Analysis of Statically Determinate Trusses

A truss is a structure composed of rod members arranged to form one or more triangles. The joints are pinned (do not transmit moments) so that the members must be triangulated. 

Perfect truss

A truss or a frame that doesn't collapse when loaded is called as a perfect truss or a perfect frame.

Example for a perfect truss or a frame is a simple triangle.

The totall number of members in the frame is m , and total joints in the frame is j, then the  condition for perfect truss or a frame is

m=2(j)-3 for simply supported truss

m=2(j) for cantilever truss

If the truss is not satisfied the truss is called imperfect truss. 

Determinacy 

Trusses are statically determinate when the entire bar forces can be determined from the equations of statics alone. Otherwise the truss is statically indeterminate. 

A truss may be statically (externally) determinate or indeterminate with respect to the reactions (less than three or more than 3 reactions )
Therefore in statically determinate pin jointed trusses only two equilibrium equations (ΣFx=0 and ΣFy=0) available at each joint, and the unknown forces are three external support reactions and one internal force in each member, by comparing the total unknowns (3 reactions + m internal forces) with the total number of available equilibrium equations (2j), we have the relation 

m+ 3= 2j for statically determinate truss.

If (m+ 3) is greater than  2j, then the truss is called statically indeterminate

Analysis of Trusses

For truss analysis, it is assumed that:

1. All members are pin-connected. 

2. All Joints are friction less hinges. 

3. External Loads are applied at the joints only. 

4. Stress in each member is constant along its length.

Sign convention

We start by assuming that all members are subjected to  tensile force. A tension member experiences pull forces at both ends of the bar and usually denoted by positive (+ve) sign (Arrow shows away form joint). When a member is experiencing a push force at both ends, then the bar is said to be in compression mode and designated as negative (-ve) sign.(Arrow shows towards to joint)

The objective of truss analysis is to determine the reactions and member forces. The methods used for carrying out the analysis with the equations of equilibrium and by considering only parts of the structure through analyzing its free body diagram to solve the unknowns.

There are two methods of determining internal forces in the members of a truss – 

1) Method of joints and 

2) Method of sections.

Method of Joints :

1. check the truss whether it is perfect or not

2. check determinant and  the external reactions

3. Select a joint with no more than two unknown forces involved and draw the Free Body Diagram (FBD)

4.Solve the unknowns by using  the two equilibrium equations. ΣFx=0 and ΣFy=0

We can assume any unknown member force will be tension. 

If negative value is obtained, this means that the force is opposite in action to that of the assumed direction ie. compression. (A positive answer indicates that the sense is correct, whereas a negative answer indicates that the sense shown on the free-body diagram must be reversed)

5. Proceed to the rest of the joints and again concentrating on joints that have very minimal of unknowns,  Once the forces in one joint are determined, their effects on adjacent joints are known. We then continue solving on successive joints until all members have been found.
6.Check member forces at unused joints with 

∑ Fx = 0 and ∑ Fy = 0, 

7. Tabulate the member forces whether it is in tension (+ve) or compression (-ve) reaction.

Zero Force Members 

Truss analysis may be simplified by determining members with no loading or zero-force. These members may provide stability or be useful if the loading changes.Zero-force members may be determined by inspection of the joints

Case 1: If two members are connected at a joint and there is no external force applied to the joint, the forces in these members will be zero.

Case 2: If three members are connected at a joint and there is no external force applied to the joint and two of the members are co linear, third member have zero force.