Load combinations, steel and reinforced concrete beams, soils and bearing capacity, stormwater runoff, Manning's equation, surveying, transportation, civil CAD and BIM, and construction administration.

Structural & Civil Design

## Why it matters

Every structural calculation starts with one question: what loads must this member carry, and in what combinations? Get the loads wrong and the most careful steel or concrete check is worthless. Load work is also where junior engineers spend much of their first years, so being fast and organised here is a real career skill.

## Load types

US building design takes its minimum loads from **ASCE/SEI 7, Minimum Design Loads and Associated Criteria for Buildings and Other Structures** (the current edition is ASCE 7-22), adopted through the International Building Code and local amendments. The common load symbols are:

- **D, dead load:** the weight of the structure and permanent parts (slabs, beams, finishes, fixed equipment).
- **L, live load:** occupants, furniture and movable contents, set by occupancy (office, assembly, storage...). **Lr** is roof live load (workers and materials during maintenance).
- **S, snow; R, rain; W, wind; E, earthquake;** plus soil pressure, fluid pressure, flood, ice and others.

Minimum live, snow, wind and seismic values come from ASCE 7's chapters and maps, and many jurisdictions modify them. Look them up in the adopted edition (ASCE's hazard tool is at https://ascehazardtool.org); never copy numbers from a course or an old project. Checked on 2026-10-23.

## From area load to line load

Floor and roof loads are given per unit area (kPa or psf). A beam carries the area halfway to the next beam on each side: its **tributary width**.

```text
line load (kN/m) = area load (kPa) * tributary width (m)
```

**Illustrative example.** Beams at 3.0 m spacing carry a dead load of 4.0 kPa (slab, finishes, ceiling, services) and an office-type live load of 2.4 kPa. Then `D = 4.0 × 3.0 = 12.0 kN/m` and `L = 2.4 × 3.0 = 7.2 kN/m`. These area loads are illustrative; real ones come from a weight takeoff and the code.

## Load combinations

Loads rarely peak at the same time, and each has a different uncertainty. **Strength design** (LRFD in steel, strength design in concrete) multiplies each load by a **load factor** and checks every combination in the code. The two you will meet first, written in the ASCE 7 style, are

```text
1.4D
1.2D + 1.6L + 0.5(Lr or S or R)
```

Others combine roof, snow, wind and seismic loads, and one family uses **0.9D** with wind or seismic to check uplift and overturning, where dead load helps. **Allowable stress design (ASD)** uses separate, unfactored-style combinations with the member's capacity divided by a safety factor. Take the full, current set from the adopted ASCE 7 edition; the factors and companion-load rules change between editions.

### Worked example (continued)

For the floor beam:

- `1.4D = 1.4 × 12.0 = 16.8 kN/m`
- `1.2D + 1.6L = 14.4 + 11.52 = 25.92 kN/m` (no snow on an interior floor)

The second **governs**. For an 8.0 m simply supported span:

```text
Mu = wu * L^2 / 8 = 25.92 * 8.0^2 / 8 = 207.4 kN*m
Vu = wu * L / 2   = 25.92 * 8.0 / 2   = 103.7 kN
```

These are the **required strengths** you compare with the member's **design strength** in the next two lessons.

### Uplift: the other direction

On a light roof, wind can pull upward. With `D = 3.0 kN/m` and a wind uplift of `W = -8.0 kN/m`, the combination `0.9D + 1.0W = 2.7 - 8.0 = -5.3 kN/m`: the beam and its connections must resist a net upward load, and the bottom flange may now be in compression. Always look for the minimum as well as the maximum.

## Doing it in code

Spreadsheets and Python handle combinations well: store loads by type, store each combination as a dictionary of factors, compute every combination, and report the maximum and minimum along with its name. Keeping the name matters, because reviewers need to see which combination governed. Commercial analysis software does the same for frames with many load cases.

## Common mistakes

- **Using the wrong tributary width** (the full spacing on an edge beam, or forgetting a cantilever).
- **Mixing service and factored loads**, for example checking deflection with factored loads.
- **Stopping at 1.2D + 1.6L** and missing wind, snow drift or uplift combinations.
- **Old or unadopted editions**: always confirm which ASCE 7 edition and local amendments apply.
- **Unit slips** between kN, kip, kPa and psf (1 kPa is about 20.9 psf).

## Professional responsibility

This lesson teaches the method. Load determination for a real structure depends on code edition, site hazards, occupancy and engineering judgement. Real designs need review and sealing by a licensed professional engineer (PE) under the codes your jurisdiction adopts.

## Recap

- Loads come from ASCE 7 (as adopted locally); area load times tributary width gives line load.
- Strength design factors and combines loads; the largest result governs, and the smallest checks uplift.
- `Mu = wL^2/8` and `Vu = wL/2` for a simple span with uniform load.

## Try it

Open the **Do** tab to compute line loads, find the governing combination and the design moment in Python.


Turn area loads into beam line loads, apply factored load combinations, and find the governing design moment.

Loads and load combinations

Intro to Structural Analysis - Loads and LRFD

Structural Analysis - Video 41:  LRFD Load Combinations Example (Ref. ASCE 7-22)

## Why it matters

Steel beams and girders frame most commercial buildings, bridges and industrial structures. A beam check combines three questions: is it strong enough in bending, strong enough in shear, and stiff enough that floors do not feel bouncy or crack finishes? This lesson walks through the simplest complete check so the structure of a real calculation is clear before you meet the many special cases.

## The governing specification

In the US, structural steel buildings are designed to **ANSI/AISC 360, Specification for Structural Steel Buildings** (the current edition is AISC 360-22), published by the American Institute of Steel Construction (https://www.aisc.org), which makes the Specification freely downloadable. Section properties come from the **AISC Steel Construction Manual** or AISC's shapes database. Bridges follow AASHTO's specifications instead. Checked on 2026-10-23.

Like concrete, steel uses **LRFD**: a factored load effect must not exceed a reduced (design) strength:

```text
Mu <= phi * Mn      Vu <= phi * Vn
```

`Mu` and `Vu` come from the load combinations of the previous lesson; `Mn` and `Vn` are the **nominal strengths** from the specification.

## Flexure: yielding

A W-shape (wide-flange) beam that is **compact** (flanges and web stocky enough not to buckle locally) and **laterally braced** at short enough intervals can reach its **plastic moment**:

```text
Mn = Mp = Fy * Zx        phi = 0.90
```

`Fy` is the yield stress (50 ksi for the common ASTM A992 grade) and `Zx` the plastic section modulus about the strong axis. Most rolled W-shapes are compact in flexure for 50 ksi steel; the specification gives the slenderness limits to check.

### Lateral-torsional buckling

If the compression flange is not braced often enough, the beam can buckle sideways and twist before it yields. AISC 360 expresses this with the **unbraced length** `Lb` compared with limits `Lp` and `Lr` for the section: below `Lp` the full plastic moment applies; beyond it, the strength drops. A slab attached to the top flange usually braces it continuously in the finished building, but **not during construction** before the slab cures, which is why erection checks matter. This lesson's example assumes full lateral bracing.

## Shear

For a rolled I-shape, shear is carried by the web:

```text
Vn = 0.6 * Fy * Aw * Cv1     Aw = d * tw
```

For most rolled I-shapes with 50 ksi steel, the specification allows `phi = 1.0` and `Cv1 = 1.0` when the web meets its slenderness limit; you confirm that condition for the section you use. Shear rarely governs simple-span floor beams, but it often does for short, heavily loaded transfer beams and at coped connections.

## Deflection

Deflection is a **serviceability** check with **unfactored (service)** loads. For a simple span with uniform load:

```text
delta = 5 * w * L^4 / (384 * E * I)
```

`E = 29,000 ksi` for steel. Limits come from the building code and the project (L/360 under live load is a common floor criterion for members supporting plaster or brittle finishes); vibration may govern long, light floors and has its own AISC design guide.

## Worked example (illustrative)

A floor beam spans 30 ft, laterally braced by the slab, with a factored load `wu = 2.0 kip/ft` and a service live load of 0.8 kip/ft. Try a section with properties close to a W18x35: `Zx = 66.5 in^3`, `Ix = 510 in^4`, `d = 17.7 in`, `tw = 0.300 in`, `Fy = 50 ksi`.

1. **Flexure.** `Mu = 2.0 × 30^2 / 8 = 225 kip-ft`. `phi*Mn = 0.90 × 50 × 66.5 / 12 = 249.4 kip-ft`. Ratio 0.90: OK.
2. **Shear.** `Vu = 2.0 × 30 / 2 = 30 kips`. `phi*Vn = 1.0 × 0.6 × 50 × (17.7 × 0.300) = 159 kips`. Ratio 0.19: OK.
3. **Live-load deflection.** `w = 0.8 / 12 = 0.0667 kip/in`, `L = 360 in`: `delta = 5 × 0.0667 × 360^4 / (384 × 29,000 × 510) = 0.99 in`. The L/360 limit is 1.00 in. Ratio 0.99: OK, just.

All three pass, but deflection is nearly at its limit. A designer might choose a slightly deeper section, or camber the beam for dead load, depending on project requirements. If the span were 32 ft, `Mu` would be 256 kip-ft and flexure would fail.

## Common mistakes

- **Assuming full bracing** when the compression flange is unbraced (cantilevers, construction stage, uplift putting the bottom flange in compression).
- **Using factored loads for deflection**, or service loads for strength.
- **Forgetting self-weight** of the beam (35 lb/ft for this one) in the dead load.
- **Mixing units**: kip-in vs kip-ft, ksi vs psi.
- **Ignoring connections.** Members are often fine and connections or coped ends govern.

## Professional responsibility

This is a teaching example. Real steel design covers local buckling, lateral-torsional buckling, combined forces, connections, stability, fire and fatigue under the adopted specification. Real designs need review and sealing by a licensed professional engineer (PE) under the codes your jurisdiction adopts.

## Recap

- LRFD check: `Mu <= phi*Mn`, `Vu <= phi*Vn`, plus a service-load deflection limit.
- Braced compact beam: `phi*Mn = 0.90 Fy Zx`; unbraced length can reduce it.
- Shear on rolled shapes: `0.6 Fy d tw` with the specification's factors; deflection `5wL^4/(384EI)`.

## Try it

Open the **Do** tab to write a beam-check function that reports moment, shear and deflection ratios.


Check a laterally braced steel beam for flexure, shear and live-load deflection, and know when lateral-torsional buckling takes over.

Steel member design

Flexural Strength of Steel Beam using LRFD and ASD|ANSI/AISC 360-16

## Why it matters

Reinforced concrete is the most used construction material on Earth: foundations, slabs, beams, columns, walls, bridges, tanks and tunnels. Concrete is strong in compression and weak in tension (roughly a tenth as strong), so steel bars are placed where tension develops. Sizing that steel for bending is the first calculation every structural engineer learns, and it shows how strength design reasons about failure.

## The governing code

US building concrete is designed to **ACI 318, Building Code Requirements for Structural Concrete** (the current edition is ACI 318-25, published in 2025), from the American Concrete Institute (https://www.concrete.org). Your jurisdiction adopts a specific edition through its building code, often an earlier one, so confirm which applies. Bridges use AASHTO; other countries use their own codes (Eurocode 2, CSA A23.3). This lesson uses the strength-design ideas common to these codes, with symbols as in ACI 318. Checked on 2026-10-23.

## How a beam carries moment

Under positive moment, the top of a beam is in compression and the bottom in tension. After the concrete cracks, the bottom steel carries all the tension `T = As * fs`, and the concrete above the neutral axis carries the compression `C`. The two forces form a couple separated by a lever arm.

At ultimate strength, the code assumes:

- concrete crushes at a strain of **0.003** at the extreme compression fibre;
- strain varies linearly through the depth (plane sections stay plane);
- the curved concrete stress distribution can be replaced by an **equivalent rectangular stress block** of intensity `0.85 f'c` and depth `a = beta1 * c`, where `c` is the neutral-axis depth and `beta1` is a code factor (0.85 for normal-strength concrete up to 4,000 psi, lower for stronger concrete);
- tension steel has yielded (`fs = fy`) in a properly designed beam.

Equilibrium (`C = T`) gives the block depth, and the couple gives the nominal moment:

```text
a  = As * fy / (0.85 * f'c * b)
Mn = As * fy * (d - a/2)
```

`b` is the beam width and `d` the **effective depth** from the top to the centroid of the tension steel.

## Strength reduction and ductility

The design strength is `phi * Mn`, and it must be at least `Mu`. ACI 318 sets `phi = 0.90` for **tension-controlled** sections, where the net tensile strain in the steel is large enough (at least 0.005) that the beam would crack and deflect visibly before it fails. Sections with less steel strain get a lower `phi`, and the code limits how much reinforcement a beam may have so it does not fail by sudden concrete crushing. The strain check is:

```text
c     = a / beta1
eps_t = 0.003 * (d - c) / c
```

The code also sets a **minimum** amount of flexural steel, so a lightly loaded beam does not fail abruptly when it first cracks. Look up the minimum, cover, spacing and bar-development rules in the adopted edition.

## Worked example (illustrative)

A rectangular beam has `b = 12 in`, overall height 24 in, and `d = 21.5 in`, with `f'c = 4,000 psi` (4 ksi) and Grade 60 steel (`fy = 60 ksi`). The required moment is `Mu = 250 kip-ft`.

1. **Required nominal moment.** `Mu / phi = 250 × 12 / 0.90 = 3,333 kip-in`.
2. **Solve for As.** Substitute `a` into `Mn`: `3,333 = 60 As (21.5 - 60 As / (1.7 × 4 × 12))`. This is a quadratic, `44.12 As^2 - 1,290 As + 3,333 = 0`, whose smaller root is `As = 2.86 in^2`.
3. **Choose bars.** Three #9 bars give `As = 3 × 1.00 = 3.00 in^2`.
4. **Check strength.** `a = 3.00 × 60 / (0.85 × 4 × 12) = 4.41 in`; `phi Mn = 0.90 × 3.00 × 60 × (21.5 - 2.21) / 12 = 260 kip-ft >= 250`. OK.
5. **Check strain.** `c = 4.41 / 0.85 = 5.19 in`; `eps_t = 0.003 × (21.5 - 5.19) / 5.19 = 0.0094 >= 0.005`: tension-controlled, so `phi = 0.90` was right.

A full design would continue: minimum steel, bar spacing and cover so the three bars fit with their stirrups, shear reinforcement, deflection, crack control, and development and splice lengths.

## Common mistakes

- **Using overall height instead of d.** The lever arm is measured to the steel centroid.
- **Mixing psi and ksi**: `f'c = 4,000` in one term and `fy = 60` in another.
- **Taking the larger root** of the quadratic, which gives an absurd amount of steel.
- **Assuming phi = 0.90** without checking the steel strain.
- **Forgetting constructability**: bars must fit, with cover and clear spacing for the aggregate to flow around them.

## Professional responsibility

This lesson teaches the mechanics of flexural design only. Real concrete design includes shear, torsion, serviceability, durability, detailing and seismic requirements. Real designs need review and sealing by a licensed professional engineer (PE) under the codes your jurisdiction adopts.

## Recap

- Stress block: `a = As fy / (0.85 f'c b)`; `Mn = As fy (d - a/2)`; design with `phi Mn >= Mu`.
- Solve a quadratic for the required steel, then round up to real bars.
- Check that `eps_t >= 0.005` (tension-controlled) and follow the code's limits and detailing rules.

## Try it

Open the **Do** tab to size reinforcement for a beam and check its strength and steel strain in Python.


Use the equivalent stress block to find a beam's flexural strength, size the tension reinforcement, and check that the section is tension-controlled.

Reinforced concrete basics

Design of Simply Supported Reinforced Concrete Beam using ACI 318

Concrete Beam Design (ACI 318-19)

Combine loads into factored design loads, then check a steel beam and size reinforcement for a concrete beam by strength design.