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Chapter 4 Deflection and Stiffness September 26, 2015 Dr. Mohammad Suliman Abuhaiba, PE 1

Chapter 4site.iugaza.edu.ps/mhaiba/files/2013/09/Ch-4-Review-D... · 2015-09-26 · Example 4-1 Dr. Mohammad Suliman Abuhaiba, PE Fig. 4–2 For the beam in Fig. 4–2, the bending

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Page 1: Chapter 4site.iugaza.edu.ps/mhaiba/files/2013/09/Ch-4-Review-D... · 2015-09-26 · Example 4-1 Dr. Mohammad Suliman Abuhaiba, PE Fig. 4–2 For the beam in Fig. 4–2, the bending

Chapter 4Deflection and Stiffness

September 26, 2015

Dr. Mohammad Suliman Abuhaiba, PE1

Page 2: Chapter 4site.iugaza.edu.ps/mhaiba/files/2013/09/Ch-4-Review-D... · 2015-09-26 · Example 4-1 Dr. Mohammad Suliman Abuhaiba, PE Fig. 4–2 For the beam in Fig. 4–2, the bending

Deflection Due to Bending

Recall Eqs. 3-3 & 3-4

Dr. Mohammad Suliman Abuhaiba, PE

September 26, 20152

Page 3: Chapter 4site.iugaza.edu.ps/mhaiba/files/2013/09/Ch-4-Review-D... · 2015-09-26 · Example 4-1 Dr. Mohammad Suliman Abuhaiba, PE Fig. 4–2 For the beam in Fig. 4–2, the bending

Example 4-1

Dr. Mohammad Suliman Abuhaiba, PE

Fig. 4–2

For the beam in Fig. 4–2, the bending moment equation,

for 0 ≤ x ≤ l, is

Using Eq. (4–12), determine the equations for slope &

deflection of the beam, slopes at ends, and max

deflection.

Page 4: Chapter 4site.iugaza.edu.ps/mhaiba/files/2013/09/Ch-4-Review-D... · 2015-09-26 · Example 4-1 Dr. Mohammad Suliman Abuhaiba, PE Fig. 4–2 For the beam in Fig. 4–2, the bending

Beam Deflection by Superposition

Table A-9

Roark’s Formulas for Stress & Strain

Dr. Mohammad Suliman Abuhaiba, PE

September 26, 20154

Page 5: Chapter 4site.iugaza.edu.ps/mhaiba/files/2013/09/Ch-4-Review-D... · 2015-09-26 · Example 4-1 Dr. Mohammad Suliman Abuhaiba, PE Fig. 4–2 For the beam in Fig. 4–2, the bending

Example 4-2

Consider the uniformly loaded beam with a

concentrated force as shown in Fig. 4–3.

Using superposition, determine the reactions

and deflection as a function of x.

Dr. Mohammad Suliman Abuhaiba, PE

Fig. 4–3

Page 6: Chapter 4site.iugaza.edu.ps/mhaiba/files/2013/09/Ch-4-Review-D... · 2015-09-26 · Example 4-1 Dr. Mohammad Suliman Abuhaiba, PE Fig. 4–2 For the beam in Fig. 4–2, the bending

Forces act on elastic systems subject to

small displacements

Displacement corresponding to any force

along its direction = partial derivative of

total strain energy wrt force

For rotational displacement, in radians,

Castigliano’s Theorem

Dr. Mohammad Suliman Abuhaiba, PE

September 26, 20156

Page 7: Chapter 4site.iugaza.edu.ps/mhaiba/files/2013/09/Ch-4-Review-D... · 2015-09-26 · Example 4-1 Dr. Mohammad Suliman Abuhaiba, PE Fig. 4–2 For the beam in Fig. 4–2, the bending

Example 4-9The cantilever of Ex. 4–8 is a carbon steel bar 10 in

long with a 1-in diameter and is loaded by a force F

= 100 lbf.

a. Find max deflection using Castigliano’s theorem,

including that due to shear.

b. What error is introduced if shear is neglected?

Dr. Mohammad Suliman Abuhaiba, PE

Fig. 4–9

Page 8: Chapter 4site.iugaza.edu.ps/mhaiba/files/2013/09/Ch-4-Review-D... · 2015-09-26 · Example 4-1 Dr. Mohammad Suliman Abuhaiba, PE Fig. 4–2 For the beam in Fig. 4–2, the bending

Utilizing a Fictitious Force

Apply a fictitious force Q at the point, and

in the direction, of the desired deflection.

Set up the equation for total strain energy

including energy due to Q.

Take derivative of total strain energy wrt Q

Set Q to zero

Dr. Mohammad Suliman Abuhaiba, PE

September 26, 20158

Page 9: Chapter 4site.iugaza.edu.ps/mhaiba/files/2013/09/Ch-4-Review-D... · 2015-09-26 · Example 4-1 Dr. Mohammad Suliman Abuhaiba, PE Fig. 4–2 For the beam in Fig. 4–2, the bending

Common Deflection Equations

Dr. Mohammad Suliman Abuhaiba, PE

Page 10: Chapter 4site.iugaza.edu.ps/mhaiba/files/2013/09/Ch-4-Review-D... · 2015-09-26 · Example 4-1 Dr. Mohammad Suliman Abuhaiba, PE Fig. 4–2 For the beam in Fig. 4–2, the bending

Example 4-10Using Castigliano’s method, determine the

deflections of points A and B due to the force F

applied at the end of the step shaft shown in Fig. 4–

10. The second area moments for sections AB and

BC are I1 and 2I1, respectively.

Dr. Mohammad Suliman Abuhaiba, PE

Fig. 4–10

Page 11: Chapter 4site.iugaza.edu.ps/mhaiba/files/2013/09/Ch-4-Review-D... · 2015-09-26 · Example 4-1 Dr. Mohammad Suliman Abuhaiba, PE Fig. 4–2 For the beam in Fig. 4–2, the bending

Procedure 1 for Statically

Indeterminate Problems1. Choose redundant reactions

2. Write equations of static equilibrium for

remaining reactions in terms of applied

loads & redundant reactions.

3. Write deflection equations for points at

locations of redundant reactions in terms

of applied loads and redundant

reactions.

4. Solve equilibrium & deflection equationsDr. Mohammad Suliman Abuhaiba, PE

September 26, 201511

Page 12: Chapter 4site.iugaza.edu.ps/mhaiba/files/2013/09/Ch-4-Review-D... · 2015-09-26 · Example 4-1 Dr. Mohammad Suliman Abuhaiba, PE Fig. 4–2 For the beam in Fig. 4–2, the bending

Example 4-14

The indeterminate beam 11 of Appendix

Table A–9 is reproduced in Fig. 4–16.

Determine the reactions using procedure 1.

Dr. Mohammad Suliman Abuhaiba, PE

Fig. 4–16


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