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 structure ends in the ground. Foundations, retaining walls, embankments, pavements and pipelines all depend on how soil carries load, settles and drains. Soil is not a manufactured material with a data sheet: it varies from one borehole to the next, so geotechnical engineers combine site investigation, lab tests and conservative judgement. This lesson covers the three basics: describing a soil sample, classifying it, and estimating how much load a footing can carry.

## Phase relations

A soil sample has three phases: solid grains, water and air. A few ratios describe it:

```text
water content    w = mass of water / mass of solids
void ratio       e = volume of voids / volume of solids
porosity         n = e / (1 + e)
degree of sat.   S = volume of water / volume of voids
specific gravity Gs (of the solids, typically 2.65 to 2.75)
```

Two relations solve most problems:

```text
S * e = w * Gs
gamma_d = Gs * gamma_w / (1 + e)     (dry unit weight; gamma_w = 9.81 kN/m^3)
```

**Worked example.** A saturated clay (`S = 1`) has `w = 20%` and `Gs = 2.70`. Then `e = 0.20 × 2.70 = 0.54`, `n = 0.54 / 1.54 = 0.35`, and `gamma_d = 2.70 × 9.81 / 1.54 = 17.2 kN/m^3`. Compaction specifications on earthworks are written in terms of dry unit weight (or dry density) and water content.

## Classification: the USCS

The **Unified Soil Classification System** (ASTM D2487) gives each soil a two-letter symbol from grain size and plasticity. The main steps for coarse soils:

1. **Coarse or fine?** If 50% or more passes the No. 200 sieve (0.075 mm), the soil is fine-grained (silt or clay) and is classified by its Atterberg limits (liquid limit and plasticity index) on the plasticity chart.
2. **Gravel or sand?** For coarse soils, compare the gravel fraction (retained on the No. 4 sieve, 4.75 mm) with the sand fraction. The larger gives the first letter: **G** or **S**.
3. **Graded well or poorly?** For clean soils (under 5% fines), use the gradation coefficients from the grain-size curve:

```text
Cu = D60 / D10            Cc = D30^2 / (D10 * D60)
well graded (W): Cu >= 4 for gravel (>= 6 for sand) AND 1 <= Cc <= 3
otherwise poorly graded (P)
```

Soils with 5 to 12% fines get dual symbols (such as SP-SM), and those with more than 12% fines are silty or clayey (GM, GC, SM, SC), which needs Atterberg limits. The full procedure is in ASTM D2487.

**Example.** 85% passes the No. 4 sieve and 3% passes the No. 200 sieve, with `Cu = 7.5` and `Cc = 1.4`. Gravel = 15%, sand = 82%, so it is a sand; clean; `Cu >= 6` and `Cc` within 1 to 3: **SW**, well-graded sand.

## Bearing capacity of a shallow footing

A footing fails in bearing when a wedge of soil shears out beneath it. Terzaghi's classic analysis, extended by later researchers, gives the **ultimate bearing capacity** of a strip footing:

```text
q_ult = c * Nc + gamma * Df * Nq + 0.5 * gamma * B * Ngamma
```

`c` is cohesion, `gamma` unit weight, `Df` embedment depth, `B` footing width. The bearing capacity factors depend on the friction angle `phi`. One widely used set (Prandtl, Reissner and Vesic) is:

```text
Nq = e^(pi tan phi) * tan^2(45 + phi/2)
Nc = (Nq - 1) * cot phi      (5.14 when phi = 0)
Ngamma = 2 (Nq + 1) tan phi
```

For `phi = 30°`: `Nq = 18.4`, `Nc = 30.1`, `Ngamma = 22.4`. Other references give somewhat different `Ngamma` values, so state which set you used.

**Illustrative example.** A 1.5 m wide strip footing sits 1.0 m deep in sand with `c = 0`, `phi = 30°` and `gamma = 18 kN/m^3`:

```text
q_ult = 0 + 18 × 1.0 × 18.4 + 0.5 × 18 × 1.5 × 22.4 = 331 + 302 = 634 kPa
```

With a factor of safety of 3 (a common choice for allowable stress design), the allowable bearing pressure is about 211 kPa. In practice, **settlement** often governs before bearing failure, groundwater reduces the effective unit weight, and shape, depth and inclination factors modify the equation. Codes and geotechnical reports also use resistance factors in LRFD.

## Common mistakes

- **Mixing up e and n**, or using `w` as a percentage in a decimal formula.
- **Classifying from grain size alone** when the soil has more than 12% fines.
- **Ignoring groundwater**, which can roughly halve the `gamma` terms below the water table.
- **Using lab-perfect parameters** without considering site variability.

## Professional responsibility

Foundation design must be based on a site-specific geotechnical investigation and report prepared by a qualified geotechnical engineer. These examples are for learning. Real designs need review and sealing by a licensed professional engineer (PE) under the codes your jurisdiction adopts. Field drilling and sampling also need site-safety training.

## Recap

- Phase relations: `S e = w Gs`; `gamma_d = Gs gamma_w / (1 + e)`.
- USCS: coarse vs fine at 50% passing No. 200; gravel vs sand; well or poorly graded from Cu and Cc.
- `q_ult = c Nc + gamma Df Nq + 0.5 gamma B Ngamma`; divide by a factor of safety and check settlement.

## Try it

Open the **Do** tab to code phase relations, bearing factors and a simplified USCS classifier.


Use soil phase relations, classify a clean coarse soil with the USCS, and estimate the bearing capacity of a strip footing.

Soil mechanics basics

Unified Soil Classification System (USCS) Explained with 3 Practical Examples

How to Calculate the Bearing Capacity of Soil? Understanding Terzaghi's bearing capacity equations

## Why it matters

When land is paved and roofed, rain that used to soak in runs off faster and in larger volumes. Without drainage design, that means flooded streets, eroded channels, overloaded pipes and polluted streams. Almost every site plan a civil engineer works on needs a stormwater calculation, and the first tool for small sites is the **rational method**.

## The rational method

```text
Q = C * i * A
```

- `Q` is the **peak** runoff rate.
- `C` is the runoff coefficient: the fraction of rainfall that runs off at the peak (near 0.9 for pavement and roofs, much lower for lawns and woods).
- `i` is the rainfall **intensity** for a storm lasting as long as the **time of concentration** of the drainage area.
- `A` is the drainage area.

Units matter:

```text
US: Q (cfs) = C * i (in/h) * A (acres)        (the 1.008 conversion is usually dropped)
SI: Q (m^3/s) = C * i (mm/h) * A (ha) / 360
```

The method assumes rain falls uniformly over the whole area at a constant intensity for at least the time of concentration, so it is meant for **small** drainage areas. Agencies set their own limits (often somewhere between tens and a couple of hundred acres); larger or complex watersheds use hydrograph methods such as NRCS (TR-55) or software like HEC-HMS and EPA SWMM.

## Composite runoff coefficient

Real sites mix surfaces. Weight `C` by area:

```text
C = sum(A_i * C_i) / sum(A_i)
```

Agencies publish `C` values by land cover and often by storm frequency; use the ones your reviewing agency requires. Values in this lesson are illustrative.

## Time of concentration and intensity

The **time of concentration** `tc` is the time for runoff to travel from the hydraulically most distant point to the outlet. Design intensity is read for a duration equal to `tc`, because that storm makes the whole area contribute at once. Several empirical formulas exist; the **Kirpich** formula, developed for small rural watersheds, is a classic:

```text
tc (min) = 0.0078 * L^0.77 * S^-0.385      L in feet, S in ft/ft
```

Many agencies prescribe a method (often the NRCS velocity method, splitting sheet flow, shallow concentrated flow and channel flow) and a minimum `tc`, frequently 5 or 10 minutes.

Intensity comes from **intensity-duration-frequency (IDF)** data for the site and the design return period (for example the 10-year storm for pipes, 100-year for overflow routes). In the US, the source is NOAA's precipitation frequency data (https://hdsc.nws.noaa.gov/pfds/), often packaged into local design manuals. Checked on 2026-10-23.

## Worked example (illustrative)

A 2.0 ha site has 0.8 ha of roofs and pavement (`C = 0.90`) and 1.2 ha of lawn (`C = 0.25`).

1. **Composite C:** `(0.8 × 0.90 + 1.2 × 0.25) / 2.0 = (0.72 + 0.30) / 2.0 = 0.51`.
2. **Time of concentration:** the longest flow path is 1,000 ft at a 2% slope. Kirpich gives `tc = 0.0078 × 1000^0.77 × 0.02^-0.385 = 7.2 min`. With an agency minimum of 10 minutes, use 10.
3. **Intensity:** the local 10-year IDF curve gives about 80 mm/h at 10 minutes (illustrative).
4. **Peak flow:** `Q = 0.51 × 80 × 2.0 / 360 = 0.227 m^3/s` (about 8 cfs).

That peak sizes the inlet and pipe (next lesson: Manning's equation). If the same site had been woodland with `C = 0.15`, the peak would have been about 0.067 m^3/s; the difference is why many jurisdictions require **detention** to hold post-development peaks to pre-development levels, and **water quality** measures (bioretention, permeable pavement, ponds) for the first flush of runoff.

## Beyond peak flow

- **Volume** matters for detention and infiltration; the rational method gives peaks, not reliable volumes (the "modified rational method" approximates them for small sites).
- **Regulation:** the US Clean Water Act drives stormwater permits (NPDES / MS4); states and cities publish design manuals. Always design to the reviewing agency's manual.
- **Climate:** some agencies now require adjusted or future rainfall depths.

## Common mistakes

- **Using the wrong units constant** (forgetting `/360` in SI).
- **Reading intensity for the wrong duration**, such as a 1-hour value when `tc` is 10 minutes.
- **Applying the rational method to a large watershed.**
- **Forgetting off-site area** draining onto the site.
- **Using `C` for the wrong storm frequency** where the manual adjusts it.

## Professional responsibility

Drainage reports are reviewed by agencies and sealed by licensed engineers. These numbers are illustrative. Real designs need review and sealing by a licensed professional engineer (PE) under the codes your jurisdiction adopts.

## Recap

- `Q = C i A` gives peak runoff for small areas; watch the unit constant.
- Composite `C` is area-weighted; intensity is read at a duration equal to `tc` for the design return period.
- Use the reviewing agency's manual for coefficients, `tc` method, rainfall data and detention rules.

## Try it

Open the **Do** tab to compute a composite C, a Kirpich time of concentration and the peak runoff in Python.


Estimate peak runoff from a small site with the rational method: composite runoff coefficient, time of concentration and design intensity.

Hydrology and stormwater

THE RATIONAL METHOD EXPLAINED IN UNDER 5 MINUTES

Where Does Stormwater Go?

## Why it matters

Once you know a peak flow, you must carry it somewhere: in a roadside ditch, a storm sewer, a culvert or a stream channel. Most of these flow with a free surface open to the air (a pipe flowing partly full counts too). **Open-channel hydraulics** answers how deep the water will be, how fast it moves, and whether the channel will erode or overflow. **Manning's equation** is the everyday tool.

## Geometry first

For a channel cross-section with flow depth `y`:

- **Flow area** `A`.
- **Wetted perimeter** `P`: the length of boundary in contact with water (not the free surface).
- **Hydraulic radius** `R = A / P`.
- **Top width** `T` at the water surface.

For a trapezoid with bottom width `b` and side slopes `z` horizontal to 1 vertical:

```text
A = (b + z*y) * y
P = b + 2*y*sqrt(1 + z^2)
T = b + 2*z*y
```

A rectangle is the special case `z = 0`. Circular pipes have their own (longer) formulas, and design charts or software handle partly full pipes.

## Manning's equation

For **uniform flow**, where depth and velocity do not change along the channel, gravity's pull down the slope balances boundary friction:

```text
V = (k / n) * R^(2/3) * S^(1/2)        Q = V * A
k = 1.0 (SI: m, m/s)      k = 1.49 (US: ft, ft/s)
```

- `S` is the channel bed slope (m/m or ft/ft); in uniform flow it equals the energy slope.
- `n` is **Manning's roughness coefficient**, an empirical value for the lining: roughly 0.011 to 0.015 for smooth concrete and pipe, 0.02 to 0.03 for earth channels, and higher for weedy or rocky streams. Published tables (for example in FHWA and USGS references, or your agency's manual) give ranges; choosing `n` is a judgement call that matters as much as the arithmetic.

## Worked example (illustrative)

A concrete-lined trapezoidal channel has `b = 2.0 m`, side slopes `z = 2` (2H:1V), `n = 0.015` and `S = 0.001`. At a depth of 1.0 m:

1. `A = (2.0 + 2 × 1.0) × 1.0 = 4.00 m^2`
2. `P = 2.0 + 2 × 1.0 × sqrt(5) = 6.47 m`
3. `R = 4.00 / 6.47 = 0.618 m`
4. `V = (1 / 0.015) × 0.618^(2/3) × 0.001^(1/2) = 66.7 × 0.726 × 0.0316 = 1.53 m/s`
5. `Q = 1.53 × 4.00 = 6.12 m^3/s`

## Normal depth

Usually you know `Q` and need the depth. Manning's equation cannot be rearranged for `y` in closed form for most shapes, so you solve it numerically: guess a depth, compute `Q`, and adjust. Because `Q` increases steadily with depth, **bisection** always works: start with a depth that is too small and one that is too large, test the midpoint, and keep the half that brackets the answer. Spreadsheets use Goal Seek; Python can use bisection or `scipy.optimize.brentq` in a full environment. The resulting depth is the **normal depth** `yn`. Add **freeboard** above it so waves, debris and uncertainty in `n` do not cause overtopping.

## Subcritical or supercritical: the Froude number

```text
Fr = V / sqrt(g * D)      D = A / T (hydraulic depth)
```

- `Fr < 1`: **subcritical** (tranquil). Most natural streams and mild channels. Downstream conditions affect the depth.
- `Fr > 1`: **supercritical** (rapid). Steep chutes and spillways. Waves cannot travel upstream.
- `Fr = 1`: critical flow; the depth is unstable and surface waves form.

For the example, `D = 4.00 / 6.0 = 0.667 m` and `Fr = 1.53 / sqrt(9.81 × 0.667) = 0.60`: subcritical. A change from supercritical to subcritical flow forms a **hydraulic jump**, which dissipates energy and must be controlled with an energy dissipator.

## Design checks

- **Capacity:** normal depth plus freeboard fits within the section.
- **Velocity limits:** high velocities erode unlined or grass channels; very low velocities let sediment settle in pipes (minimum self-cleansing velocities, often around 0.6 to 0.9 m/s, appear in many manuals).
- **Lining shear stress:** `tau = gamma * R * S` compared with the permissible shear of the lining (the method used in FHWA's HEC-15).
- **Culverts and bridges** have their own inlet and outlet control methods (FHWA HDS-5, HEC-RAS).

## Common mistakes

- **Using `k = 1.0` with US units**, or 1.49 with SI.
- **Including the free surface** in the wetted perimeter.
- **Using percent slope** (0.1) instead of decimal (0.001).
- **Treating `n` as exact**: test a range, especially for natural channels.
- **Applying uniform-flow results** near a culvert, drop or junction where flow is gradually or rapidly varied.

## Professional responsibility

Channel, culvert and storm-sewer designs follow the reviewing agency's manual and are sealed by licensed engineers. These values are illustrative. Real designs need review and sealing by a licensed professional engineer (PE) under the codes your jurisdiction adopts.

## Recap

- `R = A / P`; Manning: `V = (k/n) R^(2/3) S^(1/2)`, `Q = V A`.
- Solve normal depth numerically; bisection is simple and reliable.
- Check Froude number, velocity, shear and freeboard before calling a channel adequate.

## Try it

Open the **Do** tab to code trapezoid geometry, Manning's equation, a normal-depth solver and the Froude number.


Use Manning's equation to find the capacity of a channel or pipe, solve for normal depth, and check whether flow is subcritical.

Open-channel hydraulics

MANNING"S EQUATION EXPLAINED IN 5 MINUTES

Open Channel Flow Concepts

Soil phase relations, classification and bearing capacity, rational-method runoff, and open-channel flow with Manning's equation.