Building Design and Construction Handbook - Merritt - Ventech!

FIGURE 5.51 Dummy unit-load method applied
to a uniformly loaded, simple beam (a) to
find mid-span deflection; (b) moment diagram
for the uniform load; (c) unit load at midspan:
(d ) moment diagram for the unit load.
STRUCTURAL THEORY 5.71
FIGURE 5.52 End rotation of a simple beam
due to an end moment: (a) by dummy unit-load
method; (b) moment diagram for the end moment;
(c) unit moment applied at beam end;
(d) moment diagram for the unit moment.
5.51c), where the vertical deflection is to be determined, the moment at x is x/2,
as indicated in Fig. 5.51d. Substituting in Eq. (5.96) and taking advantage of the
symmetry of the loading gives
� �
L /2 4
wL w x dx 5wL
2
d � 2 � x � x �
0 2 2 2 EI 384EI
Beam End Rotations. As another example, let us apply the method to finding the
end rotation at one end of a simply supported, prismatic beam produced by a
moment applied at the other end. In other words, the problem is to find the end
rotation at B, � B, in Fig. 5.52a, due to M A. As indicated in Fig. 5.52b, the bending
moment at a distance x from B caused by M A is M Ax/L. If we applied a dummy
unit moment at B (Fig. 5.52c), it would produce a moment at x of (L � x)/L (Fig.
5.52d). Substituting in Eq. (5.96) gives
L xL� xdx ML
A
�B � � MA �
0 L L EI 6EI
Shear Deflections. To determine the deflection of a beam caused by shear, Castigliano’s
theorems can be applied to the strain energy in shear
2
v
V � �� dA dx
2G