Building Design and Construction Handbook - Merritt - Ventech!

5.70 SECTION FIVE
As an example of the use of the method of least work, we shall solve again for
the stress in the vertical bar in Fig. 5.50. Calling this stress X, we note that the
stress in each of the inclined bars must be (P � X)/2 cos �. With the aid of Eq.
(5.30), we can express the strain energy in the system in terms of X as
2 2
XL (P � X) L
U � � 3
2AE 4AE cos �
Hence, the internal work in the system will be a minimum when
�U XL (P � X)L
� � � 0 3
�X AE 2AE cos �
Solving for X gives the stress in the vertical bar as P/(1 � 2 cos 3 �), as before
(Art. 5.10.1).
5.10.4 Dummy Unit-Load Method
In Art. 5.2.7, the strain energy for pure bending was given as U � M 2 L/2EI in Eq.
(5.33). To find the strain energy due to bending stress in a beam, we can apply this
equation to a differential length dx of the beam and integrate over the entire span.
Thus,
L 2 M dx
U � � (5.94)
0 2EI
If M represents the bending moment due to a generalized force P, the partial derivative
of the strain energy with respect to P is the deformation d corresponding
to P. Differentiating Eq. (5.94) under the integral sign gives
L M �M
d � � dx (5.95)
0 EI �P
The partial derivative in this equation is the rate of change of bending moment with
the load P. It is equal to the bending moment m produced by a unit generalized
load applied at the point where the deformation is to be measured and in the
direction of the deformation. Hence, Eq. (5.95) can also be written
L Mm
d � � dx (5.96)
0 EI
To find the vertical deflection of a beam, we apply a vertical dummy unit load at
the point where the deflection is to be measured and substitute the bending moments
due to this load and the actual loading in Eq. (5.96). Similarly, to compute a rotation,
we apply a dummy unit moment.
Beam Deflections. As a simple example, let us apply the dummy unit-load
method to the determination of the deflection at the center of a simply supported,
uniformly loaded beam of constant moment of inertia (Fig. 5.51a). As indicated in
Fig. 5.51b, the bending moment at a distance x from one end is (wL/2)x � (w/
2)x 2 . If we apply a dummy unit load vertically at the center of the beam (Fig.