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Structural Analysis and Design of Beams Blogs

Here you will find informational articles on topics related to the Excel spreadsheets for civil and mechanical engineering calculations available from the DOWNLOADS page.  This includes articles in the clickable categories below: pipe flow calculations, open channel flow, heat transfer/heat exchangers, storm water/hydrology, continuous beam analysis and design, open channel flow measurement, and pipe flow measurement topics.  Scroll down on each category page to see all of the articles.

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List of Blog Articles - Structural Analysis and Design of Beams

Posted on Friday, December 28, 2012 at 8:26 PM

 

Scroll down for the following blog articles in this category:

  • Allowable Stress Beam Design Spreadsheet
  • Structural Analysis of Beams Spreadsheets

 

Allowable Stress Beam Design Spreadsheet

Posted on Wednesday, December 12, 2012 at 1:33 PM

 

Where to Find an Allowable Stress Beam Design Spreadsheet

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.

Background for Allowable Stress Design of Beams

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 an Allowable Stress Beam Design Spreadsheet

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:

Mmax  =  wL2/8,   where

  • Mmax  =  maximum moment in the beam
  • w  =  distributed load on the beam
  • L  =  length of span

Vmax  =  wL/2, where

  • Vmax  =  maximum shear in the beam
  • w and L are as defined above

Mallow  =  SFb,  where

  • Mallow  =  the allowable moment in the beam
  • S  =  elastic section modulus of the beam
  • Fb  =  maximum allowable stress in the beam

ymax  =  5wL4/(384EI),  where

  • ymax  =  the maximum deflection in the beam
  • E  =  elastic modulus of the beam
  • I  =  moment of inertia of the cross section of the beam

ymax  <  L/Ld,  where

  • Ld is a dimensionless number specified by code, depending on structural application and load type (typically Ld = 120, 180, 240, 360, or 600)


An Allowable Stress Beam Design Spreadsheet

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.

allowable stress beam design spreadsheet

 

Structural Analysis of Beams Spreadsheets

Posted on Wednesday, July 13, 2011 at 6:25 PM

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 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.

diagram for structural analysis of beams spreadsheets

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:

beam diagram for structural analysis of beams spreadsheet

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)  =  yo(x, 10 kN, 2 m)  +  yo(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.

 

spreadsheet for structural analysis of beams

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, 2nd Edition, American Institute of Steel Construction, Chicago, IL, American Institute of Steel Construction (1994).

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

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

 

Structural Analysis of Beams Spreadsheets

Where to find Structural Analysis of Beams Spreadsheets

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