1. Chapter 6 Section 3,4 Bending Deformation, Strain and Stress in Beams

2. 6.2 Bending Deformation and Strain
Key Points:
Bending moment causes beam to deform.
X = longitudinal axis
Y = axis of symmetry
Neutral surface – does not undergo a change in length
Figure: 06-20

3. Internal bending moment causes beam to deform.
Figure: 06-21a
Key Points:
Internal bending moment causes beam to deform.
For this case, top fibers in compression, bottom in tension.

4. Neutral surface – no change in length.
Figure: 06-22a-b
Key Points:
Neutral surface – no change in length.
All cross-sections remain plane and perpendicular to longitudinal axis.
Neglect deformation of cros section within its own plane.

5. Radius of curvature Figure: 06-23a-b Length decreases
Length = constant
Figure: 06-23a-b

6. Says normal strain is linear
Maximum at outer surface (where y = c)

7. 6.2 Bending Stress – The Flexure Formula
What about Stress????
Recall from section 6.1:
Figure: 06-24
Therefore, it follows that

8. Figure: 06-26a,b

9. This is the moment of inertia, I
Sum moments about cut:
Figure: c
This is the moment of inertia, I

10. The Flexure Formula: Or in general: Internal bending moment, lb-in
Max bending stress, psi
Distance from NA to outer fiber, in
Moment of inertia, in4
Or in general:

11. Examples: Find maximum moment Find area properties, I and c
Calculate stress

12. WHERE IS BENDING STRESS MAXIMUM???
Answer:
Outer surface (furthest away from Neutral Axis)
Value of x along length where moment is maximum!!

16. Example: The T-shape beam is subjected to the loading below.
Draw shear and moment diagram. Identify location and magnitude of Mmax.
Determine location and magnitude of maximum bending stress and draw stress profile. Is the beam safe if the material is aluminum w/ sy = 15 ksi?
What is the largest internal moment the beam can resist if sallow = 2 ksi?

17. 45K

18. Given: Typical household pliers as shown.
Statics: Example 1 - Pliers
Given: Typical household pliers as shown.
Find: Force applied to wire and force in pin that connects the two parts of the pliers.
Do this for homework.
See solution Link
Side: what is the shear stress in pin and bending stress in handle? SofM

19. Given: Crane structure as shown.
Statics: Example 2 – Crane Structure
Given: Crane structure as shown.
Find: Forces and FBD’s for cables A-B and A-E, boom DEF and post AFC.
Do this for homework.
See solution Link
Side: what is the normal stress in cables (average normal only) and normal stress in boom and post (combined loading)? SofM

20. Example 4: Determine the resultant internal loadings acting on the cross sections located through points D and E of the frame. (1-114)