Solution of Design Moment Strength of An Irregularly Shaped Beam Section

Articles > Solution of Design Moment Strength of An Irregularly Shaped Beam Section

Question:

Calculate the design moment strength of the section shown. The compressive strength of the concrete is 4000 psi, and the yield stress of the reinforcement is 60,000 psi.

Solution:

All the formulas used in this solution are obtained from Flexural Design of Reinforced Concrete Beams article.

The equivalent area of the compression zone can be found from

$A_c=frac{A_sf_y}{0.85f'_c}=frac{4.68*60}{0.85*4}=82.6in^2$

Because the equivalent area of the compression zone exceeds the areas in the rectangular regions to the left and right of the trough, the compression zone extends to some depth below the bottom of the trough. This depth is

$a_2=frac{A_c-2b_1h_1}{b_w}=frac{82.6-2*4*5}{14}=3.04in$

The equivalent compression force can be expressed in terms of a component acting in the rectangular regions adjacent to the trough, C1, and a component acting over the region below the trough, C2.

$C_1=2(0.85f'_cb_1h_1)=(2)(0.85)(4)*4*5=136kip$

$C_2=0.85f'_cb_wa_2=(0.85)*4*14*3.04=145kips$

Taking moments of the two forces about the line of action of the tension force gives the design moment strength of the section.

$M_n=phi (C_1(d-frac{h_1}{2})+C_2 (d-h_1-frac{a2}{2})) = 0.9(136*(30-5/2)+145*(30-5-3.04/2))=6430in-kip*(1/12)=536ft-kip$

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