Shear strength of reinforced concrete beams with stirrups

Abstract

This study presents alternative shear strength prediction equations for reinforced concrete (RC) beams with stirrups. The shear strength is composed of the contribution of the nominal shear strength provided by stirrups and the nominal shear strength provided by concrete. For the concrete contribution, cracking shear strength values estimated by Arslan’s equations are almost same those obtained with ACI 318 simplified equation in terms of coefficient of variation (COV). However, mean values estimated by ACI 318 tend to be more conservative comparing to the mean values obtained with Arslan’s equations. Thus, for the consideration of concrete contribution to shear strength, Arslan’s equations are used. To obtain the shear strength of RC beams, shear strength provided by stirrups is added to the concrete shear strength estimated by Arslan’s equations. Results of existing 339 beam shear tests are used to investigate how accurate proposed equation estimates the shear strength of RC beams. Furthermore, ACI 318 and TS500 provisions are also compared to the aforementioned test results. It is found that proposed equations for beams with shear span to depth ratios (a/d) between 1.5 and 2.5 are also conservative with a lower COV than ACI 318 and TS500. However, when a/d ratios exceed 2.5 (both normal and high strength concrete beams), ACI 318, TS500 and proposed equations give similar COV value.

Appendix—Estimation of concrete contribution to shear strength

In determination of concrete contribution to shear strength, cracking shear strength (v _cr,t) and dowel actions (v _cr,d) are considered [2, 3]. According to Khuntia and Stojadinovic [53], the shear stress distribution is modeled as parabolic over the effective shear depth with the maximum value at the neutral axis. Thus, the magnitude of shear resistance over the effective cross section equals τ_max = f _t = V _cr,t /(2/3b _w k ₁ d), where b _w is the width of section, τ _max is the shear stress at the neutral axis, f _t is the tensile strength of concrete, V _cr,t is diagonal tension cracking shear force and k ₁ d is the effective shear depth, respectively. The effective shear depth can be taken as k ₁ d = kd(1 + ɛ _cr /ɛ _c ), where kd is the depth of neutral axis, ɛ _c is the compressive strain in concrete and taken as 0.002. The cracking strain value in concrete is taken as ${\varepsilon _{cr} =f_t /E_c}$ , where E _c is modulus of elasticity and equal to ${4750\sqrt {f_c}}$ in MPa [54]. The tensile strength of plain concrete f _t , ranges from about 0.25 to 0.50 ${\sqrt {f_c}}$ [55–57]. The direct tensile strength is accepted as ${0.50\sqrt {f_c}}$ for normal strength concrete (NSC) and ${0.40\sqrt {f_c}}$ for high strength concrete (HSC). According to Kim and Park [11], k values can be expressed as follows, k = 0.82(nρ)^0.36. Within the practical range, i.e., ${5\leq n\leq 10}$ and ${0.005\leq \rho \leq 0.035}$; consequently, ${0.025\leq n\rho \leq 0.35}$. During the formation of primary cracks, for a reinforcing ratio ρ less than a limiting value ρ _stbl , the average strains increase until a stabilized cracking state is reached tension-softening stress in concrete at cracking. If ρ is the reinforcement ratio, and n = E _s /E _c is the modular ratio, the minimum reinforcement ratio required to maintain constant strain at the crack when the cracking load is applied to the member and held constant, is ρ _stbl = 1/(6n) [58]. According to Massicotte et al. [58], this equation can also be interpreted as the minimum steel ratio needed for a test setup to measure accurately the tension-softening branch in a plain concrete tension test under a load-controlled procedure. Since ρ _stbl is expressed by the minimum reinforcement ratio required to maintain constant strain at the crack, the corresponding limit value for the shear strength capacity of the diagonal tension crack of slender beams can also be interpreted as ρ _stbl equal to ρ. Substituting these equations into ${v_{cr,t} =\frac{2}{3}f_t k_1}$, we obtain ${v_{cr,t} =0.15\sqrt {f_c}}$ (For NSC) and ${v_{cr,t} =0.12\sqrt {f_c}}$ (For HSC).

Substituting ρ _stbl = 1/(6n) and n = E _s /E _c into ${V_{cr,d} =k_3 (f_c )^{0.5}\rho ^rb_w d}$ [11] and assuming that the modulus of elasticity of reinforcement is E _s  = 2.10⁵ MPa and r = 0.3, the dowel strength can be expressed approximately by more general terms f _c as follows v _cr,d = 0.02(f _c )^0.65. The shear strength of a diagonal tension crack of slender beams may be expressed as Eq. (3).