Regardless of their shape, length, supports or loads, all beams bend slightly. However, the extent of this

bending depends hugely on these factors. This is where beam theory comes in, to calculate the bending & deflection of beams.

## Supports

### Simply Supported Beams

- There is a vertical reaction at both ends
- Deflection at each end is zero
- Angle at each end is not necessarily zero
- Maximum deflection is in the centre (when no loads are applied)

### Bi-Support Cantilever

- One end has two vertical reaction forces next to one another, pointing in opposite directions
- Deflection at fixed end is zero
- Deflection at cantilevered end is always maximum
- Angle at fixed end is not necessarily zero

### Built-in Cantilever

- Deflection at built-in end is zero
- Angle at built in end is zero
- Maximum deflection is always at cantilevered end
- The built-in end provided both a moment and a reaction force

### Built-in at Both Ends

- Deflection at both ends is zero
- Angle at both ends is zero
- Maximum deflection is in the middle (when no loads are applied)
- There is a moment and reaction force at both ends

**Remember:** a built in end provides a moment!

## Finding Shear Force & Bending Moments

Often, shear force and bending moment diagrams need to be drawn in order to find the maximum

bending moment in a beam. To do this, the beam is split into regions – areas between two changes

in load, and each region is sectioned in the middle

Asimply supported beam with no loads on it has only one region. Add a point load in the middle,

and there are two regions, one on either side, etc.

The example above has four regions:

- From the left end to , with the distributed load acting throughout
- From to the end of the distributed load
- From the end of the distributed load to
- From to the right end

Note that the arrow labelled only represents the total weight of the distributed load, nothing

changed in terms of load at this point, so there is no new region.

### Sectioning Example & Shear Force/Bending Moment Diagrams

It is important to stick to the **sign convention** whenever solving problems that involve sectioning

beams:

Now let’s add some numbers to the example above and go through the calculations, step by step:

## 1. Taking moments about the left side to find the reaction forces

## 2. Section in the middle of region 1

## 3. Section in the middle of region 2

## 4. Section in the middle of region 3

## 5. Substitute in the region boundaries

### Shear Forces

Left Limit | Right Limit | |
---|---|---|

### Bending Moments

Left Limit | Right Limit | |
---|---|---|

The moment at the right end must also equal zero.

Now we can draw the shear force and bending moment diagrams:

We can clearly see the maximum bending moment is -32 Nm. This is needed to calculate the

maximum stress in the beam, and the deflections.

## Stress & Second Moment of Area

Often, we need to know the maximum stress in a beam, to predict where it will break. This is given

by the equation:

- is the maximum bending moment
- is the distance from the neutral axis of the stress
- is the second moment of area

The units of second moment of area are m⁴.

### Second Moment of Area,

The second moment of area is a property of a cross-section. For rectangles and circles it is given

as:

Second moments of area can be added and subtracted if the **cross sections lie on the sameneutral axis:**

However, if they do not lie on the same neutral axis, the **parallel axis theorem **must be used. Take

the T-section as an example:

First, the overall neutral axis must be found:

Then, the overall second moment of area can be found:

This equation applies for a T-section, where y₁ is above the overall neutral axis and y₂ is

below the overall neutral axis. The squared brackets should be replaced by the distance

between the section’s and the overall neutral axes.

## Deflection of Beams

A measure of the deflection of a beam is the radius of curvature, R. This brings together everything

important about beam theory, through the fundamental equation:

- is the maximum stress in the beam
- is the distance from the neutral axis
- is the maximum bending moment
- is the second moment of area
- is the Young’s modulus
- is the radius of curvature

Often, the second moment of area and Young’s Modulus are combined to give a constant the

structural rigidity, .

Rearranging this equation gives , the second derivative of deflection:

Integrating this gives the angle of deflection, also known as slope:

Integrating again gives the deflection of the beam:

There are two ways of solving for deflection, Macaulay’s method and superposition:

### Macaulay’s Method

This defines the bending moment as a step function in terms of Macaulay brackets (pointy

brackets). Each Macaulay bracket is turned on or off, depending on whether or not it applies to the

part of the beam you are looking at.

Macaulay brackets are integrated as single entities, not using reverse chain rule.

To find the Macaulay function, section the beam in the last region, using Macaulay brackets for

the distances.

Using this beam setup, sectioned at the end, we get:

- The first term in the brackets only applies when
- The second term in the brackets only applies when
- The third term in the bracket always applies

Integrating the above once gives the slope:

Integrating again gives the deflection when :

When integrating the function for the bending moment, the two constants of integration can be

found from the types of support, e.g. at a built-in support, both slope and deflection are zero,

whereas at a simple support, only deflection is zero (see Supports).

Therefore:

### Superposition

Alternatively, you can treat the different loads on a beam individually, find the deflection caused

by each one, and then add these all together. To do this, use standard results:

**Simply Supported Beams:**

**Moment-Held Beams:**

**Built-in Cantilevered Beams:**

**Built-in at both ends:**

Remember that if you want to find the end deflection of a load that does not go all the way to the

end, you need to extrapolate the deflection by multiplying the slope by the length to the end of the

beam:

- To find the shear forces and bending moments along a beam, section it and resolve forces & moments
- The second moment of area for a rectangle is given as
- The second moment of area for a circle is given as
- Second moments of area can be added and subtracted if they share a neutral axis. If they do not, the parallel axis theorem must be used.
- Deflection of beams and maximum bending moments can be calculated with the fundamental beam equation:
- Integrating the radius of curvature gives the slope
- Integrating the slope gives the deflection