Determine the components of the deflection of the point C in the frame shown in Fig. 4.16(a); consider the effect of bending only.

Question

Accepted Answer

The horizontal and vertical components of the deflection of C may be found by applying unit loads in turn at C, as shown in Figs. 4.16(b) and (c). The internal work done by this virtual force system, that is, the unit loads acting through real displacements, is given by Eq. (4.20), in which, for AB,

[latex]w_{i, M}=\int_{L} \frac{M_{ A } M_{v}}{E I} d x[/latex] (4.20)

[latex]\left.M_{ A }=W L-\left(w x^{2} / 2 ight), \quad M_{v}=1 L( ext { horiz. }), \quad M_{v}=-1 x ext { (vert. } ight)[/latex]

for BC,

[latex]M_{ A }=W y, \quad M_{v}=1 y( ext { horiz. }), \quad M_{v}=0( ext { vert. })[/latex]

Considering the horizontal deflection first, the total internal virtual work is

[latex]W_{i}=\int_{0}^{L}\left(W y^{2} / E I ight) d y+\int_{0}^{2 L }\left[W L-\left(w x^{2} / 2 ight) ight](L / 2 E I) d x[/latex]

that is,

[latex]W_{i}=(W / E I)\left[y^{3} / 3 ight]_{0}^{L}+(L / 2 E I)\left[W L x-\left(w x^{3} / 6 ight) ight]_{0}^{2 L}[/latex]

which gives

[latex]W_{i}=2 L^{3}(2 W-w L) / 3 E I[/latex] (i)

The virtual external work done by the unit load is [latex]1 \delta_{ C , H }[/latex], where [latex]\delta_{ C , H }[/latex] is the horizontal component of the deflection of C. Equating this to the internal virtual work given by Eq. (i) gives

[latex]\delta_{ C , H }=2 L^{3}(2 W-w L) / 3 E I[/latex] (ii)

Now, considering the vertical component of deflection,

[latex]W_{i}=(1 / 2 E I) \int_{0}^{2 L}\left[W L-\left(w x^{2} / 2 ight) ight](-x) d x[/latex]

Note that, for BC, [latex]M_{v}=0 .[/latex] Integrating this expression and substituting the limits gives

[latex]W_{i}=L^{3}(-W+w L) / E I[/latex] (iii)

The external virtual work done is [latex]1 \delta_{ C , V}[/latex], where [latex]\delta_{ C , V}[/latex] is the vertical component of the deflection of C. Equating the internal and external virtual work gives

[latex]\delta_{ C , V}=L^{3}(-W+w L) / E I[/latex]

Note that the components of deflection can be either positive or negative, depending on the relative magnitudes of W and w. A positive value indicates a deflection in the direction of the applied unit load, a negative one indicates a deflection in the opposite direction to the applied unit load.

Question 4.10: Determine the components of the deflection of the point C in...

Related Answered Questions