1. Analysis of Beams in Bending (5.1-5.3)
MAE 314 – Solid Mechanics
Yun Jing
Analysis of Beams in Bending

2. Bending Moment Along a Beam
In this chapter, we will learn how to find the bending moment M along the beam.
M is not necessarily constant; sometimes M is a function of x.
We will also solve for the shear force V(x), which will be used in Chapter 6.
As before, we need a new FBD every time the loading changes. 2 1 1 2 3
Analysis of Beams in Bending

3. Review of Beam Supports
3 equilibrium equations: Σ FY = 0, Σ FX = 0, Σ M = 0
Ignore the horizontal (x-direction) components, because these are axial loading. R1 R3 R1 R3 R1 M1 R2 R2 R2 R1 M1 R1 M1 R3 M2 R1 R3 R3 R4 R2 R2 R4 R2
Analysis of Beams in Bending

4. Analysis of Beams in Bending
Sign Convention
Recall the applied loading results in both a bending moment M and a shear force V.
Positive shear and bending moment
Analysis of Beams in Bending

5. Analysis of Beams in Bending

6. Analysis of Beams in Bending

7. Analysis of Beams in Bending

8. Analysis of Beams in Bending
Example Problem
Draw the shear and bending moment diagrams for the beam and
loading shown, and determine the maximum absolute value (a) of the
shear and (b) of the bending moment.
Analysis of Beams in Bending

9. Relations Between F, V, and M
For beams with more complicated loading, it is helpful to develop a relationship between load, shear and bending moment.
Sum forces in the vertical direction. or
Analysis of Beams in Bending

10. Relations Between F, V, and M
Sum moment about C’.
Neglect (Δx)2 term since it is much smaller than Δx term. or
Analysis of Beams in Bending

11. Analysis of Beams in Bending
Example Problem
Determine (a) the equations of the shear and bending moment curves
for the beam and loading shown, and (b) the maximum absolute value
of the bending moment in the beam.
Analysis of Beams in Bending

12. Analysis of Beams in Bending
Example Problem
Draw the shear and bending-moment diagrams for the beam and loading shown.
Analysis of Beams in Bending

13. Design of Beams in Bending (5.4)
MAE 314 – Solid Mechanics
Yun Jing
Design of Beams in Bending

14. Design of Beams for Bending
Recall the largest normal stress in the beam subject to bending occurs at the surface and can be defined as
A safe design requires the maximum stress is no more than the allowable stress (σmax ≤ σall), so
where
Design of Beams in Bending

15. Design of Beams in Bending
Procedure for Design
Determine the value for σall.
Draw shear and moment diagrams.
From the diagrams, determine the maximum absolute bending moment.
Determine the minimum allowable value Smin.
Use Smin to determine best cross section dimensions.
Timber beam: Smin = bh2/6
Rolled-steel beam: Use Appendix C in textbook
Design of Beams in Bending

16. Design of Beams in Bending
Example Problem
For the beam and loading shown, design the cross section of the beam,
knowing that the grade of timber used has an allowable normal stress of
12 MPa.
P 365 bottom
Design of Beams in Bending

17. Design of Beams in Bending
Example Problem
Knowing that the allowable stress for the steel used is 160 MPa, select
the most economical S-shape beam to support the loading shown.
P 338
Design of Beams in Bending

18. Design of Beams in Bending
Example Problem
A beam is to be made of steel that has an allowable bending stress of 170MPa. Select an appropriate W shape that will carry the loading shown in the figure below
P 338
Design of Beams in Bending