![]() ![]() Substitute the computed joint rotations into the equations obtained in step 3 to determine the members’ end moments.Solve the system of equations obtained simultaneously to determine the unknown joint rotations.Write the equilibrium equations at each joint that is free to rotate in terms of the end moments of members connected at that joint.Write the slope-deflection equation for the members’ end moments in terms of unknown rotations.Determine the rotations of the chord if there is any support settlement.Determine the fixed-end moments for the members of the beam.Procedure for Analysis of Indeterminate Beams and Non-Sway Frames by the Slope-Deflection Method The procedure for the analysis of indeterminate beams by the slope-deflection method is summarized below. Solving equation 11.13 for θ B and substituting it into equation 11.12 suggests the following:Įquation 11.14 is the modified slope-deflection equation when the far end is supported by a pin or roller. The slope-deflection equations for the end moments are as follows: Propped cantilever beam.Ĭonsider the propped cantilever beam shown in Figure 11.5. Using the modified equation reduces the amount of computational work, as the equation is applied only once to the span with a pin or roller at the far end. The analysis of beams or frames supported by a pin or roller at the far end of the span is simplified by using the modified slope-deflection equation derived below. The final end moments can then be computed as the summation of the moments caused by slopes, deflections, and fixed-end moments, as follows:ġ1.4 Modification for Pin-Supported End Span End moment due to end rotations ( β A and β B), chord rotation ( ψ), and fixed-end moments ( and ). Putting into equations 10.10 and 10.11 suggests the following:įig. Solving equations 11.1 and 11.2 for β A and β B and substituting them into equations 11.5 and 11.6 suggests the following: End moments due to end rotations ( β A and β B) and chord rotation ( ψ). Solving equations 11.3 and 11.4 suggests the following:įig. Similarly, taking the moment about end A to determine β B suggests the following: ![]() Thus, for the beam under consideration, the rotations β A and β B, shown in Figure 11.2, are obtained as follows: End moments due to rotations β A and β B.Īccording to the moment-area theorem, the change in slope for a particular beam equals the end shear force of the beam when it is loaded with the diagram. Ψ = chord rotation caused by settlement of end B. ![]() Β A, β B = end rotations caused by moments M AB and M BA, respectively. The rotations at the joints of the beam can be expressed mathematically as follows: The end moments are the summation of the moments caused by the rotations of the joints at the ends A and B ( θ A and θ B) of the beam, the chord rotation and the fixity at both ends referred to as fixed end moments The member experiences the end moments M AB and M BA at A and B, respectively, and undergoes the deformed shape shown in Figure 11.1b, with the assumption that the right end B of the member settles by an amount ∆. To derive the slope-deflection equations, consider a beam of length L and of constant flexural rigidity EI loaded as shown in Figure 11.1a. The rotation of the chord connecting the ends of a member ( ), the displacement of one end of a member relative to the other, is positive if the member turns in a clockwise direction.ġ1.3 Derivation of Slope-Deflection Equations The rotation θ of a joint is positive if its tangent turns in a clockwise direction. ![]() For the determination of the end moments of members at the joint, this method requires the solution of simultaneous equations consisting of rotations, joint displacements, stiffness, and lengths of members.Īn end moment M is considered positive if it tends to rotate the member clockwise and negative if it tends to rotate the member counter-clockwise. Thus, the unknowns in the slope-deflection method of analysis are the rotations and the relative joint displacements. The method accounts for flexural deformations, but ignores axial and shear deformations. Maney introduced the slope-deflection method as one of the classical methods of analysis of indeterminate beams and frames. Slope-Deflection Method of Analysis of Indeterminate Structures ![]()
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