Question 16.7: Determine the reactions and draw the shear and bending momen...

Because the beam and the loading are symmetric with respect to the vertical s axis passing through roller support C (Fig. 16.11(a)), the response of the complete beam can be determined by analyzing only the left half, AC, of the beam, with symmetric boundary conditions as shown in Fig. 16.11(b). Furthermore, from Fig. 16.11(b), we can see that the one-half of the beam with symmetric boundary conditions is also symmetric with respect to the s^′ axis passing through roller support B. Therefore, we need to analyze only one-fourth of the beam—that is, the portion AB—with symmetric boundary conditions, as shown in Fig. 16.11(c).

Since the substructure to be analyzed consists simply of the fixed beam AB (Fig. 16.11(c)), its end moments can be obtained directly from the fixed-end moment expressions given inside the back cover of the book. Thus

The shears at the ends of member AB are determined by considering the equilibrium of the member.

The shears and moments at the ends of member BC can now be obtained by reflecting the corresponding responses of member AB to the right of the s^′ axis, and the member end moments and shears on the right half of the beam can be determined by reflecting the corresponding responses on the left half to the other side of the s axis. The member end moments and shears thus obtained are shown in Fig. 16.11(d), and the support reactions are given in Fig. 16.11(e).

The shear and bending moment diagrams for the beam are shown in Fig. 16.11(f ) and (g), respectively.

As this example shows, the utilization of structural symmetry can considerably reduce the computational effort required in the analysis. The beam considered in this example (Fig. 16.11(a)) has three degrees of freedom, θ_B, θ_C, and θ_D. However, by taking advantage of the structure’s symmetry, we were able to eliminate all the degrees of freedom from the analysis.