Chapter 5: Indeterminate Structures Slope-Deflection Method
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1 Chapter 5: Indeterminate Structures Slope-Deflection Method 1. Introduction Slope-deflection method is the second of the two classical methods presented in this course. This method considers the deflection as the primar unknowns, while the redundant forces were used in the force method. In the slope-deflection method, the relationship is established between moments at the ends of the members and the corresponding rotations and displacements. The basic assumption used in the slope-deflection method is that a tpical member can flex but the shear and axial deformation are negligible. It is no different from that used with the force method. Kinematicall indeterminate structures versus staticall indeterminate structures: Redundants: 1 ( B Kinematics: 1 ( θ B Redundants: 1 ( B Kinematics: 3 ( θ A, θb, θ C Redundants: (, Kinematics: ( θ, θ B C B C 1
2 Redundants: 1 ( C Kinematics: 3 ( θ B, θc, Δ x Redundants: 6 Kinematics: 1 ( θ B Sign convention: All clockwise internal moments and end rotation are positive. Basic Idea of Slope Deflection Method The basic idea of the slope deflection method is to write the equilibrium equations for each node in terms of the deflections and rotations. Solve for the generalized displacements. Using momentdisplacement relations, moments are then known. The structure is thus reduced to a determinate structure.
3 Fundamental Slope-Deflection Equations: EI Δ M = 3 FEM θ + θ + EI Δ M = θ + θ 3 + FEM where AB A B AB BA B A BA Case A: fixed-end moments M = FEM ; M = FEM AB AB BA BA FEM AB w w = ; FEM BA =+ Example = Pab Pa b =+ Example FEM AB ; FEM BA 3
4 Case B: rotation at A M 4EI EI = θ ; M = θ A Example AB A BA Case C: rotation at B M EI 4EI = θ ; M = θb Example AB B BA Case D: displacement of end B related to end A M AB 6EI 6EI = Δ ; M BA = Δ Example Analsis of Beams Slope-Deflection Method General Procedure: Step 1: Scan the beam and identif the number of (a segments and (b kinematic unknowns. A segment is the portion of the beam between two nodes. Kinematic unknowns are 4
5 those rotations and displacements that are not zero and must be computed. The support or end conditions of the beam will help answer the question. Step : For each segment, generate the two governing equations. Check the end conditions to see whether one of the end rotations is zero or not (it is not possible for both the end rotations and other deflection components to be zero. If there are no element loads, the FEM term is zero. If there are one or more element loads, use the appropriate formula to compute the FEM for each element load and then sum all the FEMs. If one end of the segment displace relative to the other, compute the chord rotation; otherwise it is zero. Step 3: For each kinematic unknown, generate an equilibrium condition using the free-bod diagram. Step 4: Solve for all unknowns b combining all the equations from steps and 3. Now the equations are entirel in terms of the kinematic unknowns. Step 5: Compute the support reactions with appropriate FBDs. Example 1: Compute the support reactions for a beam. (Rajan s book pages , Example 5..1 Example : Compute the support reactions for a continuous beam. (Rajan s book pages , Example 5.. Example 3: Compute the support reactions for a continuous beam where the support B settles 0.5. (Rajan s book pages , Example
6 4. Analsis of Frames without Sideswa Slope-Deflection Method The analsis of frames via the slope-deflection method can also be carried out sstematicall b appling the two governing equations of beams. A sideswa will not occur if (a the frame geometr and loading are smmetric, and (b sideswa is prevented due to supports. Example: Compute the support reactions for a frame. (Rajan s book pages , Example Analsis of Frames with Sideswa Slope-Deflection Method A sideswa will occur if (a the frame geometr and loading are unsmmetrical, and (b sideswa is not prevented due to supports. Example 1: Compute the support reactions for a frame. (Rajan s book pages , Example 5..7 Example : Compute the support reactions for a frame. (Rajan s book pages 30-34, Example
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