Structural Concrete - Hassoun

5.10 Shear Stresses in Members of Variable Depth 213
Figure 5.16
Example 5.3: Distribution of stirrups.
Distance left to the free end is 2 in., which is less than 8.0 in., where no stirrups are needed.
Distribution of stirrups is shown in Fig. 5.16. Total number of stirrups is 20.
5.10 SHEAR STRESSES IN MEMBERS OF VARIABLE DEPTH
The shear stress, v, is a function of the effective depth, d; therefore, shear stresses vary along a
reinforced concrete beam with variable depth [10]. In such a beam (Fig. 5.17), consider a small
element dx. The compression force C at any section is equal to the moment divided by its arm, or
C = M/y. The first derivative of C is
ydM − Mdy
dC =
y 2
If C 1 > C 2 , then C 1 − C 2 = dC = vb dx:
ydM − Mdy
vbdx = = dM y 2 y − M y dy 2
v = 1 ( ) dM
− M ( ) dy
yb dx by 2 dx
Because y = jd, dM∕dx is equal to the shearing force V and d( jd)/dx is the slope,
v = V
bjd −
and v = V
bjd ±
M [ ] d
b( jd) 2 dx ( jd)
M (tanα) (5.26)
b( jd) 2
Figure 5.17
Shear stress in beam with variable depth.