# Truss Stability

## Overview of Truss Stability

Stability in general refers to the resistance to change, deterioration, and displacement. In the structural engineering context, it is important to make the distinction between internal and external stability.[1]

Figure 1: Simple triangle
Figure 2: Simple truss

## Internal Stability

Internal Stability of a truss can be determined by answering the question: "will the structure keep its shape when we disconect it from its supports?".[2] If the answer to this question is yes, then the truss is internally stable.

In order to help answer this question, we can use the following equation:[3]

$m \ge 2j-3$

Where $m$ is the number of members in the truss, and $j$ is the number of joints in the truss.

If the truss fails the above equation, then it is definitely internally unstable. If the truss satisfies the equation, it may be internally stable; provided it satisfies certain geometric requirements.

### Geometric Requirements for Internal Stability

A truss that is comprised solely of triangular member arrangements is known as a simple truss, and is always internally stable. A truss that is comprised of two or more member linked simple trusses is known as a compound truss and internally stable only if its geometric member arrangement provides it with rigid structure. [4] There is no definitive rule or equation for determining the rigidity of the member arrangement in a compound truss. A good method is to work your way around the truss and imagine its response to an attempt to deform its shape. If it is able to maintain its geometry and resist deformation then it is internally stable.

## External Stability

Determining external stability is more straightforward than determining internal stability. A truss is unstable externally if the number of reaction components is less than the number of equations of condition. [5]If the number of reaction components is greater than or equal to the number of equations of condition it may be externally stable provided the support reactions are not parallel or concurrent. [5] Finally with trusses we can relate the number of equations of condition with the amount of joints and members in the structure to produce the following equations:[2]

 $m+r < 2j$ Externally unstable ~Eq. 1 $m+r \ge 2j$ Externally stable provided reactions are not parallel or concurrent ~Eq. 2

Where $m$ is the number of members, $r$ is the number of reaction components, and $j$ is the number of joints in the truss.

Figure 3: Example (a) Truss

Figure 4: Example (b) Truss

## Example Problem

Determine whether the following structures are stable or not:

a)

External Determinacy:
$m + r < 2j$
$r = 5, \; m = 17, \; j = 10$
Therefore,
$22 > 20$.
Then, this structure is stable externally.

Internal Determinacy:
$m \ge 2j-3$
$m = 17, \; j = 10$
Therefore,
$m = 17$, and $2j-3 = 17$
Then, this structure is internally stable.

b)

External Determinacy:
$m + r < 2j$
$r = 3, \; m = 15, \; j = 9$
Therefore,
$18 = 18$.
However, this structure is externally unstable due to parallel reactions.
Internal Determinacy:
$m \ge 2j-3$
$m = 15, \; j = 9$
Therefore,
$m = 15$, and $2j-3 = 15$
This structure is internally stable.

c)

External Determinacy:
$m + r < 2j$ File:Truss Example c
$r = 3, \; m = 14, \; j = 9$
Therefore,
$17 < 18$.
This structure is externally unstable.
Internal Determinacy:
$m \ge 2j-3$
$m = 14, \; j = 9$
Therefore,
$m = 14$, and $2j-3 = 15$
This structure is internally unstable.

## References

1. The Free Dictionary, http://www.thefreedictionary.com/stability
2. Kassimali, A. (2011). Structural Analysis: SI Edition (4th ed.). Stamford, CT: Cengage Learning, 47-50, 103-105.
3. University of Kentucky College of Engineering, "Truss Structures" http://www.engr.uky.edu/~gebland/CE%20382/CE%20382%20Four%20Slides%20per%20Page/L5%20-%20Truss%20Structures.pdf
4. N. M. Holtz, "2: Forces in Statically Determinate Plane Trusses: Stability and Determinacy", http://lms.cee.carleton.ca/notes/3203/SDTrusses/StabilityAndDeterminacy
5. J. Erochko, Cive 3203, Class Lecture, Topic: "Structural Analysis", Carleton University, Ottawa, Ontario, 13/09/13.