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- Structural Analysis-2 - SA-2
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**Dr. A.P.J. Abdul Kalam Technical University - AKTU**- Civil Engineering
- 2 Topics
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Governing equations-compatibility equations Force displacement matrix relations- Governing equations-equilibrium equations flexibility Force displacement relations- stiffness matrix All displacement methods follow the above general procedure. The Slope-deflection and moment distribution methods were extensively used for many years before the computer era. In the displacement method of analysis, primary unknowns are joint displacements which are commonly referred to as the degrees of freedom of the structure. It is necessary to consider all the independent degrees of freedom while writing the equilibrium equations.These degrees of freedom are specified at supports, joints and at the free ends. SLOPE DEFLECTION 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 slope-deflection method can be used to analyze statically determinate and indeterminate beams and frames. In this method it is assumed that all deformations are due to bending only. In other words deformations due to axial forces are neglected. In the force method of analysis compatibility equations are written in terms of unknown reactions. It must be noted that all the unknown reactions appear in each of the compatibility equations making it difficult to solve resulting equations. The slope-deflection equations are not that lengthy in comparison. 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 moment-displacement relations, moments are then known. The structure is thus reduced to a determinate structure. The slope-deflection method was originally developed by Heinrich Manderla and Otto Mohr for computing secondary stresses in trusses. The method as used today was presented by G.A.Maney in 1915 for analyzing rigid jointed structures. Fundamental Slope-Deflection Equations: The slope deflection method is so named as it relates the unknown slopes and deflections to the applied load on a structure. In order to develop general form of slope deflection equations, we will consider the typical span AB of a continuous beam which is subjected to arbitrary loading and has a constant EI. We wish to relate the beams internal end moments in terms of its three degrees of freedom, namely its angular displacements and linear displacement which could be caused by relative settlements between the supports. Since we will be developing a formula, moments and angular displacements will be considered positive, when they act clockwise on the span. The linear displacement will be considered positive since this displacement causes the chord of the span and the span’s chord angle to rotate clockwise. The slope deflection equations can be obtained by 5 Under Revision

using principle of superposition by considering separately the moments developed at each supports due to each of the displacements & Case A: fixed-end moments , = , 6 Under Revision

, Case B: rotation at A, (angular displacement at A) Consider node A of the member as shown in figure to rotate while its far end B is fixed. To determine the moment needed to cause the displacement, we will use conjugate beam method. The end shear at A` acts downwards on the beam since is clockwise. , Case C: rotation at B, (angular displacement at B) In a similar manner if the end B of the beam rotates to its final position, while end A is held fixed. We can relate the applied moment to the angular displacement and the reaction moment 7 Under Revision

, Case D: displacement of end B related to end A If the far node B of the member is displaced relative to A so that so that the chord of the member rotates clockwise (positive displacement) .The moment M can be related to displacement by using conjugate beam method. The conjugate beam is free at both the ends as the real beam is fixed supported. Due to displacement of the real beam at B, the moment at the end B` of the conjugate beam must have a magnitude of .Summing moments about B` we have, - By our sign convention the induced moment is negative, since for equilibrium it acts counter clockwise on the member. If the end moments due to the loadings and each displacements are added together, then the resultant moments at the ends can be written as, 8 Under Revision

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