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Example 4:Flexural stress calculation using transformed area method



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=As*n
As=4*1=4in2
Transform area=4*9=36in2

location of the neutral axis shall be calculated by equalling moment of area from both parts as shown in figure 2

X(15)(X/2)-2*(4*5)*(X-2)=36*(27-X)
7.5X^2-40X+80=927-36X
7.5X^2-4X-847=0
X=10.9 in

now we will calculate the moment of inertia using parallel axis theorem 
I=(15*10.9^3)/12+10.9*15*5.45^2-2*((5*4^3)/12+4*5*8.9^2)+36*16.1^2
=12584.97in4=0.606ft4

flexural stress at extreme compression fiber where y=10.9in=0.908ft

fc=M.y/I
=70*0.908/.606
=104.92k/ft2=728.61psi

flexural stress at the center of reinforcing bars where y=16.1in=1.34ft

fs=n*M.y/I
=9*70*1.34/.606
=1393.06k/ft2=9674psi



Figure 2






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