Question 4.9: Displacements of a Stepped Cantilevered Beam by the Moment-A...
Displacements of a Stepped Cantilevered Beam by the Moment-Area Method
A nonprismatic cantilevered beam with two different moments of inertia carries a concentrated load P at its free end (Figure 4.14a). Find the slope at B and deflection at C.
The M/EI diagram is divided conveniently into its component parts, as shown in Figure 4.14b:
A_1=-\frac{P L^2}{8 E I}, \quad A_2=-\frac{P L^2}{16 E I}, \quad A_3=-\frac{P L^2}{8 E I}
The elastic curve is in Figure 4.14c. Inasmuch as \theta _{A} = 0 and υ_{A} = 0, we have \theta _{C} = \theta _{CA}, \theta_B=\theta_{B A}, \upsilon _C=t_{C A}, \text { and } \upsilon _B=t_{B A}.
Applying the first moment-area theorem,
\theta_B=A_1+A_2+A_3=-\frac{5 P L^2}{16 E I} (4.28a)
The minus sign means that the rotations are clockwise. From the second moment-area theorem,
\upsilon _{C}=A_1\left(\frac{L}{4}\right)+A_2\left(\frac{L}{3}\right)=-\frac{5 P L^3}{96 E I} (4.28b)
The minus sign shows that the deflection is downward.