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.