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Concurrent effects of the shear-lag and warping torsion on the performance of non-rectangular RC shear walls

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Abstract

This paper aimed at analytically investigating the simultaneous effects of the shear-lag and warping torsion on the performance of non-rectangular reinforced concrete (RC) shear walls. Under the concurrent action of shear and axial loadings, the induced warping deformation due to the shear-lag as well as the warping torsion has been accounted for in the elastic region. On the strength of the minimum potential energy principle, a general formulation has been derived for the stress distribution of non-rectangular RC shear walls. By introducing the appropriate geometrical assumptions, the established formulations have then been re-written for conventional T-, U-, and L-shapes RC shear walls. The veracity of the results is ascertained through a comparative study employing finite element simulations for a U-shaped wall, and good agreement has been achieved to an extent that the proposed analytical formulation is capable to, respectively, predict the axial deformation and stress distribution with an accuracy of 95 and 90%. Also, the findings for the U-shaped wall indicate that the shear-lag can significantly affect the axial stress distribution and cracking load, and neglecting the influence of this phenomenon can lead to an inaccurate and a non-conservative design. Moreover, the contribution of the shear-lag and warping torsion has separately been highlighted for the U-shaped RC wall considered in this study.

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Data availability

Data supporting this study are included within the article and supporting materials.

Abbreviations

α x, α y :

Dimensionless coefficients in warping deformation resulting from the shear-lag in x- and y-directions

β x, β y :

Constants in warping deformation resulting from the shear-lag in x- and y-directions

γ t :

Product of αx and bt

ε 50 :

Strain corresponding to the stress of \(0.50\,f^{\prime}_{{\text{c}}}\) after attaining the maximum compressive strength of concrete

ε sh :

Strain at the onset of hardening stage

ε su :

Strain corresponding to the ultimate strength of steel reinforcement

ε sy :

Strain corresponding to the yield stress of steel reinforcement

ε t :

Strain corresponding to the ultimate tensile strength of concrete

ε tu :

Ultimate tensile strain of concrete

λ b, λ c, λ t :

Constants corresponding to the shear-lag effects in x- and y-directions, and torsional moment

μ s :

The ratio of the maximum tensile stress with shear-lag and warping torsion to the maximum tensile stress of the Bernoulli–Euler assumption

μ sb, μ sc, μ sd :

Contribution of the shear-lag in x- and y-directions, and warping torsion to the additional tensile stress

ν :

Poisson’s ratio of concrete

П:

Total potential energy function

Пb, Пc :

Total potential energy function corresponding to the lateral loads in x- and y-directions

σ :

Axial stress distribution function of RC shear wall

σ a, σ b, σ c, σ d :

Axial stress distribution function under the axial load, lateral load in x- and y-directions, and torsional moment

\(\overline{\sigma }\) :

Stress distribution function with the Bernoulli–Euler assumption

\(\overline{\sigma }_{\text{b}} ,\overline{\sigma }_{\text{c}}\) :

Stress distribution function with the Bernoulli–Euler assumption in x- and y-directions

τ :

Shear stress distribution function of RC shear wall

ϕ :

Torsional moment-induced rotation angle

ψ :

Warping function

A :

Cross-sectional area of the wall

B :

Bi-moment

b fbL, b fbR, b ftL, b flR :

Widths of the left-side bottom flange, right-side bottom flange, left-side top flange, and right-side top flange

b t :

Horizontal component of the centroid coordinate of the wall in (\(\hat{x},\hat{y}\)) coordinate

C w :

Warping constant

d c :

Confined length of the section

E :

Equivalent modulus of elasticity of the wall

e c :

Clear cover

e x, e y :

Eccentricity of the lateral loads in y- and x-directions from the shear center

f u :

Ultimate stress of steel reinforcement

f y :

Yield stress of steel reinforcement

\(f^{\prime}_{{\text{c}}}\) :

Compressive strength of concrete

\(f^{\prime}_{{\text{t}}}\) :

Tensile strength of concrete

G :

Shear modulus of the wall structure

H :

Height of the wall

h :

Shear wall section height

h b, h t :

Centroid distance from the center of the bottom and top flanges

I eb, I wb, I ec, I wc :

Shear-lag constants in x- and y-directions

I x, I y :

Moment of inertia about x- and y-axes

Ῑ x, Ῑ y :

Moment of inertia about \(\overline{x}\)- and \(\overline{y}\)-axes

J :

Torsional moment of inertia

N :

Applied axial loading

\(M_{{\overline{x}}} ,M_{{\overline{y}}}\) :

Moment of the section about \(\overline{x}\) - and \(\overline{y}\) -axes

q :

Axial stress of RC shear wall

s x, s y :

Shear-lag induced additional lateral deformation in x- and y-directions

t fb t ft, t w :

Thickness of the bottom flange, top flange, and web

U fb U ft, U w :

Strain energy functions corresponding to the bottom flange, top flange, and web of the section

u :

Axial deformation function of RC shear wall

u a, u b, u c, u d :

Axial deformation function induced by axial load, lateral load in x- and y-directions, and torsional moment

u wb, u wc :

Shear-lag constants in x- and y-directions

V L :

Potential energy due to the external loads

V Lb, V Lc :

Potential energy due to the external loadings in x- and y-directions

V x, V y :

Lateral loadings in x- and y-directions

V x (max), V y (max) :

Maximum calculated lateral load in x- and y-directions before cracking, based on the Bernoulli–Euler assumption

\(V^{\prime}_{x(\max )} ,V^{\prime}_{y(\max )}\) :

Maximum calculated lateral load in x- and y-directions before cracking due to the shear-lag and warping torsion

w x w y :

Lateral deformation of the wall in x- and y-directions before cracking, based on the Bernoulli–Euler assumption

(x c,  y c):

Shear center coordinate of the section

(x G, y G):

Centroid coordinate of the section

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Acknowledgements

The authors highly appreciate the supports provided by the Center of Excellence in Composite Structures and Seismic Strengthening (CECSSS) at Sharif University of Technology (SUT).

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  1. Department of Civil Engineering, Sharif University of Technology, Tehran, Iran

    Mohammad Tabiee, Hatef Abdoos & Alireza Khaloo

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  1. Mohammad Tabiee
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Correspondence to Alireza Khaloo.

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Appendix

Appendix

The geometrical properties of a flanged RC shear wall with an arbitrary section are listed in Table 4. Moreover, as RC shear walls with T-, U-, and L-shaped sections are more commonly utilized in comparison with other non-rectangular sections, Tables 5, 6, 7 respectively summarize the coefficients required for computation of the axial deformation and stress distribution of the above-described sections.

Table 4 Coefficients of a general flanged section
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Table 5 Coefficients of T-shaped sections
Full size table
Table 6 Coefficients of U-shaped sections
Full size table
Table 7 Coefficients of L-shaped sections
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Tabiee, M., Abdoos, H. & Khaloo, A. Concurrent effects of the shear-lag and warping torsion on the performance of non-rectangular RC shear walls. Archiv.Civ.Mech.Eng 23, 138 (2023). https://doi.org/10.1007/s43452-023-00663-1

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