Flexural members - Reinforced Concrete with FRP Bars: Mechanics and Design - page 125

COMMENTARY

In their analysis, Tureyen and Frosch [34] considered a portion of a flexural
member subjected to constant shear and computed the principal stress of an
infinitesimal concrete element located in the compression zone where shear
failure may initiate. This uncracked concrete element is subjected to shear
and axial compression stresses.
When failure occurs, the principal tensile stress reaches the tensile strength
of the concrete. Using the Mohr's circle, the following can be written:

2

σ σ

(4.89)

2

f

−τ+ =−

ct

2 2

Solving for τ , the expression of the tensile stress at failure can be obtained:

+σ

(4.90)

f

2

f

τ=

ct ct

By considering the equilibrium of a member strip of infinitesimal width
chosen at a flexural crack location, the shear capacity of the section due to
the concrete itself was written in the following fashion:

2
3

σ

2 m

(4.91)

V b cf f

= +

c w ct ct

2

In fact, analyzing the free-body diagram of the member strip at a crack
location, Tureyen and Frosch [34] considered the shear stresses generated
from the flexural stresses due to Δ M = V ⋅Δ x and noted that the maximum
shear stress, achieved at the mid-depth of the compression zone, is equal to

3
2

V
bc

(4.92)

ma τ=

w

Substituting τ max = τ and σ = σ m /2 in Equation (4.90) and solving for V, Equation
(4.91) was obtained.
The design Equation (4.89) was then arrived at after substituting f

6 f =

′

ct c

in Equation (4.91), factoring out f c ′ , and rearranging terms:

4
3

σ m

(4.93)

V 16 fbcKfbc

′

= + ′ =

c cw cw

f

′

c