consider now the generic beam below

ACI writes the following rules 

At least two bars continuous top and bottom of the beam. Why?



At the face of the joint, M n+ ≥ 0.5M n−

CIE 525 Reinforced Concrete Structures 


Everywhere else, M ≥ 0.25M − / + at face of joint n

n



No lap splices within 2d either side of a section where hinging can occur or in joints. Where in the above beam?



If lap splice in a beam, splice must be enclosed in hoops or spirals with a maximum spacing of 4 in. or d /4.



Mechanical splices are permitted (and preferred)

Limits are also set on the placement of transverse reinforcement. What is the purpose of the transverse reinforcement? 

Shear reinforcement



Confinement

h

In the region a distance 2 h either side of a plastic hinge region, except only on one side at joints, 

1st hoop at 2 in. maximum from joint face

s

max



≤ d / 4 ≤ 8d b ≤ 24d tie ≤ 12′′



Seismic hoops shall restrain the beam bars as for columns

Elsewhere throughout the span, stirrups with seismic hooks 

smax ≤ d / 2

Acceptable seismic hooks are shown below.



The design shear force in a beam is calculated as the sum of 

Factored gravity load shears: w = 1.2 D + 1.0 L + 0.2S



Plastic shears: shear due to development of nominal moment capacities at member ends and ≥ 1.25 f y s

u



In a plastic hinge zone, the contribution of the concrete to the shear resistance, vc , is set equal to 0. 

Indirect way of requiring more shear reinforcement in regions subjected to cyclic nonlinear deformations



Confined core still largely intact

10.6.2 Plastic Hinge Zones and Rotation Capacities Reinforced concrete SMRFs are expected to form plastic hinges during design earthquake shaking. In the frame below, plastic hinges are shown at the ends of the beams and at the column bases. If we also assume that all of the frame deformation during an earthquake is plastic deformation

∆e < < ∆ p and ∆t = ∆ e + ∆ p ≈ ∆ p then the plastic rotation at each hinge ( θ ) can be related to the total displacement ∆t as p

θ p h = ∆ t and so, if the maximum roof displacement is known in the design earthquake, the maximum beam plastic rotation can be estimated.

∆t

h

Consider first the actual and idealized moment-curvature relationships shown in the figure below. How can this information be used to calculate m aximum rotation capacities of critical connections?



Consider now the cantilever beam shown below for which the moment and curvature diagrams are shown.



The idealized curvature distribution is shown below.

The shaded area represents the plastic rotation that occurs in addition to the elastic rotation at the ultimate stage of the beam. The inelastic area at the ultimate stage can be replaced by an equivalent rectangle of height ( φu − φ y ) and width l p , having the same area as the actual inelastic curvature distribution. 

l is the plastic hinge length over which the plastic curvature is assumed to be constant p

The plastic rotation to the left of the fixed end can be calculated as

θ p = (φu − φ y )l p So, if the plastic hinge length is known (see below), the maximum roof displacement ∆t has been estimated, and the moment-curvature relationships established for the beams and columns (including the effects of confinement), the maximum curvature demand on the cross sections can be established. The task then is to ensure that φmax ≤ φu for each hinging component. The subject of plastic hinge length has been studied (and studied and studied). The concept of a plastic hinge is introduced to simplify calculations and nothing else. Values and expressions for l p have been back calculated from experimental data. 

Presentation materials and summary by Professor Dawn Lehman (UW)

For typical beam and column proportions, use

l p = 0.5h where h is the member depth. Such an assumption should be used for calculating rotation capacities but not for the length over which confinement reinforcement is to be placed. 

What length should be used for the extent of confinement reinforcement? 

What does ACI 318 assume?

CIE 525 Reinforced Concrete Structures 10.7


Design and Detailing of SMRF Columns

10.7.1 Rules of ACI 318 ACI 318 writes rules for columns in SMRFS. What are columns? 

Modest to high axial loads: Pu > 0.1 Ag fc′



bmin ≥ 12′′





bmin bmax

≥ 0.4

0.01 ≤ ρ = g

∑ A s A g

≤ 0.06



Lower limit (0.01) is to control time-dependent deformations (creep) and to have M > M y cr



Upper limit (0.06) is to limit rebar congestion and plastic shears in columns

To limit inelastic flexural deformation in columns, the sum of the column nominal flexural strengths shall exceed the sum of the beam nominal flexural strengths (in the plane of the beam) at a beam-column connection, that is,

∑ M nc > ∑ M nb •

ACI 318 writes ∑ M c ≥ 1.2∑ M g 



Column axial load that gives the minimum column-moment strength. How to calculate? See below.

No account of strain hardening in beam rebar

CIE 525 Reinforced Concrete Structures 

Should include an appropriate slab width to calculate beam strengths. 



•


What effective width? For interior beams, ACI 318 writes b ≤ b + 16t ; l eff f

2

If the above rule is not satisfied, ACI writes that special transverse reinforcement is required over the full height of the column

ACI 352 writes ∑ M ≥ 1.4∑ M c g   

Column axial load that gives the minimum column-moment strength. How to calculate? No (explicit) account of strain hardening in beam Should include an appropriate slab width to calculate beam strengths.  



No guidance given on slab width but consider the data of Kurose et al. (SP-123, p 39) Kurose et al: effective width is dependent on level of deformation

•

At 2% drift for J2, beff = b + 2(0.7 ×

•

At 4% drift for J2, beff = b + 2(0.9 ×

94 5 94 5

)t f = b + 26t f

)t f = b + 34t f

NZ codes requires even larger multiplier than 1.4

How are columns designed for strength?

•

Using interaction curves based on design material properties

•

Confinement effects are ignored.



Consider now the generic column below (different details may be required if the column serves as a boundary element)

How about the joint regions?

Locate lap splices in middle third of column between floors; design as tension splices; enclose in transverse reinforcement

l 0

Avoid lap splices in the first story; mechanical and welded splices okay anywhere

M

M n 1.25M n

l n 4

?

ACI writes the following rules 

Special transverse reinforcement is required along length l from each joint face, where 0

l0 ≥ hmax 

≥

l n

6 ≥ 18′′

Special transverse reinforcement should be provided over a longer height at the bases of the first story columns as shown above


•


Perhaps 0.2 or 0.25 times the clear height of the column, l n

Special transverse reinforcement is intended to

•

Increase strain capacity

•

Restrain buckling of longitudinal rebar

•

Increase shear strength

•

Secondary role is to increase axial and flexural strength

Transverse reinforcement should also be provided to resist shear forces 

Calculate the plastic shear assuming nominal flexural strengths at ends of columns



To ensure that post-spalling strength exceeds pre-spalling strength

The ACI rules for special transverse rebar are



A g ′ f ′ For spirals, ρ ≥ 0.45( − 1) c ≥ 0.12 c s A f f c





y

f c′

For hoop ties, Ash ≥ 0.3( shc

f

yh

y

A g f ′ )[ − 1] ≥ 0.09 shc c A f c

yh

Equations govern for large diameter columns

Spacing of transverse rebar is as follows

s ≤ 0.25bmin

≤ 6d b ≤ s x = 4 + (

10.8

14 − h x 3

);4′′ ≤ s ≤ 6′′ x

Response of Reinforced Beam-Column Joints (per ACI)

ACI Committee 352 provides information to the code-writing committees on the design of beam-column joints for gravity and seismic actions. The report ACI 352R-91, Recommendations for the Design of Beam-Column Joints in Monolithic Reinforced Concrete Structures, is the current ACI report on the beam-column joints. ACI 352 classifies joints by type and geometry:

•

Type 1 for non-seismic applications: joint connects members designed to satisfy ACI 318 requirements and in which no significant inelastic deformations are expected

•

Type 2 for seismic applications: joint which connects members designated to have sustained strength under deformation reversals into the inelastic range

and



•

Interior: horizontal members framing into all four sides

•

Exterior: at least two horizontal members framing into opposite sides of the joint

•

Corner: all others

Gravity and lateral-load demands on beam-column joints differ as noted below.

To count as confinement as an exterior or exterior joint, a beam framing into a face must cover at least 75% of the column width. Sketches of the three joint geometries follow:



The basis steps in the design and detailing of beam-column joints are as follows:

•

Classify joints according to type and geometry

•

Define joint demands

•

Define joint capacity

•

Provide joint confinement

•

Provide reinforcement development

•

Provide adequate strength in columns

Experimental observations indicate that joint shear strength is fairly independent of the volume of transverse reinforcement if minimum amounts are provided. This is the basis of the detailing procedures in US codes. Consider the data from Kitayama et al. in SP-123 for interior joints:

These data suggest that there is no increase in joint shear strength for substantial increases in the joint lateral reinforcement ratio, beyond 0.4%.



How are joint demands calculated? Consider the interior beam-column joint shown below:

V col 1.25 A f s y

?????

V

The horizontal joint shear force is calculated as

V = T1 + T2 − Vcol = ( As [1.25 f y ] + ?????− Vcol ) Should slab contributions to the strengths of the beams be considered to estimate the demand on a beamcolumn joint?

•

What effective width should be adopted?

Should the vertical joint shear force be calculated also?

•

ACI-318 assumes that if the column rebar is correctly designed, stresses in the vertical reinforcement should be less than f and that the reserve strength in the reinforcement can serve y the function of vertical joint shear reinforcement.

ACI-318-02 and ACI 352 provide equations for the strength of beam column joints as follows:

Vu = φVn = 0.85[ γ fc′b j h] where φ = 0.85 per Section 9.3.4(c), values for γ are given in the table below, h is the depth of the column in the direction under consideration, and b is the joint width, which is also defined below.

γ Type

Interior

Exterior

Corner

1

24

20

15

2

20

15

12



The joint width is also defined in ACI 318 as follows below:

For Type 2 joints, the special transverse reinforcement in the column ends must be continued through the beam-column joint as shown below. No specific calculation for the volume of joint shear reinforcement is needed.

•

For interior joints meeting specific requirements, special transverse reinforcement can be reduced by 50% from that in column ends



ACI 318 also writes rules for anchorage of beam reinforcement in a Type 2 beam-column joint, namely,

•

For hooked bars, the required development length l dh for a bar with a standard 90-degree hook must be provided 

Value is changed from that presented in Module 08 (per Section §12.5.2) to account for load reversals (an increase) and recognizing that the hook is to be embedded in confined concrete (leading to decreases due to the provision of concrete cover [0.7] and ties [0.8]), namely, for normal weight concrete

ldh =



f y d b 65 f c′

≥ 8d b ≥ 6 in

Where is this distance measured from?

d b

h

l dh

•

The development length for a straight bar in tension is a multiple of l dh above: 2.5 times if the depth of concrete cast in one lift below the bar does not exceed 12 inches, and 3.5 times otherwise.

•

The diameter of straight bars in a beam-column joint should be smaller than 5 percent of the column dimension in the direction, that is,

h d b

≥ 20

Note that this will not prevent slip of the beam bars in the joint. To prevent such slip, h / d b ≥ 32 , which would result in very large joints.


10.9


Notes on the Design of Beam-Column Joints

A number of models have been developed for force transfer in beam-column joints. The two most popular models are the

•

Truss model: assumes perfect bond of rebar in joint and assumes all forces are transmitted to the joint by longitudinal reinforcement.

•

Diagonal strut model: assumes no bond of rebar in joint; force is applied as a compression force in concrete on opposite side of joint.

Truss model

Diagonal strut model



Forces are transferred across the joint for the two models as shown below in a figure from the text of Paulay and Priestley:

An alternate model is the strut-and-tie model shown below. This approach permits the user to select the load path: one approach is to assign all forces in longitudinal reinforcement to a truss mechanism, and all compression in the flexural compression zone to a single strut.

Which approach is correct? No single load-transfer mechanism is correct for all levels of applied load. True mechanism is likely a combination of the diagonal strut and truss models (or the strut-and-tie model) shown above.


10.10


Seismic Analysis And Design Of Structural Wall Buildings

The remainder of this module covers the design of structural walls. Procedures for the design and detailing of special structural walls using precast concrete were introduced in ACI-318-02 (Section 21.8). Such walls are not discussed in this module. The detailing provisions are intended to produce walls that respond to design (maximum) displacements essentially like cast in-situ (monolithic) special structural walls. A list of references related to structural walls is presented below. 1. ACI, 2002, Building Code Requirements for Structural Concrete, ACI 318-02, American Concrete Institute, Farmington Hills, MII 2. Aktan, A. E. and Bertero, V. V., 1985, “RC Structural Walls: Seismic Design for Shear”, Journal of Structural Engineering , ASCE, Vol. 111, No. 8, pp 1775-1791 3. ATC, 1996, Seismic Evaluation and Retrofit of Concrete Buildings , Report ATC-40, Applied Technology Council, Redwood City, California. 4. FEMA, 2000, Prestandard and Commentary for the Seismic Rehabilitation of Buildings, Report FEMA 356, Federal Emergency Management Agency, Washington, D.C 5. Khan, F. R. and Sbarounis, J. A., 1964, “Interaction of Shear Walls and Frames”, Journal of the Structural Division, ASCE, Vol. 90, ST3, pp 285-335 6. Moehle, J. P., 1984, “Seismic Response of Vertically Irregular Structures”, Journal of Structural Engineering , ASCE, Vol. 110, No. 9, pp 2002-2014 7. Moehle, J. P., 1992, “Displacement-Based Design of RC Structures Subjected to Earthquakes”, Earthquake Spectra , EERI, Vol. 8, No. 3, pp 403-428 8. Paulay, T., 1986, “The Design of Ductile Reinforced Concrete Structural Walls for Earthquake Resistance”, Earthquake Spectra , EERI, Vol. 2, No. 4, pp 783-823 9. Paulay, T. and Priestley, M. J. N., 1992, Seismic Design of Reinforced Concrete and Masonry Buildings, Wiley 10. Seneviratna, G. D. and Krawinkler, H., 1994. “Strength and Displacement Demands for Seismic Design of Structural Walls”, Proceedings, Fifth U.S. National Conference on Earthquake Engineering, Chicago, IL 11. Sozen, M. A. and Moehle, J. P., 1993, “Stiffness of Reinforced Concrete Walls Resisting In-Plane Shear”, EPRI TR-102731 , Electric Power Research Institute, Palo Alto, CA 12. Wallace, J. W. and Moehle, J. P., 1992, “Ductility and Detailing Requirements of Bearing Wall Buildings”, Journal of Structural Engineering , ASCE, Vol. 118, No. 6., pp 1625-1644 13. Wallace, J. W., and Moehle, J. P., 1993, “An Evaluation of Ductility and Detailing Requirements of Bearing Wall Buildings Using Data from the March 3, 1985 Chile Earthquake”, Earthquake Spectra, EERI, Vol. 9, No. 1, pp 137-156.



14. Wallace, J. W, 1994, “New Methodology for Seismic Design of RC Shear Walls”, Journal of Structural Engineering , ASCE, Vol. 120, No. 3, pp 863-884 15. Wallace, J. W, 1995, “Seismic Design of RC Structural Walls: Part 1: A New Code Format”, Journal of Structural Engineering , ASCE, Vol. 121, No. 1, pp 75-87 16. Wallace, J. W. and Thomsen, J. H., 1995, “Seismic Design of RC Structural Walls: Part II: Applications”, Journal of Structural Engineering , ASCE, Vol. 121, No. 1, pp 88-101 17. Wallace, J. W., 1996, “Evaluation of UBC-94 Provisions for Seismic Design of RC Structural Walls”, Earthquake Spectra , EERI, Vol. 12, No. 2, pp 327-348 18. Wight, J. K., Wood, S. L., Moehle, J. P., and Wallace, J. W., 1996, “On Design Requirements for Reinforced Concrete Structural Walls, ACI Special Publication SP-162, American Concrete Institute, pp. 431-456 19. Wood. S. L., 1991, “Performance of Reinforced Concrete Buildings During the 1985 Chilean Earthquake: Implications for the Design of Structural Walls”, Earthquake Spectra, EERI, Vol. 7, No. 4., pp 607-638 20. Wood, S. L., 1989, “Minimum Tension Reinforcement Requirements in Walls”, ACI Structural Journal , American Concrete Institute, Vol. 86, No. 5, pp. 582-591 21. Wood, S. L., 1990, “Shear Strength of Low-Rise Reinforced Concrete Walls”, ACI Structural Journal , American Concrete Institute, Vol. 87, No. 1, pp. 99-107 10.11

Structural Walls

10.11.1 Introduction Structural walls are commonly used to resist lateral forces in reinforced concrete buildings

•

Key advantage is that walls provide vertical continuity in the lateral system 

Excellent performance (limited collapses) in past earthquakes



Damage in past earthquakes has included 

Cracking at base of walls and in coupling beams



Loss of tension capacity: loss of anchorage of rebar in foundation; fracture of rebar; failure of tension splices





Do not offset walls in plan



Can reduce volume of walls over the height of a building but must do so with care

•

Moehle (1984) showed that walls could be terminated below the top of a building frame without negative consequences

• No design rules for terminating all walls in a building so extend some walls over the full height.

•

Locations of walls in plan can 

Lead to substantial torsional response



Mitigate torsional response


•


Must ensure that the diaphragms can transfer the inertial loads to the walls 

How? Consider the diaphragms shown shaded below

V M



Conventional practice is to design the diaphragms as flexure-dominated beams 

Improved approaches?

10.11.2 Wall Classification by Elevation and Openings Structural walls are often characterized by their geometry as either

•

Flexural ( hw / l w ≥ 2 ; design controlled by flexure: high ratio of M /V )

•

Squat ( hw / l w ≤ 1 or 2; design controlled by shear: low ratio of M /V )

•

Coupled (overturning moment converted into a T-C couple)

•

Punched (analyze and design using strut-and-tie models?)



Examples of these four types of walls are shown below

10.11.3 Wall Classification by Plan Structural walls are also characterized by their location and function in a building. Three common examples are

•

•

Bearing walls in which the walls support a substantial percentage of the gravity loads 

Common in apartment buildings because walls used as party walls to separate apartments



Often flanged walls (see below)

Frame walls 

•

Support only a small percentage of the gravity loads

Core walls in which the walls enclose the vertical transportation and mechanical shafts 

Often flanged walls (see below)

Examples of these three types of walls are shown below.



10.11.4 Frame-Wall Interaction For the 2D frame wall system shown above, how are loads distributed between the moment-resisting frames and the structural walls? This subject was studied first for wind-resistant frame-wall buildings

•

Fazlur Khan of Skidmore, Owings, and Merrill (see Khan et al., 1964)

Consider now the 2D multi-story frame-wall building shown below


•


Such a system would be termed a dual system in the US codes of practice and seismic design guidelines

What is the effect of wall or frame yielding on the above force, shear, and displacement distributions?



Structural walls and moment frames also interact in a 3-D sense. Consider the frame-wall building below (for which only one wall is drawn) and earthquake-induced loads in the plane of the wall. Assume here that the frame is a 3-D space frame.

B A C

Under the lateral loading shown, the wall at A will tend to uplift. Such uplift cannot occur without developing moments and shears in the beams BAC at the second and third floor levels. Such 3D interaction can substantially alter the force-displacement response of a building.



10.11.5 Flanged Walls In bearing wall buildings such as that shown in Section 16.2.3, walls are often joined at the corners by perpendicular walls. Another example is shown below.

l w

hw

10.11.6 Internal Forces Controlling Wall Behavior The behavior of structural walls can be classed as follows: 1. Flexural behavior where the response is governed by the yielding of the flexural rebar 2. Flexural-shear behavior where some yielding of the flexural rebar precedes shear failure 3. Shear behavior where the wall fails in shear with no flexural yielding of rebar

•

Diagonal tension shear failure

•

Diagonal compression shear failure



Note that flexural and shear strength and stiffness are very sensitive to the co-existing axial force

•

If the axial load is less than the balanced load (but greater than 0), the larger the axial force, the larger the flexural and shear strength and stiffness 

P

Implications for coupled walls?

M 10.12

Behavior of Flexural Walls

10.12.1 Introduction The discussion below presents fundamental issues in the behavior and design of reinforced concrete wall buildings of moderate height (between 65 and 250 feet in height). For walls designed per current US practice, nonlinear response is expected in the design and maximum earthquakes. For a cantilever wall such as that shown on page 36, the nonlinear response should take the form of plastic rotation near the base of the wall. Subjects discussed in this section include

•

Distribution of vertical rebar

•

Fracture of tension reinforcement

•

Plastic moment strength

•

Confinement

•

Flexural instability

•

Shear strength

10.12.2 Distribution of Vertical Rebar The longitudinal rebar in flexural walls can be designed and detailed a number of ways. Consider two structural walls with identical concrete cross sections and rebar area but with the rebar

•

Lumped at each end of the wall

•

Distributed uniformly over the length of the wall

Moment-curvature relationships for the two walls are shown below. Both walls were 120 inches long by 12 inches thick. Wall 1 (solid line below) included 4 #9 bars at each end of the wall and #4 bars each face at 18 inches on center between the boundary elements; the total area of vertical rebar is 12.71 in 2. Wall 2



(dashed line below) included 32 #4 bars in each face for a total area of vertical rebar is 12.56 in 2. Grade 60 A706 rebar and unconfined concrete with f ′ = 3000 psi were assumed in both cases. c

50000

) n i p i k ( t n e m o M

40000 30000 20000 10000 0 0.00e0

5.00e-5

1.00e-4

1.50e-4

2.00e-4

2.50e-4

Curvature (1/in)

Conventional practice is to place a minimum area of distributed rebar along the wall length and to concentrate the reinforcement required for flexural strength at the boundaries (ends) of the wall, that is, Wall 1 above. For rectangular walls, similar strength and deformation capacity can be achieved by distributing the vertical reinforcement along the length of the wall, that is,

•

Flexural strength is not substantially reduced

•

Deformation capacity (measured here in terms of curvature) is only modestly reduced

Further,

•

Resistance to sliding along construction joints may be improved using distributed reinforcement

•

Construction may be simplified with the use of distributed reinforcement

10.12.3 Fracture of Tension Reinforcement Wood (1991) reported that fracture of wall flexural rebar likely contributed to the near collapse of a building during the 1985 Chile earthquake, noting that where the amount of wall rebar is small, strain can concentrate and accumulate at cracked sections under reversed cyclic loading, leading perhaps to rebar fracture. Wood (1991) recommended that the following relationship be met to avoid fracture of tension reinforcement based on results of tests of symmetrically reinforced walls:

CIE 525 Reinforced Concrete Structures ρ f + t y

f c′


P A > 0.15

where ρ is the total area of vertical wall reinforcement divided by the gross cross-sectional area of the t wall, is the rebar yield strength, P is the axial load on the wall, A is the gross cross-sectional area of y the wall, and f c′ is the compressive strength of the concrete. The data studied by Wood showed fracture of the tension rebar at drift ratios exceeding 0.02 (or displacements exceeding 2% of the story height). As such, the above equation should be satisfied whenever drifts are expected to exceed 0.010 to 0.015 (likely large drifts for most structural walls).

10.12.4 Plastic Moment Strength The plastic moment strength of a cantilever wall at its base may significantly exceed its nominal design moment because

•

More vertical rebar are provided that what is required by analysis

•

Expected yield strength or rebar will exceed the nominal yield strength

•

Strain hardening of rebar in the wall boundaries

Unless better information is available, the plastic moment strength of a wall can be estimated assuming the following for the wall boundary rebar

• •

f yexp = 1.25 f ynom

max

= 1.25 f yexp ≅ 1.6 f y

Should the effect of increased concrete strength (due to concrete strength exceeding the design value and some confinement) be considered? Typically, no.

•

A smaller value of concrete strength will lead to a larger value of the depth to the neutral axis and increased requirements for confinement rebar

Once the flexural strength at the base of the wall has been calculated, how should the minimum strengths of wall cross sections above the base be established? Paulay and P riestley (1992) provide guidance on this subject. Consider the cantilever wall shown below of length l w .



The base moment in the above figure is equal to the plastic moment strength of the wall calculated above for the assumed axial load. The shaded portion of the moment diagram shows the moments that would result from the application of the code-based lateral static forces with the plastic moment strength developed at the base of the wall. The straight dashed line represents the minimum flexural strength that should be provided putting aside the effect of diagonal tension. When curtailing vertical bars, the effect of diagonal tension on the internal flexural tension forces (refer back to the previous discussion on shear in reinforced concrete beams) must be c onsidered.

•

Paulay and Priestley recommend that a tension shift of l w be assumed for the curtailment of vertical reinforcement: see the shaded bilinear envelope in the figure above.

•

Bars to be curtailed must be extended a development length beyond (above) the shaded bilinear envelope.

10.12.5 Confinement at Wall Boundaries Confinement is often provided at the boundaries of flexural walls to avoid loss of compressive capacity after spalling of cover concrete under design or maximum earthquake shaking. What is the threshold strain for spalling?

•

Tests of walls, beams, and columns show spalling strains ranging between 0.004 and 0.005

•

Assume a spalling strain of 0.004 for this discussion



Once the spalling strain is selected, how is the need for confinement determined? Consider the figure below from Paulay and Priestley.

•

Confinement must be provided over the length of the wall for which the concrete strain exceeds 0.004.

•

What is the effect of axial load?

The need to confine the wall boundary can be established by calculation of the depth to the neutral axis. Assume for this (crude) calculation that

•

The total displacement at the tip of the wall, δ , is equal to the plastic displacement

•

The length of the plastic hinge at the base of the wall is 0.5l

u

w

Then, the depth to the neutral axis associated with an extreme fiber compression strain of 0.004 can be calculated as follows: δ u ≅ θ p hw δ u

hw

= ϕ p (0.5l w )

ϕ p =

c=

ε c

=

0.004

c l w

500(

δ u

hw

c

)



So, if the calculated depth to the neutral axis exceeds the threshold value calculated above, the extreme fiber strain exceeds 0.004, and confinement of the wall boundary is required. If spalling of the concrete in the wall boundary is anticipated in design or maximum shaking, transverse reinforcement given by equations previously presented for columns in special moment-resisting space frames must be provided:



A g ′ f ′ For spirals, ρ s ≥ 0.45( − 1) c ≥ 0.12 c A f f c



y

For hoop ties, A ≥ 0.3( sh sh

c

f c′ f yh

y

A f ′ g )[ − 1] ≥ 0.09 shc c A f c

yh

The transverse rebar also serves to restrain the longitudinal bars against buckling. Cyclic loading tests have shown that a longitudinal spacing of 6 d is sufficient to achieve a compression strain of 0.02. Such b spacing should be used in boundary elements. To provide reasonable confinement pressures, the maximum spacing should not exceed 6 inches or h/4 in the longitudinal direction and 10 inches in the transverse direction. ACI provide two methods for calculating whether confinement is required in wall boundaries. The first and preferred method involves calculation of the depth to the neutral axis in the wall and a check of the calculated value against the threshold value of

l w

c=

600(

δ u

h

)

w

where δ / h ≥ 0.007 . u

w

The second method involves calculation of a nominal axial stress on the gross section of the wall under the action of gravity and factored code forces. If the calculated stress exceeds 0.2 f ′ , confinement c (special transverse reinforcement) is required. There is little (or no) relationship between these stresses on the gross section and strains in the walls. A study by Wallace and Moehle (1991) demonstrated that this code provision is very conservative and results in confinement for most walls.

10.12.6 Instability of Wall Boundaries A minimum thickness for boundary elements of first story walls has been included in selected seismic design codes but not in ACI 318. A value of 10% of the story height is used in the NZ code based on experimental data. (There is no evidence of flexural walls failing due to out-of-plane buckling in the field.) Paulay and Priestley (1992) present a detailed treatment of the subject and offer deformation-based equations for calculating the minimum thickness of wall elements.



10.12.7 Shear Strength Similar to beams and columns in special moment-resisting frames, walls should be designed for shear strength in excess of that associated with flexural plastic hinging. Design against shear failure in structural walls force requires information on

•

Plastic moment strength of the wall

•

Variations in the distribution of lateral load

•

Wall shear strength

The plastic moment strength of a wall is often substantially greater than the required strength for the reasons cited earlier. The plastic moment strength should be calculated by moment-curvature analysis considering

•

Expected yield strengths of vertical reinforcement

•

The vertical reinforcement placed in the wall boundaries and not that required to resist the code based lateral forces

•

All web reinforcement in the wall regardless of what was assumed for the design

•

Strain hardening of vertical reinforcement

The plastic moment strength may exceed 150% to 200% of the moment calculated under the action of the code lateral forces. The distribution of lateral inertia forces on a building changes continuously during earthquake shaking and may differ substantially from that assumed for design. Shown below are the code-required strength and plastic strength of the wall over its height. What are plausible variations in the lateral forces and what is the effect of the maximum shear force in the wall?



In the above figure, φ is the ratio of the plastic moment strength to the required (code) strength and ω v 0,w is the ratio h / h . 1

2

Bertero recommended that the inverted triangular load profile used for design of flexural reinforcement be replaced by a uniform load (constant acceleration) profile for design for shear. What is the effect of this substitution?

•

For the same flexural strength at the base of the wall, the shear force is increased by a factor of 1.33, that is, ω = 1.33 v

•

Others have proposed values up to 1.8: see Paulay and Priestley for information.

The calculation of the maximum expected shear force can be made using the above information, namely,

Vu = ωvφ 0,wV code where V is the design base shear per the code and all other terms are defined above. code The shear strength of a reinforced concrete wall is typically calculated using equations that are effectively identical to those used for beams except that both the concrete and the rebar contribute to the nominal strength of the wall in the plastic hinge zone. Setting the concrete contribution to the nominal strength equal to zero will produce a more conservative design but this does not appear to be warranted. Experimental studies by Aktan and Bertero (1985) indicated that brittle failure modes such as diagonal compression are possible when the nominal shear stresses are high. To avoid such failure, the maximum nominal shear stress in a wall under maximum expected shear forces (and not those calculated per the code) should not exceed 6 f c′ psi, that is,

V

u

Aw

≤ 6 f c′


10.13


Behavior of Squat Structural Walls

10.13.1 Introduction Squat structural walls with a aspect ratios of less than 1 or 2 are widely used in low- and medium-rise buildings. Such walls can be classed as either

• Elastic walls: walls remain fully elastic in maximum earthquake shaking; often requires large foundations, tension piles, etc

• Ductile walls Standard approaches for predicting the flexural strength of squat walls can be followed because at the maximum strength much of the flexural reinforcement has yielded

•

Violates the plane sections hypothesis

Evenly or uniformly distributed vertical reinforcement is preferred in squat walls

•

Deeper compression zone

•

Improved conditions for sliding resistance (shear friction and dowel action)

Is confinement needed in squat walls?

•

Extreme fiber strains?

How should the design shear force, V u , be calculated:

•

Code forces to determine the design base shear

•

Establish flexural strength corresponding to the design base shear

•

Back-calculate V =

M u

p

h

10.13.2 Modes of Failure in Squat Walls Three modes of failure are considered for squat walls

•

Diagonal tension

•

Diagonal compression

•

Sliding shear

These failure modes are shown in parts (a) through (e) of the figure below from Paulay and Priestley.



Diagonal Tension Failure A corner-to-corner crack such as that shown in part (a) of the figure above is most critical but is unlikely to form. For such a case, the ultimate shear force that can be applied at the top of the wall is given by

h Vu = Ash f y ( w ) sh where A

sh

is the area of horizontal rebar of yield strength f that is vertically spaced at s , where the y

h

height of the wall is hw . A more likely crack is that shown in part (b) of the figure, where the crack forms at a steeper angle to the horizontal: say 45 degrees. In this case, the horizontal rebar need only resist that portion of V u in the shaded region, that is,

Vu (

hw lw

) = Ash f y (

hw sh

)

If a load path is available to transfer the shear force to the remainder of the wall, such as a tie beam as indicated in the above figure, yielding of the horizontal rebar may not result in failure. What happens to the remainder of the shear force, that is, the portion of the shear force in the unshaded region?

•

Load transfer to the foundation by diagonal compression



Diagonal Compression Failure When the average shear stress in the web of a wall is high and adequate horizontal rebar has been provided to prevent diagonal tension failure, the concrete may fail in diagonal compression; see part (c) of the figure above. Reversed cyclic loading may result in the formation of two sets of diagonal shear cracks as shown in part (d) of the figure. Such cracking will substantially reduce the strength of the concrete struts. Limitations on maximum shear stress at the walls flexural strength are needed to avoid compression failure. Consider a single compression strut

Vertical rebar needed to resist the overturning moment on the compression strut

The compression strut can crush if the ultimate shear force exceeds 10

′ A , where A is the crossw w

c

sectional area of the wall

•

Some have recommended that V ≤ 6 f ′ A for ductile walls u

c

w

Sliding Shear Failure As flexural cracks open and close and vertical rebar alternately yields in tension and compression in a squat wall under reversed cyclic loading, a shear plane can develop as shown in part (e) of the figure on the previous page and in the figure from Paulay and Priestley below.

How are shear forces transferred during sliding?

•

Dowel action of the vertical reinforcement along the length of the crack


•


Compression struts in the concrete at the toe of the wall 

Only mobilized after the rebar at the toe of the wall yields in compression

The figures below from Paulay and Priestley show the measured response of a squat wall. In part (a) of the figure, the lateral resistance of the wall is plotted versus the total displacement (including flexural deformation, shear deformation, and slip at the base of the wall). Part (b) of the figure presents the lateral resistance versus the slip at the base of the wall. Much of the total displacement is slip. Part (c) of the figure shows the improvement in response obtained with the addition of diagonal reinforcement that is shown in the figure at the bottom of the page. 

Diagonal reinforcement is not commonly used in US practice

Response following the addition of diagonal reinforcement; see below

Resistance to sliding shear can be provided by

•

Dowel action

•

Shear friction in the compression zone

•

Diagonal reinforcement



fc′ Aw and provide diagonals for 0.3V u



Should be added if V ≥ 7 u



Method for calculation (and affects on diagonal tension and flexural resistances) is presented in Paulay and Priestley

On the basis of test data, Paulay and Priestley recommend the following equation for the resistance due to dowel action

Vdo = 0.25 Asw f y where A sw is the total area of vertical reinforcement in the web of a squat wall. For resistance due to shear friction in the compression zone,

V f = 0.25 f c′ A f

where the effective area of shear friction, A , is shown below. f


10.14


Behavior of Coupled Structural Walls

10.14.1 Introduction Coupled walls resist lateral loads through a combination of

•

Shear and moment at the base of each wall

•

Internal couple due to axial loads developed in the walls

The primary purpose of the beams between the coupled walls during earthquake shaking is to transfer shear from one wall to the other, producing the internal couple identified above, as shown below in the figure from Paulay and Priestley.

For this coupled wall,

•

The shear resistance is Vn = Vn1 + V n 2

•

l is the distance between the centroids of The flexural resistance is M n = M1 + M 2 + Tl , where the walls.

How is the magnitude of the force T controlled?

•

By limiting the strength of the coupling girders 

Limit the strength so that the wall remains in compression if possible



How are the individual walls designed?

•

Using standard procedures for bounded values of axial and flexural loads

How are the moments in the walls determined? The answer is not straightforward.

•

Compression wall is stiffer than the tension wall 

Carries more shear than the tension wall and perhaps most of the shear

In the past coupling beams have been designed as conventional flexural members with stirrups and with some shear resistance assigned to the concrete. As shown in part (a) of the figure below, such a beam may fail in diagonal tension for the reasons discussed earlier in the presentation on squat walls.

If conventionally reinforced for shear using capacity design principles, some deformation can be accommodated. However, after a few cyclic load reversals, the cracks at the wall boundaries may join and a sliding shear failure may result. What are the limits?

•

ln / h < 2 ) then conventional shear reinforcement can be used If V ≤ 4 f ′b d (for

•

How is V u calculated?

u

c w

Under reversed cyclic loading, extremely high bond stresses are developed to accommodate the high rate of change of moment along the short span. Such bars often develop tension over the entire span so the shear is transferred across the coupling beam primarily by a single compression strut as shown in part (c) of the figure above. This observation has led to the use of a rebar layout that utilizes diagonal reinforcement in coupling beams as shown in the figure below.

•

Such diagonal rebar are either in tension or compression over their entire length so bond problems do not arise.

•

Transfer of diagonal tension and compression to the rebar produces very ductile behavior



How is the required area of rebar in the diagonal, A sd , calculated?

A sd =

T b φ

y

=

Q

1

2sin α φ f y

where φ = 0.85 per Section 9.3.4(c), and all other terms were defined previously. Note that this rebar must be anchored or developed in tension in the walls at each end. A development length of 1.5 l d is recommended because of the concentration of anchorage forces in the wall boundaries. The diagonal rebar must be enclosed in ties or spirals as shown in the section at the bottom of the previous page. A nominal amount of conventional shear reinforcement should be placed in the beam to bind the concrete together and minimize the likelihood of falling concrete. Because diagonal compression forces are carried entirely by the confined diagonal bars, no limitations on the maximum shear stress need be imposed, however ACI limits the nominal shear strength of a coupling girder to 10 fc′ Acp .


10.15


Analysis of Squat Walls with Large Openings

Squat walls in low- and medium-rise buildings often contain openings for windows, doors, mechanical penetrations and so on. Documents such as ACI 318 provide little guidance for the analysis and design of such walls. For walls such as that shown below from Paulay and Priestley, strut-and-tie models can be used to provide admissible load paths to the foundation.

Capacity design procedures can aid in the design of such walls. 1. Choose selected tension chords or tie to y ield in the design or maximum earthquake 2. Back-calculate forces in other struts and ties and design these struts and ties to remain elastic for lateral loads sufficient to yield the selected chords and ties of 1.


10.16


Practice of Seismic Analysis and Design of Structural Wall Buildings Per ACI 318

The key steps in the IBC force-based seismic analysis of reinforced concrete structural (shear) wall buildings follow the steps listed above for moment-frame buildings. Values for R and C d from the IBC for reinforced concrete shear wall buildings are presented below. In CIE 525, attention is focused on the special reinforced concrete shear walls in building frames. Values for special reinforced concrete shear walls in other types of Basic Seismic-Force-resisting Systems are listed below (from Table 1617.6 of the 2000 International Building Code):

R

Ω0

C d

5.5

2.5

5

Dual systems with special moment frames

8

2.5

6.5

Dual systems with intermediate moment frames

6

2.5

5

Basic seismic-force-resisting system

Bearing wall system

ACI 318 writes rules regarding the design and detailing of special reinforced concrete structural walls and coupling beams. Some of the key rules are listed below. Web Reinforcement

•

The distributed web reinforcement ratios, ρ v and ρ n (vertical and horizontal rebar, respectively) shall not be less than 0.0025 unless the design shear force, V u , is less than Acv

f c′



For squat walls, hw / l w ≤ 2.0 , ρv ≥ ρ n



Do not include chord or boundary element reinforcement in the calculation of ρ v and ρ n .



Shear reinforcement should be uniformly distributed and at as small a spacing as practical (and economical)

•

Maximum spacing of rebar is 18 inches

•

At least two curtains (layers) of rebar shall be used if the design shear force in the wall exceeds 2 Acv

•

f c′

All continuous rebar must be fully developed (anchorage and splices)



Shear Strength of Walls and Wall Piers

•

The nominal shear strength, V , of a structural wall shall not exceed (Equation 21-7): n

Vn = Acv (α c

f c′ + ρ n f y )

where α is 3.0 for h / l ≤ 1.5 and 2.0 for h / l ≥ 2.0 , and varies linearly for values of the c w w w w aspect ratio between 2 and 3. 

Recognizes the higher shear strength of squat walls



For a rectangular section without openings, Acv is the product of the width and length of the wall, in units of in 2.



For wall piers and the calculation of V n , use a value of the aspect ratio that is the larger of the value for the pier and the value for the entire wall. 



Ensures that a pier is not assigned a unit strength larger than that for the entire wall.

The nominal shear strength of all wall piers resisting a common lateral force (that is, several walls in a given direction or several piers in a punched wall) shall not be assumed to exceed 8 A cv 

Average unit strength for the total cross-sectional area is limited to 8 Acv



Preventing what type of failure?



•

Diagonal tension

•

Diagonal compression

Nominal shear strength for a single pier must not exceed 10 Acp

• •

f c′ f c′

f ′ c

Also applies to coupling girders in coupled shear walls

Check that V ≤ φ V u n 

φ = 0.60 for a wall in which the nominal shear strength is less than the shear corresponding to development of its nominal flexural strength.



Otherwise, see Section §9.3 of ACI-318-02.

Design for Flexure and Axial Loads

•

Design as a beam-column element (by hand calculation or UCFyber/Xtract [or equivalent])

Module 10

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