Structural Analysis of Beams and Frames Structures using ... · Structural Analysis of Beams and...
Transcript of Structural Analysis of Beams and Frames Structures using ... · Structural Analysis of Beams and...
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Structural Analysis of Beams and Frames Structures using Stiffness
Matrix
Dr. Nasrellah Hassan Ahmed
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• 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).
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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.
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MEMBER STIFFNESS RELATIONS
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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
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• 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,
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• By using same procedure followed, the all stiffness matrix coefficients can be obtained.
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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
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Fixed End Moments
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• Determine the fixed-joint force vector and the equivalent joint load vector for the propped-cantilever beam shown in the Fig
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• 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
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• 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
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PLANE FRAMES
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