• ## Potential energy

To see these principles applied, we analyze a very simple structure, a taut rope (see Fig. 1.5).
Imagine that the rope is pulled taut by a horizontal force H and that it carries a distributed load p. The distribution of the vertical force V within the rope and the deflection w of the rope are to be calculated. The deflection w is the solution of the boundary value problem

because it encapsulates the structural mechanics of the rope.
To approximate the deflection w(x) of the rope, the rope is subdividedÂ into four linear elements: see Fig. 1.5. The first and the last node are fixedÂ so that only the three internal nodes can be moved. Between the nodes theÂ deflection is linear, that is the rope is only allowed to assume shapes thatÂ can be expressed in terms of the three unit displacements Ďi(x) of the threeÂ internal nodes (see Fig. 1.5).

The nodal deflections, w1, w2, w3, play the role of weights. They signal how much of each unit deflection is contained in wh.
All these different shapesâlet the numbers w1, w2, w3 vary from ââ to +ââconstitute the so-called trial space Vh.
The space Vh itself is a subset of a greater space, the deformation space V of the rope. The space V contains all deflection curves w(x) that the rope can possibly assume under different loadings during its lifetime. It is obvious that the piecewise linear functions wh in the subset Vh represent only a very small fraction of V .

The next question then is: what values should be chosen for the three nodal deflections w1, w2, w3 of the FE solution? What is the optimal choice?
According to the principle of minimum potential energy, the true deflection w results in the lowest potential energy on V

wins the competition on the small subset Vh â V . Then Î (wh) is as close as
possible to Î (w) on Vh.Â Because each function wh in Vh is uniquely determined by the nodal deflections
wi at the three interior nodes, i.e. the vector w = [w1, w2, w3]T, theÂ potential energy on Vh is a function of these three numbers only

is the best approximation on Vh.

Â

• ## Key points of the FE method

FE method = restriction
Analyzing a structure with finite elements essentially amounts to constraining the structure (see Fig. 1.1), because the structure can only assume those shapes that can be represented by shape functions.

This is an important observation, because the accuracy of an FE solution depends fundamentally on how accurately a program can approximate the influence functions for stresses or displacements. Influence functions are displacements:
they are the response of a structure to certain point loads. The more flexible an FE structure is, the better it can react to such point loads, and hence the better the accuracy of the FE solution; see Fig. 1.2.

FE method = method of substitute load cases It is possible to interpret the FE method as a method of substitute loadings or load cases, because in some sense all an FE program does is to replace the original load with a work-equivalent load, and solve that load case exactly.
The important point is that structures are designed for these substitute loads not for the original loads.

â˘ FE method = projection method
The shadow of a 3-D vector is that vector in the plane with the shortest distance to the tip of the vector.
The FE method is also a projection method, because the FE solution is the shadow of the exact solution when it is projected onto the trial space Vh, where

Vh contains all the deformations the FE structure can undergo. The metric applied in the projection is the strain energy: one chooses that deformation uh in Vh whose distance to the exact solution u measured in units of strain energy is a minimum.
Let u denote the exact equilibrium position of a plate (subjected to some load), and let uh be the FE approximation of this position. Now to correct the FE position, that is, to force the plate into the correct shape, a displacement field
e = u â uh must be added to uh.
Let Ď^e ij and Îľ e^e ij denote the stresses and strains caused by this displacement field e. The FE solution guarantees that the energy needed to correct the FE solution is a minimum

This is equivalent to saying1 that the work needed to force the plate from its position uh into the correct position u is a minimum. The effort cannot be made any smaller on the given mesh.
In a vertical projection the length of a shadow is always less than the length of the original vector (see Besselâs inequality [232]); this implies that The situation is different if a support of a structure is displaced. Then the FE projection is a skew projection (see Sect. 1.38, p. 187), that is, the shadow is longer than the original vector. This means that a greater effort is needed to displace a support of a more rigid structure than of a more flexible structure.
The situation is different if a support of a structure is displaced. Then the FE projection is a skew projection (see Sect. 1.38, p. 187), that is, the shadow is longer than the original vector. This means that a greater effort is needed to displace a support of a more rigid structure than of a more flexible structure.
But it will be seen later that even then a minimum principle still applies.
Because the FE solution is the shadow of the true solution, it cannot be improved on the same mesh. This is also why some load cases cannot be solved on an FE mesh. Each projection has a blind spot; see Fig. 1.3. The equivalent nodal forces at the free nodes cancel and so Ku = 0.

FE method = energy method
An FE program thinks in terms of work and energy. Loads that contribute no work do not exist for an FE program. Nodal forces represent equivalence classes of loads. Loads that contribute the same amount of work are identical for an FE program.
In modern structural analysis, zero is replaced by vanishing work. In classical structural analysis a distributed load p(x) is identical to a second load ph(x) if at each point 0 < x < l of the beam the load is the same:

This is the weak equal sign. If all really means all then of course the weak equal
sign is identical to the strong equal sign. But in all other cases there remains
a specific difference, in that equivalence is established only with regard to a
finite set of virtual displacements Î´w.
Because the FE method is an energy method, problems in which the strain energy is infiniteâtheoretically at leastâcannot be solved with this method;
see Fig. 1.4.
â˘ FE method = method of approximate influence functions
We will see that a mesh is only as good as the influence functions that can be generated on that mesh. According to Bettiâs theorem, the displacement u(x) or the stress Ďx(x) at a point x is the L2-scalar product of the applied load p and the corresponding influence function (Greenâs function)

All an FE program does is to replace the exact Greenâs functions with approximate
Greenâs functions G^h 0 and G^h 1 , respectively. Therefore the error in an FE solution is proportional to the distance between the approximate and the exact Greenâs function:

• ## Introduction to finite elements

In this introductory chapter various aspects of the FE method are studied, initially highlighting the key points.