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Strain energy, shape functions, minimum potential energy principle

Why systems of nature assume states with the lowest energies is a question that even automotive engineers have to contemplate with. The minimum potential energy principle is used in Finite Element structural Analysis.

Pierre de Fermat’s least-time principle for light states that light will travel through an optical system in such a way as to pass from starting to ending point in the least amount of time. Maybe the light has a mind of its own?

Example no.2 Tautochrone curve is the curve on which a ball sliding without friction, under the influence of gravity, takes the same exact amount of time no matter which point you place it on the curve. (Wiki) This is also a consequence of the minimum potential energy principle.

What is the mathematical formulation that describes this principle of least action?

\prod = U- W

Strain energy:

When an external force is applied on a body and the body deforms, the energy of the body is said to have reached a higher level. This higher energy level is termed as strain energy by mechanical engineers. Note that there exists two types of strain energies:

  • Elastic strain energy
  • Inelastic strain energy

 

Strain energy (2)

 

When a bar is loaded upto a certain point B on the load-displacement curve and the force then slowly removed (static load implies no inertial effects), a certain part of the elasticity is still retained in the bar. This is referred to as ‘elastic strain energy’.

 

Potential energy:

As far I understood it, in structural analysis, any body that is elastic is said to have potential energy.

\prod = U- W – equation 1

where,

U- strain energy

W- potential energy of loading

W comes from the principle of ”conservation of energy” and is usually of 3 types:

  • Body forces a.k.a distributed forces. It is the force per unit Volume, just like the self-weight of a bar under gravity.
  • Traction forces are frictional forces.
  • Concentrated point loads.

W has to be subtracted from equation 1 because this part of work potential contributes no longer to the potential energy of a system. Just like, you drop a book and it has no more potential energy.

Note:

There are many ways to express strain energy, a few examples are shown below:

Strain energy U = \sum \frac{1}{2}\int \sigma ^{T} \varepsilon A dx

Strain energy U = \frac{1}{2}\left \langle q_1~q_2 \right \rangle \bigl(\begin{smallmatrix}  k_1 & k_2\\  k_3 & k_4  \end{smallmatrix}\bigr) \begin{Bmatrix}  q_1\\  q_2  \end{Bmatrix}

The second equation form above is called the quadratic matrix form. q1,q2 are the displacements at the nodes.

 

 

equations

What do you observe from the above figure?

It is the fact that I haven’t used the ‘method of sections’ to break-up the system and perform the force balance. This is the primary advantage of using the PMPE. It requires no force balance. One arrives at the familiar FEM equation F=KX just by using the strain energy equations.

There is one disadvantage of the above method though. You get to calculate the stiffness matrix of the system from the above method, but how accurate is this stiffness matrix? Notice that the deflections of the elements(spring-mass) are calculated only at the nodal points. What about the deflections happening within the element?

 

Shape functions:

I have always found myself hitting a virtual brick learning wall when encountered with the theory of shape functions. As a student, one first encounters shape functions when reading into in the Rayleigh-Ritz method.

As pointed out in the previous section, the disadvantage of finding the stiffness matrix using the PMPE discretization approach is that deflections are computed only at the nodes. Hence, if one wanted to arrive at a more accurate solution, the number of finite element discretizations would have to be a very large amount. One can make life simpler using something known as shape functions. These are basically mathematical analytical functions that vary between 0 and 1. Why they vary between 0 and 1 is a result of the local coordinate system one chooses (phi from the diagram below). The advantage of using a local coordinate system is that irrespective of what x1 and x2 is, your shape functions would have the same form/shape. (refer to a FEM textbook for the equations on the local coordinate system phi, which in-turn is a function of x.) Equations of the shape functions N are basically functions of phi.

 

shape

Everything begins with the local coordinate system ‘phi’

 

 

So now we have a way of representing the deflections in an element not only at its nodal points, but also at any point in between. Notice that the shape function N1 has the value of 0, exactly where the shape function N2 has the value of 1. This pattern, you can observe even for quadratic shape functions also.

Notice that the above element is a 2node element. In cases of a 3 node element, the shape functions take the form of a quadratic function i.e N1, N2, N3.

 

 

example

Crude example of FE discretization for a beam under axial load

The split is made-up exactly where the tractional force T acts. The areas of the 2 sections can be calculated as averages.

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