In the real world, beams and shafts are often given more support than necessary which causes it to be indeterminate. A common example is the beam in steel frames of buildings, where it’s fixed on both ends:

In the real world, beams and shafts are often given more support than necessary which causes it to be indeterminate. A common example is the beam in steel frames of buildings, where it’s fixed on both ends:

Adding supports in this manner is actually good (even though its indeterminate), because it makes structures stronger. Therefore knowing how to analyse statically indeterminate beams/shafts will be important for you as future engineers.

In indeterminate scenarios we have more forces than necessary to ensure static equilibrium. To overcome this, we use the *superposition method*,
which states that the net displacement on a point is just the sum of the displacements due to the individual forces acting at that point.

Sounds complicated? Not to worry, let’s look at a simple example to understand this:

The beam above has an extra support at B which is redundant. To fully analyse the beam, we need to reactions A_{y}, B_{y} and C_{y}. Using our standard *equations of equilibrium*,
we can get A_{y} and C_{y} in terms of B_{y} but we still need to solve for B_{y}.

Here’s where superposition comes in. We isolate the support B as such:

That is, for (i) we remove the redundant support at B and consider the resulting displacement (Δ_{Bi}↓). Then in (ii) we consider the displacement caused by the reaction B_{y}
(Δ_{Bii}↑).
Since in the real scenario there is a pin at B (i.e. Δ_{B} = 0), we have our *compatibility equation*:

Using this, we can solve for B_{y} and then get our A_{y} and C_{y}.

The displacements Δ_{Bi}↓ and Δ_{Bii}↓ can be obtained either using double integration or
virtual work. For simplicity, we provide here a list of displacement expressions for common loading conditions:

Alright! Looks like we’re ready. Let’s look at an example now.

Adding supports in this manner is actually good (even though its indeterminate), because it makes structures stronger. Therefore knowing how to analyse statically indeterminate beams/shafts will be important for you as future engineers.

In indeterminate scenarios we have more forces than necessary to ensure static equilibrium. To overcome this, we use the *superposition method*,
which states that the net displacement on a point is just the sum of the displacements due to the individual forces acting at that point.

Sounds complicated? Not to worry, let’s look at a simple example to understand this:

The beam above has an extra support at B which is redundant. To fully analyse the beam, we need to reactions A_{y}, B_{y} and C_{y}. Using our standard *equations of equilibrium*,
we can get A_{y} and C_{y} in terms of B_{y} but we still need to solve for B_{y}.

Here’s where superposition comes in. We isolate the support B as such:

That is, for (i) we remove the redundant support at B and consider the resulting displacement (Δ_{Bi}↓). Then in (ii) we consider the displacement caused by the reaction B_{y}
(Δ_{Bii}↑).
Since in the real scenario there is a pin at B (i.e. Δ_{B} = 0), we have our *compatibility equation*:

Using this, we can solve for B_{y} and then get our A_{y} and C_{y}.

The displacements Δ_{Bi}↓ and Δ_{Bii}↓ can be obtained either using double integration or
virtual work. For simplicity, we provide here a list of displacement expressions for common loading conditions:

Alright! Looks like we’re ready. Let’s look at an example now.