Kong and Rangan (1998)

The theory developed by Kong and Rangan (1998) to calculate the shear strength of reinforced concrete beam is based on the stress analysis of the web portion of a beam and is adopted from work by Hsu (1988, 1993) and Vecchio and Collins (1982, 1993).

In this model, the shear response and shear strength of a region of a beam can be evaluated by performing a stress analysis of a cracked concrete element, as shown in Figure 2.4. This element is presented in the form of a strut-and-tie model comprising a concrete strut inclined at an angleθ , tied in place by reinforcing bars in the longitudinal and transverse direction. This concrete strut develops a compressive stress σ_dalong its axis (d-direction) and a tensile stress σ_rin the orthogonal direction (r-direction), which are taken as principal stresses. These stresses can be transformed into longitudinal l- and transverse t-directions using Mohr’s stress circle, and then superimposed on the stresses in the reinforcement.

Figure 2.4: Stress Analysis of a Reinforced Concrete Element (Kong and Rangan, 1998)

The stress analysis of the element can be solved by using equilibrium, strain compatibility and constitutive laws for stress and strain relationships of concrete and steel. The method of analysis is described below.

Equilibrium

The equilibrium equations are:

l

σ , = normal stress in l- and t-directions respectively and are positive for tension

r

d σ

σ , = principal stresses in the d-and r-directions respectively and are positive for tension

p l = smeared longitudinal tensile reinforcement ratio attributed to shear

= ^b_v

(

)

p t = smeared transverse reinforcement ratio

= b s A

v sv

A sv = total area of all legs of vertical stirrups across the width of the beam s = spacing of stirrups along the longitudinal axis of a beam

st sl f

f , = stresses in longitudinal and transverse reinforcement respectively

Strain Compatibility

The principal strain directions are assumed to coincide with the corresponding principal stress directions. The average strains in the l- and t-directions may be related to principal strains by using Mohr’s strain circle, as below:

θ

ε , = average strains in the element in l- and t-directions respectively and are positive for tension

r d ε

ε , = average principal strains in the element in d- and r- directions respectively and are positive for tension

γlt = average shear strain in the element in the l- and t-coordinate system

Stress and Strain Relationships of Concrete

• Softened concrete in compression

The stress and strain curve of softened concrete in compression is adopted from Vecchio and Collins (1993), where the effective compressive strength of a strut in a reinforced concrete element is less than the uniaxial concrete compression strength due to the presence of tensile strains in the perpendicular directions. This softening effect is taken into account by means of a softening factor.

The stress and strain curve of a softened concrete in compression may be described as propose a flat region throughout this range of ε_d):

' c

d ξf

σ =− (2.31)

For ε_d ≤ε_o (the post-peak branch where only stress softening is applied):

)

fc = concrete cylinder compressive strength in MPa '

εo = strain corresponding to the peak concrete compressive stress

= ⎟

E c = modulus of elasticity of concrete (from Carrasquillo et al., 1981) = 3320 f_c^' +6900

ζ = softening factor applicable for all grades of concrete, proposed by Vecchio and Collins (1993)

The stress and strain relationship of concrete in tension is given by Collins et al. (1996) as follows:

cr

ε = concrete cracking strain

=

c cr

E f

cr

f = concrete cracking stress = 0.33 f_c^'

Stress and strain relationship for steel

The stress and strain relationship of longitudinal and transverse steel reinforcement is represented by elasto-plastic curves as follows:

l

f = s E_sε_lwhen ε_l ≤ f_sly/E_s (2.35a)

y

fs_l

= when ε_l > f_sly/E_s (2.35b)

f = st E_sε_twhen ε_t ≤ f_svy/E_s (2.36a)

= f_svywhenε_t > f_svy/E_s (2.36b)

where

y svy

s f

f_l , = yield stresses of the longitudinal and transverse steel reinforcement respectively

E s = modulus of elasticity of steel = 200 x 10³ MPa

Solution

The stress analysis involves thirteen unknowns, which are σ_l, σ_t, σ_d,σ_r , ν_lt, ε_l, , ε_t ε

ε , ,γ , f , f . From the equilibrium, strain compatibility and stress and strain

relationships for concrete and steel, ten equations are obtained. Three more equations are needed to obtain a solution.

The axial force N at a certain region of the beam is assumed to produce a uniform stress on the beam cross section. The intensity of this stress in the web of the beam in the l-direction is equal to N/Ag , where Ag is the gross concrete area of the beam cross section.

This assumption is not entirely true as the stress distribution is non-uniform because of flexural cracks. In the case of a reinforced concrete beam, N/Ag is zero and the accuracy of this assumption does not affect the stress analysis of the beam. Therefore,

Ag

= N

σl (2.37)

As the beam is not subjected to any axial force in the transverse direction, it is assumed that the resultant tensile stress in that direction is zero:

=0

σt (2.38)

In order to trace the load-deformation response of the beam region in terms of average shear stress,ν_lt, and average shear strain,γ_lt, the strainε_dcan be specified for each load stage. This requires the area of the longitudinal tensile steel, A_slV, which resists the shear force, to be defined as below:

M

A s = total longitudinal steel in the tension zone

M

and M is the bending moment co-existing with the shear force V. Also, A_slVis always positive and taken as greater than zero.

For the simplification of the solution process, some of the equations are rearranged as follows:

The longitudinal strain ε_l can be expressed as

s

The transverse strain ε_t can be expressed as

s

The principal concrete tensile strain ε_r is obtained from combining Equations 2.27 and 2.28, which yield

d t

r ε ε ε

ε = _l+ − (2.45)

The angle of inclination of the concrete compressive strut θ is given by

)

For simplicity, the value of A_slMis calculated at the load stage corresponding to the peak of the ν_lt-γ_lt curve, which represents the shear strength Vu of the region. Since V_u is unknown in the beginning, some iteration is required. Initially, a trial value of Vu from the initial stress analysis is selected and A_slV is calculated using Equations 2.39 and 2.40 for a known value of moment to shear ratio, M/V. The stress analysis of the model is then performed to establish the peak of the ν_lt-γ_ltcurve, and hence V_u. Using this new value of Vu, A_slVis calculated and the stress analysis is repeated. The entire process is continued until convergence is reached.