1. Deflection and Stiffness
MTE 427 Machine Design
Deflection and Stiffness
3rd teaching
Pichet PINIT

2. What will be learned today?
Spring and stiffness
Spring rate of Tension, Compression and Torsion
Deflection of Beam due to bending
Double Integration method
Superposition
Strain energy and Castigino’s Theorem
Columns

3. Spring rate and stiffness1
Hooke’s law states that
Spring is a mechanical element that exerts force when deformed.
Materials have their property – elasticity. They act as if they were spring.

4. Spring rate and stiffness2
Equivalent spring
Linear spring
Nonlinear stiffening spring
Nonlinear softening spring

5. Spring rate and stiffness3
Equation
For linear spring, k is constant; then
As seen, the unit of k is the unit of force per distance (N/m).

6. Spring rate and stiffness4
Torsion
Tension
Distance in term of loads
Spring rate or constant

7. Deflection of beam due to bending1
Beam is a member carrying the transversal load
Shafts, axles, cranks, levers, and spring are often treated as beam when designed.

8. Deflection of beam due to bending2
Displacement and load relationship

9. Deflection of beam due to bending3
Double integration method starts from moment equation and then integrates this equation twice and uses the boundary conditions to solve for integration constants to get the deflection equation.
See Ex 4.1 and proof the deflection equation of Table A-9-5

10. Deflection of beam due to bending4
Superposition method resolves the effect of combined loading on a structure by determining the effects of each load separately and adding the result algebraically.
Table A-9-6
Table A-9-7

11. Strain energy and Castiglino’s Theorem1
The external work done on an elastic member in deforming it is transformed into strain energy.
The strain energy is equal to the product of the average load and the deflection.

12. Strain energy and Castiglino’s Theorem2
Tension and compression
Torsion
Direct shear

13. Strain energy and Castiglino’s Theorem3
Bending
Bending shear

14. Strain energy and Castiglino’s Theorem4
Catiglino’s Theorem states that when forces act on elastic system subject to small displacement, the displacement corresponding to any force, in the direction of the force, is equal to the partial derivative of the total strain energy with respect to that force.
Mathematical formula
where is the displacement of the point of application of the force in the direction of This is true for and

15. Columns1
Column is a member carrying the compressive force that is parallel to the axis of the column.
Types of columns
Long column with central loading
Column with eccentric loading
Struts or short column with eccentric loading

16. Columns2 Long column with central loading (Euler column formula)
where is the slenderness ratio.

17. Columns3 Long column with central loading
If is the actual slenderness ratio and is greater than , then use the Euler column formula.

18. Columns4 Column with eccentric loading (Secant column formula)
where is the eccentric ratio.

19. Columns5 Struts or short column with eccentric loading
If is the actual slenderness ratio and is greater than , then use the Secant column formula; otherwise use equation above.

20. Class work
Problem 4.45

21. Homework
Problem 4.66