Moment Expression and Redundant Reaction for Indeterminate Beams

1. From Determinate to Indeterminate Structures

So far, we've dealt with statically determinate beams, where the number of unknown reactions is equal to the number of static equilibrium equations (3 for a 2D beam: $\sum F_x=0,\sum F_y=0,\sum M=0$).

An indeterminate structure has more unknown reactions than available equilibrium equations. For example, a propped cantilever (one fixed end, one roller end) has 4 unknown reactions (3 at the fixed end, 1 at the roller), but still only 3 equilibrium equations. It is "indeterminate to the first degree".

To solve such a problem, we need an additional equation. This is where Castigliano's theorem comes in. The strategy is to select one of the "extra" reactions and treat it as an unknown load. This extra reaction is called the redundant.

The following resource explains this concept well.

The key takeaway is this: we simplify the problem by removing the redundant support and replacing it with an unknown force (the redundant reaction, which we'll call R). This turns the indeterminate structure into a familiar, statically determinate one, but with an unknown load R acting on it.

This image perfectly illustrates the process for a propped cantilever beam.

2. Setting Up M(x) and ∂M/∂R for a Redundant Reaction

Once we have our primary structure with the redundant R acting as a load, the process is very similar to what we did in the last lesson. The main difference is that our unknown variable is now R, not a dummy load P.

Let's walk through a classic example: a propped cantilever of length L with a uniformly distributed load (UDL) w.

Step 1: Identify Indeterminacy and Choose the Redundant

• The beam has a fixed support (3 reactions) and a roller support (1 reaction). Total = 4 reactions.
• Since 4 > 3, the beam is indeterminate to the first degree.
• We will choose the vertical reaction at the roller support as our redundant, R.

Step 2: Define the Primary Structure

• The primary structure is a simple cantilever beam of length L.
• It is subjected to the original UDL w and an unknown upward point load R at the free end.

Step 3: Write the Bending Moment Expression, M(x)

• Let's set our origin x=0 at the propped (roller) end. We make a cut at a distance x from this end.
• The moment at the cut consists of two parts:
  • The moment from the redundant reaction R is +Rx. (Causes tension at the bottom, so it's positive).
  • The moment from the UDL w is -w * x * (x/2) = -wx²/2. (Causes tension at the top, so it's negative).
• The total moment expression is:
  $M(x)=Rx-\frac{w{x}^2}{2}$
  This expression is a function of both the position x and the unknown redundant R.

Step 4: Find the Partial Derivative, ∂M/∂R

• Now, we differentiate M(x) with respect to R. We treat x and w as constants.
  $\frac{\partial M}{\partial R}=\frac{\partial }{\partial R}\left(Rx-\frac{w{x}^2}{2}\right)$
• The derivative of Rx with respect to R is x.
• The derivative of -wx²/2 with respect to R is 0, since it does not contain R.
• So, the partial derivative is:
  $\frac{\partial M}{\partial R}=x$

We have now found the two essential components, M(x) and ∂M/∂R, needed to solve for the redundant R.

3. Example with Multiple Segments

For more complex beams, you might need to write multiple moment expressions for different segments. The procedure is the same, but it requires more bookkeeping. A table is an excellent way to organize your work.

Let's look at an example from a set of lecture notes that demonstrates this systematic approach for a continuous beam over two spans.

In this example, they again choose a vertical reaction (By) as the redundant. The key steps you should have noticed are:

1. Express all reactions in terms of R: Before writing moment expressions, they use the equations of static equilibrium on the whole beam to find Ay and Dy as functions of By (our R). This is a critical first step.
2. Divide the beam into segments: The loading and supports create discontinuities, so the beam is divided into segments AB, BC, and CD.
3. Systematic setup: The table format is extremely useful. For each segment, it clearly defines the origin, limits of integration, the moment expression M(x), and the partial derivative ∂M/∂R. For segment AB (0 ≤ x ≤ 10), for instance:
   • M = (245 - 0.5By)x - 15x²
   • ∂M/∂By = -0.5x

This systematic approach is exactly what is needed for the exam.

The Bridge to the Solution

Why are we doing this? Remember from the last lesson that Castigliano's theorem gives us the deflection at the point of a load: $\delta =\int \frac{1}{EI}M\frac{\partial M}{\partial P}dx$.

In our case, the "load" is the redundant reaction R. We also know something about the deflection at the location of R. For a roller support on a rigid foundation, the vertical deflection is zero. This is our compatibility equation.

So, we can set the deflection equal to zero:
${\delta }_R=\int \frac{1}{EI}M\frac{\partial M}{\partial R}dx=0$
By substituting the expressions for M(x) (which contains R) and ∂M/∂R that we just learned to derive, we get an equation where R is the only unknown. Solving this integral equation gives us the value of the redundant reaction.

Conclusion

Today, we've built the crucial bridge between understanding Castigliano's theorem and applying it to solve indeterminate structures. The ability to correctly set up the problem is the most important part of the process.

Key Takeaways:

• An indeterminate structure has more reactions than static equilibrium equations.
• We make it solvable by choosing one reaction as the redundant (R) and treating it as an unknown load on a determinate primary structure.
• The bending moment expression, M(x), must be written as a function of the external loads and the unknown redundant R.
• The partial derivative, ∂M/∂R, is found by differentiating M(x) with respect to R.
• These two expressions, M(x) and ∂M/∂R, are the key inputs for the compatibility equation $\int M(\partial M/\partial R)dx=0$, which we will use to find the value of R.

In our next lesson, we will put this into practice. We will take the M(x) and ∂M/∂R for a propped cantilever, plug them into the compatibility equation, perform the integration, and solve for the numerical value of the redundant reaction.