Load Calculation

Part of Bridges

Estimating the loads a bridge must carry and checking whether your design is strong enough.

Why This Matters

Every bridge fails because the actual stress in its materials exceeded what those materials can withstand. Load calculation is the practice of estimating those stresses before you build, while you can still change the design. It does not require advanced mathematics — the basic principles are arithmetic and geometry — but it does require systematic thinking about everything that might load the bridge and everything that might weaken it.

Intuition and tradition are partial guides. Experienced bridge builders develop feel for proportions that work, and traditional structures often embody accumulated empirical knowledge. But intuition fails when you are building something larger, spanning farther, or carrying heavier loads than was common before. A cart bridge needs to carry much more than a footbridge. A bridge that occasionally carries loaded wagons needs to be designed for that load specifically, not assumed to be adequate because it carried pedestrians safely for years.

In a rebuilding context, the effort spent on load calculation is a form of resource conservation. Under-designing a bridge wastes the community’s investment when it fails prematurely. Over-designing wastes materials and labor that could be used elsewhere. Good calculation steers between these failures.

Types of Loads

Dead load is the permanent weight of the bridge itself — the deck, the structural beams or arch, the fill, the railings. Dead load is constant and calculable. Estimate it by calculating the volume of each component and multiplying by the material’s weight per unit volume:

• Stone masonry: ~2,200 kg/m³
• Solid timber: ~700–900 kg/m³ depending on species
• Compacted gravel fill: ~1,800 kg/m³
• Earth fill: ~1,600–1,800 kg/m³

Sum all components to get total dead load. Divide by the span to get dead load per meter of bridge length.

Live load is the variable load of users: people, animals, carts, wagons. This is what the bridge was built to carry. Live load changes with each crossing, may occur simultaneously with other crossings, and sometimes includes impact forces from moving loads.

Standard working estimates:

• Pedestrians: 400–500 kg/m² of deck area
• Pack animals in a line: 200–300 kg/m
• Loaded farm cart (horse-drawn, 2 wheels): 1,500–2,500 kg total
• Loaded wagon (4 wheels): 3,000–6,000 kg total
• Ox cart heavily loaded: 2,000–4,000 kg

Always design for the maximum realistic load. If the community has large wagons, design for them even if most crossings are pedestrian.

Dynamic load accounts for the fact that moving loads create larger stresses than the same load applied statically. A horse trotting across a beam bridge bounces the deck, creating momentary stresses larger than its static weight would generate. For simple timber bridges carrying animals and carts, use a dynamic amplification factor of 1.3–1.5 — multiply your live load estimates by this factor before calculating stresses.

Wind load is significant for long spans and tall structures. For most small bridges under 10 m span, wind load on the structure itself is modest. However, wind load on vehicles, animals, or canvas-covered wagons crossing a high exposed bridge can be significant. For bridges over 5 m above the waterway in open, windy locations, include a lateral wind force of at least 50 kg/m of bridge length.

Flood load includes both the hydrostatic pressure of backed-up water and the hydrodynamic force of flowing water against piers. This is covered in flood planning, but remember it as a structural load as well.

Simple Beam Calculation

For a simply supported beam (supported at both ends, free to rotate at supports) carrying a uniform load w (kg/m) over span L (meters):

Maximum bending moment at mid-span: M = wL²/8 (in kg·m)

Maximum shear force at supports: V = wL/2 (in kg)

To check if a timber beam can carry this moment, you need the section modulus of the beam’s cross-section. For a rectangular beam of width b and depth d:

Section modulus Z = bd²/6 (in m³)

The required section modulus to carry moment M: Z_required = M / f_allow

where f_allow is the allowable bending stress of the timber in kg/m² (convert from standard values in kg/cm² or MPa).

Typical allowable bending stresses for structural timber (working stress, not failure stress):

• Oak, ash, beech: 80–120 kg/cm² (8–12 MPa)
• Pine, spruce, fir: 60–90 kg/cm² (6–9 MPa)
• Defective or wet timber: reduce by 30–50%

Example: A timber beam bridges 4 m, carries dead load of 100 kg/m and live load of 200 kg/m (× 1.4 dynamic factor = 280 kg/m). Total design load = 380 kg/m.

M = 380 × 4² / 8 = 760 kg·m = 76,000 kg·cm

Using oak at 100 kg/cm²: Z_required = 76,000 / 100 = 760 cm³

For a beam of width b = 15 cm, required depth: d = √(6Z/b) = √(6 × 760 / 15) = √304 = 17.4 cm

So a 150 × 180 mm oak beam is adequate (180 mm ≥ 174 mm). Use 150 × 200 mm for a comfortable margin.

Arch Thrust Calculation

For a semicircular arch of span S carrying a uniform load per unit length of w (kg/m), the horizontal thrust H at the abutment is approximately:

H = wS²/(8r)

where r is the rise of the arch (for a semicircle, r = S/2, giving H = wS/4).

This thrust must be resisted by the abutment mass. A rough check: the resultant force line (combining thrust and vertical load) must fall within the middle third of the abutment base for stability. If it falls outside the middle third, the abutment will tend to rotate.

For simple stone masonry in compression under the arch thrust: check that the compressive stress in the arch ring does not exceed the allowable stress for the masonry (typically 10–20 kg/cm² for rubble masonry, 30–50 kg/cm² for good ashlar). Arch ring compressive stresses are almost always far below this limit for modest spans — arches are very efficient in compression.

Safety Factors and Conservative Design

Never design to exactly the calculated capacity. Always include a safety factor — a multiplier that the structure’s capacity must exceed the calculated demand by:

• Timber structures: safety factor of 2.5–3.0
• Masonry arches: safety factor of 3.0–4.0
• Novel or uncertain designs: safety factor of 4.0 or more

Apply safety factors to account for: timber variability and hidden defects, uncertainty in load estimates, material deterioration over time, errors in construction, and loads you didn’t think of.

In practice: if your calculation says a beam can just carry the load, it isn’t safe enough. The calculation should show the beam has more than twice the capacity needed, preferably three times, before you feel confident.