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

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

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.

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

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).

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

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.

Deflection of beam due to bending2 Displacement and load relationship

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

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

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.

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

Strain energy and Castiglino’s Theorem3 Bending Bending shear

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 .

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

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

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

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

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.

Class work Problem 4.45

Homework Problem 4.66