Forces and Loads

How Bridges Work 🏆

Chapter 1 · Forces and Loads

Within sight of each other, on the Firth of Forth, stand three bridges that carry traffic across the same water in three completely different ways: a steel rail bridge from 1890 built as a series of balanced cantilevers, a suspension road bridge from 1964, and a cable-stayed crossing from 2017. All three do the same job. They take a load that is out in the middle of the air and pass it, step by step, to the ground on either side. This course follows that idea from the simplest beam to the Forth bridges themselves, and this first chapter gives you the vocabulary: the five kinds of force, the two kinds of load, and the picture of a load path that every later chapter builds on.

What was run

Every Python snippet in this chapter and in the three exercise solutions was executed, and the output shown in the comments is what Python printed. You only need Python 3 and its standard library.

The Job of a Bridge

A bridge has one rule: whatever weight lands on it must end up in the ground. Nothing else matters if that fails. Everything in this course, from the shape of an arch to the paint on the Forth Bridge, is about making that journey safe and economical. Engineers describe the journey with two ideas: loads (what pushes on the bridge) and forces (what happens inside the structure as it resists them).

Two Kinds of Load

A load is anything that pushes on the structure. They are sorted by how they behave:

Load	What it is	Example	Behaviour
Dead load	the weight of the bridge itself	steel, concrete deck, road surface, railings	always there, never changes
Live load	the weight of things that use the bridge	cars, lorries, trains, pedestrians	moves and varies
Environmental load	weather and nature	wind, temperature change, snow, water flow, earthquakes	varies, sometimes suddenly

The surprise for beginners is how big the dead load is. For a long bridge, carrying its own weight is the hardest job: a longer span has to support more steel, and that extra steel has to be supported too. Much of bridge history, including the cantilever design of the Forth Bridge, is a story of using material more cleverly so the bridge wastes less strength carrying itself.

Mass Is Not Force

Before any calculation, one trap. Mass (kilograms) is how much matter there is. Force (newtons, N, or kilonewtons, kN) is how hard that mass pulls down under gravity. To convert, multiply by the gravitational acceleration, about 9.81 m/s²: a 1000 kg mass weighs about 9.81 kN. Engineers also speak loosely about "tonnes" of load, but the calculation must be in force units, and mixing the two is the commonest error in a first structures course.

G = 9.81                       # m/s2, turns mass into weight

def kn(kg):
    return kg * G / 1000      # kg -> kilonewtons

deck_area = 12 * 8             # a 12 m x 8 m deck, in m2
slab_mass = deck_area * 0.25 * 2400   # 0.25 m of concrete at 2400 kg/m3
dead = kn(slab_mass)
live = 5 * deck_area              # assume 5 kN on every m2
print(round(slab_mass), round(dead, 1), live, round(dead + live, 1), round((dead + live) / 4, 1))
# 57600 565.1 480 1045.1 261.3
    

Read the output as: the slab has a mass of 57,600 kg, which is a dead load of 565.1 kN; the assumed live load adds 480 kN, giving 1045.1 kN in total; shared between four supports that is 261.3 kN each. (The live load of 5 kN/m² is a round figure chosen for this example. Real design codes set the values for each bridge type.)

Five Kinds of Force

Loads come from outside. Forces are what they cause inside each member of the structure. There are five basic kinds, and every bridge in this course is an arrangement of them:

Force	What it does	Everyday picture	Where in a bridge
Tension	pulls the material apart	a tug of war rope	suspension cables, stay cables, hangers, bottom chord of a truss
Compression	squeezes the material together	standing on a can	stone arches, towers, piers, top chord of a truss
Shear	slides one part past the next	scissors cutting paper	near supports; bolts and rivets joining members
Bending	curves the member: top squeezed, bottom stretched	a plank across a ditch	every beam and deck
Torsion	twists the member	wringing out a cloth	curved decks, off-centre loads, boxes in wind

Bending is really tension and compression working together: when a plank sags, its lower face is stretched and its upper face is squeezed. That is why a deep beam is stronger than a shallow one of the same weight (How Bridges Work 2). Materials differ in which force they prefer. Stone and concrete handle compression well and tension badly; steel handles both; a rope handles only tension. Much of bridge design is choosing the right material for the right force.

The Load Path

Follow a lorry as it crosses a bridge. Its wheels press on the road surface. The surface passes the weight to the deck, the deck to the beams or girders under it, the beams to the supports (piers, towers, abutments), and the supports to the foundations, and finally into the rock or soil. That chain is the load path, and a bridge fails if any link is too weak or missing.

lorry wheel
   │
   ▼
road surface
   │
   ▼
deck slab
   │
   ▼
beams / girders          ← bending and shear
   │
   ▼
piers or towers          ← compression
   │
   ▼
foundations
   │
   ▼
rock or soil
    

Each type of bridge is a different answer to the question "which path should the load take?". A beam sends it sideways along the member and down through its ends. An arch turns the load into compression along a curve (How Bridges Work 3). A suspension bridge hangs the deck from cables in tension, which run to towers and anchorages (How Bridges Work 7). A cable-stayed bridge runs straight lines of tension from deck to tower (How Bridges Work 8). Learn to ask "where does this weight go next?" and any bridge becomes legible.

Balance: Reactions at the Supports

A structure that is standing still has forces that balance: what pushes down must be matched by what pushes up. At the two ends of a simple beam the ground pushes up with reactions, and they must add up to the load. Where the load sits decides how the reaction is shared: the closer the load is to a support, the more of it that support carries. For a single point load on a beam of span L, with the load a distance a from support A, taking moments about A gives the reaction at B, and the reaction at A is what is left:

def reactions(span, load, a):
    rb = load * a / span          # moments about A
    ra = load - rb                # the rest goes to A
    return ra, rb

print(reactions(10, 60, 3))        # (42.0, 18.0)  load near A: A carries more
print(reactions(10, 60, 5))        # (30.0, 30.0)  load at midspan: shared equally
    

A 60 kN load three metres from A on a 10 m beam gives 42 kN to A and 18 kN to B, which add up to 60 kN. At midspan the load splits evenly. These two numbers are the first thing an engineer finds when checking any beam, because everything else (bending, shear, support design) follows from them.

Stress: How Hard the Material Works

A force alone does not tell you whether a member will fail: a 100 kN pull is nothing to a thick steel bar and fatal to a thin one. What matters is the force per unit of cross-section, called stress: stress = force ÷ area. With force in newtons and area in square millimetres, the answer comes out directly in megapascals (MPa), the unit in which material strengths are quoted. Ordinary structural steel yields (starts to deform permanently) at roughly 250 to 350 MPa, depending on the grade. In design you never aim for the limit: you divide the strength by a factor of safety (say 2) so that the real working stress is only half of what the material can take.

import math

def stress_mpa(force_kn, area_mm2):
    return force_kn * 1000 / area_mm2     # N per mm2 = MPa

area = math.pi * (20 / 2) ** 2              # a 20 mm diameter rod
s = stress_mpa(120, area)
print(round(area, 1), round(s, 1), round(250 / s, 2))
# 314.2 382.0 0.65

need = 120000 * 2 / 250                   # area for a factor of safety of 2
d = 2 * math.sqrt(need / math.pi)
print(round(need), round(d, 2))
# 960 34.96
    

The 20 mm rod would be stressed to 382 MPa under a 120 kN pull, well beyond a steel of 250 MPa (the "factor" of 0.65 means it would fail). To reach a safety factor of 2 the rod needs an area of 960 mm², which is a diameter of about 35 mm. Doubling the diameter would multiply the area by four, so the cost of strength grows quickly, and this is why engineers care so much about shape and not just size.

Common mistakes

Mixing mass and force (kg against kN); forgetting that the dead load is always there, so the live load is an addition and not the whole story; and using the diameter instead of the area in a stress calculation (stress depends on the area, which grows with the square of the diameter).

Where This Leads

You now have the toolkit: five forces, three kinds of load, a load path from wheel to rock, reactions that balance the load, and stress as force over area. The next chapters put these to work on one structure type at a time, beginning with the simplest of all: a beam across a gap. The three Forth bridges will return in the later chapters as worked examples of the cantilever, the suspension bridge and the cable-stayed bridge.

Exercise 1

A footbridge deck is 20 m long and 3 m wide, made of a 0.2 m thick concrete slab (2400 kg/m³), and is designed for a live load of 4 kN on every square metre. It rests on two supports, one at each end. Write a program that works out the mass of the slab, the dead load and the live load in kN, the total, and how much each support carries. Which is bigger, the dead load or the live load, and what does that suggest about the design?

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Exercise 2

A beam of span 12 m carries two point loads: 30 kN at 4 m from the left support A and 50 kN at 9 m from A. Write reactions(span, loads) that takes a list of (force, distance) pairs and returns the reactions at A and B by taking moments about A. Check your answer by confirming that the reactions add up to the total load, and by taking moments about B instead. Predict first which support carries more.

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Exercise 3

A steel hanger carries a steady tension of 200 kN. The steel yields at 355 MPa and the design needs a factor of safety of 1.5. Work out the minimum cross-sectional area, then test stock rods of diameter 25, 30, 32, 36 and 40 mm: print the area, the stress and the factor of safety for each, and report the smallest stock rod that is safe.

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Quick Reference

Loads: dead (the bridge itself), live (traffic), environmental (wind, temperature, water)
Forces: tension, compression, shear, bending, torsion
Load path: wheel → surface → deck → beams → piers or towers → foundations → ground
Reactions: they add up to the total load; the nearer support carries more
Stress: force ÷ area; N/mm² = MPa
Factor of safety: strength ÷ working stress
Units: kg is mass; kN is force; multiply kg by 9.81 and divide by 1000

Coming next

How Bridges Work 2 takes the simplest structure of all, a beam across a gap: why bending makes depth matter more than width, why the I-beam has its shape, and how the span limits what a plain beam can do.