Mechanics of Materials Goal:Load Deformation Factors that affect deformation of a structure P PPP Stress: intensity of internal force

Normal Stress (  ) Definition: stresses that act in a direction perpendicular to the cut surface P  n P  n tensile stresses (+) compressive stresses (-) Uniformly distributed stresses: P: normal force acting at the cut surface A: cross sectional area Non-uniformly distributed stresses :

Normal Stress – Example 1 Find stresses at cross sections AA and BB. The cross sectional areas of AA and BB are S AA and S BB respectively. P1P1 P3P3 P2P2 A A B B

Deformation and Normal Strain Deformation: P P L0L0 L  Normal Strain: Strain at one point: change in size change in shape

Normal Strain – Example 2 Find the total deformation of the structure shown below if the values in the strain gauges are and P1P1 P2P2 L1L1 L2L2 P3P3 strain gauge 1 strain gauge 2

Stress-Strain Relationship Review: load stressstrain deformation

Constitutive Law - Stress and Strain Relationship Tensile Test: - Apply load - Measure strain - Plot stress vs. strain curve

Stress - Stain Diagram - Ductile Material (structural steel) True DiagramPartially Enlarge Diagram Proportional limit Yield stress Ultimate stress

Stress - Stain Diagram - Ductile Material Proportional Limit: the largest value of stress for which Hooke’s law may be applied for a given material. Yield Point  ): a critical point, after the yield point, the specimen undergoes a large deformation with a relatively small increase in the applied load. Plastic Deformation: deformation that remains after the load is applied. Ultimate Stress (  ): the maximum stress developed in a material before rupture. Breaking Stress (  ): stress at rupture.

Stress - Strain Diagram - Aluminum Alloy - no noticeable yield point - offset method - yield occurs at offset.

Stress - Strain Diagram - Brittle Material - rupture occurs without noticeable any prior change in the rate of elongation. - no difference between and.

Linear Elasticity Hooke’s Law: Poisson’s ratio: the ratio of the lateral or perpendicular strain to the longitudinal or axial strain.

Stress-Strain Relationship – Example 3 Find the total deformation of the structure shown below. Express the answer in terms of P’s, S’s, L’s and E. P1P1 P2P2 L1L1 L2L2 P3P3

Shear Stress Shear force: force that acts tangential to the surface. Average shear stress: V V

Shear Stress - double shear Shear stress: Bearing stress: F.B.D. of bolt

Shear Stress - single shear F.B.D. of bolt Shear stress: Bearing stress:

Shear Strain Shear strain: changes in shape (angle). V d   L If deformation is small, i.e.,  is small,

Sign Conventions for Shear Stress Positive faces: outward normal direction is in the positive direction of a coordinate axis. Negative faces: the opposite faces Positive shear stress: on a positive face, acts in the positive direction of one of the coordinate axes. on a negative face, acts in the negative direction of one of the coordinate axes. Negative shear stress: on a positive face, acts in the negative direction of one of the coordinate axes. on a negative face, acts in the positive direction of one of the coordinate axes. x y z

Sign Conventions for Shear Strain Positive shear strain: Negative shear stress: If the angle between two positive faces (or two negative faces) are reduced. If the angle between two positive faces (or two negative faces) are increased.

Shear Stress vs. Strain - Hooke’s Law in Shear G: shear modulus or modulus of rigidity

Shear Stress and Strain – Example 4 A punch for making holes in steel plates is shown in the following figure. Assume that a punch having diameter d = 20 mm. is used to punch a hole in a 8 mm. plate, as shown in the cross-sectional view. If a force P = 110kN is required to create the hole, what is the average shear stress in the plate and the average compressive stress in the punch?