The Finite Difference Method

Finite Difference Approximation to Determine the Deflection of a Simply Supported Beam Subjected to a Uniformly Distributed Load, q N/m.

A beam (length L, second moment of area I, Young's modulus E) is divided into 5 elements by 6 nodes (1 at each end and 4 equally spaced along the beam length).
The derivatives are approximated by using the differences of the nodal displacements yi as shown below.

As well as solving with Excel, these equations can also be solved using Matlab:
The two matrices can be input as follows:

A=[5 -4 1 0; -4 6 -4 1; 1 -4 6 -4; 0 1 -4 5];
B=[1; 1; 1; 1]; and the equations solved by the left division command: A\B

ans=

5.0000
8.0000
8.0000
5.0000

As a specific example, the deflections of a steel beam for nodes 1 and 2 (0.2m and 0.4m from the left hand end - by symmetry the deflections at nodes 4 and 3 are the same) E = 210 GPa, 1 m long, 20 mm wide by 40 mm deep, (I = 106.67E-9 m4). with a uniformly distributed load of q = 100 N/m (so Ri = 20 N) were determined using the above equations and also modelled in Pro/Mechanica Structure FEA and later with Solid Works and analysed with COSMOSWorks FEA. The problem was also solved by substituting values in the equation for deflection derived from the analytical solution:

y = q x (l3 - 2 l x2 + x3)/(24 E I)
The results from these four evaluation methods are shown in the table below:

Method of evaluation: Finite differences Pro/Mechanica FEA COSMOSWorks FEA Analytical
Deflection at node 1, mm 0.0357 0.0364 0.037 0.036
Deflection at node 2, mm 0.0571 0.0584 0.0583 0.0554

It should be noted that E for steel in Pro/Mechanica is 209 GPa, 200GPa in COSMOSWorks, and 210 GPa was used in the other two methods. The results are in good agreement - the Pro/Mechanica solution involved 394 equations, whereas the FD approximation used only 4. Such a good results from a very simple FD model are only obtainable for a very simple configuration. Normally far more equations would be needed. The COSMOSWorks model contained over 5000 elements.

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David J Grieve, updated 11th January 2007, original 30th April 2002.