Truss Analysis using Method of Joints

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Trusses are used to support roof, strengthen bridges, or support towers. In this article we will learn how to analyze a truss.

Detailed description of the video lecture is given below.

Why Truss Analysis

Why should we analyze a truss?.This is because if you want to design truss members, and its joints properly, you should have clear knowledge of what is the load carried by each members of the truss, under a given loading condition.

Fig.1 Determination of internal forces help in proper design of truss joints and members

Assumptions behind Truss Analysis

Before going to truss analysis,let’s see the main assumptions behind it.

  • Truss members are connected at their ends only, and they are connected by friction-less pins. So you don’t have to consider any secondary bending moment induced do to force of friction.
  • Truss is loaded only at joints.
  • Weight of truss members can be neglected, compared to load acting on the truss.

Nature of load in truss members

Force developed in a truss member is always axial. It can be either tensile, or compressive.

Fig.2 Forces in truss members either tensile or compressive

If a member is under tensile load, this will be the direction of internal force developed .So you can notice that, under tensile load, internal force developed in the member is directed away from the joint. Similarly in case of compressive force, the internal force developed in the member is directed towards the joint.

Fig.3 Direction of internal forces developed in tensile and compressive case

Method of Joints

The most common way to determine forces inside a truss is method of joints. The basic concept of method of joints is that, since the truss is in equilibrium, each joint in truss will also be in equilibrium. The procedure for method of joints is as follows.

  1. Determination of reaction forces

  2. For this purpose we can use 3 equilibrium equations of truss. That is sum of horizontal force is zero, sum of vertical force is zero, and moment acting at any point in truss is zero.

    Fig.4 First step in truss analysis, Determination of reaction forces
  3. Applying equilibrium of joint concept

  4. After determining the reaction forces, next step is to apply concept of equilibrium of joints. Consider a joint, where there are not more than 2 members, in which forces are unknown. 2 unknowns, because, we have got only 2 equations of equilibrium to solve them. Joint is in equilibrium in x direction, and joint is in equilibrium in y direction. So we can solve for both the unknown forces.

    Please note that we cannot use equilibrium of moment at a joint because, moment produced by member forces in a joint is zero, since all forces are passing through same point.

    Fig.5 Concept of equilibrium is applied to a joint, where not more than 2 forces are unknown
  5. Moving to the Next Joint

Once you are done with one joint, you can move to next joint. And do the same analysis there. The procedure is repeated till we have solved all the unknown forces in truss.

Fig.6 After solving forces around one joint, you can move to next joint as shown

Method of Joints – Example

Now let’s do a sample problem. Stress analysis of following structure, which is used to lift weight. One end of that is connected to a roller and other end to a pin.

Fig.7 A sample problem

So first step, determination of reaction forces from free body diagram. Since one end is connected by a roller, reaction force will be purely vertical. At the other end both, vertical and horizontal reaction forces are present. As the whole structure is under static equilibrium, we can use 3 equilibrium equations to solve for 3 components of reaction.

Fig.8 Free-body diagram and determination of reaction forces

Now analysis at each joint. Let’s assume that all members are under tension. So forces are moving away from joint. Internal forces developed and number of unknown forces around each joint is as shown here.

Fig.9 Number of unknown forces around each joint

It is clear that, we could start our analysis, either from joint A, or from joint D, both are having 2 unknowns. Other joints are having 3 unknown forces. Let’s start with joint A. Forces in both members can be solved using the equilibrium formulae.

Fig.10 Forces around Joint A is solved using the 2 equilibrium equations

If sign of any force comes negative, that means that member is under compression.
Now we can move to point B, there number of unknowns are 2 now. Using the same concept we can solve for forces in members 3, and 5. Now the only unknown force remaining is in member 4. Which can be easily solved by considering equilibrium at point C or point D.You have to just by consider any of one equilibrium equation for this.