Determine the equation of the deflection curve of the beam shown in Fig. P.16.12. The flexural rigidity of the beam is EI.

Answer:

$ν=-{\frac{1}{E I}}\left({\frac{125}{6}}z^{3}-50[z-1]^{2}+{\frac{50}{12}}[z-2]^{4}-{\frac{50}{12}}[z-4]^{4}-{\frac{525}{6}}[z-4]^{3}+237.5z\right)$

Question

Determine the equation of the deflection curve of the beam shown in Fig. P.16.12. The flexural rigidity of the beam is EI.

Answer:

[latex]ν=-{\frac{1}{E I}}\left({\frac{125}{6}}z^{3}-50[z-1]^{2}+{\frac{50}{12}}[z-2]^{4}-{\frac{50}{12}}[z-4]^{4}-{\frac{525}{6}}[z-4]^{3}+237.5z\right)[/latex]

Accepted Answer

Taking moments about D,

[latex]R_{\mathrm{A}} imes4+100-100 imes2 imes1+200 imes3=0[/latex]

from which

[latex]R_{\mathrm{{A}}}=-125\,\mathrm{N}[/latex]

Resolving vertically,

[latex]R_{\mathrm{{D}}}-125-100 imes2-200=0[/latex]

Therefore,

[latex]R_{\mathrm{B}}=525\,\mathrm{N}[/latex]

The bending moment at a section a distance z from A in the bay DF is given by

[latex]M=+125z-100[z-1]^{0}+\frac{100[z-2]^{2}}{2}-525[z-4]-\frac{100[z-4]^{4}}{2}[/latex]

in which the uniformly distributed load has been extended from D to F and an upward uniformly distributed load of the same intensity applied from D to F.
Substituting in Eqs (16.31),

[latex]u^{\prime\prime}=-{\frac{M_{y}}{E I_{y y}}},\quad ν^{\prime\prime}=-{\frac{M_{x}}{E I_{x x}}}[/latex] (16.31)

[latex]E I\left({\frac{\mathrm{d}^{2}v}{\mathrm{d}z^{2}}} ight)=-125z+100[z-1]^{0}-50[z-2]^{2}+525[z-4]+50[z-4]^{2}[/latex]

[latex]E I\left({\frac{\mathrm{d}\upsilon}{\mathrm{d}z}} ight)={\frac{-125z^{2}}{2}}+100[z-1]^{1}-{\frac{50[z-2]^{3}}{3}}+{\frac{525[z-4]^{2}}{2}}+{\frac{50[z-4]^{3}}{3}}+C_{1}[/latex]

[latex]E I v={\frac{-125z^{3}}{6}}+50[z-1]^{2}-{\frac{50[z-2]^{4}}{12}}+{\frac{525[z-4]^{3}}{6}} +\frac{50[z-4]^{4}}{12}+C_{1}z+C_{2}[/latex]

When z = 0, υ = 0 so that [latex]C_{2}=0,[/latex] and when z = 4 m, υ = 0, which gives [latex]C_{1}=237.5.[/latex] The deflection curve of the beam is the

[latex]\upsilon=\frac{1}{E I}\left(\frac{-125z^{3}}{6}+50[z-1]^{2}-\frac{50[z-2]^{4}}{12}+\frac{525[z-4]^{4}}{12}+\frac{50[z-4]^{4}}{12}+237.5z ight)[/latex]

Question 16.12: Determine the equation of the deflection curve of the beam s......

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