1. Beam Deflection Review (4.3-4.5)
MAE 316 – Strength of Mechanical Components
NC State University Department of Mechanical and Aerospace Engineering
Beam Deflection Review

2. Deflection Due to Bending (4.3)ds dy θ dx y
Slope of the deflection curve
Beam Deflection Review

3. Deflection Due to Bending (4.3)
Assumption 1: θ is small. 1. 2.
Assumption 2: Beam is linearly elastic.
Thus, the differential equation for the deflection curve is:
Beam Deflection Review

4. Deflection Due to Bending (4.3)
Recall:
So we can write:
Deflection curve can be found by integrating
Bending moment equation (2 constants of integration)
Shear-force equation (3 constants of integration)
Load equation (4 constants of integration)
Chosen method depends on which is more convenient.
Beam Deflection Review

5. Method of Superposition (4.5)
Deflection and slope of a beam produced by multiple loads acting simultaneously can be found by superposing the deflections produced by the same loads acting separately.
Reference Appendix A-9 (Beam Deflections and Slopes)
Method of superposition can be applied to statically determinate and statically indeterminate beams.
Beam Deflection Review

6. Method of Superposition (4.5)
Consider the following example:
Find reactions at A and C.
Method 1: Choose MC and RC as redundant.
Method 2: Choose MC and MA as redundant.
Beam Deflection Review

7. Example Problem
For the beam and loading shown, determine (a) the deflection at C, and (b) the slope at end A.
Beam Deflection: Method of Superposition

8. Example Problem

9. Example Problem For the beam shown, determine the reaction at B.
Beam Deflection: Method of Superposition

10. Example Problem

11. Castigliano’s Theorem (4.7-4.10)
MAE 316 – Strength of Mechanical Components
NC State University Department of Mechanical and Aerospace Engineering
Castigliano’s Theorem

12. Castigliano’s Theorem (4.8)
This method is a powerful new way to determine deflections in many types of structures – bars, beams, frames, trusses, curved beams, etc.
We can calculate both horizontal and vertical displacements and rotations (slopes).
There are actually two Castigliano’s Theorems.
The first can be used for structures made of both linear and non-linear elastic materials.
The second is restricted to structures made of linear elastic materials only. This is the one we will use.
Castigliano’s Theorem

13. Castigliano’s Theorem (4.8)
“When forces act on elastic systems subject to small displacements, the displacement corresponding to any force, in the direction of the force, is equal to the partial derivative of the total strain energy w.r.t. that force.”
Where: Fi = Force at i-th application point δi = Displacement at i-th point in the direction of Fi U = Total strain energy
Castigliano’s Theorem

14. Castigliano’s Theorem (4.8)
We can also use this method to find the angle of rotation (θ).
Where:
Mi = Moment at i-th application point
θi = Slope at i-th point resulting from Mi
U = Total strain energy
Castigliano’s Theorem

15. Castigliano’s Theorem (4.8)
General case δ1
U stored in structure F1 F2 F3 Fn δ2 δ3 δn
Castigliano’s Theorem

16. Strain Energy in Common Members (4.7)
Spring k F δ
Note: 
Check:
Castigliano’s Theorem

17. Strain Energy in Common Members (4.7)
Bar subject to axial load F L
A,E
Castigliano’s Theorem

18. Strain Energy in Common Members (4.7)
Shaft subject to torque L
J,G T
Castigliano’s Theorem

19. Strain Energy in Common Members (4.7)
Beam subject to bending L
I,E M
Castigliano’s Theorem

20. Strain Energy in Common Members (4.7)
Beam in direct shear
Castigliano’s Theorem

21. Strain Energy in Common Members (4.7)
Beam in transverse shear L
I,E V
Correction factor for transverse shear (see table 4-1 in textbook)
Castigliano’s Theorem

22. Strain Energy in Common Members (4.7)
For structures with combined loading (or multi- component structures) add up contributions to U.
Castigliano’s Theorem

23. Castigliano’s Theorem - Frame (4.8)
For the structure and loading shown below, determine the vertical deflection at point B. Neglect axial force in the column. P B A L1 L2
E, I
Castigliano’s Theorem

24. Castigliano’s Theorem - Frame (4.8)
For the structure and loading shown below, determine the vertical and horizontal deflection at point B. Neglect axial force in the column. w B A L1 L2
E, I
Castigliano’s Theorem

25. Castigliano’s Theorem – Curved Beam (4.9)
For the structure and loading shown below, determine the vertical and horizontal deflection at point B. Consider the effects of bending only. B A
E, I Fv Fh R
Castigliano’s Theorem

26. Castigliano’s Theorem - Trusses
For the structure and loading shown below, determine the vertical deflection at D and horizontal deflection at C. Let L = 16 ft, h = 6 ft, E = 30 x 103 ksi, P = 18 kips, Atens = 2.5 in2, and Acomp = 5 in2. P A C B D
L/2 h
Castigliano’s Theorem

27. Statically Indeterminate Problems (4.10)
For the structure and loading shown below, find the fixed end reactions. L A B w
Castigliano’s Theorem

28. Statically Indeterminate Problems (4.10)
A curved frame ABC is fixed at one end, hinged at another, and subjected to a concentrated load P, as shown in the figure below. What are the horizontal H and vertical F reactions? Consider bending only.
Castigliano’s Theorem

29. Special Cases: Hollow Tapered Beam
Find the tip deflection for the structure and loading shown below. P dA
dB= 2dA t L x
Castigliano’s Theorem

30. Special Cases: Beam With Spring
For the beam-spring system below, find the deflection at end C. L P A C B k a x
E, I
Castigliano’s Theorem