Disclaimer: Please note that quadratic shape functions as described below aren't the way you actually usually would model a beam (I'll return to this in a later post). Still, I think they demonstrate the principle nicely.
The concept of a shape function is one of the most important concepts which needs to be understood when doing finite element analysis of continuous problems. It really is a relatively simple concept to grasp (although I've always hated it when textbooks call something simple). To clarify the meaning of shape functions, I will only show the meaning of the shape functions, and leave finite element analysis out of this post.
The next picture shows three shape functions, by which we can approximate the deflection of a beam. It's important to note that the shape functions have values of 1 at one node, and 0 at the other nodes. This means that when we sum up the shape functions, the value at each of the nodes is purely determined by a single shape function at a time. This is really important to understand, so take a moment to think about it.
To illustrate the principle without finite element analysis: we know how the beam will deflect (in real world cases of finite element analysis, we usually won't). This is assuming we know the material properties. This means that we also know the deflections at points i, j and k. We multiply each shape function by the deflection value at each respective node (note that one of the shape functions will be multiplied by 0). By summing the multiplied shape function back together, we will get a close approximation of the correct deflection.
Important properties of the shape functions
The single most important consequence of using shape functions to approximate some continuous function is that we now have a group of known functions (which are multiplied by some unknown scalar). If we ingrate a function times a constant, the constant stays the same: . This means that we can integrate the continuous function without actually knowing what this constant is, thus converting a continuous problem to a discrete one!
I think an analogy can be drawn to Fourier analysis, or something of the likes (for people involved in DSP). I won't delve into this any deeper, but there's one important point to recognize: the shape function discretization will only take into account low frequencies, leaving high frequency data out. This is important when simulating acoustics or vibration; the shape functions will only be able to carry vibrations accurately up to some frequency. This can be remedied by shrinking the elements or using higher order shape functions. For acoustic propagation in air (for example), a proper element size is often given as being the shortest wavelength examined divided by five or six.