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For an allowable stress beam design spreadsheet, click here to visit our spreadsheet store. Obtain a convenient, easy to use beam design spreadsheet using allowable stressl design at a reasonable price. Read on for information about the use of deflection limits and serviceability requirements for simply supported beam design.
Design of a simply supported beam with uniform distributed load can be carried out as follows. Based on inputs of span length, elastic modulus, live load, dead load, allowable bending stress, deflection limit for live load and deflection limit for live load and dead load acting simultaneously, the equations in the next section can be used to calculate maximum moment, maximum shear, elastic section modulus, and minimum moments of inertia required to satisfy the constraints on deflection. The equations can also be used to check on whether a known design satisfies strength and deflection requirements.
Equations for the first step in an allowable stress beam design spreadsheet are as follows for a simply supported beam subject to a uniform distributed load:
M_{max} = wL^{2}/8, where
V_{max} = wL/2, where
M_{allow} = SF_{b}, where
y_{max} = 5wL^{4}/(384EI), where
y_{max} < L/L_{d}, where
The image below shows a screenshot of an allowable stress beam design spreadsheet. Based on inputs of span length, elastic modulus, live load, dead load, allowable bending stress, deflection limit for live load and deflection limit for dead load, the spreadsheet can be used to calculate maximum moment, maximum shear, elastic section modulus, and minimum moments of inertia required to satisfy the constraints on deflection.
For low cost, easy to use spreadsheets to make these calculations in S.I. or U.S. units, as well as checking with a known design to see if strength and deflection requirements are met, click here to visit our spreadsheet store.
Where to Find Structural Analysis if Beams Spreadsheets
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
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 w_{1}(x) acts on the beam. Then the deflection y_{1}(x) of the beam is governed by Eq.1:
Next, remove the load w_{1}(x) and apply a different load, w_{2}(x). Then the deflection y_{2}(x) of the beam is also governed by Eq.1:
Adding Eqs. 3 and 4 and defining a new function, y_{3}(x) ≡ y_{1}(x) + y_{2}(x), gives
In words, y_{3}(x) satisfies the differential equation for a beam subjected to the combined loading, w_{1}(x) plus w_{2}(x), and, furthermore, y_{3}(x) can be found by simply adding the deflection equations corresponding to w_{1}(x) and w_{2}(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 y_{3}(x)? Answer: Superposition is in fact not worth bothering about, unless tabulated solutions exist for y_{1}(x) and y_{2}(x). Because if someone else has already solved the differential equations for y_{1}(x) and y_{2}(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 y_{3}(x).
Example Structural Analysis of Beams Calculations
As an illustration, consider the beam shown in the figure below.
For concreteness, let a_{1} = 2 m, a_{2} = 3 m, L = 12 m, P_{1} = 10 kN, P_{2} = 14 kN, E = 200 GPa, and I = 600 000 cm^{4}.
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
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