Chapter 4site.iugaza.edu.ps/mhaiba/files/2013/09/Ch-4-Review-D... · 2015-09-26 · Example 4-1 Dr....

Chapter 4Deflection and Stiffness

September 26, 2015

Dr. Mohammad Suliman Abuhaiba, PE1

Deflection Due to Bending

Recall Eqs. 3-3 & 3-4

Dr. Mohammad Suliman Abuhaiba, PE

September 26, 20152

Example 4-1

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.

Beam Deflection by Superposition

Table A-9

Roark’s Formulas for Stress & Strain

September 26, 20154

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.

Fig. 4–3

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

September 26, 20156

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?

Fig. 4–9

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

September 26, 20158

Common Deflection Equations

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.

Fig. 4–10

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

Example 4-14

The indeterminate beam 11 of Appendix

Table A–9 is reproduced in Fig. 4–16.

Determine the reactions using procedure 1.

Fig. 4–16

Chapter 4site.iugaza.edu.ps/mhaiba/files/2013/09/Ch-4-Review-D... · 2015-09-26 · Example 4-1 Dr....

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