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Structural Analysis of Beams and Frames Structures using ... · Structural Analysis of Beams and Frames Structures using Stiffness Matrix Dr. Nasrellah Hassan Ahmed •The term “beam”

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  • Structural Analysis of Beams and Frames Structures using Stiffness

    Matrix

    Dr. Nasrellah Hassan Ahmed

  • • The term “beam” is used herein to refer to a long straight structure, which is supported and loaded in such a way that all the external forces and couples (including reactions) acting on it lie in a plane of symmetry of its cross-section, with all the forces perpendicular to its centroidal axis. Under the action of external loads, beams are subjected only to bending moments and shear forces

    (but no axial forces).

  • ANALYTICAL MODEL

    • For analysis by the matrix stiffness method, the continuous beam is modeled as a series of straight prismatic members connected at their ends to joints, so that the unknown external reactions act only at the joints.

  • MEMBER STIFFNESS RELATIONS

  • Derivation of Member Stiffness Matrix k

    • Various classical methods of structural analysis, such as the method of consistent

    deformations and the slope-deflection equations, can be used to determine

    the expressions for the stiffness coefficients kij in terms of member length and its flexural rigidity, EI

  • • Since the foregoing equation describes the variation of displacement along the member’s length due to a unit value of the end displacement u1, while all other end displacements are zero, it represents the member shape function N1; that is,

  • • By using same procedure followed, the all stiffness matrix coefficients can be obtained.

  • MEMBER FIXED-END FORCES DUE TO LOADS

    • As the foregoing relationship indicates, the total forces Q that can develop at the ends of a member can be expressed as the sum of the forces ku due to the end displacements u, and the fixed-end forces Qf that would develop at the member ends due to external loads if both member ends were fixed against translations and rotations

  • Fixed End Moments

  • • Determine the fixed-joint force vector and the equivalent joint load vector for the propped-cantilever beam shown in the Fig

  • • Determine the joint displacements, member end forces, and support reactions for the

    • three-span continuous beam shown in Fig. below, using the matrix stiffness method

  • • Determine the joint displacements, member end forces, and support reactions for the

    • three-span continuous beam shown in Fig. below, using the matrix stiffness method

  • PLANE FRAMES