Page 75 - Handbook of Civil Engineering Calculations, Second Edition

1.58 STRUCTURAL STEEL ENGINEERING AND DESIGN

FIGURE 39

section; w x load intensity at the given section; M and m bending moment at the giv-
en section induced by the actual load and by the unit moment, respectively.
2. Evaluate the moments in step 1
2
Evaluate M and m. By proportion, w x w(L x)/L; M (x /6)(2w w x )
2
2
(wx /6)[2 (L x)/L] wx (3L x)/(6L); m 1.
3. Apply a suitable slope equation
L
L
2
Use the equation A 0 [Mm/(EI)] dx. Then EI A 0 [wx 3L x)/(6L)] dx [w/(6L)]
2
3
L
3
4
L
3
4
4
1
0 (3Lx x ) dx [w/(6L)](3Lx /3 x /4)] 0 [w/(6L)](L L /4); thus, A /8wL /
(EI) counterclockwise. This is the slope at A.
4. Apply a unit load to the beam
Apply a unit downward load at A as shown in Fig. 39c. Let m
denote the bending mo-
ment at a given section induced by the unit load.
5. Evaluate the bending moment induced by the unit load;
find the deflection
L
L
3
Apply y A 0 [Mm
/(EI)] dx. Then m
x; EIy A 0 [wx (3L x)/(6L)] dx [w/(6L)]
L
4
3
0 x (3L x) dx; y A (11/120)wL /(EI).
The first equation in step 3 is a statement of the work performed by the unit moment at
A as the beam deflects under the applied load. The left-hand side of this equation express-
es the external work, and the right-hand side expresses the internal work. These work
equations constitute a simple proof of Maxwell’s theorem of reciprocal deflections, which
is presented in a later calculation procedure.
DEFLECTION OF A CANTILEVER FRAME
The prismatic rigid frame ABCD (Fig. 40a) carries a vertical load P at the free end. Deter-
mine the horizontal displacement of A by means of both the unit-load method and the
moment-area method.