This post shows a sample linear structural analysis of a 2D determinate simple truss with static loads using the classical approach.
This example shows units in SI and English. The English system is in parentheses.
In terms of computations, however, it will primarily be in the SI system. The author converted the answer from the SI solution to English to reflect the latter system. It only applies to definite values, not equations.
We will update this post to reflect solutions in the English system soon.
Structural Model
Let's consider a three-segment Warren truss example. All members are assumed to have a uniform section of the same material.
Structural Loads
The truss has the following static loads:
- A \(20kN (4.5kip)\) concentrated load.
- A \(40kN (9.0kip)\) concentrated load.
- A \(10kN (2.25kip)\) concentrated load.
We can see the position and direction of these loads in the following figure. We can talk more about this in preparation.
Structural Analysis
Type of Analysis: Classical Approach, Linear Analysis, Static Loads, Plane Truss, 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:
- \(A(0.0m, 0.0m)\) \(A(0.0ft, 0.0ft)\). Hinge support and truss joint.
- \(B(1.5m, 2.6m)\) \(B(4.9ft, 8.5ft)\). Truss joint and location of \(20kN (4.5kips)\) horizontal load to the right.
- \(C(3.0m, 0.0m)\) \(C(9.8ft, 0.0ft)\). Truss joint and location of \(40kN (9kips)\) vertical downward load.
- \(D(4.5m, 2.6m)\) \(D(14.8ft, 8.5ft)\). Truss joint.
- \(E(6.0m, 0.0m)\) \(E(19.7ft, 0.0ft)\). Roller support and truss joint.
- \(F(7.5m, 2.6m)\) \(F(24.6ft, 8.5ft)\). Truss joint.
- \(G(9.0m, 0.0m)\) \(G(29.5ft, 0.0ft)\). Truss joint and location of \(10kN (2.25kips)\) vertical downward load.
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 a 2D truss, it is:
\(D=(m+r)-2j\)
For this truss example, there are 11 members: \(AB\), \(BC\), \(CD\), \(DE\), \(EF\), \(FG\), \(AG\), \(CE\), \(EG\), \(BD\), \(DF\), 7 joints: \(A\), \(B\), \(C\), \(D\), \(E\), \(F\), \(G\), and 3 reaction components: \(A_h\), \(A_v\), \(R_E\); hence \(D=0\)
\(D=(11+3)-2(7)=0\)
A determinacy of zero indicates that the structure can be 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:
- The reaction components \(A_h, A_v, R_E\) are not collinear, parallel, or concurrent with each other.
- 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
- \(A_v\) and \(R_E\) are acting upwards.
Then, applying the equilibrium equations:
\(\sum{F_h}=0]\space{\rightarrow_+}\)
\(20-A_h=0\)
\(A_h=20kN (4.5kip)\)
\(\sum{M_A}=0]\space{\circlearrowright_+}\)
\(40(3)-R_E(3+3)+10(3+3+3)+20(2.6)=0\)
\(R_E=43.67kN(9.82kip)\)
\(\sum{F_v}=0]\space{\uparrow_+}\)
\(43.67-40-10+A_v=0\)
\(A_v=6.33kN(1.42kip)\)
From our calculations, the reaction components of the structure are:
\(A_h=20kN (4.5kip)\)
\(A_v=6.33kN(1.42kip)\)
\(R_E=43.67kN(9.82kip)\)
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.
We use two primary methods to solve the member forces:
- The method of joints and,
- The method of sections.
Simple trusses, like our example, can be solved using either of the two. Below are links showing the solution to these bar forces:
- Full solution using method of joints
- Full solution using method of sections
The following section shows the summarized results for each truss member.
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:
