Example 3: Flexural stress calculation using transformed area method for rectangular beam

Assume the section for beam shown in figure 1 cracked calculate the flexural stresses for the given load or moment using transformed area method.

Figure 1
Transform area=n*As
As=4*.79=3.16 in2
Transform area=3.16*10=31.6in2
 
now we will determine the location of the neutral axis as shown in figure 2
 
12*X*(X/2)=(17.5-X)*31.6
6*X^2+31.6X-553=0
X=7.32 in
 
moment of inertia using parallel axis theorem 
I=Ic+Ad^2
I=(12*7.32^3)/12+12*7.32*3.66^2+31.6*10.18^2=4843.67in4
=0.234ft4
 
now we will calculate the moment applied to beam
Maximum moment for simply supported beam
M=(W*L^2)/8
=(1.5*24^2)/8
=108K-ft
 
now we will calculate flexural stress at extreme compression fiber. y=7.32 in =0.61ft
 
fc=M*y/I
=108*0.61/0.234
=281.538k/ft2=1955.125psi
 
 
now we will calculate flexural stress at center of reinforcing steel. y=10.18 in =0.85ft
 
fs=nM*y/I
=10*108*0.85/0.234
=3923.076k/ft2
=27243.58psi
Figure 2