Simplified Assessment of Bending Moment Capacity

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SIMPLIFIED ASSESSMENT ASSE SSMENT OF BENDING B ENDING MOMENT CAPACITY CAPACITY FOR RC MEMBERS WITH CIRCULAR CIRCULAR CROSS-SECTION CROS S-SECTION Article

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Giuseppe Maddaloni

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Parthenope University of Naples

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Cosenza, Galasso and Maddaloni

3rd fib International Congress - 2010

SIMPLIFIED ASSESSMENT OF BENDING MOMENT CAPACITY FOR RC MEMBERS WITH CIRCULAR CROSS-SECTION Edoardo Cosenza, Dept. of Structural Engineering, University of Naples Federico II, Naples, Italy Carmine Galasso, Dept. of Structural Engineering, University of Naples Federico II, Naples, Italy Giuseppe Maddaloni, Dept. of Technology, University of Naples Parthenope, Naples, Italy

ABSTRACT Reinforced concrete (RC) members with circular cross-section are widely used in structural and geotechnical engineering (e.g. columns in frame structures, foundation piles, contiguous pile walls). Generally, for such members, the analysis is more complex than for rectangular cross-section and the problem is not sufficiently investigated in literature. Circular shape and uniform distribution of reinforcement along the perimeter cause some difficulties for a simple assessment of bending moment capacity. In this study, for RC members with circular cross-section, simplified methods for the evaluations of bending moment of resistance are presented. The performed analyses demonstrate that the design value of moment capacity, determined by the proposed approach, is very close to the results obtained applying rigorous methods. Furthermore, analysis results prove that in bending condition without axial load, the flexural strength depends, on the geometry of the section (i.e. radius and concrete cover) and on mechanical ratio of steel reinforcement by a very simple formula.

Keywords: Circular cross-section, Flexure capacity, Simplified formulae.

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INTRODUCTION

Reinforced concrete (RC) columns of circular cross-section are widely used in the building of framed structure (e.g. in high rise buildings, Fig. 1) or in bridges. Circular cross-section members are frequently used also in geotechnical engineering, e.g. in pile foundation systems. Columns are basically axial load carrying element; however, as a result of lateral load due to wind pressure or seismic excitation, they are subjected to considerable shear and bending load. For a circular RC cross-section with symmetric arrangement of steel, the loading axis always coincides with a symmetry axis of the element cross-section and then the member is always subjected to uniaxial bending (i.e. no biaxial bending arises) and compression. It is nevertheless true that analysis of such members is more complicated than that of rectangular cross-section members. Circular shape of the cross-section and uniform distribution of reinforcement along the perimeter of cross-section create some peculiarities in assessment of state of stresses and deformations and hence in assessment of carrying capacity of such members. In the engineering literature this problem is not 1,2 enough investigated. Some studies on rectangular and circular concrete cross-sections analysis/design focused on integration methods using analytical and numerical algorithms. In this case a computer program implementing these procedures is necessary and a quick “handmade” calculation is not possible. Furthermore, the majority of design codes do not distinguish distinctly between the design of rectangular section and the one of a circular section.

Fig. 1 Examples of RC columns of circular cross-section at School of Engineering of University of Naples. The objective of this paper is to present a quite simple method for analysis of RC circular section. The developed equations are based on the assumption that the entire steel area is lumped into equivalent steel ring. This assumption simplifies the calculations but it affects the results of the analysis. An example for the verification of the results of the proposed formulae indicates a very good approximation of the values obtained with other methods widely used in practice. Then the proposed approach may be considered simple and more straightforward for professional engineers.

2



RC MEMBERS ULTIMATE FLEXURE CAPACITY

*

On 14 January 2008 the Italian Minister for Infrastructures signed a Decree containing the 3,4 new Building Code (NTC), published on the Official Gazette no. 29 (4 February 2008) . Sec. 4.1.2.1.2 of NTC gives principles and rules for the evaluation of the RC member flexure capacity (with or without axial force). When determining the ultimate moment resistance of RC cross-sections, the following 5 assumptions are made, consistently with Eurocode 2 (EC2) Sec. 6.1 : 1. plane cross-sections remain plane after deformation up to failure; 2. the strain in bonded reinforcement, whether in tension or in compression, is the same as that in the surrounding concrete (i.e. perfect bond exists between steel and concrete); 3. the tensile strength of the concrete is neglected; 4. the stresses in the concrete in compression are derived from the design stress/strain relationship given in Sec. 4.1.2.1.2.2 of NTC (in Sec. 3.1.7 of EC2); 5. the stresses in the reinforcing are derived from the design curves given in Sec. 4.1.2.1.2.3 of NTC (in Sec. 3.2.7 of EC2); f yk f 6. design strength for concrete and steel are defined as f cd = 0.85 ck and f yd = γ s γ c respectively (Secc. 4.1.2.1.1.2 and 4.1.2.1.1 3 of NTC and Secc. 3.1.6 and 3.2.7 of EC2), where f ck is specified compressive strength of concrete (cylinder strength) and f yk is specified yield stress of steel,

Eurocode-like LRDF; 7. material safety factor are takes as

c

c

and

s

are material safety factor according to

= 1.5 for concrete and

s

= 1.15 for steel (Secc.

4.1.2.1.1.2 and 4.1.2.1.1 3 of NTC and Sec. 2.4.2.4 of EC2).

For the analysis and design of cross-sections at Ultimate Limit State (ULS), simplified stress-strain relationships may be used. For instance, in order to simplify the calculations, the axial force and bending moment analysis usually idealizes the stress-strain behavior of the concrete with a rectangular stress block with a depth equal to some fraction of the neutral axis depth and a magnitude equal to some fraction of the concrete compressive design strength. According to EC2, if the width of the compression zone decreases in the direction of the extreme compression fiber the value of the effective strength should be reduced by 10% . More detailed moment curvature analysis may be performed with more complex stressstrain relationships (e.g. parabola-rectangle diagram). For a given section, the position x of the neutral axis (from the extreme compression fiber) is calculated based on the force equilibrium as in Eq. 1. N c + N s ' − N s = N Ed

(1)

N c and N s ' represent compressive forces in concrete and steel reinforcement of the

compression zone, respectively, N s represents the tensile force in reinforcement of the tension zone; N Ed is design value of the applied axial force (compression). *

In the rest of the article, all calls and verbatim citations of NTC and EC2 will be simply indicated in italic.

3



When assuming a simplified elasto-idealplastic stress-strain diagram for reinforcing steel (i.e. with a horizontal top branch without a strain limit), the flexural failure occurs due to concrete crushing. Any strain diagram corresponding to such failure mode has its fixed point at the limit value of ε cu (maximum concrete compressive strain) with a linear strain 6

distribution over the depth of the section . The member flexural capacity (i.e. the design value of moment of resistance) M Rd can be determinated by summing the moment due to internal (and external) forces about the axis through the center of cross-section of the member. CIRCULAR CROSS-SECTION ANALYSIS In this section equations for computing the flexure capacity of circular cross-section with longitudinal steel bar arranged (equally spaced) in a circle of radius r are presented ( r = R − c , where R is the radius of the circular cross-section and c is the concrete cover). Note that the equations of equilibrium are complex because of the cross-section type and the discrete position of bars, i.e. utilization of reinforcement strength depends on its location in the cross-section 7. The compressive force in concrete is given in Eq. 2 using the formula of area for a circular segment (Fig. 2); the relationship between concrete compressive stress distribution and concrete strain is assumed to be rectangular. The factor λ , defining the effective height of the compression zone and the factor η, defining the effective strength, are assumed equal to 0.8 and 0.9 respectively (according to NTC and EC2). N c =

R 2

2

(2θ − sin 2θ ) f cd

(2)

θ is one half of the angle subtended at the center of the cross-section by the concrete compression stress block, f cd is the design value of concrete compressive strength. The compressive force in concrete is applied at a distance d c from center of crosssection, see Fig. 2. R − 0.8 x θ = arccos x 2 R (2θ − sin 2θ ) Ac = 2 4 sin 3 θ d c = R 3 2θ − sin 2θ

Fig. 2 Circular segment formulae. The compressive and tensile forces in reinforcement are given in Eq. 3 and 4; the bars with the same ordinate from the neutral axis are subject to the same stresses (Fig. 3). nc

N s ' =

∑ n A i

s ,i

f s ,i

(3)

1

4



nt

N s =

∑ n A i

s ,i

f s ,i

(4)

1

As,i is the cross sectional area of each reinforcing bar, ni is the number of reinforcing bar in each row parallel to the neutral axis, nc is the number of rows of reinforcing bars in the compression zone, nt is the number of rows of reinforcing bars in the tension zone, f s,i is the stress in each row parallel to the neutral axis. Assuming that the maximum concrete compressive strain ε cu is reached, the assumption

of plane cross-section after deflection allows the calculation of the strain in any row of reinforcing bars ε s , i as in Eq. 5, where yi is the vertical distance of the i-th row of reinforcing bars from the neutral axis. ε s ,i = ε cu

y i x

(5)

The determination of location of the neutral axis (i.e. x or, equivalently, θ) is carried out by iteration methods.

Fig. 3 Diagrams for analysis of circular cross-section. The design flexural capacity M Rd is equal to the sum of the design flexural strength due to concrete and the design flexural strength due to steel.

PROPOSED METHOD

In the proposed method longitudinal bar arrangements are replaced with a thin steel ring equivalent to steel total area ( As ) , Fig. 4. This assumption does not account for vertical bar location with respect to the neutral axis affecting the r esults of the analysis. Moreover, the diagram of concrete and reinforcement strength are superseded by rectangular ones with effective strength equal to f cd ' = 0.9 f cd and f yd respectively, as in Fig. 4. The value of θ defining the compressive part of the cross-section may be determined from the condition of equilibrium that the sum of all (internal and external) forces is equal to zero, Eq. 6.

5

Cosenza, Galasso and Maddaloni R 2

2


θ   π − θ  (2θ − sin 2θ ) f cd ' +    As f yd −   As f yd = N Ed

 π 

 π 

(6)

Fig. 4 Diagrams for analysis of circular cross-section by proposed method.

 θ   π − θ   As and   As are approximately cross-sectional areas of longitudinal π π    

In Eq. 6, 

reinforcement in compression and tension correspondingly (the value of angle defining the compressive part of reinforcement in the cross section, θ s, does not coincide with θ, see Fig. 5).

Fig. 5 Difference between θ and θ s. Dividing each terms by

2 R 2 f cd '

, Eq. 6 may be formulated as in Eq. 7.

(2θ − sin 2θ ) + 2ωθ − 2ω (π − θ ) = 2νπ In Eq. 7 ω =

As f yd

π R 2 f cd '

and ν =

N Ed 1

π R 2 f cd '

(7)

are the dimensionless parameter of cross-

section, i.e. mechanical steel ratio and design axial force normalized to the total crosssectional concrete area of the member. Setting ϕ = 2θ , condition of equilibrium can be rewritten as in Eq. 8. f (ϕ ) = 0 ⇔ ϕ (1 + 2ω ) − sin ϕ − 2π (ω + ν ) = 0

6

(8)


3rd fib International Congress - 2010 †

The solution of Eq. 8 may be found by Newton's method since an analytical expression for the derivative of f (ϕ ) may be easily obtainable, Eq. (9). f ' (ϕ ) = (1 + 2ω ) − cos ϕ

(9)

Flexural capacity of circular cross-sections is determined from condition of equilibrium that the sum of the moments due to external and internal forces about the axis t hrough the center of cross-section of the member as in Eq. 10. 4 2 M Rd = M Rd ,c + M Rd , s = R 3 sin 3 θ f cd ' + ( R − c ) As sinθ f yd π 3

(10)

ILLUSTRATIVE EXAMPLE

The values of ultimate bending moment capacity of a circular cross-section of radius 30 cm, concrete cover of 4 cm, for reinforcement ratio (ρ) between 0.01 and 0.04 and for ν values between 0 and 0.5 are given in Table 1 using the presented approaches; concrete and steel are characterized by f ck = 25 MPa and f yk = 450 MPa respectively. In Table 1, M Rd 1 is the value of flexural capacity of cross-section computed by Biaxial software (freely available at the website of the Italian network of earthquake engineering university labs http://www.reluis.it/index_eng.html) using parabola-rectangle diagram for concrete under compression and elasto-idealplastic stress-strain diagram for reinforcing steel; the software, based on the well-known fiber method, allows to account for the effective vertical bar location with respect to the neutral axis. M Rd 2 is the value of flexural capacity of cross-section computed using stress-block diagram for concrete under compression and elasto-idealplastic stress-strain diagram for reinforcing steel ( accounting for the effective vertical bar location with respect to the neutral axis). M Rd 3 is the value of flexural capacity of cross-section computed by proposed method. Table 1 reveals that the design value of load carrying capacity of eccentrically compressed RC members of circular cross-section determined by the proposed method is very close to that determined using more rigorous methods of analysis. The comparison shows that the average ratio between design load carrying capacity of eccentrically compressed RC members of circular cross-section determined by the proposed method ( M Rd 3 ) and the rigorous ones ( M Rd 1 ) is 1.031 with a coefficient of variation (CoV, the ratio of the standard deviation to the mean) of 1.4%; the maximum M Rd 3 value of is 1.058. M Rd 1

†

According to Newton's method, given a function ƒ( x) and its derivative ƒ '( x), we begin with a first guess

x0 (e.g.,

ϕo = π is a reasonable value for the first guess of the Newton’s method for the case on

examination). A better approximation x1 is x1

= x0 −

f ( x0 ) f ' ( x 0 )

. Newton's method is perhaps the best

known method for finding successively better approximations to the zeroes (or roots) of a real-valued function. It can often converge remarkably quickly.

7



Table 1 also shows a conservative good approximation of M Rd 1 using a rectangular stress distribution for concrete under compression: the minimum value of

M Rd 2 M Rd 1

is 0.953 (mean

0.984, CoV 1 %) if the value of the effective strength of stress block is reduced by 10% according to EC2 prescription. Note that, because of errors in the realization phase, the location of steel bars in the cross section cannot perfectly match the distribution used in the calculations. To take into account this problem, 100 possible configurations of bar arrangement have been simulated using the Monte Carlo method. The simulated cross-sections have been obtained by varying the angular position of bar according to a Normal probability model centered on the design value ("true", α = 0) with a standard deviation of π /10, as in Fig. 6.

Fig. 6 Simulation of eccentricity in bars arrangement. It is interesting to note that the value of flexure capacity of cross-section obtained by proposed methods practically coincides with the mean value of flexure capacity values obtained by rigorous method for the 100 simulated cases (e.g. Fig. 7).

Fig. 7 Examples of M Rd distribution accounting for eccentricity in bars arrangement (data: R = 30 cm, c = 4 cm, ρ = 1%, ν = 0, f ck = 25 MPa and f yk = 450 MPa ).

8



Table 1 Results of illustrative example. ν ν

ρ 1% 2% 3% 4% 1% 2% 3% 4% 1% 2% 3% 4% 1% 2% 3% 4% 1% 2% 3% 4% 1% 2% 3% 4%

0

0.1

0.2

0.3

0.4

0.5

MRd1 [kNm] 278 462 680 848

MRd2 [kNm] 276 459 676 844

330 505 715 875 375 539 737 895 404 560 754 908 418 571 762 913 426 573 760 909

327 502 708 868 370 530 729 886 393 551 743 899 406 557 748 899 406 555 742 893

MRd2 /MRd1 0.993 0.994 0.994 0.995 0.991 0.994 0.990 0.992 0.987 0.983 0.989 0.990 0.973 0.984 0.985 0.990 0.971 0.975 0.982 0.985 0.953 0.969 0.976 0.982

MRd3 [kNm] 279 472 699 876

335 516 735 907 378 551 763 931 409 576 783 948 427 591 795 958 433 596 799 962

MRd3 /MRd1 1.004 1.022 1.028 1.033 1.015 1.022 1.028 1.037 1.008 1.022 1.035 1.040 1.012 1.029 1.038 1.044 1.022 1.035 1.043 1.049 1.016 1.040 1.051 1.058

BENDING WITHOUT AXIAL FORCE

The circular shape is not widely used in members subject to simple bending but it may be encountered in special cases, like contiguous pile walls. For this case, analysis results prove that the flexural strength depends, on the geometry of the section (i.e. radius and concrete cover) and on mechanical ratio of steel reinforcement by a very simple formula, Eq. 11. A M Rd = kM Rd , max = k s f yd (2 R − 2c ) = k As f yd r 2 As

(11)

f yd (2 R − 2c ) = As f yd r is the flexure capacity when concrete is neglected 2 and the longitudinal bars arrangement is replaced with two steel points equivalent to steel total area ( As ) , as in Fig. 8.

M Rd , max =

9



Fig. 8 Schematic of proposed method for bending without axial force.

Fig. 9 Regressions for k coefficient of Eq. 11. The coefficient k depends on mechanical ratio of steel reinforcement and on

c R

ratio as in

Fig. 8. Figure refers to a circular cross-section of radius 30 cm with different values of

c

R ratio (0.10, 0.15 and 0.20) and with five different steel ratio ( 0.4%, 1%, 2%, 3% and 4%); concrete and steel are characterized by f ck = 25 MPa and f yk = 450 MPa

respectively. Using a non-linear regression, the observational data may be modeled by the function of Eq. 12.  

− 0.3

k = 0.75 ω

c R

 

+ 0.08 

(12)

The curves of Fig. 8 show a low sensitivity of k to

c

c

ratio is neglected as R R explanatory variable, the very simple regression function of Eq. 13 may be used.

10

ratio; if the



k = 0.75 ω −0.13

(13)

The comparison of actual and predicted values of k coefficient has indicated that the maximum deviation is below 1% using both Eq. 12 and Eq. 13.

CONCLUSIONS

Generally, the columns of RC buildings or bridges are subjected to axial load and uniaxial or biaxial bending moments as a result of their geometry, the shape of the cross-section and the type of external actions (e.g. wind and seismic forces). For this type of structures the cross-section are typically rectangular or circular. Generally, for RC members of circular cross-section, the structural analysis (and design) is more complex than for rectangular cross-section member and the problem is not sufficiently investigated in literature. Circular shape and uniform distribution of reinforcement along the perimeter cause some difficulties for a simple assessment of bending moment capacity. In this study, simple formulae were proposed for the ultimate analysis and design of circular cross-section subjected to axial loads combined with uniaxial bending. An extensive example has been carried out to determine the degree of accuracy of the proposed design formulae. The results obtained for a wide range of design case (corresponding to the most frequently used in practice) have shown a very good approximation of the values computed by more rigorous methods. Furthermore, analysis results prove that in bending condition without axial load, the flexural strength depends, on the geometry of the section (i.e. radius and concrete cover) and on mechanical ratio of steel reinforcement by a very simple formula. Acknowledgements

The authors thank Eng. Antonio Naclerio for the precious collaboration on the development of numerical examples. REFERENCES

1.

2. 3. 4. 5. 6. 7.

Bonet, J. L., Barros, M. H. F. M., Romero, M. L. (2006). Comparative study of analytical and numerical algorithms for designing reinforced concrete section under biaxial bending. Computers & Structures, 84, pp. 2184-2193. Davalath, G. S. R., Madugula, M. K. S. (1988). Analysis/Design of Reinforced Concrete Circular Cross Sections . ACI Structural Journal, 85(6). CS.LL.PP. (2008). DM 14 Gennaio, Norme tecniche per le costruzioni. Gazzetta Ufficiale della Repubblica Italiana 29 ( in Italian). CS.LL.PP. (2009). Istruzioni per l’applicazione delle norme tecniche delle costruzioni. Gazzetta Ufficiale della Repubblica Italiana 47 ( in Italian). CEN, European Committee for Standardisation (2004). Eurocode 2: Design of concrete structures. Part 1-1: General rules and rules for buildings. Cosenza, E., Manfredi, G., Pecce, M. (2008). Strutture in cemento armato. Basi della progettazione. Hoepli (in Italian). Ghersi, A. (2005). Il cemento armato. Dalle tensioni ammissibili agli stati limite: un approccio unitario. Dario Flaccovio editore ( in Italian).

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Simplified Assessment of Bending Moment Capacity

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