# Problem 709 | Propped Beam with Spring Support

**Example 06**

The beam in Figure PB-006 is supported at the left by a spring that deflects 1 inch for each 300 lb. For the beam E = 1.5 × 10^{6} psi and I = 144 in^{4}. Compute the deflection of the spring.

**Solution 06**

Assume there is no spring support at the left end

$\delta = \dfrac{w_oL^4}{8EI} = \dfrac{200(12^4)(12^3)}{8(1.5 \times 10^6)(144)}$

$\delta = \dfrac{w_oL^4}{8EI} = \dfrac{200(12^4)(12^3)}{8(1.5 \times 10^6)(144)}$

$\delta = 4.1472 \, \text{ in}$

Considering the spring reaction

$\delta - \delta_{spring} = \delta_R$

$4.1472 - \dfrac{R}{k} = \dfrac{RL^3}{3EI}$

$4.1472 - \dfrac{R}{300} = \dfrac{R(12^3)(12^3)}{3(1.5 \times 10^6)(144)}$

$4.1472 = 0.0079413R$

$R = 522.23 \, \text{ lb}$

$\delta_{spring} = \dfrac{R}{k} = \dfrac{522.23}{300}$

$\delta_{spring} = 1.74 \, \text{ in} \,$ *answer*

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