Question 7.7: Determine the slope at point B of the cantilever beam shown ...
Determine the slope at point B of the cantilever beam shown in Fig. 7.12(a) by the virtual work method.
The real and virtual systems are shown in Figs. 7.12(b) and (c), respectively. As shown in these figures, an x coordinate with its origin at end B of the beam is selected to obtain the bending moment equations. From Fig. 7.12(b), we can see that the equation for M in terms of the x coordinate is
0 < x < 5 m M = -60x
Similarly, from Fig. 7.12(c), we obtain the equation for M_v to be
0 < x < 5 m M_v = -1
The slope at B can now be computed by applying the virtual work expression given by Eq. (7.31), as follows:
1(θ) = \int_{0}^{L}{\frac{M_vM}{EI} dx} (7.31)
1(θ_B) = \int_{0}^{L}{\frac{M_vM}{EI} dx}1(θ_B) = \frac{1}{E} \int_{0}^{5}{-1(-60x) dx}
(1 kN-m)θ_B = \frac{750 kN^2-m^3}{EI}
Therefore,
θ_B = \frac{750 kN-m^3}{EI} = \frac{750}{70(10^6)600(10^{-6})} = 0.0179 rad.
The positive answer for θ_B indicates that point B rotates clockwise, in the direction of the unit moment.
θ_B = 0.0179 rad.
