In this article you will learn how to calculate the forces in the members of a statically determinate truss using the method of joints.

Method of joints
Truss solution manual
Method of Joints - example solution of the task

Method of joints

Method of joints is one of the methods used to solve truss tasks. It is an analytical method considered by many to be the easiest, but at the same time it is labor-intensive with a lot of calculations, especially if the truss has a large number of nodes and members.

Other methods include:

Ritter's analytical-diagram method
Cremona method - plotting

The method involves calculating the normal forces in the bars of a truss by successively separating the nodes ( that is, the points at which the bars come together).

It is important that the node we calculate has a maximum two bars For which we do not know the normal forces. Since we have two equilibrium equations (horizontal forces and vertical forces), we can calculate up to two unknown forces from these equations.

Truss solution manual

Below you will find the recipe, instructions in points for solving the truss:

Designation of nodes - consecutive numbers (1,2,3...) or letters of the alphabet (A,B,C...)
Designation of bars - usually consecutive digits (1,2,3...)
Determination of reaction in supports
Calculation of support reactions from equilibrium equations (Fx, Fy, Mi)
Separation of nodes and calculation of forces in rods from equilibrium equations (Fx, Fy)
Check for the last node - optional
Summary in the table (Rod No. -> Normal force).
Drawing of the truss diagram with the values of the forces in the bars plotted

Method of Joints - example solution of the task

Below I have included a diagram of the truss we will solve. The truss consists of 6 nodes and 9 bars. It is loaded with three concentrated forces P1=2 kN, P2=6 kN and P3=3 kN.

I have labeled the nodes with the numbers 1 through 6. In the supports, I have added the support reaction forces R1 for the sliding pivot support at node 1. H4 and V4 for the non-sliding pivot support at node 4. As a reminder, you will find the types of supports and support reactions in this entry

In the next step, we will calculate the support reactions from the three equilibrium equations.

All examples used in this post were created in my Truss Calculator. In this application you will determine support reactions and normal forces in truss members. Step-by-step calculations, drawings of forces at each node and analytical calculation. Try it !

Once we have calculated the values of the support reactions, we can proceed to the next step, which is to separate the nodes. In our case, we will start with node No. 1. In this node we have two unknown bars bar N1-5 and N1-2. The angle of 45 degrees is due to the geometry of the bar system.

Node 1

The figure above shows the drawing of forces for node 1 and the calculation of forces in bars 1-2 and 1-5. These forces are calculated from the equilibrium conditions sum of horizontal force projections and sum of vertical force projections must be zero. Bar 1-2 is stretched because the value of the force is positive. On the other hand, bar 1-5 is compressed because the value of the force is negative.

The next node we will deal with is node 4. At this node we also have two unknown forces for bar 4-3 and 4-6. Also we use the equilibrium conditions sum of the projections of horizontal forces and sum of the projections of vertical forces.

Node 4

When drawing the forces for subsequent nodes, remember the reactions or external forces applied to the node. A common mistake in node balancing solutions is to omit these forces.

We proceed in the same way with the next nodes, remembering that the maximum number of unknown forces in the bar is two.

Node 3

Node 2

Node 5

After calculating all the normal forces in the bars, we prepare tables with the bar-force combination in the bar - the mode of work tension/compression.

In addition, it is useful to draw the entire truss diagram with the values of the forces in the bars plotted. The red color denotes tension bars and the blue color denotes compression bars. The black color is zero bars - if they exist.