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# Note of MATRIX ANALYSIS OF STRUCTURES by GARIKAPATI RAMBABU

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Lecture 11: The Stiffness Method Washkewicz College of Engineering Introduction Although the mathematical formulation of the flexibility and stiffness methods are similar, the physical concepts involved are different. We found that in the flexibility method, the unknowns were the redundant actions. In the stiffness method the unknown quantities will be the joint displacements. Hence, the number of unknowns is equal to the degree of kinematic indeterminacy for the stiffness method. Flexibility Method: • Unknown redundant actions {Q} are identified and structure is released • Released structure is statically determinate • Flexibility matrix is formulated and redundant actions {Q} are solved for • Other unknown quantities in the structure are functionally dependent on the redundant actions Stiffness Method: • Unknown joint displacements {D} are identified and structure is restrained • Restrained structure is kinematically determinate, i.e., all displacements are zero • Stiffness matrix is formulated and unknown joint displacements {D} are solved for • Other unknown quantities in the structure are functionally dependent on the displacements. 1

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Lecture 11: The Stiffness Method Washkewicz College of Engineering Actual Beam Restrained Beam #1 Restrained Beam #2 Neglecting axial deformations, the beam to the left is kinematically indeterminate to the first degree. The only unknown is a joint displacement at B, that is the rotation. We alter the beam such that it becomes kinematically determinate by making the rotation θB zero. This is accomplished by making the end B a fixed end. This new beam is then called the restrained structure. Superposition of restrained beams #1 and #2 yields the actual beam. 2

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Lecture 11: The Stiffness Method Washkewicz College of Engineering Due to the uniform load w, the moment 1MB 1 MB wL2   12 is developed in restrained beam #1. The moment 1MB is an action in the restrained structure corresponding to the displacement θB in the actual beam. The actual beam does not have zero rotation at B. Thus for restrained beam #2 an additional couple at B is developed due to the rotation θB. The additional moment is equal in magnitude but opposite in direction to that on the loaded restrained beam. 2MB  4 EI B L Imposing equilibrium at the joint B in the restrained structure M  wL2  12  4 EI B L  0 yields B  wL3 48 EI 3

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Lecture 11: The Stiffness Method Washkewicz College of Engineering In a manner analogous to that developed for the flexibility method, we seek a way to consider the previous simple structure under the effect of a unit load. We also wish to utilize the superposition principle. Both will help develop a systematic approach to structures that have a higher degree of kinematic indeterminacy. The effect of a unit rotation on the previous beam is depicted below Here the moment applied mB will produce a unit rotation at B. Since mB is an action corresponding to the rotation at B and is caused by a unit rotation, then mB is a stiffness coefficient for the restrained structure. The value of mB is mB  4 EI L 4