A beamis supported at both ends and has the central half of its span reinforced such that its flexural rigidity is 2EI; the flexural rigidity of the remaining parts of the beam is EI. The beam has a span L and carries a vertically downward concentrated load, W, at its mid-span point. Assuming a

Question

A beamis supported at both ends and has the central half of its span reinforced such that its flexural rigidity is 2EI; the flexural rigidity of the remaining parts of the beam is EI. The beam has a span L and carries a vertically downward concentrated load, W, at its mid-span point. Assuming a deflected shape function of the form

$v=\frac{4 v_{ m } x^{2}}{L^{3}}(3 L-4 x) \quad(0 \leq x \leq L / 2)$

in which $v_{ m }$ is the deflection at the mid-span point, determine the value of $v_{ m }$ .
Ans. $0.00347 WL ^{3} / EI$ .

[latex]v=\frac{4 v_{ m } x^{2}}{L^{3}}(3 L-4 x) \quad(0 \leq x \leq L / 2)[/latex]

in which [latex]v_{ m }[/latex] is the deflection at the mid-span point, determine the value of [latex]v_{ m }[/latex].
Ans. [latex]0.00347 WL ^{3} / EI[/latex].

Accepted Answer

The method of solution used in S.15.22 may be used for this problem. The strain energy of the beam is, from Eqs (9.21) and (13.3)

[latex]U=\int_{L} \frac{M^{2}}{2 E I_{z}} d x[/latex] (9.21)

[latex]\frac{ d ^{2} v}{ d x^{2}}=\frac{M}{E I}[/latex] (13.3)

[latex]U=2\left(\frac{E I}{2}\right) \int_{0}^{L / 4}\left(\frac{ d ^{2} v}{ d x^{2}}\right)^{2} d x+2\left(\frac{2 E I}{2}\right) \int_{L / 4}^{L / 2}\left(\frac{ d ^{2} v}{ d x^{2}}\right)^{2} d x[/latex]

which gives, when the equation for the assumed displaced shape is substituted

[latex]U=\frac{144 E I v_{ m }^{2}}{L^{3}}[/latex]

The total potential energy is then

[latex]TPE =\frac{144 E I v_{ m }^{2}}{L^{3}}-W v_{ m }[/latex]

From the principle of the stationary value of the total potential energy

[latex]\frac{\partial( TPE )}{\partial v_{ m }}=\frac{288 E I v_{ m }}{L^{3}}-W=0[/latex]

from which

[latex]v_{ m }=\frac{0.00347 W L^{3}}{E I}[/latex] .

Question 15.P.23: A beamis supported at both ends and has the central half of ...

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