Mathcad Community Challenge September 2026: Valencian Towers

The September 2026 Mathcad Community Challenge was based on civil engineering from the 14th century around the Gothic arch of Valencia’s Torres de Serranos:

Introduction

The challenge was as follows:

Challenge 1. Calculate the following:

• the vertical reaction forces at each supporting wall.
• the horizontal thrust pushing outward against the masonry buttresses.
• the total resultant force that the foundations must withstand at the base.
• the angle of the resultant force with respect to horizontal.

Challenge 2. Based on the results of the first challenge, calculate the following:

• the parallel shear force acting directly across the plane of the mortar joint.
• the average and maximum shear stress across the contact area.

Challenge 3. Use XY Plots or Chart Components to generate a bending moment diagram and/or a shear force diagram for the main gate pointed arch.

Challenge 4. Can you calculate a geometrical factor of safety for the Gothic pointed arch of the main gate? Jacques Heyman’s thrust line theory may help in this situation.

The submissions

This month had three submissions, two from Terry Hendicott and one from Fred Kohlhepp.

Terry’s first worksheet

Terry’s first worksheet was extremely thorough and went into much more detail than I expected. It examined each of the 22 bricks that formed each side of the arch. Terry constructed quite a few algebraic equations for the geometry as well as the weight of the wall and the arch itself.

The first challenge used an approach based on equilibrium of horizontal force, vertical force, and moment. This resulted in 66 equations for 69 variables. The Solve Block spanned 13 pages of the worksheet! Constructing that was certainly a Herculean effort.

Editor’s Note: This screenshot was produced by changing the Page Size to A1 paper, a new selection as of PTC Mathcad Prime 12. Click here to learn more about the enhancement. Terry’s paper size, when submitted, was A4.

The worksheet documentation included several additional images that helped explain his approach. He referenced material properties for limestone and included information on the shortcomings of medieval mortar.

In challenge 2, Terry created an XY Plot that displays the geometry and the shear stresses. For calculating the values, he makes use of the min and max functions, multiplying by units, vector mathematics, and the vectorize function.

In challenge 3, his bending moment and shear force diagram in an XY Plot show how complicated the forces are through the arch. To calculate the vectors to plot, he defines functions that are also properly vectorized for element-by-element calculations.

Finally, Terry calculates the factor of safety as 1.506. This helps explain why the Torres de Serranos has remained standing for over 600 years.

Terry’s second worksheet

Wow, I applaud Terry’s daring and ambitious approach: solving the problem via Finite Element Analysis with triangular plate elements. In fact, it was 3200 elements and 1717 nodes!

To construct the mesh of nodes, Terry made use of a range variable and the relatively new vec function. He used function definitions with range variables to construct the elements. To graph the mesh on an XY Plot, he created another set of matrix functions. These techniques could be used by engineering students.

To apply the loads, Terry used functions to identify the top nodes and apply the loads horizontally. This includes the weight of the tower’s center section above the arch as well as the weight of the bricks themselves. This is all much easier to describe than to implement mathematically.

Terry references the equations for a plane stress triangle from a book on the Finite Element Method. He constructs the stiffness matrix, which again is very easy to describe in words, but is extremely large and complicated in reality. He uses a technique of “infinite” stiffness for a mirror symmetry constraint.

The really interesting part is that the solution is a somewhat simple looking equation on page 8: the displacements are equal to the inverse of the stiffness matrix times the force vector.

Terry plots the scaled deflections, which is very common in FEA due to the actual displacements usually being too small to see. Some of the displacements wouldn’t be possible because of the constraints of the rest of the structure, so Terry uses a trial and error approach to fix that.

Once you have the displacements, you can calculate the stresses. Again, it’s much more complicated than it sounds, especially when there are XY Plots and 3D Plots to support presentation of the reactions and stresses.

This worksheet is a masterpiece in executing a complex analysis in Mathcad that is typically handled in more complicated and dedicated tools (in this, Computer Aided Design pre- and post- processors and Finite Element Analysis solvers).

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Fred Kohlhepp’s worksheet

Fred submitted a worksheet that incorporates plenty of documentation to explain the approach, as well as supporting images. Some of the complex functions are placed in collapsed areas. These are worth a look, and involve integrals and double integrals.

I always enjoy the use of different fonts, text color, and region highlighting to draw the reader’s eye exactly where you want it to go. This worksheet excels at that.

There are several XY plots that display the geometry (with neutral line), moment of inertia, shear force, moment, angular deflection, and displacement. Very nice. These are followed by the shear and moment diagrams. This is a great display of the structural analysis capabilities within Mathcad for civil engineering.

(Side note: to encourage the challenge, I also submitted a couple worksheets with some fairly straightforward calculations for the first two challenges.)

Engineering calculations for civil and structural engineers

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What can we learn?

This month’s submissions show how Mathcad can be used for civil engineering for a real world example. It’s representative of the kinds of calculations that go into structural analysis.

Speaking of which, Terry went above and beyond with his two thorough worksheets. As someone who has both worked in and taught Finite Element Analysis, I thought his second worksheet was absolutely amazing. This worksheet can be used by students and professionals to further their understanding of the subject. As Terry states in his worksheet, “one point of the monthly challenges is to show the power of PTC Mathcad Prime,” and he certainly does that.

Join us in November 2026 for an electrical engineering themed challenge!