*This post shows a sample **linear** structural analysis of a **2D** **determinate** **rigid gable frame** 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 rigid gable frame example. All members are assumed to have a uniform section of the same material.

## Structural Load

The frame has the following static loads:

- A \(20kN (4.50kip)\) concentrated load
- A uniform load of \(10kN/m (0.69kip/ft)\) - acts on a 4m (13.12ft) horizontal projection
- A \(60kN (13.49kip)\) concentrated load
- A \(5kN (1.12kip)\) 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 Frame, Determinate Frame*

### 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.*

##### Global References

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 point of the frame. Consequently, the x-axis will run horizontally with the y-axis perpendicular to it.

We also need to identify the location of all points of interest: the location of supports, change in loads, and differences in the frame's cross-sectional properties. Starting from \(A\) and going around the structure, we identify the following points of interest:

- \(A(0.0m, 0.0m), A(0.0ft, 0.0ft)\). The hinge support.
- \(B(0.0m, 5.0m), B(0.0ft, 16.40ft)\). Frame joint and the \(20kN (4.50kip)\) concentrated load acting to the right.
- \(C(4.0m, 8.0m), C(13.12ft, 26.25ft)\). Frame joint.
- \(D(10.0m, 5.0m), D(32.81ft, 16.40ft)\). Frame joint.
- \(E(10.0m, 0.0m), E(32.81ft, 0.0ft)\). The roller support.
- \(F(6.0m, 7.0m), F(19.69ft, 22.97ft)\). The \(60kN (13.49kip)\) vertical downward concentrated load.
- \(G(10.0m, 2.5m), G(32.81ft, 8.20ft)\). The \(5kN (1.12kip)\) concentrated load acting to the left.
- The uniform distributed load of \(10kN/m (0.69kip/ft)\)
**↓**acts as a projection between joints \(B\) and \(C\).

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

##### Local References

When dealing with structures composed of elements with varying properties and internal forces along their length, we need to create a localized reference system for each. It is a common step, especially when dealing with structural frames.

The first step would be to break the frame into its members.

Then, we analyze each member separately and assign a location of the local origin. Usually, we place it at the left-most part of the member when looking inside the frame. You can put it on the right-most side - there is no hard rule. What's important is consistency, and it must be understandable. The x-axis would go along the length of the member while the y-axis would run perpendicular to it.

Finally, from the selected origin, we label each point of interest.

Let's illustrate how we can create our local system for each member. In our frame example, we have four members: \(AB\), \(BC\), \(CD\), and \(DE\).

In member \(AB\), let the origin be joint \(A\), the local coordinates of the member are:

- \(A(0.0m, 0.0m), A(0.0ft, 0.0ft)\).
- \(B(5.0m, 0.0m), B(16.40ft, 0.0ft)\).

In member \(BC\), let the origin be joint \(B\), the local coordinates of the member are:

- \(B(0.0m, 0.0m), B(0.0ft, 0.0ft)\).
- \(C(5.0m, 0.0m), C(16.40ft, 0.0ft)\).

In member \(CD\), let the origin be joint \(C\), the local coordinates of the member are:

- \(C(0.0m, 0.0m), C(0.0ft, 0.0ft)\).
- \(F(2.236m, 0.0m), F(7.34ft, 0.0ft)\).
- \(D(6.71m, 0.0m), D(22.0ft, 0.0ft)\).

In member \(DE\), let the origin be joint \(D\), the local coordinates of the member are:

- \(D(0.0m, 0.0m), D(0.0ft, 0.0ft)\).
- \(G(2.5m, 0.0m), G(8.20ft, 0.0ft)\).
- \(E(5.0m, 0.0m), E(16.40ft, 0.0ft)\).

#### Determinacy

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

For a 2D frame, it is:

\(D=(3m+r)-(3j+c)\)

For this frame example, there are four members: \(AB\), \(BC\), \(CD\), \(DE\), five joints: \(A\), \(B\), \(C\), \(D\), \(E\) and three reaction components: \(A_h\), \(A_v\), \(R_E\); hence \(D=0\)

\(D=(3(4)+3)-(3(5)+0)=0\)

*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 frame is externally stable. In terms of its internal stability, the frame 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 (to the global system):

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

\(20-5-A_h=0\)

\(A_h=15kN (3.37kip)\)

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

\(20(5)+10(4)(\frac{1}{2}\times4)+60(4+2)-5(2.5)-R_E(4+6)=0\)

\(R_E=52.75kN (11.86kip)\)

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

\(A_v+52.75-60-10(4)=0\)

\(A_v=47.25kN (10.62kip)\)

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

\(A_h=15kN (3.37kip)\)

\(A_v=47.25kN (10.62kip)\)

\(R_E=52.75kN (11.86kip)\)

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 2D frames, we limit to axial, shear, and moment only.

##### Modeling Axial, Shear, and Moment Behavior

We can use axial, shear, and moment equations and graphs to describe the internal behavior of each frame member.

Creating these equations and diagrams deserves a separate section. Learn more about how to construct these using these individual posts:

##### Local System

The following section shows each frame member's axial, shear, and moment behavior in the local system. *You can view the complete solution in this **link**.*

##### Member AB

##### Member BC

##### Member CD

##### Member DE

##### Global System

The following section shows all frame members' combined axial, shear, and moment diagrams.

#### 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 the following:

*The author will post the mathematical solution for this example soon. In the meantime, we can describe its deflection **qualitatively**.*

- When the member experiences a positive moment, the member concaves upward, locally speaking
- When the member experiences a negative moment, the member concaves downward, locally speaking
- The hinge support restricts the frame's translation at that point but is free to rotate.
- The roller support restricts vertical translation at that point but is free to move horizontally or rotate.
- The joints turned but maintained their rigidity.