Section 1: Types of Force
Forces impact structures in a variety of different ways depending on how they are applied. Forces can move a structure slightly or cause damage by changing its shape. Sometimes the changes are almost impossible to see — for example, a bridge will sag slightly when a vehicle drives over it. Forces can also make an object move or change direction.
There are five types of force you need to know:
Compression
The supporting columns of the Queensferry Crossing experience a huge downward force from the weight of the platform and traffic crossing it. The ground exerts an upward reaction force on the bases of the columns. These two opposing forces try to squash or shorten the column — the column is said to be in compression. A weightlifter's body experiences compression in the same way: the weights push down, the ground pushes up.
Tension
Tension is the opposite of compression. A structure under tension is being pulled apart. In a tug of war, both teams pull the rope in opposite directions — the rope is in tension. Wire ropes, cables and chains are typically in tension.
Bending
Structures that carry loads across their length are subject to bending forces. A weightlifting bar bends due to the downward forces of the weights at each end. Beams in buildings and bridges experience bending.
Shear
Shear forces can be described as tearing or cutting forces. A pair of scissors cuts a ribbon using shear — the two blades push in opposite directions across the material. A lawn mower blade shears the grass.
Torsion
Torsion (also called torque) forces twist or turn a structure. A screwdriver being twisted to drive a screw applies torsion to the screw. A spanner turning a bolt applies torsion to the bolt.
Section 2: Free-Body Diagrams
Before working through levers, moments and beams, it helps to have one tool that will be used again and again: the free-body diagram. It is simply a simplified sketch that shows all the forces acting on a structure or body, with everything else stripped away. It isolates the object from its surroundings and replaces real components (hinges, supports, ropes, the ground) with arrows representing the forces they exert.
To draw a free-body diagram:
- Draw a simple outline of the structure (or just a line or box)
- Replace all connections and supports with arrows showing the force they exert
- Label each force with a letter or value
- Show downward forces (loads, weight) acting through their point of application
- Show upward reaction forces at support points
A crate sits still on flat ground. Two forces act on it: its own weight (W), pulling straight down, and the reaction from the ground (R), pushing straight up. Because the crate is not moving, these two forces must be equal and opposite.
- The amount of matter in an object
- Measured in kilograms (kg)
- Stays the same everywhere: on Earth, on the Moon, or in space
- The force of gravity pulling an object down
- Measured in newtons (N), like every other force
- Changes with gravity: the same object weighs less on the Moon
On Earth, g = 9.8 m/s², so every kilogram of mass has a weight of about 9.8 N. For example, a 50 kg crate has a weight of 50 × 9.8 = 490 N.
In everyday speech people say “I weigh 60 kg”, but that is really their mass. On a free-body diagram, always draw weight, a force in newtons, never mass in kilograms.
Think of a free-body diagram as a simplified way of seeing a structure. When you look at a TV sitting on a table, you see the wood, the screen and the cables. Put on your engineering glasses and all of those physical details disappear, leaving only a skeleton of the hidden pushes and pulls that keep everything perfectly still.
In the free-body diagram, the table top becomes a single beam. The weight of the TV (FTV) and the weight of the games console (FConsole) push down on it, exactly where each one sits.
The two table legs react against those downward forces by pushing up on the table top. These upward forces are the reactions RA and RB. The table is not moving, so the legs must push up with exactly as much force as the TV and console push down:
The legs don't share the load equally. The TV sits in the middle, so each leg carries about half of its weight, but the console sits much closer to leg B, so leg B carries most of that. The distances d1, d2 and d3 are what let you work out exactly how much each leg carries, using moments in Section 5.
Section 3: Levers
A lever is a simple machine that can amplify force. It is made up of a rigid bar that turns about a pivot point called the fulcrum. Forces act on the lever at different points along its length, and the lever's job is to turn one of those forces into a bigger (or more useful) one somewhere else.
You already know from free-body diagrams that every force acts at a specific point. With a lever, what matters most is not what each force is called, but where it acts relative to the fulcrum. Moving the fulcrum, or moving a force closer to or further from it, changes how the lever behaves — this is the idea you'll use again in the next section on moments.
The fulcrum can sit in different places
The fulcrum does not have to be positioned halfway between the two forces. Depending on where it sits, a lever can behave very differently. Here are three arrangements you will meet often — pay attention to the order the fulcrum and the two forces appear in, left to right.
The fulcrum sits in the middle, with one force acting on each side of it.
Examples: a seesaw (a child sits at each end, the pivot block is in the middle); a pair of scissors (the pivot screw is between the finger holes and the blades); a crowbar resting over a fixed edge, lifting something on the other side.
The fulcrum sits at one end. The force being moved acts somewhere in the middle, and the force you apply acts at the far end.
Examples: a wheelbarrow (fulcrum at the wheel, the weight of what's inside acts partway along, your hands apply force at the handles, furthest from the wheel); a bottle opener (fulcrum at the lip resting on the bottle rim, the force lifting the cap acts in the middle, your hand applies force at the far end); a nutcracker.
The fulcrum sits at one end. This time, the force you apply acts somewhere in the middle, and the force being moved acts at the far end.
Examples: a fishing rod (fulcrum at the hand nearest the reel, the other hand applies force further up the rod, the fish pulls at the tip — the furthest point away); a pair of tweezers; a broom (top hand acts as the fulcrum, the lower hand applies force, the bristles act at the far end).
Levers in everyday objects
A door is also a lever — the hinge is the fulcrum. Push near the hinge and the door is hard to move; push at the handle, as far from the hinge as possible, and it swings open easily with far less force. This is exactly the same idea you'll use in the next section: the further a force acts from the fulcrum, the bigger its turning effect.
When a counterbalance weight is attached to one side of a lever to assist the user, it is known as a counterbalance. Cranes and diggers use counterbalances to prevent them toppling over.
Examine the photos below to identify: (a) where the fulcrum is, (b) where each force acts, and (c) which arrangement it matches — fulcrum in the middle, or fulcrum at one end with the object being moved or the applied force in the middle.
Section 4: Moments
A force does not always make an object move in a straight line — very often it makes the object turn instead. The turning effect of a force about a point is called a moment. You will also meet the word torque: torque is a turning force, and it is the word engineers usually use when something is twisted about its own axis, such as a spanner turning a nut or a motor turning a shaft. A moment and a torque are worked out in exactly the same way (force × distance). Every moment acts around a fixed point called the pivot (or fulcrum): the point about which the object is free to rotate. On a see-saw, the pivot is the block in the middle. On a spanner tightening a bolt, the pivot is the bolt itself. On a door, the pivot is the hinge. Moments and torque are both measured in newton-metres (Nm).
The MFd triangle. Cover up the quantity you want to find, and what is left shows you how to work it out.
- Cover M: F and d sit side by side, so M = F × d
- Cover F: M sits over d, so F = Md
- Cover d: M sits over F, so d = MF
To open a heavy garden gate, a person pushes on the edge furthest from the hinge with a force of 25 N. The hinge (the pivot) is 0.6 m from where the force is applied, measured perpendicular to the push. Calculate the moment used to open the gate.
Equilibrium and the Principle of Moments
When a system is in a steady, balanced state it is said to be in equilibrium. For a lever or beam to be in equilibrium, the clockwise moments must equal the anticlockwise moments — one side has no more turning effect than the other, so nothing rotates.
Sometimes there is more than one force turning a beam in the same direction — for example, two children sitting on the same side of a see-saw. Rather than writing out every individual moment, we use the symbol ∑ (the Greek capital letter sigma). In maths and science, ∑ simply means "add up all of the following." So ∑CWM means: add up every clockwise moment acting on the beam, however many there are, to get one total.
In words, the Principle of Moments says:
(the dots show that if more than one force acts on a side, you add all of their moments together first)
A lever has a force of 400 N acting 1 m from the fulcrum on the right, and a force of 200 N acting 2 m from the fulcrum on the left. Is the lever in equilibrium?
A gardener uses a crowbar and a wooden block to lift a heavy paving slab. She exerts a downward force of 400 N, 800 mm from the fulcrum. The slab is 400 mm from the fulcrum on the other side. Calculate the force of the slab on the end of the crowbar when all the forces are in equilibrium.
The distances can stay in millimetres here, because both sides of the equation use the same unit.
A wooden plank is balanced across a fixed support so that it can pivot freely. A toolbox weighing 12 N is placed 2.4 m from the pivot. A tin of paint weighing 8 N is placed on the other side. How far from the pivot must the tin of paint be placed for the plank to balance?
A see-saw is pivoted at its centre. On the right-hand side, one child applies a downward force of 300 N at 1 m from the pivot, and a second child applies 200 N at 2 m from the pivot. What downward force must an adult apply at 1.4 m from the pivot, on the left-hand side, to balance the see-saw?
Use the Principle of Moments to find the missing force or distance in each problem. Show all working.
Section 5: Beams and Reactions
A beam is a structural member that carries loads across its length. For a horizontal beam to be in equilibrium, two conditions must be satisfied simultaneously:
The upward forces at the support points are called reactions (RA and RB). To find both reactions, we use the Principle of Moments and the force balance equation together. A free-body diagram of the beam — showing every downward force and both reactions — is the best starting point for any of these problems.
Method for finding beam reactions
- Take moments about one reaction (RA) — this eliminates RA from the equation, allowing you to solve for RB
- Use the force balance (∑ up = ∑ down) to find RA
- Check: RA + RB should equal the sum of all downward forces
A 5 m beam is supported at each end (RA at the left, RB at the right). Two downward loads are applied: 10 000 N at 2 m from RA, and 6 000 N at 4 m from RA. Beam weight is ignored. The free-body diagram below shows every force acting on the beam, with all distances measured from RA.
∑CWM = ∑ACWM
(10 000 × 2) + (6 000 × 4) = RB × 5
44 000 = RB × 5
RB = 8 800 N
RA + RB = 10 000 + 6 000 = 16 000 N
RA = 16 000 − 8 800 = 7 200 N
Checking your answers with the NoStress simulator
NoStress is a beam simulator. You build a beam, add its supports and loads, then press Solve to see the reactions at each support. Show Working then sets out the moments step by step, so you can compare the simulator's method with your own. Use it to check answers you have already worked out, not to replace the working: in an assessment you need to show every step yourself.
This simulator is loaded with the worked example above. Press Solve and check that RA = 7 200 N and RB = 8 800 N (the simulator shows these in kilonewtons, as 7.20 kN and 8.80 kN), then press Show Working to see the moments. Then try changing one of the loads, or where it sits, and solve again: which reaction changes most?
Show all working for each problem. Remember: take moments about one reaction first, then use the force balance to find the other.
This opens with an empty 5 m beam. For each question, set the beam length, add the supports and loads in the right places, then press Solve to check your reactions against your own working.
Section 6: Vectors and Equilibrium
Force is a vector quantity — it has both magnitude (size) and direction. Vectors can be represented as drawn lines where:
- The length of the line (drawn to scale) represents the magnitude of the force
- The direction and arrowhead show the direction of the force
Drawn to the same scale, the 600 N vector is exactly twice the length of the 300 N vector.
When multiple forces act along the same line, they can simply be added or subtracted. For example: a cyclist pedalling with 800 N, assisted by a tailwind of 400 N, but with 200 N road friction, has a net force of 800 + 400 − 200 = 1 000 N.
The same forces can also be added by drawing them tip to tail: draw the first vector, then start the next one from the tip (arrowhead) of the one before, and so on. The resultant is a single arrow drawn from the very start of the first vector to the very end of the last one. This works for any number of forces, and it is the same idea used for angled forces below.
Resultant of two angled forces
When two forces act at an angle to each other, their combined effect is a single force called the resultant. The resultant can be found graphically using the parallelogram of forces:
- Choose a suitable scale and draw both force vectors from a common point C
- From the tip of each vector, draw a line parallel to the other vector
- The point where these lines intersect (D) completes the parallelogram
- Draw the line from C to D — this is the resultant
- Measure the length of C to D and convert using your scale to find the magnitude
- Measure the angle to find the direction
Conditions for Equilibrium
For a structure or body to be in equilibrium (a balanced, stable state), three conditions must all be satisfied:
| Condition | Meaning |
|---|---|
| ∑ upward forces = ∑ downward forces | Vertical forces balance — no net vertical movement |
| ∑ leftward forces = ∑ rightward forces | Horizontal forces balance — no net horizontal movement |
| ∑ CWM = ∑ ACWM | Moments balance — no net rotation |
Graphical solution for forces in members
For a structure such as a crane or frame, the forces in individual members can be found graphically by drawing a triangle of forces:
- Identify the joint where the known load acts
- Draw the known force to scale
- From each end of this line, draw lines parallel to the unknown force members
- Where the lines intersect completes the triangle
- Measure each side to find the forces (following arrowheads round the triangle indicates compression or tension)