Written By Brent Maxfield
  • 12/9/2022
  • Read Time : 5 mins

Calculate and Plot Shear and Bending Moments in Mathcad

ptc mathcad shear bending moment diagram civil engineering

Editor's note: This blog article was originally written as a PTC Mathcad Prime worksheet. For the best reading experience, please download the Mathcad Prime 8 worksheet here. If you need a Mathcad Prime worksheet viewer, download Mathcad Prime for free here.

A truck driving on a bridge causes the supporting beams to deflect, resulting in various stresses in the beams. Similarly, as you walk across the floors in your home, the beams or joists deflect under your weight. The deflection is a result of the loading, which is related to the shear, the bending moment, the beam slope, and the beam deflection. In this blog and several that follow, I will discuss how Mathcad can be used to calculate, plot, and derive these relationships.

Drawing shear and bending moment diagrams of simply supported beams is one of the tasks in a basic engineering statics course. Mathcad can be used to calculate shears and moments and also draw the shear and bending moment diagrams. 

In this blog, I will show how to do this for simply supported beams with varying loading conditions.

I am a proponent of creating functions so that they can be used multiple times, rather than just solving a single problem. So, first, let's create some functions to help solve for and plot the shear and bending moment diagrams for a uniformly loaded, simply supported beam.

The figures in this blog were created in Microsoft PowerPoint as embedded objects. If you download the linked Mathcad file and double click on the figure it will open in PowerPoint.

A uniformly loaded simply supported beam made in PowerPoint

Let's first calculate the left reaction, RA, by summing the moment about point B, and then calculate the right reaction, RB, by summing the vertical loads.

Once these functions are written, the results will be tested. A clockwise moment will be considered positive.
The variable "Span" is the length of the beam, and the variable "w" is a uniform load (force/length) along the length of the beam. Because Mathcad is unit aware, the span and load can be in any units of length and force. 

The right arrow in the below functions is the Symbolic Evaluation Operator from the Symbolics section on the Math tab. It provides a symbolic solution rather than a numeric solution.

Sum moments about point B, and then solve for RA to calculate the left reaction. Sum vertical loads and solve for RB to calculate the right reaction: Note: The function for the right reaction uses the function for the left reaction, RA. Test for a 20 foot beam and a 2 kip/ft load. Test for a 10 m beam and a 5kN/m load.

Next, write a function for shear at a distance x by summing the vertical loads. The shear at a distance x from the left is equal to the left reaction less the load applied to the beam between the left and the distance x (w*x).

Mathcad shear function on a uniformly loaded beam with accompanying figure made in PowerPoint, with test cases

Now, write a function for the moment at a distance x by summing the moments about x.

Mathcad calculation for the moment on a uniformly loaded beam

Now that the equations are written, calculate and plot the shear and moment diagrams using the XY plot features. Insert the plot from the Plots menu in the Traces section. Select XY Plot from the Insert Plot button.

In order to create the plots, define the values for span and load, and then create a range variable for the points to plot. The range variable is defined by a starting value with the second value setting the step size. The final number is the ending value. It is created just like it looks. Type the first value, then type a comma and enter the second value, which sets the step size. Then, type . . and enter the ending value.

The equation for shear or moment will be on the y-axis.

Mathcad created ranges for XY Plot shear bending moments feet

Mathcad Prime XY Plot uniformly loaded beam shear bending moment in feet

Test again using metric units. Define values of span and uniform load, and the range variable, z, which will set the points to plot.

Mathcad created ranges for XY Plot shear bending moments meters

Mathcad Prime XY Plot uniformly loaded beam shear bending moment in meters

Let's now create shear and moment diagrams for a point load, P, placed anywhere along the beam at a distance "a" from the left end. Create functions to describe the loading, shear, and moment. Ignore the beam weight for this exercise. For this example, I use the keyword solve with the Symbolic Evaluation Operator to demonstrate how to solve for RA.

Diagram of beam with a point load placed anywhere made in PowerPoint

Mathcad creating and solving for shear on a beam with a point load anywhere along the beam with a step function using if logic

Mathcad Prime calculate bending moment on a beam with a point load

Now that the equations for a point load are written, calculate and plot the shear and moment diagrams using the XY plot features. 

In order to create the plots, define the values for span, load, and distance, and then create a range variable for the points to plot. 

Mathcad Prime defined values for span load distance range variable for XY plot

Shear and bending moment diagram for a beam with a point load Mathcad Prime XY plot

This final example derives the functions for a triangular loading.

Similar to the previous examples, create functions for the reactions, and then create functions for shear and moment.

This example uses the solve keyword with the Symbolic Evaluation Operator to derive the functions for reactions.

Beam with triangular loading diagram made in PowerPoint

Mathcad Prime create functions for triangular loaded beam civil engineering

Mathcad Prime shear function triangular loaded beam

Mathcad Prime Moment function for triangular loaded beam

Define values for span, loading, and the range variable, z, which will set the points to plot. Display the values of the range variable and the calculated shears and moments that will be included in the plot.

Mathcad Prime span loading range variables for triangular loaded beam XY plot

Mathcad Prime shear bending moments XY plot triangularly loaded beam diagram

Now that functions for uniform load, point load, and triangular load have been derived, use these functions to combine multiple loading conditions.

For these examples of combining loads, numeric results will be used and values of the loading conditions must be provided. Range variables will be used to define the points to plot and locations to calculate shears and moments. It is important to evaluate the range variable (using the = sign) to convert the range variable to a vector of values.

Note the use of the Vectorization operator in these examples. It is found on the Matrices/Tables tab in the Matrices and Tables section, in the Vector/Matrix Operators button. The Vectorization operator is the bottom operator. (It is also found in the Operators section of the Math tab in the Vectors and Matrix area. The keyboard short cut is CTRL + SHIFT + ^.) It is a right pointing arrow above the expression. It tells Mathcad to perform the operation on an element-by-element basis.

The following examples show a combination of point loads, uniform loads, and triangular loads.

beam combination point loads uniform distribution diagram in PowerPoint

Mathcad Prime formulas civil engineering loads conditions combination loaded beam shear bending moment

Shear and bending moment diagrams for combination-loaded beam XY plot Mathcad Prime

Combination point, uniformly, and triangularly loaded beam diagram made in PowerPoint

Mathcad Prime civil engineering beam triangular point uniform loaded combination beam values solving function reactions

Shear and bending moment plots for a combination (uniform, triangular, and point) loaded beam XY Plot Mathcad Prime

Combination two points, uniformly, and triangularly loaded beam diagram made in PowerPoint

Mathcad Prime civil engineering beam triangular two point uniform loaded combination beam values solving function

Shear and bending moment plots for a combination (uniform, triangular, and two points) loaded beam XY Plot Mathcad Prime

In this blog, I have used shear and bending moment diagrams to illustrate many Mathcad features. In addition, I have shown how to combine functions for uniform load, triangular load, and point loads to calculate and plot reactions, shears, and bending moments. These examples illustrate:

  • The significant use of functions
  • The power of using the Symbolic Evaluation Operator, including the use of keywords
  • The use of Microsoft PowerPoint component to create graphics
  • Using XY plots
  • Defining range variables for plotting
  • Evaluating range variables to create a vector of values
  • Using the Vectorization operator to do element-by-element operations

My next blog expands on these topics, showing how to derive functions for maximum beam bending moment for various loading conditions.


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About the Author

Brent Maxfield is from Salt Lake City, Utah. This is an ideal location for him because of his love for outdoor activities. He loves hiking and skiing in the nearby mountains, and also loves to explore the red rock canyons and deserts found in Southern Utah.

Brent Maxfield is a registered Professional Structural Engineer in the State of Utah. He graduated Magna Cum Laude from Brigham Young University with a degree in Civil Engineering and earned a Master of Engineering Management degree from BYU. He has been a practicing structural engineer for 36 years.

He was awarded the 2012 Utah Engineer of the Year by the Utah Engineers Council. He is active in professional associations having served on the Board of Directors of the Structural Engineers Association of Utah and the EERI Utah Chapter. He has also served on the Structural Advisory Committee to the Utah Uniform Building Codes Commission.

He has used PTC Mathcad extensively for 20 years. He is the author of “Essential PTC® Mathcad Prime® 3.0: A Guide for New and Current Users”, available on Amazon.com.

Calculate and Plot Shear and Bending Moments in Mathcad
Learn to calculate and plot shear and bending moment diagrams in PTC Mathcad Prime under a variety of loading conditions, including uniform, point, triangular, and combination loads.