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Mechanics of Materials

Mechanics of Materials. Deformation. Goal:. Load. Factors that affect deformation of a structure. P. P. P. P. Stress: intensity of internal force. Normal Stress ( s ). Definition: stresses that act in a direction perpendicular to the cut surface. s. s. n. n. P. P.

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Mechanics of Materials

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

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

  3. Normal Stress – Example 1 Find stresses at cross sections AA and BB. The cross sectional areas of AA and BB are SAA and SBB respectively. B A P2 P1 P3 A B

  4. Deformation and Normal Strain change in size Deformation: change in shape L0 d P P L Normal Strain: Strain at one point:

  5. Normal Strain – Example 2 Find the total deformation of the structure shown below if the values in the strain gauges are and P2 strain gauge 2 strain gauge 1 P1 P3 L2 L1

  6. Stress-Strain Relationship Review: load deformation stress strain

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

  8. Stress - Stain Diagram - Ductile Material (structural steel) True Diagram Partially Enlarge Diagram • Proportional limit • Yield stress • Ultimate stress

  9. 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.

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

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

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

  13. 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. P2 P1 P3 L2 L1

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

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

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

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

  18. Sign Conventions for Shear Stress Positive faces: outward normal direction is in the positive direction of a coordinate axis. Negative faces: the opposite faces y 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. x z Positive 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. Negative shear stress:

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

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

  21. 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?

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