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• Introduction to slope-deflection method. • General procedure of slope-deflection method of analysis. • Derivation of slope-deflection equations. • Work examples on slope-deflection method of analysis: beams and
frames.
Method of Analyzing Indeterminate Structure
• Force Method• In 530314• Primary unknowns Forces and Moments
• Displacement method• The slope-deflection method• Moment distribution method• Primary unknown “Displacment”• Most computer programs used to analyze a wide range of
indeterminate structures.
Displacement method of Analysis• Satisfy equilibrium equations for the structure• Unknown displacements are written in terms of loads by using the
load-displacement relations, then solved for displacements
Degree of Freedom• When a structure is loaded, the node will undergo unknown
displacements.• These displacements are referred to as the “degree of freedom”.
Specify degree of freedom is a necessary 1st step when apply displacement method
• In 2D, each node can have at most 2 linear displacements & 1 rotational displacement.
Degree of Freedom
• The number of these unknowns are referred to as the degree in which the structure is kinematically indeterminate
• Any load applied to the beam will cause node A to rotate.• Node B is completely restricted from moving.
Degree of Freedom
• The beam has nodes at A, B & C. There are 4 degrees of freedom θA, θB, θC, ∆C
• The frame has 3 degrees of freedom θB, θC, ∆B
Slope-Deflection Method
• The slope-deflection method uses displacements as unknowns and is referred to as a displacement method.
• In the slope-deflection method, the moments at the ends of the members are expressed in terms of displacements and end rotations of these ends.
• An important characteristic of the slope-deflection method is that it does not become increasingly complicated to apply as the number of unknowns in the problem increases.
• In the slope-deflection method the individual equations are relatively easy to construct regardless of the number of unknowns
Derivation of Slope-Deflection Eqs
• To derive the general form of the slope-deflection equation, let us consider the typical span AB of the continuous beam shown below when subjected to arbitrary loading.
• The slope-deflection equation can be obtained using the principle of superposition
• By considering separately the moments developed at each support due to θA, θB, ∆, and P
• Assume clockwise is positive
Derivation of Slope-Deflection Eqs
• Moment due to angular displacement @A, θA
• เพื่อท่ีจะหา MAB จะหา disp เราจะใชว้ธีิ conjugate bm method
4
2
AB A
BA A
EIMLEIML
Derivation of Slope-Deflection Eqs
• Moment due to angular displacement @B, θB
• MBA ท่ี apply เพื่อท่ีจะหา angular disp และ reaction MAB ท่ี support เราสามารถเขียนความสมัพนัธ์ไดว้า่