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In all our investigations of the stresses and deflections of beams having two supports, we have supposed that the supports exercise no constraint on bending of the beam, i.e. the axis of the beam has been assumed free to take up any inclination to the line of supports. This has been necessary because, without knowing how to deal with the deformation of the axis of the beam, we were not in a position to find the bending moments on a beam when the supports constrain the direction of the axis. We shall now investigate this problem. When the ends of a beam are fEed in direction so that the axis of the beam has to retain its original direction at the points of support, the beam is said to be built-in or direction fmed. Consider a straight beam resting on two supports A and B (Figure 14.1) and carrying vertical loads. If there is no constraint on the axis of the beam, it will become curved in the manner shown by broken lines, the extremities of the beam rising off the supports.
Figure 14.1 Beam with end couples.
In order to make the ends of the beam lie flat on the horizontal supports, we shall have to apply couples as shown by MI and M2. If the beam is finny built into two walls, or bolted down to two piers, or in any way held so that the axis cannot tip up at the ends in the manner indicated, the couples such as MI and M2 are supplied by the resistance of the supports to deformation. These couples are termed fuced-end moments, and the main problem of the built-in beam is the determination of these couples; when we have found these we can draw the bending moment diagram and calculate the stresses in the usual way. The couples M I and M2 in Figure 14.1 must be such as to produce curvature in the opposite direction to that caused by the loads.
14.2
Built-in beam with a single concentrated load
We may deduce the bending moments in a built-in beam under any conditions of lateral loading from the case of a beam under a single concentrated lateral load.
Built-in and continuous beams
340
W ( l - f ) + ( V )
Figure 14.2 Built-in beam carrying a single lateral load.
Consider a uniform beam, of flexural stiffness EI, and length L, which is built-in to end supports C and G, Figure 14.2. Suppose a concentrated vertical load Wis applied to the beam at a distance a from C. If M, and M, are the restraining moments at the supports, then the vertical reaction is at C is
w
( ); ; 1 - -
+-(M,-M,)
The bending moment in the beam at a &stance z from C is therefore c _ - - - - _ - - _
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