Structures

Part of the Statics course offered by the Division of Applied Mechanics, School of Engineering and the Engineering and Technology Portal

Lecture

Structural engineering relies heavily on the strengths of materials and their ability to withstand forces of tension or compression. When used in conjunction with each other, as in the case of a truss, individual load bearing members both share and transmit loads, enabling the structure to accomplish much more than any individual member could alone.

Plane Trusses

One way of distributing a force across a large distance is by building a Plane Truss, which takes advantage of the principle of Equilibrium to translate forces along a system of interconnecting members.

A Warren Truss

A Pratt Truss

File:Simpletruss.PNG

A Simple Triangle Truss

The simplest truss is a triangle made of three points: A, B and C, and three members: AB, BC and CA. The method of distributing forces amongst many members relies on the engineer's ability to place a tensile or compressive force at any particular location. To properly sum forces at a particular point, one must be able to sum forces into that point and also away from it. Thus, AB and AC are in compression while BC is in tension.

Method of Joints

The principle of equilibrium may be used to solve the loads impinging upon a truss by analyzing the effect of each load at a single point on the truss. Each load is broken down into its <math>\ x</math> and <math>\ y</math> component vectors and these are summed to equal zero at a particular point on the truss (e.g. A). Similarly, each load creates a moment rotating about that same point A and these may also be summed together to equal zero. Therefore, given enough information about the loads applied, any of the other loads may be calculated.

Therefore, the Method of Joints may also be applied to a truss joint to determine the force (compressive or tensile) in each member of the truss. A place where the forces and members meet may become the joint to be examined (A), and the principle of equilibrium is applied, thus solving the question of where the forces are transmitted in that truss around that joint.

Example:
Consider a truss bridge under loads <math>\vec F_1</math>, <math>\vec F_2</math> and <math>\vec F_L</math>. What resultant forces <math>\vec R_1 </math> and <math>\vec R_2 </math> act upon the bottom the bridge if <math>\vec F_1 = 3N</math> and <math>\vec F_2 = 7N</math>? Assume that <math>\vec R_1 </math> and <math>\vec R_2 </math> are <math>\ L</math> meters apart, and <math>\vec F_L</math> is <math>\frac{2}{3}L</math> from <math>\vec R_1 </math>.

File:LoadedTrussBridge.JPG

Solution:
Equilibrium dictates that <math> \sum \vec F = 0 </math> and <math> \sum \vec M = 0 </math>.

We may sum the forces around an arbitrary point A. Lets select the location of <math>\vec F_1</math> for A...

There are no <math>\ x</math> components of the force vectors, so <math> \sum \vec F_x \ = 0 </math>.

However, <math> \sum \vec F_y \ = \vec R_1 + \vec R_2 - \vec F_1 - \vec F_2 - \vec F_L = 0 </math>.

Therefore, <math> \sum \vec F_y \ = \vec R_1 + \vec R_2 - 3N - 7N - \vec F_L = 0 </math> and...

<math> \vec R_1 + \vec R_2 = 10N + \vec F_L</math>    (1)

We may then sum the moments around point A...

Therefore, <math> \sum \vec M_A \ = \vec R_1*\frac{L}{2} - \vec R_2*\frac{L}{2} + 7N*\frac{L}{4} + \vec F_L*\frac{L}{6} = 0</math>. However, this leaves us with too many unknowns to solve independently.

Alternately, we may sum the forces around an arbitrary point B at the location of <math>\vec F_2</math>. This has no effect on the force equilibrium, so next we sum the moments around point B...

Substituting in the values we already have...

Re-arranging...

<math> \vec F_L = 9\vec R_1 - 3\vec R_2 + 9N</math>    (2)

We can combine (2) with (1) from above and get

<math> \vec F_L = \vec R_1 + \vec R_2 - 10N = 9\vec R_1 - 3\vec R_2 + 9N</math>

or

<math>  \vec R_2 = 2\vec R_1 + \frac{19N}{4}</math>

Method of Sections

Space Trusses

Assignments

Activities:

Create an activity

Readings:

Peruse the appropriate sections of Wikibooks:Statics
Introduction to Structural Design, Virginia Tech.

Study guide:

Wikipedia article:Plane Truss
Wikipedia article:Tension
Wikipedia article:Compression
Wikipedia article:Method of Joints
Wikipedia article:Method of Sections