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# Trusses Frame Machines Analysis

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### Trusses Frame Machines Analysis

1. 1. ES220 Statics Exam III Review Notes
2. 2. Analysis of Structures <ul><li>Trusses </li></ul><ul><ul><li>Designed to support loads </li></ul></ul><ul><ul><li>Consist entirely of two-force members </li></ul></ul><ul><li>Frames </li></ul><ul><ul><li>Designed to support loads </li></ul></ul><ul><ul><li>Include one or more multi-force members </li></ul></ul><ul><li>Machines </li></ul><ul><ul><li>Designed to transmit and/or modify forces </li></ul></ul><ul><ul><li>Include one or more multi-force members </li></ul></ul>
3. 3. Two-Force Members <ul><li>Pinned at both ends (both joints) </li></ul><ul><li>No applied forces between joints </li></ul><ul><li>No applied moments </li></ul><ul><li>Line of action of forces is directed along a line drawn between the two joints </li></ul>
4. 4. Trusses <ul><li>Consist entirely of straight two-force members, connected at joints </li></ul>
5. 5. Trusses <ul><li>Example on board: decompose truss into two-force members and joints, and show how forces meet at joints </li></ul>
6. 6. Trusses: Method of Joints <ul><li>Typically used to find forces in all or several of the members </li></ul><ul><li>Each joint is a particle </li></ul><ul><li>Particle equilibrium in 2D: </li></ul><ul><li>For each joint, we have 2 equations, therefore, we can solve for 2 unknowns </li></ul><ul><li>Must start process at a joint with only 2 unknown forces </li></ul>
7. 7. Trusses: Method of Joints <ul><li>Find a joint with only two unknown forces </li></ul><ul><ul><li>First, may need to draw an FBD of the entire truss and find support reactions </li></ul></ul><ul><li>Draw FBD of selected joint </li></ul><ul><ul><li>Draw each force along the direction of the member </li></ul></ul><ul><ul><ul><li>Draw in tension (away from joint) </li></ul></ul></ul><ul><ul><li>Find angle of force from truss geometry </li></ul></ul><ul><ul><li>Resolve angled forces into x, y components </li></ul></ul>
8. 8. Trusses: Method of Joints <ul><li>Apply equilibrium, to solve for 2 unknown forces </li></ul><ul><ul><li>A + sign: the force is tension (T) </li></ul></ul><ul><ul><li>A – sign: the force is compression (C) </li></ul></ul><ul><li>Find the next joint that has only 2 unknown forces and repeat the process </li></ul><ul><ul><li>Typically this is adjacent to the prior joint </li></ul></ul><ul><li>Repeat with additional joints until all member forces are known. </li></ul><ul><li>Remember to specify (T) or (C) for each force! </li></ul>
9. 9. Special Cases Zero Force Members:
10. 10. Trusses: Method of Joints <ul><li>Example </li></ul>
11. 11. Trusses: Method of Joints <ul><li>Example </li></ul>
12. 12. Trusses: Method of Sections <ul><li>Typically used when only 1 or a few member forces are needed </li></ul><ul><li>Since a section consists of multiple joints and multiple members, it is a rigid body </li></ul><ul><li>Rigid body equilibrium in 2D </li></ul><ul><li>For each section, have 3 equations, therefore, can solve for 3 unknown forces </li></ul>
13. 13. Trusses: Method of Sections <ul><li>Identify the member(s) for which we will determine the forces </li></ul><ul><ul><li>Can determine up to 3 forces for each section </li></ul></ul><ul><li>Create a separable section by cutting through the member(s) </li></ul><ul><ul><li>May need to cut additional members to cut loose a section </li></ul></ul><ul><ul><li>Decide which portion of the truss to keep </li></ul></ul>
14. 14. Trusses: Method of Sections <ul><li>Draw a free body diagram of the section </li></ul><ul><ul><li>Replace cut members with forces directed along the direction of the cut members </li></ul></ul><ul><ul><ul><li>Draw unknown forces in tension (pointed away from the joint) </li></ul></ul></ul><ul><ul><li>Include externally applied forces and support reactions </li></ul></ul><ul><ul><ul><li>First, you may need to draw an FBD of the entire truss to find support reactions </li></ul></ul></ul><ul><ul><li>Include dimensions needed to sum moments </li></ul></ul>
15. 15. Trusses: Method of Sections <ul><li>Determine angles from truss geometry </li></ul><ul><li>Resolve angled forces into x, y components </li></ul><ul><li>Apply equilibrium, to solve for 3 unknown forces </li></ul><ul><ul><li>A + sign: the force is tension (T) </li></ul></ul><ul><ul><li>A – sign: the force is compression (C) </li></ul></ul>
16. 16. Trusses: Method of Sections <ul><li>Cut the member for which you are finding the force </li></ul><ul><li>If possible, choose the other cuts such that the extra unknown forces (but not the force you need) pass through a single joint </li></ul><ul><ul><li>Sum the moment about that joint </li></ul></ul>Special Case: Finding only one force
17. 17. Trusses: Method of Sections <ul><li>Example </li></ul>
18. 18. Frames <ul><li>Contain at least one multi-force member </li></ul><ul><li>Objective: Find some or all forces acting on members and/or find support reactions </li></ul>
19. 19. Frames: Method of Analysis <ul><li>Draw a FBD of the entire frame, showing applied loads and support reactions </li></ul>
20. 20. Frames: Method of Analysis <ul><li>Draw a FBD for each member of the frame </li></ul><ul><li>Find any two-force members </li></ul><ul><ul><li>Pinned at both ends, no applied moments or loads </li></ul></ul><ul><ul><li>Draw forces directed along line connecting the two joints </li></ul></ul>
21. 21. Frames: Method of Analysis <ul><li>Multi-force members: </li></ul><ul><ul><li>Pin connections: x and y reaction forces </li></ul></ul><ul><ul><li>Where two members connect, draw forces equal and opposite </li></ul></ul>
22. 22. Frames: Method of Analysis <ul><li>Draw forces on connecting members equal and opposite </li></ul><ul><li>Draw forces due to two-force members in the known direction </li></ul>
23. 23. Frames: Method of Analysis <ul><li>FBD of entire frame is a rigid body </li></ul><ul><li>A multi-force member is a rigid body </li></ul><ul><li>For each rigid body, can solve for three unknowns, using 2D equilibrium </li></ul><ul><li>Using 2 or more of these rigid bodies, can find all the desired forces and reactions </li></ul>
24. 24. Frames: Method of Analysis <ul><li>Example </li></ul>
25. 25. Machines <ul><li>Transmits or modifies forces </li></ul><ul><li>Contain one or more multi-force members </li></ul>
26. 26. Machines: Method of Analysis <ul><li>Similar to method for frames </li></ul>
27. 27. Machines
28. 28. Machines
29. 29. Machines
30. 30. Machines