Question 9.9: Determine the slope and deflection at end B of the prismatic...
Determine the slope and deflection at end B of the prismatic cantilever beam AB when it is loaded as shown (Fig. 9.48), knowing that the flexural rigidity of the beam is EI = 10 MN · m².
We first draw the free-body diagram of the beam (Fig. 9.49a). Summing vertical components and moments about A, we find that the reaction at the fixed end A consists of a 50 kN upward vertical force R_A and a 60 kN · m counterclockwise couple M_A. Next, we draw the bending-moment diagram (Fig. 9.49b) and determine from similar triangles the distance x_D from the end A to the point D of the beam where M = 0:
\frac{x_{D}}{60}=\frac{3-x_{D}}{90}=\frac{3}{150} x_{D}=1.2 m
Dividing by the flexural rigidity EI the values obtained for M, we draw the (M/EI) diagram (Fig. 9.50) and compute the areas corresponding respectively to the segments AD and DB, assigning a positive sign to the area located above the x axis, and a negative sign to the area located below that axis. Using the first moment-area theorem, we write
u _{B / A}= u _{B}- u _{A}= area from A to B = A_{1}+A_{2}
\begin{aligned}=-\frac{1}{2}(1.2 m ) &\left(6 \times 10^{-3} m ^{-1}\right) \\&+\frac{1}{2}(1.8 m )\left(9 \times 10^{-3} m ^{-1}\right)\end{aligned}
=-3.6 \times 10^{-3}+8.1 \times 10^{-3}
=+4.5 \times 10^{-3} rad
and, since u _{A}=0,
u _{B}=+4.5 \times 10^{-3} rad
Using now the second moment-area theorem, we write that the tangential deviation t_{B / A} is equal to the first moment about a vertical axis through B of the total area between A and B. Expressing the moment of each partial area as the product of that area and of the distance from its centroid to the axis through B, we have
t_{B / A}=A_{1}(2.6 m )+A_{2}(0.6 m )=\left(-3.6 \times 10^{-3}\right)(2.6 m )+\left(8.1 \times 10^{-3}\right)(0.6 m )
=-9.36 mm +4.86 mm =-4.50 mm
Since the reference tangent at A is horizontal, the deflection at B is equal to t_{B / A} and we have
y_{B}=t_{B / A}=-4.50 mmThe deflected beam has been sketched in Fig. 9.51.
