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Example 1:Flexural stress calculation using transformed area method for rectangular beam with double layer of reinforcing bars

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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

first, we will calculate the transformed area of steel. the concrete below the neutral axis will not contribute to resisting of tension. figure 2 shows the section after transformation 
As=6*1=6.0in2
Transformed area for first layer=As*n=2*8=16in2
Transformed area for second layer=As*n=4*8=32in2


figure 2

Neutral axis can be calculated by equalling moment of area from both parts as shown below 
16*(18-x)+32*(21-x)=(x/2)*(14*x)
960-48x=7x^2
7x^2+48x-960=0
x=8.77 in
moment of inertia for transformed area=
=(14*8.77^3)/12+8.77*14*4.385^2+16*9.23^2+32*12.23^2=9297.20in4
=0.44835ft4

flexural stress for compression zone. y at extreme compression equal 8.77in=0.73ft

fc=(M*y)/I
fc=110*0.73/0.44835=179.10k/ft2=1243psi


flexural stress for tension zone for y at bottom layer equal 12.23in=1.019ft

fc=n(M*y)/I
=8*(110*1.019)/0.44835
=2000.04k/ft2=13889.194psi


flexural stress for tension zone for y at the center of reinforcing steel layer equal 20-x=20-8.77=11.23in=0.9358ft

fc=n(M*y)/I
=8*(110*0.9358)/0.44835
=1836.743k/ft2=12755.138psi






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