structural analysis of beams and frames structures using ... · structural analysis of beams and...

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