ES220 Statics

Exam III Review Notes

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

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

Trusses

• Consist entirely of straight two-force members, connected at joints

Trusses

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

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

0, 0x yF F

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

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!

0, 0x yF F

Special Cases

Zero Force Members:

Trusses: Method of Joints

• Example

Trusses: Method of Joints

• Example

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

0, 0, 0x y zF F M

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

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

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)

0, 0, 0x y zF F M

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

Trusses: Method of Sections

• Example

Frames

• Contain at least one multi-force member

• Objective: Find some or all forces acting on members and/or find support reactions

Frames: Method of Analysis

• Draw a FBD of the entire frame, showing applied loads and support reactions

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

Frames: Method of Analysis

• Multi-force members:– Pin connections: x and y reaction forces– Where two members connect, draw forces

equal and opposite

Frames: Method of Analysis• Draw forces on connecting

members equal and opposite• Draw forces due to two-force

members in the known direction

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

0, 0, 0x y zF F M

Frames: Method of Analysis

• Example

Machines

• Transmits or modifies forces• Contain one or more multi-force members

Machines: Method of Analysis

• Similar to method for frames

Machines

Machines

Machines

Machines