# How to Draw V and M Diagrams – A Tutorial

## Where to Find a Tutorial on How to Draw V and M Diagrams

To obtain a tutorial program on  how to draw V and M diagramsclick here to visit our download store.  Look for “Understanding V and M Diagrams” in the “Strength of Materials” category to find it.  This standalone program provides practice for learning the principles underlying shear diagrams and bending moment diagrams.  You can buy this convenient tutorial beam diagrams program on how to draw shear and bending moment diagrams for a very reasonable price (only \$4.95).

## A Description of the Tutorial Program on How to Draw V and M Diagrams

This standalone program contains over 50 exercises, which have been chosen to represent all common boundary conditions and load cases for structural beams.  The difficulty of the exercises ranges from easy to challenging.

The program is highly interactive: starting with a point at the origin of the shear diagram, the user calculates the coordinates of the next point where the diagram changes slope or direction.  Then the user employs the mouse cursor to plot this point on the screen.  After plotting the point, the user specifies the type of curve (for example, straight line or curved line concave up), connecting the two points.  The user proceeds in a similar manner to plot the next point on the diagram and connect it to the most recently plotted point.  Continuing in this fashion, the user completes the shear diagram and then goes on to the bending moment diagram.

Five kinds of Help screens are available:

• A step-by-step tutorial on how to operate the program is provided which guides the user through a simple example on how to draw V and M diagrams.
• Diagrams defining sign conventions for V (shear) and M (bending moment) are given.
• Techniques for calculating V and M are described in terms of specific numerical calculations.
• The answer for the current step in the diagram construction is given—the value of V or M or type of curve connecting two points on the diagram.  How the answer is calculated is explained in detail.
• A complete numerical example showing both shear and bending moment diagrams with key values labeled on the diagrams is provided.  When the user positions the mouse cursor over any part of the diagram, a pop-up window opens containing an explanation of how that particular part of the diagram was constructed.  For example, positioning the cursor over the point of maximum moment on the beam (which is loaded with a distributed load) brings up the comment that the maximum moment occurs where the shear is zero.  Positioning the cursor over a jump in the shear diagram brings up the comment that the jump is caused by the support reaction acting on the beam.

A quiz facility is also provided for use of the program in a classroom setting.

## Screenshot of Tutorial Program on How to Draw V and M Diagrams

The beginning of the tutorial on How to Draw V and M Diagrams is shown in the image below.  This program can be used to help understand shear and bending moment diagrams and how to construct them.

Figure 1. A Tutorial Program on How to Draw V and M Diagrams

## Reference

Rossow, M., Shear and Bending Moment Diagrams – A Tutorial, a blog article at: http://www.engineeringexceltemplates.com/blog.aspx

# Spreadsheets for Allowable Stress Design of Beams

## Where to Find an Excel Spreadsheet for Allowable Stress Design of Beams

For an Excel spreadsheet for allowable stress design of beamsclick here to visit our spreadsheet store.  Obtain a convenient, easy to use spreadsheet for allowable stress design of beams 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 Allowable Stress Design of Beams

Equations for the first step in allowable stress design of beams calculations 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)

## A Spreadsheet for Allowable Stress Design of Beams

The screenshot below shows an Excel spreadsheet for allowable stress design of beams.  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.

# Using Superposition in a Continuous Beam Analysis Spreadsheet

## Where to Find a Continuous Beam Analysis Spreadsheet

To obtain a continuous beam analysis spreadsheet, Click Here to go to our spreadsheet store.    Also, check out our Free Android App for analyzing a simply supported beam with a concentrated load.  Read on for information about performing  continuous beam analyses via superposition and how Excel spreadsheets can be used in this procedure.

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 in a Continuous Beam Analysis Spreadsheet

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 Calculations with Beam Formulas

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)  =  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.

## Screenshot for Continuous Beam Analysis Spreadsheet Calculations

The screenshot image below shows an Excel spreadsheet to 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 twoconcentrated 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 and additional workbooksfor other boundary conditions are available in either U.S. or S.I. units at low cost in our spreadsheet store.

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, “Structural Analysis of Beams Spreadsheets,”  an online blog article