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Deformation of Bars due to Axial Load

A prismatic b...

Question

A prismatic bar $A B C$ is subjected to an axial load of $25 k N$ ; the reactions $R_{A}$ and $R_{C}$ will be

R_{A} = - 10 k N

and

R_{C} = - 15 k N

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R_{A} = 10 k N

and

R_{C} = - 35 k N

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R_{A} = - 15 k N

and

R_{C} = - 10 k N

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R_{A} = 15 k N

and

R_{C} = - 40 k N

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Solution

The correct option is C $R_{A} = - 15 k N$ and $R_{C} = - 10 k N$
The free body diagram of the given figure is shown below.

$∴$ Equilibrium condition for the bar is given by
$R_{A} + R_{B} = 25$ ... (i)
Compatibility condition for the bar is given by
$Δ_{A B} + Δ_{B C} = 0$ .. (ii)
But $Δ_{A B} = \frac{R_{A} \times 2 L}{A E}$ (tensile)
$Δ_{B C} = \frac{R_{B} \times 3 L}{A E}$ (compressive)
Substituting values of $Δ_{A B}$ and $Δ_{B C}$ in (ii), we get
$\frac{R_{A} \times 2 L}{A E} - \frac{R_{B} \times 3 L}{A E} = 0$ [from (i)]
$\Rightarrow R_{A} \times 2 L - (25 - R_{A}) \times 3 L = 0$
$\Rightarrow 2 R_{A} - 75 + 3 R_{A} = 0$
$\Rightarrow R_{A} = 15 k N$
ALso, $R_{B} = 25 - 15 = 10 k N$
Since the forces left to right directions are taken as positive, therefore, the reactions which are acting from right to left are negative.

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