If you want to obtain Excel spreadsheets for structural analysis of beams, click here to visit our download page.  Read on for information about performing beam analyses via superposition and how Excel spreadsheets can be used for structural analysis of beams.

The equation giving the deflection of a beam with a complicated loading can often be found relatively easily by superposing two or more deflection equations corresponding to simple loadings.  Superposition can be used, however, only if the beam deflections are small, say less than 1/500-th of the beam span.  Fortunately the vast majority of beams designed by structural and mechanical engineers involve deflections this small or smaller, and thus superposition is applicable to a wide range of practical problems.

Background on Superposition for Structural Analysis of Beams Spreadsheets

 Background on Superposition for Structural Analysis of Beams

The theoretical justification for superposition is straightforward.  Consider the differential equation for beam deflection, y(x)

in which w(x) is the load acting on the beam, E isthe elastic modulus of the beam material, I is the moment of inertia of the cross section, and x is a horizontal coordinate, measured from the left end and locating points on the beam.  The deflection function y(x) must satisfy Eq. 1 and also the boundary conditions.  For example, for a beam fixed at both ends, the boundary conditions would be

in which L is the length of the beam.

Now suppose that a load w₁(x) acts on the beam.  Then the deflection y₁(x) of the beam is governed by Eq.1:

Next, remove the load w₁(x) and apply a different load, w₂(x).  Then the deflection y₂(x) of the beam is also governed by Eq.1:

Adding Eqs. 3 and 4 and defining a new function, y₃(x) ≡ y₁(x) + y₂(x), gives

In words, y₃(x) satisfies the differential equation for a beam subjected to the combined loading, w₁(x) plus w₂(x), and, furthermore, y₃(x) can be found by simply adding the deflection equations corresponding to w₁(x) and w₂(x) acting alone (Note that boundary conditions, such as Eq. 2, also are satisfied after superposition).

So why bother with superposition?  Why not just solve Eq. 5 directly for y₃(x)?  Answer: Superposition is in fact not worth bothering about, unless tabulated solutions exist for y₁(x) and y₂(x).  Because if someone else has already solved the differential equations for y₁(x) and y₂(x) (and the solutions are available to you, typically through a published table of solutions) then all you have to do is add their results—you completely avoid the time-consuming, error-prone process of solving the differential equation for y₃(x).

Example Structural Analysis of Beams Calculations

 Example Structural Analysis of Beams Calculations

As an illustration, consider the beam shown in the figure below.

For concreteness, let a₁ = 2 m, a₂ = 3 m, L = 12 m, P₁ = 10 kN, P₂ = 14 kN, E = 200 GPa, and I = 600 000 cm­⁴.

The general result for a single load is given by equation (6) below, which is found in all tables of beam deflection formulas:

The deflection equation is

This equation can be used to give the deflection equation y(x) for our two-load problem through superposition

y(x)  =  y_o(x, 10 kN, 2 m)  +  y_o(x, 14 kN, 9 m)           .                                                          (7)

That is, we apply Eq. 6 twice, once for the 10-kN load acting a = 2 m from the left end, and once for the 14-kN load acting a = 12 m − 3 m = 9 m from the left end.

The forms of Eqs. 6 and 7 are well-suited for implementation in a spreadsheet.  We only have to program a single formula (with an “If” statement) representing Eq. 6, and then we can superpose  the results of that formula once for each concentrated load acting on the beam—no matter how many loads act or where they act.  The same superposition approach can be used to calculate the shear and moment diagrams.Obviously, a similar approach can be used for other tabulated solutions, such as those corresponding to a concentrated moment or distributed load acting on the beam.

Structural Analysis of Beams Spreadsheet

Structural Analysis of Beams Spreadsheet

The screenshot image below shows an Excel spreadsheet for structural analysis of beams.  Specifically, it will calculate the shear and moment diagrams and deflections for two concentrated loads acting on a simply-supported beam.  Note that only the absolute minimum of information is required: the magnitude and location of the loads and the values of E and I.  No nodal numbering, element numbering, boundary condition specification, output specification, and load type must be entered.

 

The workbook of which this spreadsheet is a part contains tabs for one and two concentrated forces, one and two concentrated moments, one and two linearly varying distributed loads, and a combination of all three types of loadings.  The procedure to extend the analysis to other load cases is also presented in a tab.  Because all formulas used in each tab are visible and can be unlocked, userspossessing only a basic knowledge of Excel may easily customize the spreadsheet to meet particular needs and recurrent applications.  This Excel workbook for structural analysis of beams and additional workbooks for other boundary conditions are available in either U.S. or S.I. units at low cost from our download page.

References

1. Manual of Steel Construction, Load & Resistance Factor Design, Volume I, Structural Members, Specifications & Codes, 2^nd Edition, American Institute of Steel Construction, Chicago, IL, American Institute of Steel Construction (1994).

2. Egor P. Popov, Engineering Mechanics of Solids, 2^nd Edition, Prentice Hall, New York, NY (1998).

3. Rossow, Mark, "Using Superposition in a Continuous Beam Analysis Spreadsheet," an online blog article