How to Analyze Special Compound Trusses?

Structural Analysis

Type of Analysis: Classical Approach, Linear Analysis, Static Loads, Plane Trusses, Determinate Truss

Preparation

Before analyzing a structure, we'll need to make some preparations first. That includes setting up our references and finding their determinacy.

Set-Up References

An excellent structural analysis must have a uniform mathematical understanding of the structure. It ensures that other people can easily understand your results.

We first place a Cartesian grid with its origin defined by our preference. In this case, let's assign the origin \(A(0, 0)\) at the bottom-left joint of the truss. Consequently, the x-axis will run along the length of the truss with the y-axis perpendicular to it at the origin.

We also need to identify the location of all points of interest: the location of supports, joints, and loads:

• \(A(0.0m, 0.0m)\) \(A(0.0', 0.0')\). Hinge support and truss joint.
• \(B(0.0m, 1.83m)\) \(B(0.0', 6.0')\). Truss joint.
• \(C(3.66m, 4.57m)\) \(C(12.0', 15.0')\). Truss joint.
• \(D(7.32m, 4.57m)\) \(D(24.0', 15.0')\). Truss joint and location of \(106.8kN (24kips)\) horizontal load to the right.
• \(E(10.98m, 1.8m)\) \(E(36.0', 6.0')\). Truss joint.
• \(F(10.98m, 0.0m)\) \(F(36.0', 0.0')\). Roller support and truss joint.

You can label each joint according to your preference. The most important thing is that its coordinates must be defined appropriately.

Determinacy

We need to find the structure's determinacy \(D\) to know our approach.

For 2D trusses, it is:

\(D=(m+r)-2j\)

For this truss example, there are 9 members: \(AB\), \(BC\), \(CD\), \(DE\), \(EF\), \(AD\), \(AE\), \(BF\), \(CF\), 6 joints: \(A\), \(B\), \(C\), \(D\), \(E\), \(F\), and 3 reaction components: \(A_h\), \(A_v\), \(R_F\); hence \(D=0\)

\(D=(9+3)-2(6)=0\)

A determinacy of zero indicates that the structure can be thoroughly analyzed using only the equilibrium equations.

Main Analysis

Stability

The first requirement is to know if our structure is externally and internally stable.

Let's examine its external stability first: 

1. The reaction components \(A_h\), \(A_v\), \(R_F\) are not collinear, parallel, or concurrent with each other.
2. The determinacy is equal to zero.

From these observations, we can conclude the truss is externally stable. 

In terms of its internal stability, the truss arrangement doesn't pose any risks of excessive deformation or immediate collapse; hence, it's internally stable.

Since the structure is externally and internally stable, we can proceed with the analysis. If it is unstable, we may have to adjust its model before proceeding.

Reactions

‍The second requirement for a complete analysis is to compute the support loads of the structure. Solving for the components enables us to understand the transfer of loads.

For a determinate structure, solving for its reactions is straightforward. To solve it, always remember that the whole model must obey the laws of equilibrium.

As a demonstration, we first break the support loads into their components (not their resultant) along the axes. In our example, we have three components:

• We assume \(A_h\) to be acting towards the left, and 
• \(A_v\) acting downward, and
• \(R_F\) acting upward.

Then, applying the equilibrium equations:

\(\sum{F_h}=0]\space{\rightarrow_+}\)

\(106.8-A_h=0\)

\(A_h=106.8kN (24kip)\)

\(\sum{M_F}=0]\space{\circlearrowright_+}\)

\(106.8(2.74+1.83)-A_v(3.66+3.66+3.66)=0\)

\(A_v=44.48kN (10kip)\)

\(\sum{F_v}=0]\space{\uparrow_+}\)

\(R_F-44.48=0\)

\(R_F=44.48kN (10kip)\)

From our calculations, the reaction components of the structure are: 

\(A_h=106.8kN (24kip)\)

\(A_v=44.48kN (10kip)\)

\(R_F=44.48kN (10kip)\)

If our answer is negative, the assumed direction is wrong, and the correct one is the opposite.

Force Analysis

The third requirement for a complete analysis is understanding the internal force and stress developed on the structure due to the applied loads.

We have four types of stresses to analyze: axial, shear, moment, and torsion. Typically, we do these by modeling the behavior using functions and diagrams.

In our example, the predominant forces the members will experience are axial.

Modeling Axial Behavior

Our task now is to determine the force developed for each member.

Recall that we can use two methods to solve the member forces:

• The method of joints and,
• The method of sections. 

Normally we would proceed to either of the two methods, as we have demonstrated for a simple truss. However, if we look at our example, we'll discover that each joint has three unknowns making it impossible to use the method of joints from the very start. Likewise, it is hard to identify three truss members that we can cut to use the method of sections. 

To analyze such compound trusses, we first have to find the simple trusses that make up the compound truss and investigate the elements of connection.

Connection Elements

Recall that a compound truss is simply a combination of two or more simple trusses connected by elements of connection. These connections can either be a link or a hinge.

Again, identify all simple trusses first. There are two simple trusses: \(ADE\) and \(BCF\). It leaves us with three elements of connection: members \(AB\), \(CD\), and \(EF\).

Stability of Connection Elements

In identifying the connection elements, we must also analyze if the connection between simple trusses is stable. The connection elements must also obey the basic external stability rules:

1. All component forces must not be collinear, parallel, or concurrent.
2. There must be at least 3 component connections.

In our truss example, our links do not violate these rules; hence, it is stable. This step ensures that we keep key concepts intact when solving our truss.

Use the Method of Sections for Connection Elements

After identifying the elements of connection and analyzing if it is stable, we use the method of sections to solve for the bar forces developed in these connection elements. Place the cutting plane so that at least we reveal one of these internal forces.

Continue with the solution and analyze one part of the truss and solve for the bar forces:

\(\sum{M}_G=0]\circlearrowleft_+\)

\(106.8(2.74+1.83)+F_{EF}(3.66+3.66+3.66)=0\)

\(F_{EF}=-44.45kN (C)\)

\(\sum{M}_H=0]\circlearrowleft_+\)

\(F_{AB}(3.66+3.66+3.66)-44.45(3.66+3.66+3.66)+106.8(2.74+1.83)=0\)

\(F_{AB}=0kN\)

\(\sum{F_h}=0]\space{\leftarrow_+}\)

\(106.8-106.8-F_{DC}=0\)

\(F_{DC}=0kN\)

Once you have solved for the internal force on one connection element, it becomes possible to solve for the bar forces of the entire compound truss structure. You can use the method of joints or sections to continue with the solution.

The following section shows the summarized results for each truss member.The author will post a complete solution soon.

Axial

Table

Usually, we summarize the results in a tabular format:

‍

For easy reference, it is best to categorize each member - top chord, bottom chords, and web members.

Diagram

Sometimes, we may have a visual aid of the bar forces acting per member using axial diagrams:

‍

These diagrams are helpful if we deal with members with varying cross sections or material along its length, for example.

Deflections

The final requirement is to analyze the structure's deflection. In this part, we analyze the translations and rotations of the object from their original position.

We have two types to analyze: rotation and translation. As in the previous part, we can describe their behavior using functions and diagrams.

The topic of deflection deserves a separate section. There are many ways how to explain a structure's movement, such as:

• Virtual Work Method

The author will post the mathematical solution for this example soon.