Question 13.11: Determine the reactions and draw the shear and bending momen...
Determine the reactions and draw the shear and bending moment diagrams for the four-span continuous beam shown in Fig. 13.16(a) using the method of consistent deformations.
Symmetry As the beam and the loading are symmetric with respect to the vertical s axis passing through roller support C (Fig. 13.16(a)), we will analyze only the right half of the beam with symmetric boundary conditions, as shown in Fig. 13.16(b). The response of the left half of the beam will then be obtained by reflecting the response of the right half to the other side of the axis of symmetry.
Degree of Indeterminacy The degree of indeterminacy of the substructure (Fig. 13.16(b)) is 2. Note that, since the degree of indeterminacy of the complete continuous beam (Fig. 13.16(a)) is three, the utilization of structural symmetry will reduce the computational effort required in the analysis.
Primary Beam The vertical reactions D_y and E_y at the roller supports D and E, respectively, of the substructure are selected as the redundants. The roller supports at D and E are then removed to obtain the cantilever primary beam shown in Fig. 13.16(c).
Compatibility Equations See Fig. 13.16(b) through (e).
Δ_{DO} + f_{DD} D_y + f_{DE} E_y = 0 (1)
Δ_{EO} + f_{ED} D_y + f_{EE} E_y = 0 (2)
Deflections of the Primary Beam By using the deflection formulas, we obtain
Δ_{DO} = -\frac{17wL^4}{24EI} Δ_{EO} = -\frac{2wL^4}{EI}
f_{DD} = \frac{L^3}{3EI} f_{ED} = \frac{5L^3}{6EI}
f_{EE} = \frac{8L^3}{3EI}
By applying Maxwell’s law,
f_{DE} = \frac{5L^3}{6EI}Magnitudes of the Redundants By substituting the deflections of the primary beam into the compatibility equations, we obtain
-\frac{17wL^4}{24EI} + (\frac{L^3}{3EI})D_y + (\frac{5L^3}{6EI})E_y = 0 (1a)
-\frac{2wL^4}{EI} + (\frac{5L^3}{6EI})D_y + (\frac{8L^3}{3EI})E_y = 0 (2a)
which can be simplified to
8D_y + 20E_y = 17wL (1b)
5D_y + 16E_y = 12wL (2b)
Solving Eqs. (1b) and (2b) simultaneously for D_y and E_y, we obtain
D_y =\frac{8}{7}wL ↑ E_y =\frac{11}{28}wL ↑
Reactions The remaining reactions of the substructure, obtained by applying the equations of equilibrium, are shown in Fig. 13.16(f ). The reactions to the left of the s axis are then obtained by reflection, as shown in Fig. 13.16(g).
Shear and Bending Moment Diagrams By using the reactions of the continuous beam, its shear and bending moment diagrams are constructed. These diagrams are shown in Fig. 13.16(h).
