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Structural Analysis of Beams and Frames Structures using Stiffness Matrix Dr. Nasrellah Hassan Ahmed
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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”

Nov 03, 2019

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Page 1: 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”

Structural Analysis of Beams and Frames Structures using Stiffness

Matrix

Dr. Nasrellah Hassan Ahmed

Page 2: 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”

• 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).

Page 3: 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”

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.

Page 4: 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”
Page 5: 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”
Page 6: 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”

MEMBER STIFFNESS RELATIONS

Page 7: 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”
Page 8: 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”
Page 9: 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”
Page 10: 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”
Page 11: 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”

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

Page 12: 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”
Page 13: 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”
Page 14: 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”
Page 15: 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”
Page 16: 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”

• 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,

Page 17: 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”

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

Page 18: 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”

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

Page 19: 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”

Fixed End Moments

Page 20: 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”
Page 21: 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”

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

Page 22: 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”

• 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

Page 23: 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”
Page 24: 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”
Page 25: 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”
Page 26: 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”

• 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

Page 27: 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”
Page 28: 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”
Page 29: 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”
Page 30: 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”
Page 31: 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”
Page 32: 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”

PLANE FRAMES

Page 33: 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”
Page 34: 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”
Page 35: 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”
Page 36: 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”