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Example 6:flexural stress for U shape beam using transform area method

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determine the flexural stresses for the beam shown in figure 1 using the transformed-area method.
Figure 1






Transform area for each layer=n*As
As=2*0.79=1.58in2
Transform area=8*1.58=12.64in2

we will determine neutral axis location by equalling moment of area from both parts as shown in figure 2
2*X(5)(X/2)+3*28*(X-1.5)=12.64*(15.5-X)+12.64*(17.5-X)
5X^2+84X-126-195.92+12.64X-221.2+12.64X=0
5X^2+109.28X-543.12=0
X=4.17in
Figure 2         

moment of inertia using parallel axis theorem 
I=Ic+Ad^2
=2*((5*4.17^3)/12+5*4.17*2.085^2)+(28*3^3)/12+28*3*2.67^2+12.64*11.33^2+12.64*13.33^2
=4763.522in4=0.2297ft4

flexural stress at extreme compression for y=4.17in=0.3475ft
fc=(M*y)/I
=(130*0.3475)/0.2297=196.669k/ft2=1365psi

flexural stress at the center of reinforcing steel bars. Y=16.5-4.17=12.33in=1.0275ft

fs=(n*M*y)/I
=(8*130*1.0275)/0.2297=452.155k/ft2=32306psi





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