A compound beam ABC has a roller support at A, an internal hinge at B, and a fixed support at C (Fig. 9-20a). Segment AB has length a and segment BC has length b. A concentrated load P acts at distance 2a/3 from support A and a uniform load of intensity q acts between points B and C.
Determine the deflection $\delta_{B}$ at the hinge and the angle of rotation $\theta_{A}$ at support A (Fig. 9-20d). (Note: The beam has constant flexural rigidity EI.)

Question

A compound beam ABC has a roller support at A, an internal hinge at B, and a fixed support at C (Fig. 9-20a). Segment AB has length a and segment BC has length b. A concentrated load P acts at distance 2a/3 from support A and a uniform load of intensity q acts between points B and C.
Determine the deflection [latex]\delta_{B}[/latex] at the hinge and the angle of rotation [latex] heta_{A}[/latex] at support A (Fig. 9-20d). (Note: The beam has constant flexural rigidity EI.)

Accepted Answer

For purposes of analysis, we will consider the compound beam to consist of two individual beams: (1) a simple beam AB of length a, and (2) a cantilever beam BC of length b. The two beams are linked together by a pin connection at B.
If we separate beam AB from the rest of the structure (Fig. 9-20b), we see that there is a vertical force F at end B equal to 2P/3. This same force acts downward at end B of the cantilever (Fig. 9-20c). Consequently, the cantilever beam BC is subjected to two loads: a uniform load and a concentrated load. The deflection at the end of this cantilever (which is the same as the deflection [latex]\delta_{B}[/latex] of the hinge) is readily found from Cases 1 and 4 of Table G-1, Appendix G:

[latex] u=-\frac{qx^{2}}{24EI}(6L^{2}-4Lx+x^{2})[/latex] [latex] u^{\prime}=-\frac{qx}{6EI}(3L^{2}-3Lx+x^{2})[/latex]

[latex]\delta_{B}=\frac{qL^{4}}{8EI}[/latex] [latex] heta_{B}=\frac{qL^{3}}{6EI}[/latex]

[latex] u=-\frac{Px^{2}}{6EI}(3L-x)[/latex] [latex] u^{\prime}=-\frac{Px}{2EI}(2L-x)[/latex]

[latex]\delta_{B}=\frac{PL^{3}}{3EI}[/latex] [latex] heta_{B}=\frac{PL^{2}}{2EI}[/latex]

[latex]\delta_{B}=\frac{qb^{4}}{8EI}+\frac{Fb^3}{3EI}[/latex]

or, since F = 2P/3,

[latex]\delta_{B}=\frac{qb^{4}}{8EI}+\frac{2Pb^3}{9EI}[/latex] (9-56)

The angle of rotation [latex] heta_{A}[/latex] at support A (Fig. 9-20d) consists of two parts: (1) an angle BAB′ produced by the downward displacement of the hinge, and (2) an additional angle of rotation produced by the bending of beam AB (or beam AB′) as a simple beam. The angle BAB′ is

[latex]( heta_{A})_{1}=\frac{\delta_{B}}{a}=\frac{qb^{4}}{8aEI}+\frac{2Pb^3}{9aEI}[/latex]

The angle of rotation at the end of a simple beam with a concentrated load is obtained from Case 5 of Table G-2. The formula given there is

[latex] u=-\frac{Pbx}{6LEI}(L^{2}-b^{2}-x^{2})[/latex] [latex] u^{\prime}=-\frac{Pb}{6LEI}(L^{2}-b^{2}-3x^{2}) [/latex] (0 ≤ x ≤ a)

[latex] heta_{A}=\frac{Pab(L+b)}{6LEI}[/latex] [latex] heta_{B}=\frac{Pab(L+a)}{6LEI}[/latex]

If a ≥ b, [latex]\delta_{C}=\frac{Pb(3L^{2}-4b^{2})}{48EI}[/latex] If a ≤ b, [latex]\delta_{C}=\frac{Pa(3L^{2}-4a^{2})}{48EI}[/latex]

If a ≥ b, [latex]x_{1}=\sqrt{\frac{L^{2}-b^{2}}{3}}[/latex] and [latex]\delta_{\max}=\frac{Pb(L^{2}-b^{2})^{{3}/{2}}}{}[/latex]

[latex]\frac{Pab(L+b)}{6LEI}[/latex]

in which L is the length of the simple beam, a is the distance from the left-hand support to the load, and b is the distance from the right-hand support to the load. Thus, in the notation of our example (Fig. 9-20a), the angle of rotation is

[latex]( heta_{A})_{2}=\frac{P\left(\frac{2a}{3} ight)\left(\frac{a}{3} ight)\left(a+\frac{a}{3} ight)}{6aEI}=\frac{4Pa^{2}}{81EI}[/latex]

Combining the two angles, we obtain the total angle of rotation at support A:

[latex] heta_{A}=( heta_{A})_{1}+( heta_{A})_{2}=\frac{qb^{4}}{8aEI}+\frac{2Pb^3}{9aEI}+\frac{4Pa^2}{81EI}[/latex] (9-57)

This example illustrates how the method of superposition can be adapted to handle a seemingly complex situation in a relatively simple manner.

Question 9.8: A compound beam ABC has a roller support at A, an internal h...

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