Slope deflection method for indeterminate beam

In summary, the author of the given link discusses the equation of the displacement of a beam, which includes terms such as 3 Φ, 2EI / L, 2θa, and 2θB. The purpose of these terms is to explain the equilibrium and moments of the beam, with the equation ultimately being used for solving problems related to beams.
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Homework Statement




In the link at 12:35 , the author show the equation of the displacement of the beam without explaining , can someone try to explain why we need to use 3 Φ , 2EI / L , 2θa and 2θB ? I have no idea at all .

Homework Equations

The Attempt at a Solution

 
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I would guess this was covered in an earlier lecture.
In the left hand picture, we have a beam subject to a moment M at x=0.
For equilibrium, there would also be an upward force F at x=0 and downward at x=L. To balance the moments we have M=FL.
Consider a point at x. The moments from the left of that are Fx-M, so EI y" = Fx-M. Integrating, EI y' = Fx2/2-Mx+constant.
For small angles, tan θ≈θ, so from the boundary condition at x=0, the constant is EI θ11.
Integrating again, EI y = Fx3/6-Mx2/2+ EI θ11x+constant.
The boundary condition (L, 0) gives θ11=ML/(3EI).
 
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What is the slope deflection method?

The slope deflection method is a structural analysis technique used to determine the bending moments and rotations at various points along an indeterminate beam. It takes into account the flexibility of the structure and uses equilibrium equations to calculate the unknown forces and displacements.

When is the slope deflection method used?

The slope deflection method is typically used for analyzing continuous beams, frames, and other indeterminate structures. It is most effective when the number of unknowns is less than or equal to the number of equilibrium equations.

How does the slope deflection method differ from other structural analysis methods?

The slope deflection method differs from other structural analysis methods, such as the moment distribution method, in that it considers the rotation at each joint in addition to the bending moments. This allows for a more accurate analysis of indeterminate structures.

What are the advantages of using the slope deflection method?

One advantage of using the slope deflection method is that it can handle structures with multiple supports and varying stiffness. It also takes into account the actual geometry of the structure, rather than assuming a fixed end rotation. Additionally, it can be used to analyze structures with both fixed and pinned supports.

What are the limitations of the slope deflection method?

Some limitations of the slope deflection method include the assumption of small displacements and rotations, neglecting shear deformations, and the need for manual calculations which can be time-consuming for complex structures. It is also not suitable for structures with changing cross-sections or for structures with large displacements or rotations.

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