Mechanics & Structures University Engineering

Basic Structures

Chief Noter

July 31, 2021
4 min read
Beams Buckling Stress Analysis Structures Supports

This notes sheet looks at structures that are static: they do not move. The equilibrium of forces and
moments are analysed, and their effects on stresses, strains, and deformations.

Joints & Connections
Statical Determinacy
Sectioning Structures
Maximum Loads
Summary

To begin with, a recap of Newton’s Laws of Motion:

An object will stay at rest or constant velocity unless acted upon by a net resultant force.
The resultant force on an object is directly proportional to the rate of change of momentum
of the object, and acts in the same direction (F = ma)
For every action, there is an equal but opposite reaction.
Static Equilibrium
One condition for static equilibrium is that there is no net resultant force acting on the particle, but
another is that there is no net rotational moment:

Static Equilibrium

One condition for static equilibrium is that there is no net resultant force acting on the particle, but
another is that there is no net rotational moment:

$\sum F = 0\quad \quad \sum M = 0$

Free body diagrams are used to analyse systems of moments and forces.

Before any analysis is done, the coordinate system must be defined.

This is to ensure the directions are all dealt with correctly, to avoid sign errors, and helps to
separate systems into horizontal, vertical, and normal components.

Joints & Connections

Different types of supports can withstand different moments and forces:

Built-in Supports

Built in support, built in beam at one end, forces and moments of basic structures

Horizontal and vertical reaction forces
Support moments without rotation

Pin Supports

Horizontal and vertical reaction forces
Cannot resist rotation

Roller/Sliding Support

Rolling support, sliding contact support, basic structures

Can only support a reaction force in one direction (normal to the surface)
Free to move along the surface
Cannot resist rotation

Pin jointed structures, pin joints, roller and pin joints, bride structures, basic structuresg

Pin joints are generally used to represent the connections between different beams in a ‘pin jointed structure’, for example a truss bridge. Therefore, when analysing such structures, we assume the beams are free to rotate at each connection.

Statical Determinacy

If for any particular structure enough forces and moments are known to calculate the remaining unknown forces from equilibriums only, it is said to be statically determinate. If more information is required, the system is statically indeterminate.

To find out of a system is statically determinate, count the number of bars, b, reactions, n, and pins, j. There are three scenarios:

$b + n = 2j$ The system may be statically determinate, though this depends on the positioning of the beams
$b + n > 2j$ The system is statically indeterminate
$b + n < 2j$ The structure is a mechanism

In the system below, there:

4 bars, $b$
6 end reactions, $n$
5 pins, $j$

$4+6=2(5)$ : the system is statically determinate.

Sectioning Structures

Sectioning basic structures, sectioning pin jointed frames

To simplify issues with complex pin jointed structures, section the system across the beams that
you want to find the tension in. This allows you to treat the new, smaller system, as a body with
only external forces.

Maximum Loads

Often, the maximum allowable load on a frame needs to be calculated. This will depend on a
number of factors: tensile/compressive yield in the beams, failure of pins, buckling etc.

Generally, there will be one beam which experiences a greater tensile/compressive force than any
other. Provided that all beams have the same elastic modulus and cross-section (so the same
second moment of area, see beam theory), this beam will be the most likely to fail due to
tensile/compressive yielding.

The beam most likely to fail due to buckling, however, is the compressed beam with the greatest value of $TL^2$ .

The beam is in tension if the force points away from the pin, and compression if the force
points towards the pin.

Tabulating the compressive, tensile and TL² values shows which beam will fail in which condition
for a given load application.

A structure is in static equilibrium if there is neither a net resultant force nor moment acting about it.
Built in supports cannot move or rotate.
Pin supports can rotate
Sliding Supports can rotate and move in the direction of the surface they are on
A system may be statically determinate if
- $b$ is the number of bars
- $n$ is the number of end reactions
- $j$ is the number of pins
Complex structures should be sectioned to simplify and solve them
Failure in beams:
- The beam with the highest tensile/compressive force in it is most likely to fail due to tensile/compressive yielding
- The beam with the highest value of $TL^2$ is the most likely to fail due to buckling