Question 7.6: Determine the slope and deflection at point A of the beam sh...

Real System See Fig. 7.11(b).

Slope at A, θ_A The virtual system consists of a unit couple applied at A, as shown in Fig. 7.11(c). From Fig. 7.11(a) through (c), we can see that there are no discontinuities of the real and virtual loadings or of EI along the length of the beam. Therefore, there is no need to subdivide the beam into segments. To determine the equation for the bending moment M due to real loading, we select an x coordinate with its origin at end A of the beam, as shown in Fig. 7.11(b). By applying the method of sections described in Section 5.2, we determine the equation for M as

0 < x <L              M = -\frac{1}{2}(x)(\frac{wx}{L})(\frac{x}{3}) = -\frac{wx^3}{6L}

Similarly, the equation for the bending moment M_{v1} due to virtual unit moment in terms of the same x coordinate is

0 < x <L             M_{v1} = 1

To determine the desired slope θ_A, we apply the virtual work expression given by Eq. (7.31):

1(θ) = \int_{0}^{L}{\frac{M_{v}M}{EI} } dx                (7.31)

The negative answer for θ_A indicates that point A rotates counterclockwise, in the direction opposite to that of the unit moment.

θ_A = \frac{wL^3}{24EI} 

Deflection at A, Δ_A The virtual system consists of a unit load applied at A, as shown in Fig. 7.11(d). If we use the same x coordinate as we used for computing θ_A, then the equation for M remains the same as before, and the equation for bending moment M_{v2} due to virtual unit load (Fig. 7.11(d)) is given by

0 < x <L                M_{v2} = -1(x) = -x

By applying the virtual work expression given by Eq. (7.30), we determine the desired deflection Δ_A as

1(Δ) = \int_{0}^{L}{\frac{M_{v}M}{EI} } dx                   (7.30)

The positive answer for Δ_A indicates that point A deflects downward, in the direction of the unit load.