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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
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.
Beam Deflection by Superposition
Table A-9
Roark’s Formulas for Stress & Strain
Dr. Mohammad Suliman Abuhaiba, PE
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.
Dr. Mohammad Suliman Abuhaiba, PE
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
Dr. Mohammad Suliman Abuhaiba, PE
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?
Dr. Mohammad Suliman Abuhaiba, PE
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
Dr. Mohammad Suliman Abuhaiba, PE
September 26, 20158
Common Deflection Equations
Dr. Mohammad Suliman Abuhaiba, PE
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
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.
Dr. Mohammad Suliman Abuhaiba, PE
Fig. 4–16