# Shape Functions of beam element with 3 nodes (quadratic element)

For a two node beam element there are four shape functions for four degree of freedom: For a straight three node beam element how shape functions are?

Please note that beam is straight and not curved.

• Is the three-node beam straight or curved? – alephzero Feb 10 '19 at 14:05
• This is straight 3 node beam, also added definition to the question. thanks – epsi1on Feb 10 '19 at 14:15

## 1 Answer

If the nodes are at $$\xi = -1, 0, +1$$ you can find the shape functions using Lagrangian polynomial interpolation.

In fact you don't need to work through the general procedure, since you can write down the general form the shape functions must take with only a few unknown parameters, and then solve for the unknown values.

Consider the shape functions that are non-zero, and have non-zero derivative, at $$\xi = 1$$. They must be zero, and have zero derivatives, at $$\xi = -1, 0$$. Therefore they must have the general form $$N(\xi) = (\xi+1)^2 \xi^2 (a\xi + b)$$ for some values of $$a$$ and $$b$$. Using the product rule, the derivative is $$N'(\xi) = 2(\xi+1)\xi(2\xi + 1)(a\xi + b) + (\xi+1)^2 \xi^2 a.$$ At $$\xi = 1$$ we therefore have \begin{align}N(1) &= 4a + 4b \\ N'(1) &= 16a + 12b\end{align}

For one shape function we want $$N(1) = 1, N'(1) = 0$$ and for the other, $$N(1) = 0, N'(1) = 1$$.

Solving the simultaneous equations for $$a$$ and $$b$$ gives the two shape functions as $$\begin{gather}(\xi+1)^2 \xi^2 (-3\xi+4)/4 \\ (\xi+1)^2 \xi^2 (\xi-1)/4 \end{gather}$$ You can get the shape functions that are nonzero at $$\xi = -1$$ simply by changing $$\xi$$ to $$-\xi$$ (be careful with the sign of the non-zero slope shape function!)

The shape functions for the middle node can be found in the same way, but if you see what is going on you can just write them down by inspection: $$\begin{gather}(\xi-1)^2(\xi+1)^2 \\ (\xi-1)^2 \xi (\xi+1)^2 \end{gather}$$

• thanks for answer, is it Lagrangian or Hermit interpolation? – epsi1on Feb 10 '19 at 17:48
• i'm updating my question based on your answer. – epsi1on Feb 10 '19 at 17:57
• I suppose the nearest thing in Wikipedia is actually en.wikipedia.org/wiki/…. But personally I don't bother too much about what numerical methods are called! – alephzero Feb 10 '19 at 19:22