WORCESTER POLYTECHNIC INSTITUTE

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1 WORCESTER POLYTECHNIC INSTITUTE MECHANICAL ENGINEERING DEPARTMENT STRESS ANALYSIS ES-2502, C 2012 Lecture 03: Stress 17 January 2012

2 General information Instructor: Cosme Furlong HL-151 (508) Teaching Assistants: Morteza Khaleghi HL-150 (508) Tatiana Popova

3 Free-body diagrams Operator applies 20-lb to pedal stretching spring by 1.5 in. Schematic representation: Free-body diagram: Actual mechanism

4 Force analysis. Free-body diagrams Number of unknowns? Equilib. Equations? Is this a statically indetermined case? Overall FBD

5 Force analysis. Free-body diagrams

6 Force analysis. Free-body diagrams 3D Number of unknowns? Equilib. Equations? 2D Overall FBD Individual FBD s

7 Arbitrary component under load Internal resultant loading Component is in equilibrium Virtual section

8 M F R F R o M o O Internal resultant loading Equivalent force and moment at the section

9 Internal resultant loading Orthogonal components of the equivalent force and moment

10 Internal resultant loading Equivalent forces and moments at the section Component is in equilibrium

11 Internal resultant loading: example A Determine resultant internal loading acting on the cross section at C of the cantilever shown: Approach: 1) Define free-body diagrams 2) Apply equilibrium equations

12 Internal resultant loading: example A Free-body diagrams (FBDs): Section to the right of C Section to the left of C Use either section to determine internal loadings

13 Internal resultant loading: example B Determine resultant internal loading acting on the cross section at C of the machine shaft shown. Shaft is supported by bearings at A and B, which only exert radial forces on the shaft Approach: 1) Define free-body diagrams 2) Apply equilibrium equations: reactions at bearings 3) Apply equilibrium equations: internal loading

14 Internal resultant loading: example B Overall free-body diagram (FBD) FBD is in2d, why? y x C

15 Internal resultant loading: example B Select and define free-body diagram of section (FBD) C Compute internal loading y x

16 Internal resultant loading: example C Determine the resultant internal torque acting on the cross sections through points C and D. Supports A and B allow free turning of the shaft Approach: 1) Define free-body diagram 2) Apply equilibrium equations

17 Internal resultant loading: example D The force F= 80 lb f acts on the gear tooth. Determine the resultant internal loadings on the root of the tooth, i.e., at the centroid point A of section a-a Approach: 1) Define free-body diagram 2) Apply equilibrium equations

18 Stress. Definition: intensity of internal force: acting on a specific plane passing through a point M F R F R o M o Equivalent force and moment at the section O

19 Stress. Definition: intensity of internal force: acting on a specific plane passing through a point Finite force F acting on a finite area A Definition Normal stress: z lim A0 F z A

20 Stress. Definition: intensity of internal force: acting on a specific plane passing through a point Finite force F acting on a finite area A A Definitions Shear stresses: zx zy lim A0 lim A0 F x A F y A

21 Stress. Definition: intensity of internal force: acting on a specific plane passing through a point Finite force F acting on a finite area A Normal and shear stresses on plane x

22 General state of stresses State of stresses of a finite size cube Further sectioning leads to a stress cube

23 Average normal stress in an axially loaded bar

24 Average normal stress in an axially loaded bar Bar subjected to axial load Internal distribution of forces

25 Average normal stress in an axially loaded bar Internal distribution of forces F Rz F z df da P A A Average normal stress: P A

26 Average normal stress in an axially loaded bar Tensile average normal stress Compressive average normal stress

27 Reading assignment Chapter 1 of textbook Review notes and text: ES2001, ES2501

28 Homework assignment As indicated on webpage of our course

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