Determine the flexural stiffness of the beam in Figure P7.35 (see Example 7.16 for criteri...

Determine the flexural stiffness of the beam in Figure P7.35 (see Example 7.16 for criteria) for (a) moment applied at A and (b) moment applied at C. E is constant.

Figure P7.35:

Example 7.16:

Compare the magnitude of the moment required to produced a unit value of rotation (θ_A = 1 rad) at the left end of the beams in Figure 7.29a and c. Except for the supports at the right end—a pin versus a fixed end—the dimensions and properties of both beams are identical, and EI is constant. Analysis indicates that a clockwise moment M applied at the left end of the beam in Figure 7.29c produces a clockwise moment of M/2 at the fixed support.

The conjugate beam for the pin-ended beam in Figure 7.29a is shown in Figure 7.29b. Since the applied moment Mʹ produces a clockwise rotation of 1 rad at A, the reaction at the left support equals 1. Because the slope at A is negative, the reaction acts downward.

To compute the reaction at B, we sum moments about support A.

Summing forces in the y direction, we express Mʹ in terms of the properties of the member as

The conjugate beam for the fixed-end beam in Figure 7.29c is shown in Figure 7.29d. The M/EI diagram for each end moment is drawn separately. To express M in terms of the properties of the beam, we sum forces in the y direction.

NOTE. The absolute flexural stiffness of a beam can be defined as the value of end moment required to rotate the end of a beam—supported on a roller at one end and fixed at the other end (Figure 7.29c)—through an angle of 1 rad. Although the choice of boundary conditions is somewhat arbitrary, this particular set of boundary conditions is convenient because it is similar to the end conditions of beams that are analyzed by moment distribution—a technique for analyzing indeterminate beams and frames covered in Chapter 11. The stiffer the beam, the larger the moment required to produce a unit rotation.

If a pin support is substituted for a fixed support as shown in Figure 7.29a, the flexural stiffness of the beam reduces because the roller does not apply a restraining moment to the end of the member. As this example shows by comparing the values of moment required to produce a unit rotation (Equations 1 and 2), the flexural stiffness of a pin-ended beam is three-fourths that of a fixed-end beam.

Figure 7.29: Effect of end restraint on flexural stiffness. (a) Beam loaded at A with far end pinned; (b) conjugate structure for beam in (a) loaded with M/EI; (c) beam loaded at A with far end fixed; (d) conjugate structure for beam in (c) loaded with M/EI.