In this whitepaper, we outline six tips that every civil engineer should apply when creating calculation sheets in engineering calculation software. Alongside these topics are shown smart ways that modern engineering calculation software can save time or make the task easier. The examples cover engineering principles and provide insight into what can be done with these powerful tools.

Each example has a link to a PDF with the fully disclosed equations, documented design procedures and references. They were developed in Maple Flow, an engineering calculation tool published by Maplesoft. Maple Flow has a freeform interface that gives flexibility when creating print-ready calculation sheets and is backed by a units-aware math engine.

**Engineering calculations are at the heart of civil engineering design.** The calculation sheets that accompany a project can provide evidence of the design competency and that the specifications are met.

Common building calculations are formalized in design guides, such as Eurocodes, AASHTO, IBC etc. The equations tend to be mathematically simple — only needing basic arithmetic operations on the design parameters — but they still require engineering experience to determine the *which*, *why*, and *when*.

The calculations for other civil engineering projects can demand more mathematical insight, and may need more computing horsepower to complete the iterations.

In all cases, civil engineers need to create the design using validated, reliable software to help them do their calculations. The typical options available are:

**Spreadsheet software**(Microsoft Excel or Google Sheets) - has a low cost at the point of access, but has hidden costs, measured in terms of maintainability and the risk of error during manipulation.**Application-specific software**- will be appropriate for a limited number of modelling scenarios. These often integrate well with software from the same category and are ideal if the user can stay within that restricted framework.**Engineering calculation software**(e.g., Maple Flow) - can offer greater flexibility for novel designs, but demand more technical knowledge or mathematical experience to use. These tools, however, satisfy the broader requirements of engineering calculations, documentation, and auditability.

Many civil engineers choose to use engineering calculation software, simply because they

- need the technical freedom for unique designs,
- want to become familiar with a specific design code or engineering principle,
- require interactive design reports with live math,
- or simply don't like working with "black-box" software.

In this whitepaper, we outline six tips that every civil engineer should apply when creating calculation sheets in engineering calculation software. Alongside these topics are shown smart ways that modern engineering calculation software can save time or make the task easier. The examples cover engineering principles and provide insight into what can be done with these powerful tools.

Each example has a link to a PDF with the fully disclosed equations, documented design procedures and references. They were developed in Maple Flow, an engineering calculation tool published by Maplesoft. Maple Flow has a freeform interface that gives flexibility when creating print-ready calculation sheets and is backed by a units-aware math engine.

Engineering calculation sheets are more than the sum of their arithmetic operations. They display properties, assumptions, and requirements beyond the basic number crunching. As part of the project documentation, they communicate compliance with internationally recognized standards such as Eurocodes or IBC/IRC. They may show reviewer approvals, and they tie the design to the requested design outcomes.

An engineering equation is derived from physical laws that are inherently dimensional. Lengths, forces, stresses, and moments all have units, and the units are important.

As a part of the broader requirements, it is vital to document and communicate any design calculations, and make sure all the assumptions are recorded.

TIP 1:Gather your references in one place

Clearly documenting any source material is always important. In the rush to deliver project designs, the larger aims of the design sheets are sometimes overlooked. The engineer should consider the following:

- Midway through a project, the design engineer may discover a particular use case or requirement that will need another design angle to be covered. Leave space for additional design notes to clarify assumptions that led you to the design.
- Give citations and position the references to be easy to find (using a URL or DOI link). A colleague can retrace the design steps more easily if you copy related images into the document to add context.
- Include enough detail to find codes and standards again - include the year, section, page.
- Worked examples from design guides should be included for later use. These can be used to test out any formulas entered manually into the project calculation sheet.
- Lock down your units early on — this avoids mistakes from incorrect factors of magnitude or using feet instead of inches. NASA lost a $125M Mars orbiter spacecraft in 1999, which was attributed to a conversion error from English to metric units.

**Smart use of calculation software:**

- Using free-form calculation software will allow the engineer to move elements around quickly and easily without disrupting the embedded linked formulas.
- Set the preferred units as part of your project global settings, and use the software to display them for any summary print layouts.

From an accountability perspective the design document will possibly be reviewed many times and must be accurate and error-free.

The calculation sheets will often be included into a design package for permits or a report set for delivery to a client or regulatory body.

TIP 2:Prepare the document for readability

Consider the audience when preparing the document. This may extend beyond your reviewer and the customer. Your coworker may need to step in if you are on vacation. You may want to revisit and reuse your design as a base for a future project, and so any assumptions need to be carefully captured.

Here are steps to take as you create your calculation sheets:

- If a formula cannot be seen in the calculation sheet there is a high chance of error, by the creator or by someone updating the document for future use. Avoid the use of hidden formulas where possible using named variables or include "self- check calculations". Break up large equations into preliminary steps and be consistent in your use of variables and subscripts.
- Add images to ensure the orientation of the design area is correctly assigned, with angles and distances taken from the correct reference point.
- Add notes about assumptions or calculation decisions in a different color.
- Build the document with access control that matches the required data privacy and security. Exporting a finished design sheet into PDF will lock the document at an instance in time for safe reviewing. Recent electronic signature features in Adobe Acrobat or Docusign allows the calculation sheet to be integrated into a company's document management system.
- Re-read your finished document after 1-2 days as if you are reviewing it — a refocus can help you notice errors.

An engineered design (as opposed to a crafted design) is characterized by rigorous calculations based upon the principles of math, science and empirical observation.

Civil and structural engineering projects will require engineered design calculations to demonstrate the accuracy, reliability, and competence of the design. Depending on how widespread the application of the design is, the engineer may be fortunate enough to have the common requirement scenarios compiled into a design guide and shared amongst the engineering community.

TIP 3:Recognize the limits of a standard design

Engineering excellence requires the ability to assess when a standardized design is not suitable for the situation, and other factors need to be considered. These can show up when differentiating between live loads and dead loads, and when evaluating the points on the structure that loads will be applied.

Once the design goes beyond the described situation in a design guide, the engineer has a choice — they can proceed with the custom design (assembling the calculations from other sources, pulling in equations from industrial research or from other engineering domains), or can choose to rework the design back to fit the standard from the design guide.

When calculations will be used as part of an approval process for building permits and inspections, there may not be the option to customize beyond the tried-and- tested designs, but it is common for extra calculations to be added to an existing design sheet once a new use case is brought to light.

**EXAMPLE:** The civil engineer will commonly be called on to calculate loading on beams. The load on beams in steel-framed structures is usually arranged intentionally to act through the shear center. This means that torsional effects can be safely ignored.

However, this is not always the case. A torque may result from an eccentrically applied load, resulting in torsion (and hence a twist angle). Torsion may be significant enough to control the overall design, and in severe cases can result in failure from torsional buckling.

Civil engineers must recognize when torsion dictates the design, and apply the appropriate engineering rules.

This application analyzes a simply-supported beam with torsional and lateral loading. The beam is a W10X54 shape (as defined by the AISC Steel Shapes Database). The torque is applied at the mid-span, and thus results in a twist angle and lateral displacement.

The application can be found here.

TIP 4:Customizing a standard design using iterative methods

There are instances where the design needs pre- calculated results from a data table in an industry publication or design guide. These are generated for a pre-defined set of parameter values, and may be derived from equations that describe the underlying theory.

If you need data that isn't explicitly tabulated, linear interpolation between known values may be required. This, however, is prone to error.

A better approach would be to solve theoretical equations in engineering calculation software.

**EXAMPLE:** When designing bolt groups, the bolt group coefficient is an important quantity: that is the ratio of the factored force (or available strength) of the bolt group and the shear capacity of a single bolt. Once the coefficient is known, a bolt group can be designed for any load.

Bolt group coefficients are tabulated in design guides — in the US, in the AISC Manual of Steel Construction: Load and Resistance Factor Design. These tables are generated from a theoretical approach called the Instantaneous Center of Rotation method (or the IC Method). The theory assumes that

- the sum of the bolt forces in the vertical and horizontal directions are equal to the applied shear and axial loads
- the moment of the bolt forces about the instantaneous center of rotation is equal to the moment of the applied load

This results in a set of equations that need to be numerically solved.

The AISC bolt coefficient tables are limited to common bolt patterns, and a set of pre-defined load eccentricities and angles. Non-tabulated values must be extracted by using linear interpolation. That is labor-intensive and if performed manually, introduces risk to the accuracy of the calculation.

The IC Method involves several assumptions, but ultimately leads to three simultaneous equations featuring three unknowns. The equations are implicit — so solving them can be performed using a numeric iterator.

Once the evaluation of the bolt spacing and bolt coefficient is found for a particular design, the design criteria can be optimized by choosing higher or lower parameter values (via new iterations) to improve the outcome.

When working in a spreadsheet this relies on an app add-on such as the Goal Seek or Solver tool in Excel. The engineer manually selects parameters to minimize the desired bolt design values by varying the other parameters, subject to certain constraints. For bolt groups, the objective is to set the sum of the forces in the x and y direction on the bolt group to zero by varying the instantaneous center of rotation.

Setting the spreadsheet software to automate the parameter iterations across combinations of variables will require using Visual Basic.

The alternative is to use calculation software like Maple Flow that can numerically solve the equation and apply various combinations of initial parameters without having to change multiple cells of data. This application implements the theory used to generate the AISC tables. By changing a few parameters, you can quickly compute the bolt group coefficient for any bolt and load configuration (all without thumbing through a manual.)

The application can be found here.

**Smart use of calculation software:**

- For simpler load equations, calculation software will allow the rapid change of several parameters (via handy dials or toggles) to get close to the desired result.

Where it is not practical to reduce a parameter further (e.g., in Bolt Groups, there is a physical limit on how close together the bolts can get), the engineer should be ready to consider other support elements (adding braces for beams, specifying multiple supports spaced more regularly etc.). Working to stay within the confines of a design in this way typically comes more easily with experience.

TIP 5:Using a Parameter Sweep to hit the design sweet-spot

For complex designs with requirements that are interdependent on a series of environmental factors, there may be some factors that are given a higher priority by the stakeholders. The engineer could be given guidance in the project requirements from the purchasing team based on availability and price. The customer may require a certain structure size or performance outcome from the design. Even so, there can still remain many alternatives to choose from, and the best fit will sometimes be a convergence on a sweet-spot based on the choice of materials and the specific use cases.

**Smart use of calculation software:**

- Calculation software with advanced math engine libraries, such as Maple Flow, can create a 3D plot of results across several parameters at once (often called a Parameter Sweep). This can help identify the sweet-spot where multiple requirements are met in the best way possible.

**EXAMPLE:** When selecting lumber for a construction project, there is the choice of tree species, size of cut, and the commercial grade to consider, along with specific design factors relating to repeating boards, wet-use or flat use. This leads well to the use of a matrix for collecting multiple material qualities for comparison calculations.

The material properties and design factors under use conditions can be found in the US from the American Wood Council publication "National Design Specification® (NDS) for Wood Construction, 2018 edition."

Starting a design calculation based on the initial use of Douglas Fir-Larch, Maple Flow will allow a table to be created (or loaded from an external data source) that can be drawn from during calculations. If the design is later amended to respond to a change in wood species selection (e.g., to Alaska Cedar), the table can be reapplied without changing the rest of the live equations.

The design example is continued here.

TIP 6:Knowledge of solving Differential Equations

There are many environmental applications where the physics model maps the design elements using differential equations. These exceed the capabilities of spreadsheet cell-based formula and require a powerful math engine to evaluate the solution.

Whilst traditionally the differential equations only appeared in the academic or research fields, the more widespread use of environmental models has brought this into more civil engineering activities.

Some engineers who work on a lot of specialist projects will look for the matching software packages that are tailored to that genre. Hydrology project teams may look for software that addresses the scenarios of wastewater and storm water management, and provides additional modeling options such as FEM or CAD. StormCAD or WaterCAD are two such products packages that give greater integration options for planners already tied to using GIS-models. However, there are times the complex differential equations appear in other types of civil engineering projects or the project finances will not cover the expense of specialist modeling software.

In these situations, an engineer familiar with calculation software such as Maple Flow can still solve the differential equations, with the bonus that the workflow and the formatting of the software will align well with the look and feel of their day-to-day calculation sheets.

**Smart use of calculation software:**

- Design engineers who are familiar with the functions available in the math-engine Maple can leverage those within Maple Flow. Maple contains powerful 'solver' functions that can evaluate numerical solutions to differential equations.
- Adding specific formatting commands in the plots and charts (to set colors and fonts) can be used to create a single look and feel across projects, without having to reformat again ahead of printing.

**EXAMPLE:** Here is a case study from a flood risk assessment - the water depth and surface profile along a channel need to be established during peak flow rates. Roof gutters and channel spillways are cases of spatially varied flow where a long rectangular channel has an inflow of pouring water along its length.

The profile of the water surface varies with distance upstream. This can be solved with a numerical solution of the governing differential equation.

For a subcritical flow, downstream conditions determine the water surface profile. The water depth reaches the critical height (i.e. the minimum energy height) near the free fall - this is the boundary condition on the differential equation.

After solving, the water surface profile can be visualized using the Maple plot function, as shown in Figure 5.

The engineer may need to consider design changes where the channel cross-section changes based on design (maybe to avoid ice expansion issues in winter) and having the full equation will allow those scenarios to be played out.

Civil engineers can benefit from matching calculation software tools to their needs, and understanding the underlying math considerations when encountering a new design requirement. The efficiency and accuracy of project design sheets will also improve when the engineer makes use of a workflow that lines up with the way they choose to gather and communicate the information. Maple Flow stands out as calculation software that is aligned to the way civil and structural engineers approach project design sheets. It mimics the traditional hand calculation format with a paperlike format but gives access to a powerful math engine when needed.

It is hoped that this document can be used by civil engineers and design reviewers to assess how many of the productivity tips they are using in their existing processes today, and to see how creating cleaner calculation sheets can also improve readability and lower the error rate in documents.

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