Cremona method

From this text, you will learn more about the graphical Cremona Method for calculating forces in truss members and you will find examples of solutions for trusses using this method.

The graphical Cremona Method (Cremona's force diagram) is a technique used for analyzing internal forces in truss members, particularly useful for solving statistically determinate trusses.

Necessary condition for static determinacy

Before we delve into the details of this method, let’s clarify how to check the necessary condition for the static determinacy of a truss.

For forces converging at a single joint, we can write two equilibrium equations: ( \( \sum x=0 , \sum y=0 \) ). Denoting the number of joints by w, the total number of equilibrium equations for the entire truss is 2w.

Three of these equations are used to determine the reactions at the support points, which means that to calculate the forces in the members, there remain 2w−3 equations.

For the static equations to be sufficient to solve the truss, meaning that the truss is statically determinate, the number of internal forces we want to determine must equal 2w−3. Since the number of internal forces is equal to the number of members in the truss, denoting the number of members by p, we obtain the relationship presented in capital letters above.

What the Cremona Method is based on

Solving the truss involves determining the support reactions at the support points and internal forces, which may act in the members in the form of compression or tension.

Each joint of the truss can be considered as a point where a specific number of external and internal forces converge (active forces - externally applied, support reactions, forces in the members).

In the case of a planar system, we have two types of equilibrium conditions:

• Analytical – the sum of the projections of all forces on the x and y axes must equal zero.
• Graphical – the polygon of forces present in the system must close.

The Cremona Method is based on the graphical condition.

Main assumptions and steps of the Cremona Method

To make it easier to understand the subsequent steps, let’s refer to an example, let’s take a look at this truss:

Of course, we check the necessary condition for static determinacy. Is p=2w-3?

p = 9 – number of members,

w = 6 – number of joints,

thus:

9 = 2 * 6 - 3

9 = 9 -> condition met.

Additionally, the truss must also be geometrically stable, but we will not delve into this issue now.

1. Drawing the structure and marking external forces

We start by drawing the truss, marking the external forces (loads and support reactions). Each member in the truss should be marked with symbols to make it easier to refer to during the analysis.

2. Calculating support reactions

From the conditions of statics:

\( \sum x = 0,\quad \sum y = 0,\quad \sum M = 0 \)

Based on these equations, we calculate the support reactions.

For the considered example:

Reactions: \begin{aligned} &\sum{M_A}=0 &-20\cdot 4+H_B\cdot 4-0 && H_B=20 \ kN\\ &\sum{X}=0 & M_A+H_B-15=0 && H_A=-5 \ kN\\ &\sum{Y}=0 & V_A-10+20=0 && V_A=-10 \ kN\\ \end{aligned}

3. Describing the external and internal fields of the truss

We describe (e.g., A, B, C, etc.) the external fields of the truss that separate passive forces (reactions) and active forces – applied from outside, as well as the internal fields of the truss. It does not matter where we place the field "A".

4. Creating the external polygon of forces

We create a polygon of forces consisting of external forces. We check if the polygon of forces closes. We move from field A to field B, C, etc.

Moving from field A to field B, we mark the vector between these fields, i.e., 10 kN in the chosen scale. The start of this vector is described as point A, and the end as point B.
Next, between field B and C, we have a vector of 20 kN. The start of the vector is at point B, and the direction is towards point C.
We mark the vector of 15 kN from point C to point D.
The vector of 20 kN from point D to E.
The vector of 10 kN downwards, from point E to F.
The vector of 5 kN to the left, which closes the polygon of forces – from point F we return to point A. SUCCESS!

5. Creating internal polygons of forces for subsequent joints

For subsequent joints, we create the so-called polygon of forces (vector diagram). We start from the joint where the external forces or support reactions are known, and the unknowns are the forces in only two members that reach this joint.

We draw the forces to scale, in the appropriate directions. A polygon is formed, whose successive segments correspond to the forces acting in the truss members. We continue building the force diagram for subsequent joints until all forces in the truss are determined.

The forces in the members are proportional to the lengths of the segments on the Cremona force diagram. Based on this diagram, you can determine which members are in compression and which are in tension. This is determined by the direction in which we had to move to go from one field to another; if the direction is "towards the joint," then the member is in compression, if "away from the joint," then the member is in tension.

6. Drawing the axial (normal) force diagram

It is best to draw the axial force diagram on the truss immediately after creating the polygon for each joint.

It is hard to understand without an example, so it looks like this in the example:

The basis for building the equilibrium of subsequent joints is the external polygon of forces that we have already drawn. There are only two joints from which we can start analyzing the truss, i.e., only in two joints are there no more than two unknown forces in the members. These are the joints at the bottom on the left and right side, marked as A and F in the first drawing.

Let’s start from the joint at the bottom on the left side

We move through the successive fields constructing the polygon of forces, moving clockwise between the fields.

1. Between field A and G, we have a member whose direction is vertical, so we can mark a dashed line/light gray from point A in a vertical direction, and somewhere on this line must be point G.
2. Between field G and E, we have a horizontal direction, so we can mark the horizontal direction from point E, which we know where it is, and thus find point G.
3. We draw vectors from A to G, from G to E, the lengths of these vectors are, in the accepted scale, the values of the forces in the members, and the direction in which we moved around this joint shows us whether the member is in tension or compression.
4. We mark the calculated (found graphically correctly) forces on the diagram.

Both members are in tension.

Next will be the joint on the left side at the top

We already know the force in the vertical member, the unknowns are the forces in the diagonal and horizontal members, so we can analyze this joint.

1. From field B to H, we mark the horizontal direction, we know where point B is, we look for point H.
2. From field H to G, we mark the direction at the angle at which the diagonal member is between these fields; from the previous step, we know where point G is, so a straight line passes through this point, and thus we find point H.
3. We draw vectors from B to H and from H to G, the lengths of these vectors are, in the accepted scale, the values of the forces in the members, and the direction in which we moved around this joint shows us whether the member is in tension or compression.
4. We mark the calculated (found graphically correctly) forces on the diagram.

And so we proceed through the subsequent joints until we determine the forces in all members.

This is what a complete diagram should look like:
Normal force diagram [kN]



In summary, the Cremona Method is a graphical tool primarily used in the analysis of forces in statically determinate trusses. It involves creating force diagrams for each joint, which allows determining the forces acting in the members.