# User Manual

Updated February 2020

This documentation applies to the full edition of Deflection, version 5.8.1.

## Introduction

**Deflection** is a graphical software application that simplifies and streamlines the structural analysis of beams. The full version includes a database of built-in materials and shapes, and it also allows the user to customize common cross sections. Reactions, diagrams, equations, and basic internal stress analysis is automatically performed.

Unlike other software tools, **Deflection** aims to be an integrated software solution that is simple and lightweight while powerful and reliable at the same time. The app is currently available on every major software platform (iOS, Android, macOS, and Windows).

**Deflection** is designed to be a learning aid as well as a long-term reference tool for mechanical engineering, civil engineering, structural engineering, and architecture.

## Quick Start

Solve a simple cantilever in seconds!

- If there are any loads on your beam, you can either drag them off the beam or clear the beam completely.
- Drag loads onto the beam.

Achievement Unlocked!You generated shear and bending moment diagrams by simply placing loads on the beam!

- Drag a fixed support onto the beam.

Achievement Unlocked!You generated slope and deflection diagrams by simply placing loads on the beam!

- Customize the beam properties as needed.
- You are done! Review your results or iterate on your design.

## Loads

Any number of loads can be applied on the beam. Three basic types of loads are supported:

- Forces (also known as "point loads" or "concentrated loads")
- Moments
- Distributed forces (also known as "distributed loads").

#### Adding loads

To add a load:

- Go to the main design screen.
- If necessary, unselect everything by tapping on an empty area to show the load icons.
- Tap (or click) and hold any load icon from the load toolbar.

- Drag and place the load on the beam.

#### Load values

Forces, moments, and distributed forces are defined by two types of values:

- Location along the beam
- The location can be modified by dragging the load along the beam using your touch our mouse.
- The location can also be set by selecting an editable label and entering a value using the screen numpad or an external keyboard.

- Magnitude
- A magnitude value can be set by selecting an editable label and modifying value using the numpad or the keyboard in desktop editions.
- A magnitude sign (direction) can be set in two different ways: You can quickly change the sign of a force by flipping the force vertically across the beam using touch our mouse. You can also specify the sign of a force using the numpad or the minus "-" key on a keyboard. button in the

#### Load sign conventions

To simplify the representation of direction of forces and moments, instead of defining **positive** and **negative** sign conventions in the free body diagram, the app displays direction visually using icons and other graphics.

The direction of loads is displayed in two ways:

The

**load icon**itself illustrates the direction of the applied load. For example, upward applied forces are represented by arrows that point up, and counter-clockwise applied moments are represented by circular arrows that point in a counter-clockwise rotation.The direction of the load is also displayed in the

**magnitude value label**for each load. For example, next to the "1 kN-m" value for a moment magnitude, there is a circular arrow that can point clockwise or counter-clockwise, depending on the direction of the applied moment.

### Force loads

**Forces** (point loads) are represented by a vertical arrow.

Forces are defined by two values:

- Location along the beam (with units of length, such as meters, inches, etc.)
- Force magnitude (with units of force, such as Newtons, kilograms, pound-force, etc.)

**Note:** Normally, kilograms are considered to be units of mass rather than units of force. The app, however, assumes that kilograms and pounds are units of force by multiplying internally by the gravitational constant. We use the conversion factor of 9.80665 N/kg to convert between force and mass.

### Moment loads

**Moments** are represented by a pair of circular arrows.

Moments, like forces, are defined by two values:

- Location along the beam (with units of length, such as meters, inches, etc.)
- Moment magnitude (with units of force-distance, such as Newton-meter, lb-ft, etc)

### Distributed force loads

**Distributed forces** (distributed loads) are represented by an array of vertical arrows.

Distributed forces are slightly more complex than point forces and moments because they are defined by four values:

- Two independent positions along the beam (with units of length, such as meters, inches, etc.)
- Two independent force magnitudes (with units of force, such as Newtons, kilograms, pound-force, etc.)

Since all of these four values are independent of each other, there is a lot of flexibility when they are being specified and modified: The left position and the right positions can be moved independently of each other. The force magnitudes at each endpoint can have opposite or equal signs, and it's also possible to set one magnitude to be zero and the other one to be non-zero.

Although distributed loads are defined by their two endpoints, the app also allows you to specify the distance between the endpoints and computes the right-side endpoint for you.

When you drag a distributed load using the graphic editor, it matters where you press on the load:

- If you select the left side of the load, you will only drag that left-most end-point of the load.
- If you select the center of the load, you will drag both end-points of the load.
- If you select the right side of the load, you will only drag that right-most end-point of the load.

## Supports

When a support is placed on the beam, the app automatically enforces constraints on the internal system of equations that defines the problem being solved.

Four different types of supports can be used. The following table summarizes each type of support as well as the constraints that characterize each one.

Type | Force (F) | Moment (M) | Slope (Θ) | Deflection (δ) |
---|---|---|---|---|

Fixed supports | F | M | Θ(x) = 0 | δ(x) = 0 |

Simple supports | F | M → 0 | continuous | δ(x) = 0 |

Fixed hinges | F | M → 0 | discontinuous | δ(x) = 0 |

Floating Gerber hinges | F → 0 | M → 0 | discontinuous | continuous |

#### Specifying supports

Defining supports is simpler than defining loads because supports only require a single value to be specified (i.e., their location along the beam). Supports can be placed and dragged along the beam in the same way as loads.

### Fixed supports

**Fixed** supports can be used for cantilevers and statically-indeterminate cases.

When a **fixed** support is used, the app attempts to solve for the its vertical force (F) and moment (M) reactions. Fixed supports impose the following constraints on the system:

- The slope Θ(x) at the support location must be continuous and equal to 0.
- The deflection δ(x) at the support location must be continuous and equal to 0.

### Simple supports

A **simple** support is sometimes also referred to as a **pin** support or a **middle** support. A **simple** supports is different from a **fixed** support because

**simple**supports do not have a moment (M) reaction, and**simple**supports require the slope at their location to be continuous but not necessarily equal to 0.

When a **simple** support is used, the app attempts to solve for the support's vertical force (F) reaction. This type of support imposes these constraints on the system:

- The slope Θ(x) at the support location must be continuous.
- The deflection δ(x) at the support location must be continuous and equal to 0.

### Fixed hinge supports

A **fixed hinge** support is sometimes also referred to as a **hinged** support. This type of support is different from a **simple** support because

- a
**fixed hinge**support does not require the slope Θ(x) at its location to be continuous, whereas**simple**supports do.

When a **fixed hinge** support is used, the app attempts to solve for the support's vertical force (F) reaction. This type of support imposes this constraint on the system:

- The deflection δ(x) at the support location must be continuous and equal to 0.

### Gerber hinge supports

A **Gerber hinge** support is sometimes also referred to as a **floating hinge** support. This type of support is different from a **fixed hinge** support because

- a
**Gerber hinge**support does not require the deflection δ(x) at its location to be 0 (but it does have to be continuous).

When a **Gerber hinge** support is used, the app does not solve for a force or moment reaction on the support. This type of support only imposes a very relaxed constraint on the system:

- The deflection δ(x) at the support location must be continuous.

## Beam properties

### Beam length

Like other quantities in the main design editor, the beam length is specified directly in-place.

- Go to the main design view and scroll to the beam editor.
- Tap or click on the beam length label under the beam visualization.
- Enter the desired value and units using the on-screen numpad or keyboard.

If you enter a beam length shorter than the position of an existing load or support, that load or support will be rendered in red, and it will be displayed on the side of the screen. It will be ignored in any calculations until it is moved to a valid beam position.

### Cross section

The moment of inertia and area are both a function of the beam's cross section geometry.

The cross section's moment of inertia is required to compute the deflection of the beam and solve for the beam reactions.

The cross section's area is only used to compute the beam weight for custom materials and shapes if self-weight is enabled.

There are three different ways to provide the moment of inertia and area values:

- Specify the values directly
- Specify a cross section from the built-in library to import its properties
- Customize a cross section from the library to automatically calculate its properties

#### Set moment of inertia and area directly

When you enter a custom value directly, it will erase any geometry that was previously applied.

- Go to the main design view and scroll to the moment of inertia section.
- Tap or click on the moment of inertia label.
- Enter the desired value and units using the on-screen numpad or keyboard.

#### Import a cross section from the library

When a cross section is selected from the library, the moment of inertia and area are automatically imported from built-in reference tables.

- Go to the main design view and scroll to the moment of inertia section.
- Tap on the search icon next to the moment of inertia label.
- Select a cross section from the library.
- In the cross section details screen, additional details for the selected cross section will be displayed.
- Click on the button.
- The app will return to the main design screen, which will have the relevant properties for the cross section applied.

#### Customize a cross section shape

When you customize a shape, its moment of inertia, area, and any other relevant properties are automatically calculated and applied to the beam analysis.

- Select cross section from the library
- Customize the dimensions by tapping and editing the labels around the cross section.
- Rotate the section if needed by tapping on the button in the center of the cross section.

### Material

The elastic modulus and density are properties of the material.

The material's elastic modulus is required to compute the deflection of the beam and solve for the beam reactions.

The material's density is only used to compute the beam weight for custom materials and shapes if self-weight is enabled.

There are two different ways to specify the material properties of a beam:

- By entering specific elastic modulus and density values
- By selecting a material from the built-in library

#### Specify elastic modulus value directly

If you enter an elastic modulus value directly, it will override any material that was previously selected from the library.

- Go to the main design view and scroll to the elastic modulus section.
- Tap or click on the elastic modulus label.
- Enter the desired value and units using the on-screen numpad or keyboard.

#### Import material from library

When a beam material is selected from the built-in library, the elastic modulus and the density for that material are automatically imported and applied to the design.

- Go to the main design view and scroll to the elastic modulus section (top of screen).
- Tap on the search icon next to the elastic modulus label.
- Select a material from the library.

The color of the beam will change to match the color coding of the material. The colors used for materials are for visualization purposes, and they are not meant to be realistic or accurate.

## Results

The app automatically generates a solution and outputs results every time any changes are made.

In order to compute reactions and deflection, the beam equations need to be fully determined. These are some valid support configurations:

- 2 simple supports (ie, simple supported beam)
- 1 fixed support (ie, cantilever)
- 1 fixed support, 1 hinge, 1 simple support
- 1 fixed support, 1 floating Gerber hinge, and another fixed support

Different results are displayed in several ways by the app:

**Labeled support icons**To see the reaction values for a support on the beam, simply select its icon, and its calculated force and moment will be displayed in a label.**Diagrams**Diagrams for shear, bending moment, slope, deflection, internal shear stress, and internal bending stress are generated. The software automatically determines and labels interesting diagram values by searching for minima, maxima, and in some cases, zero crossings.**Diagram ruler**To obtain a value at a specific location along the beam, you can tap on the diagrams area and move the ruler to display the diagram values at that specific location.**Equations**All of the step equations needed to generate the diagrams are displayed under the diagrams area.**The beam visualization**The beam itself in the free body diagram visualiation will bend to mirror the calculated deflection curve. Note that the deflection of the beam is exagerated for visualization purposes, and it is not drawn to scale with respect to the beam's dimensions.

### Diagrams

The app produces up to 8 different diagrams (plots).

Diagram | Units |
---|---|

Shear | force |

Moment | moment |

Slope | angle* |

Deflection | length |

Internal shear stress | pressure |

Internal bending stress | pressure |

Internal shear stress profile | pressure |

Internal bending stress profile | pressure |

*Assumes that a ≈ tan(a) for small values.

The **shear** and **bending moment** diagrams are generated whenever any loads are placed on the beam.

The **slope** and **deflection** diagrams are only generated when supports are placed on the beam and the system of equations is fully determined. Slope and deflection diagrams require both loads and supports to be present on the beam. If no supports are present in the beam OR if insufficient support conditions are specified, a warning message is displayed to remind the user to add at least one support to compute these diagrams.

If the geometry for the cross section is known, the **internal shear stress** and the **internal bending stress** diagrams can be displayed by pressing the toggle button next to the shear and moment diagrams. The **shear** and **bending** stress profiles for the cross section will be displayed on the sides of the cross section.

### Equations

All of the corresponding equations for the shear, moment, slope, and deflection diagrams are presented beneath the diagrams section.

Since these equations are "step" equations, the app only shows equations for one region of the beam at a time. To see equations for a different region:

- Either swipe the equations horizontally to move to a different region of the beam,
- Or tap on a region in the diagrams, and the equations beneath will automatically scroll to that region.

Equations can be displayed using two different frames of reference: (1) absolute position within the beam, or (2) relative position within the step functions's region. To toggle between these 2 frames of reference, use the toggle button right above the equations.

## App screens and controls

### Design file list screen

From the design file list screen you can either quickly start a new design file from a template or you can also access previously created files. Your work is saved automatically any time you make changes, and you can return to it from this screen next time you start the app.

**iOS only:** To delete an old design file, swipe the row to the left to display the delete button. This functionality has not yet been ported to other platforms (Android, macOS, and Windows).

### Main design screen

The main design screen integrates many sophisticated modules to help you enter design values directly and to help you visualize and interact with the results from a single place.

#### Material

The first section is for material properties. Here you can load a material from the library by pressing the search button in the left, or you can enter elastic modulus or density directly. Tap on the square checkbox button to the right of the density value to apply beam self weight. When you enable beam self weight, the density value will highlight, and the beam visualization will display a short distributed load in the background.

#### Cross Section

The second section is for cross section properties. Here you can load a material from the library by pressing the search button in the left, or you can enter moment of inertia and area directly. Similar to density, the area value is grayed-out when beam self-weight is not being applied since this value is not used for any other purpose.

When you load a shape from the library, you can edit its dimesions in-place from this module. The internal stress profiles for the geometry will update automatically, and they will be displayed to the sides.

#### Beam editor

The third section is the interactive beam free body diagram. Use this section to apply and move loads and reactions around.

#### Diagrams

The fourth section displays the diagrams for internal reactions, deflection, and stresses. You can tap on any point to read exact values. If the app has enough input data to compute internal stresses, a toggle button will be displayed to the left of the shear and bending moment diagrams to allow you to obtain internal stress values instead of shear force and bending moment.

#### Equations

The fifth section displays the equations for the internal reactions and deflection diagrams. These equations are step polynomials and they apply only to their respective spans on the beam. You can swipe the equations sideways to reveal the equations for other regions, or you can tap on the diagrams above to display the equations for a specific region.

### Design screen components

The main design screen combines a lot of complex controls to allow you to design and obtain results seamlessly. This section describes some of the most important ones that you should be familiar with.

As a general convention, any interactive widgets that accept input values will have blue backgrounds, whereas read-only output controls will have white backgrounds.

#### Loads and supports toolbar

The loads and supports toolbar allows you to drag and drop loads and supports directly onto the beam. When you tap and hold on any icon, the app will display the name of the load and you can drag it around.

#### Editable Label

By convention, editable labels on visualizations have white rounded backgrounds as this screenshot shows. Labels without a background are read-only and cannot be modified unless they have a pencil icon next to them.

#### Numpad

The on-screen numpad allows you to enter values on your phone or tablet without an attached keyboard. It also allows you to specify the units for the label that you are editing, and it has a convenience scroller to the side that allows you to slide values up and down.

### Cross section library screen

The cross section library screen allows you to browse and search for specific reference table shapes. You can browse the catalogue by swiping the horizontally or by tapping on the shape icons in the toolbar area.

You can search for specific shapes by using the input textbox. The search algorithm performs a fuzzy search to be as flexible as possible, so entering "5050" is the same as entering "50 50" or "50x50". The library by default displays data for worldwide shapes, but if you want to restrict the database used, you can do so by configuring the app Settings.

When you select a shape from the catalogue, you can see reference properties for it and you can select it to apply it to your beam design file.

### Material library screen

The material library is not as extensive as the cross section library. In this screen you can simply tap on a row to select the material and apply it to your design file.

Each material is color coded for visualization purposes. The elastic modulus and densities shown in this view are meant to be representative values for the material type and are provided for convenience only. The user should always obtain more precise and accurate values from other sources depending on the use case.

### Settings screen

The settings screen allows you to enable and disable the databases used in the cross section catalogue as well as configure the default units used throughout the app.

Since the best units to use depend on context, you can specify different units for some unit types in this screen. For length, you can configure general lengths separately from deflection lengths because these 2 measurements will usually vary by many orders of magnitude. Likewise, for pressure, you can configure the units of elastic modulus separately from internal stress.

### Exporting files

Design files can be exported by pressing the Export button right next to the Settings button. The app will generate a permanent link that you can open in a browser or send via text or email. The exported document requires an active internet connection for interaction, and it can be printed into a PDF from the browser.

### Clear loads

Loads and supports can be easily removed from the beam by dragging them off of it. On desktop editions, you can press the delete or backspace button in the keyboard to delete a load.

#### Using the device shake gesture (iOS only)

If you have many loads and supports, removing them one at a time may feel unnecessarily tedious. You can quickly remove all loads and supports from the beam by simply shaking your phone or tablet.

- Shake your device.
- Select in the prompt.
- All loads and supports will fly off the beam!

## Solver Methods and Assumptions

This software integrates a large and complex set of formulas and algorithms that work together to streamline and simplify structural beam analysis. Like all engineering tools, it is important for the engineer to understand the capabilities and limitations of the tools being used.

In the following sections we list some relevant assumptions we made, but this is not meant to be a comprehensive listing of the many implicit and explicit assumptions used througout the app.

### Reactions and deflection

Reaction forces and moments are obtained by solving the Euler–Bernoulli equations, assuming linear elastic bending. The equations are solved using local and global stiffness matrices.

We take some measures to mitigate numerical error related to the matrix numerical solver, and we make our best effort to indicate to the user when numerical error may be inteferring with the solver algorithm. However, our numerical error detection algorithms are heuristic and may not be able to catch all unstable systems of equations. For example, the system of equations may be unstable when a support is placed extremly close to another support or to the edge of the beam.

Diagrams are generated by linear additive superposition of the individual contributions of all loads and reactions once all reactions are known. Each diagram is the superposition of polynomial step functions. The polynomial equations used to generate each diagram match exactly with the equations displayed by the app.

### Internal stresses

Internal stresses are computed after the stiffness method is used to solve for all reactions. In general, the internal stress results are approximations, and they assume that the beam is long, narrow, and that the amount deflection is relatively small.

#### Assumptions

- Longitudinal axis on neutral surface of beam bends, but it does not change in terms of length.
- All cross sections remain plane and perpendicular to longitudinal axis.
- Deformation in cross section is insignificant.

### Bending Stress

The bending stress distribution across the cross section is assumed to be linear and it's based on the Flexure Formula. In essence, we assume that the bending stress at any vertical point y of the cross section is linearly proportional to the internal moment at that point of the beam, multiplied by y and divided by the moment of inertia.

`stress(y) = -M / I * y`

#### Assumptions

- Bending only occurs symmetrically along the vertical axis.
- The material is linearly elastic, so Hooke's law applies.
- The material is homogeneious.
- The cross section is constant along the length of the beam.

### Shear Stress

The shear stress distribution along the cross section vertical axis of the beam is obtained from the Shear Formula.

```
shear_stress(x,y)
= shear_force(x)
* distance_to_centroid_above(y)
* area_above(y)
/ (moment_of_inertia * width(y))
```

#### Assumptions

- Average shear stress is constant across width of section
- Approximation is only valid for narrow width cross sections with respect to their height.
- Cross section width does not change abruptly across its height
- Cross section side walls are vertical. Therefore the formula is
**not accurate for circular cross sections, plates, or wide flanges, etc.**

### Shape properties

Properties (moment of inertia, area, etc) for custom shapes are computed by generating closed 2-dimensional paths for the cross section and then applying generalized path algorithms to yield moments of area for the path. In the case of rounded shapes, the path is approximated using dense sets of straight line segments, so the computed values are reasonable approximations but not exact.

Properties for custom shapes may not match table reference values exactly because the cross section editor does not currently consider rounded corners or angled flanges and walls.

### Built-in database values

Built-in database values are compiled and derived from many public online sources and are provided for reference only. The end user should always double check values against relevant local and manufacturer standards and datasheets.

In the case of section profiles, we do our best to provide reasonable values and we perform validation by comparing reference values against our own calculations.

In the case of material properties, we chose what we think are reasonable representative values for average cases. The user should be aware that material properties vary significantly based on many factors (especially for non-metals) and other sources should be consulted for more accuracy.

## Software limitations

We continuously deliver new software functionality, but at present, these are some of the current limitations and missing features in the software. We definitely plan to add some of these missing features in future releases. Let us know if we missed something or if there is a feature in particular that you think is essential for you.

The following is not meant to be a comprehensive list of design limitations of the software and is provided for reference only.

- Yield strength not taken into account.
- Shear strength not taken into account.
- Shear flow is not taken into account.
- Internal stress output are not accurate for circular shapes, wide shapes, or wide flanges.
- Applying horizontal loads is not supported.
- Cannot set moment of inertia and section modulus for different spans independently.
- Reinforced concrete is not modeled. Concrete is treated as a homogeneous material.
- Variations in lumber properties are not modeled. Lumber is treated as a homogeneous material.

## Contact

If you have any questions or comments about the software or this documentation, please email contact@ketchep.com