Three rods are hinged together to support a vertical load Was shown in Fig. 1.29. Find the force carried by each rod if A, L and E denote their respective cross-sectional area, length and modulus of elasticity. Outer rods AD and CD are identical.

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Accepted Answer

Equilibrium condition: For static eq-uilibrium of joint D,

[latex]\sum F_x=0: F_{A D} \sin heta=F_{C D} \sin heta[/latex]

or, [latex] F_{A D}=F_{C D}=F_1[/latex]

[latex]\sum F_y=0: 2 F_1 \cos heta + F_2=W[/latex] (i)

Compatibility condition:The deformed shape of three members is shown by dotted line. It is assumed that the angle [latex] heta[/latex] does not change appreciably, then the compatibility equation,

[latex]\Delta L_1= \Delta L_2 \cos heta[/latex] (ii)

or, [latex]\frac{F_1 L_1}{A_1 E_1}=\frac{F_2 L_2}{A_2 E_2} \cos heta [/latex]

But, [latex]L_2= L_1 \cos heta [/latex]

Therefore, [latex]F_1 = F_2 \frac{A_1 E_1}{A_2 E_2} \cos^{2} heta [/latex] (iii)

From Eqs (i) and (ii),
[latex]F_2=\frac{W}{1+\frac{2 A_1 E_1}{A_2 E_2} \cos ^3 heta}[/latex]

and [latex]F_1=\frac{W \cos ^2 heta}{\frac{A_2 E_2}{A_1 E_1}+2 \cos ^3 heta}[/latex]

Question 1.19: Three rods are hinged together to support a vertical load Wa...

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