Slope Deflection Method
Slope deflection method
From Wikipedia, the free encyclopedia
The slope deflection method is a structural analysis method for beams and frames introduced in 1914 by George A. Maney.[1] The slope deflection method was widely used for more than a decade until the moment distribution method was developed.
Contents
[hide]
1 Introduction
2 Slope deflection equations
2.1 Derivation of slope deflection equations
3 Equilibrium conditions
3.1 Joint equilibrium
3.2 Shear equilibrium
4 Example
4.1 Degrees of freedom
4.2 Fixed end moments
4.3 Slope deflection equations
4.4 Joint equilibrium equations
4.5 Rotation angles
4.6 Member end moments
5 See also
6 Notes
7 References
Introduction[edit]
By forming slope deflection equations and applying joint and shear equilibrium conditions, the rotation angles (or the slope angles) are calculated. Substituting them back into the slope deflection equations, member end moments are readily determined.
Slope deflection equations[edit]
The slope deflection equations can also be written using the stiffness factor and the chord rotation :
Derivation of slope deflection equations[edit]
When a simple beam of length and flexural rigidity is loaded at each end with clockwise moments and , member end rotations occur in the same direction. These rotation angles can be calculated using the unit dummy force method or Darcy's Law.
Rearranging these equations, the slope deflection equations are derived.
Equilibrium conditions[edit]
Joint equilibrium[edit]
Joint equilibrium conditions imply that each joint with a degree of freedom should have no unbalanced moments i.e. be in equilibrium. Therefore,
Here, are the member end moments, are the fixed end moments, and are the external moments directly applied at the joint.
Shear equilibrium[edit]
When there are chord rotations in a frame, additional equilibrium conditions, namely the shear equilibrium conditions need to be taken into account.
Example[edit]
Example
The statically indeterminate beam
From Wikipedia, the free encyclopedia
The slope deflection method is a structural analysis method for beams and frames introduced in 1914 by George A. Maney.[1] The slope deflection method was widely used for more than a decade until the moment distribution method was developed.
Contents
[hide]
1 Introduction
2 Slope deflection equations
2.1 Derivation of slope deflection equations
3 Equilibrium conditions
3.1 Joint equilibrium
3.2 Shear equilibrium
4 Example
4.1 Degrees of freedom
4.2 Fixed end moments
4.3 Slope deflection equations
4.4 Joint equilibrium equations
4.5 Rotation angles
4.6 Member end moments
5 See also
6 Notes
7 References
Introduction[edit]
By forming slope deflection equations and applying joint and shear equilibrium conditions, the rotation angles (or the slope angles) are calculated. Substituting them back into the slope deflection equations, member end moments are readily determined.
Slope deflection equations[edit]
The slope deflection equations can also be written using the stiffness factor and the chord rotation :
Derivation of slope deflection equations[edit]
When a simple beam of length and flexural rigidity is loaded at each end with clockwise moments and , member end rotations occur in the same direction. These rotation angles can be calculated using the unit dummy force method or Darcy's Law.
Rearranging these equations, the slope deflection equations are derived.
Equilibrium conditions[edit]
Joint equilibrium[edit]
Joint equilibrium conditions imply that each joint with a degree of freedom should have no unbalanced moments i.e. be in equilibrium. Therefore,
Here, are the member end moments, are the fixed end moments, and are the external moments directly applied at the joint.
Shear equilibrium[edit]
When there are chord rotations in a frame, additional equilibrium conditions, namely the shear equilibrium conditions need to be taken into account.
Example[edit]
Example
The statically indeterminate beam