The document discusses analysis methods for trusses, frames, and machines. Trusses consist of two-force members and can be analyzed using the method of joints or method of sections. The method of joints uses particle equilibrium at joints to solve for forces. The method of sections uses rigid body equilibrium to solve for forces by cutting the truss into sections. Frames contain multi-force members and can be analyzed by drawing free body diagrams of individual members and applying equilibrium equations. Machines contain multi-force members and are analyzed similarly to frames.

1. ES220 Statics Exam III Review Notes

2. Analysis of Structures Trusses Designed to support loads Consist entirely of two-force members Frames Designed to support loads Include one or more multi-force members Machines Designed to transmit and/or modify forces Include one or more multi-force members

3. Two-Force Members Pinned at both ends (both joints) No applied forces between joints No applied moments Line of action of forces is directed along a line drawn between the two joints

4. Trusses Consist entirely of straight two-force members, connected at joints

5. Trusses Example on board: decompose truss into two-force members and joints, and show how forces meet at joints

6. Trusses: Method of Joints Typically used to find forces in all or several of the members Each joint is a particle Particle equilibrium in 2D: For each joint, we have 2 equations, therefore, we can solve for 2 unknowns Must start process at a joint with only 2 unknown forces

7. Trusses: Method of Joints Find a joint with only two unknown forces First, may need to draw an FBD of the entire truss and find support reactions Draw FBD of selected joint Draw each force along the direction of the member Draw in tension (away from joint) Find angle of force from truss geometry Resolve angled forces into x, y components

8. Trusses: Method of Joints Apply equilibrium, to solve for 2 unknown forces A + sign: the force is tension (T) A – sign: the force is compression (C) Find the next joint that has only 2 unknown forces and repeat the process Typically this is adjacent to the prior joint Repeat with additional joints until all member forces are known. Remember to specify (T) or (C) for each force!

9. Special Cases Zero Force Members:

10. Trusses: Method of Joints Example

11. Trusses: Method of Joints Example

12. Trusses: Method of Sections Typically used when only 1 or a few member forces are needed Since a section consists of multiple joints and multiple members, it is a rigid body Rigid body equilibrium in 2D For each section, have 3 equations, therefore, can solve for 3 unknown forces

13. Trusses: Method of Sections Identify the member(s) for which we will determine the forces Can determine up to 3 forces for each section Create a separable section by cutting through the member(s) May need to cut additional members to cut loose a section Decide which portion of the truss to keep

14. Trusses: Method of Sections Draw a free body diagram of the section Replace cut members with forces directed along the direction of the cut members Draw unknown forces in tension (pointed away from the joint) Include externally applied forces and support reactions First, you may need to draw an FBD of the entire truss to find support reactions Include dimensions needed to sum moments

15. Trusses: Method of Sections Determine angles from truss geometry Resolve angled forces into x, y components Apply equilibrium, to solve for 3 unknown forces A + sign: the force is tension (T) A – sign: the force is compression (C)

16. Trusses: Method of Sections Cut the member for which you are finding the force If possible, choose the other cuts such that the extra unknown forces (but not the force you need) pass through a single joint Sum the moment about that joint Special Case: Finding only one force

17. Trusses: Method of Sections Example

18. Frames Contain at least one multi-force member Objective: Find some or all forces acting on members and/or find support reactions

19. Frames: Method of Analysis Draw a FBD of the entire frame, showing applied loads and support reactions

20. Frames: Method of Analysis Draw a FBD for each member of the frame Find any two-force members Pinned at both ends, no applied moments or loads Draw forces directed along line connecting the two joints

21. Frames: Method of Analysis Multi-force members: Pin connections: x and y reaction forces Where two members connect, draw forces equal and opposite

22. Frames: Method of Analysis Draw forces on connecting members equal and opposite Draw forces due to two-force members in the known direction

23. Frames: Method of Analysis FBD of entire frame is a rigid body A multi-force member is a rigid body For each rigid body, can solve for three unknowns, using 2D equilibrium Using 2 or more of these rigid bodies, can find all the desired forces and reactions

24. Frames: Method of Analysis Example

25. Machines Transmits or modifies forces Contain one or more multi-force members

26. Machines: Method of Analysis Similar to method for frames

27. Machines

28. Machines

29. Machines

30. Machines