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ME323 LECTURE 23
Alex Chortos
Superposition for Quick Solutions to Beams
FBD + Force Balance
Graphical Method
Moment Balance + Integration
Stresses in Beams Deflections in Beams
Superposition Method
This Lecture
▪ Simple determinate problem
▪ Complicated determinate
problem
▪ Indeterminate with one
redundant reaction
▪ Indeterminate with two
redundant reactions.
Method of Superposition
The slope, deflection, reactions, internal shear and bending moment of a beam that simultaneously supports several different loads can be obtained by linear superposition, that is, by addition of the effect of the loads acting separately.
𝑣𝑎 𝑥 = 𝑣𝑏 𝑥 + 𝑣𝑐(𝑥)
Two Point Sources
Appendix EFind the deflection at B.
δ𝐵 = 𝑣𝑏 𝐿 + 𝑣𝑐(𝐿)
𝑣𝑎 𝑥 = 𝑣𝑏 𝑥 + 𝑣𝑐(𝑥)
δ𝐵 =𝑃𝐿3
3𝐸𝐼+
𝑃𝐿2
2
3𝐿 −𝐿2
6𝐸𝐼=21
48
𝑃𝐿3
𝐸𝐼
Example 11.17
Calculate the beam deflections.
Statically Indeterminate Beams
1. Determine the degree of static indeterminacy
2. Break the problem into statically determinate subproblems
3. Write compatibility equations
4. Write force deformation equations
5. Substitute force-deformation equations into the compatibility
equations and solve for unknown reactants.
6. Write superposition equations.
Solving for One Redundant Reaction Force
Calculate the reaction forces.
Reaction Forces and Displacements
𝐴𝑦 =𝑤𝑜𝐿
2
𝑀𝐴 =𝑤𝑜𝐿
2
8
𝑣 𝑥 = 𝑣𝑏 𝑥 + 𝑣𝑐(𝑥)
𝑣 𝑥 =𝑤𝑜
768𝐸𝐼27𝑥2𝐿2 + 57𝐿𝑥3 − 32𝑥4
Solving for Two Redundant Reaction Forces
What constraints do we need to impose to reach the boundary conditions?
𝑣 𝐴 = 0 𝑣 𝐵 = 0
𝑣′ 𝐴 = 0 𝑣′ 𝐵 = 0
Beam Deflections Summary
▪ Write down boundary conditions
▪ Draw a free body diagram of the structure and write
down the equilibrium equations. If determinate, solve
the equilibrium equations.
▪ Divide the beam into sections based on loading
conditions.
▪ For each section, draw a FBD that enables calculation
of the internal resultant (M(x)) in that section of the
beam.
▪ Integrate once to find the slope and twice to find the
deflection.
▪ Implement continuity conditions.
▪ Implement boundary conditions.
Collections of Bending Beams
Huang et al, Nature Communications, 11:2233, 2020.
So Far in This Course
• Axial deformation• Torsional deformation• Bending in beams
Connolly et al, PNAS, 114:51, 2017.