Forces and Structures

A 20 kg bag of feed sits in the middle of a plank laid across two sawhorses, and nothing happens. Slide the same bag toward one end and, before it gets halfway there, the far end lifts off its sawhorse. Nothing was added and nothing broke. The loads and the supports simply stopped balancing, and the plank began to rotate.

That balance is what every standing structure maintains, from a bookshelf to a bridge. Statics is the study of things that stay still, and it runs on exactly two rules. This lesson covers the rules, the three ways a member carries a load, and the shape trick that keeps trusses standing. It ends with a question statics cannot answer on its own, which is where the next lesson picks up.

Two rules for standing still

A structure at rest satisfies two conditions at once. First, the forces on it must sum to zero in every direction. If they did not, the structure would accelerate, which is exactly what standing still forbids. Second, the torques on it must sum to zero about any point. If they did not, it would start to spin even with every force balanced. Every statics problem is some version of enforcing these two rules with algebra.

The first rule explains the book on the table. Gravity pulls a 2.5 kg book down with a force of 2.5 times 9.8, about 24.5 newtons. The table pushes back up with exactly 24.5 newtons, and the two cancel. That upward push from a support is called a reaction force, and finding the reactions is most of what statics asks for. A table under a heavier book pushes harder without being asked; the load sets the reaction.

The second rule matters the moment a load sits off-center. A 4.0 m plank weighing 160 N rests on supports at its two ends, and a 600 N worker stands 1.0 m from the left end. Take the sum of torques about the left support, where the left reaction creates no torque because its lever arm is zero. The plank's own weight acts at its center, 2.0 m out, and the worker acts at 1.0 m. Balance demands R_right times 4.0 = 160 times 2.0 plus 600 times 1.0, so R_right = (320 + 600) / 4.0 = 230 N. The remaining share of the total 760 N lands on the left support: 530 N.

The near support carries more than twice the far one. Anyone who has steered a heavy couch through a doorway knows this by feel: the end nearest the pivot point does the real work. Structural engineers run this same calculation for every beam and column, because the reaction forces are what the bolts, welds, and foundations below must actually hold.

Statics also needs one shared convention: where the weight of an extended object acts. Every bit of a plank has mass, but the math stays simple because the whole weight behaves as if it acted at a single point, the center of mass. For a uniform plank that point is the geometric center, which is why the plank's 160 N was placed at 2.0 m. A roof's snow load spreads across the whole surface instead of sitting in one spot; engineers sum such a distributed load into one equivalent force and then apply the same two rules unchanged.

Torque, like force, bookkeeps with a sign: a torque that would turn the plank one way cancels one that would turn it the other. In the plank problem the worker and the plank's own weight both turned the plank the same direction about the left support, and the right support's reaction turned it back. Once every push has been converted into torque about a chosen point, the whole problem collapses into one line of arithmetic.

Torque: force times lever arm

Torque is the turning effect of a force, and it equals the force times the lever arm, the perpendicular distance from the pivot to the line along which the force pushes. Its unit is the newton-meter. Push with 200 N on a wrench held 0.30 m from the bolt and the bolt sees 200 times 0.30, or 60 N m of torque. Slide a 0.60 m breaker bar over the handle and the same 200 N of hand delivers 120 N m. The force never changed. The lever arm did.

Everyday design leans on this constantly. A door handle sits at the edge far from the hinges because torque about the hinge grows with distance; the same push near the hinge barely turns anything. A long screwdriver handle turns a stubborn screw more easily than a short one for the same grip. When a bolt will not move, the honest options are a longer bar or a bigger push, and the longer bar is cheaper.

Torque is also the second equilibrium rule in everyday clothes. A seesaw balances when the torques match, not when the weights match. A 45 kg child can balance a 30 kg child by sitting closer to the pivot, because what has to cancel is weight times distance, not weight alone. The code below finds the balance point and then checks it:

g = 9.8
m1, d1 = 30.0, 1.6      # 30 kg child, 1.6 m left of the pivot
m2 = 45.0               # 45 kg child, seat distance unknown
d2 = m1 * d1 / m2       # torque balance: m1*d1 = m2*d2

left = m1 * g * d1
right = m2 * g * d2
print(f"45 kg child sits {d2:.2f} m from the pivot")
print(f"left torque {left:.1f} N*m, right torque {right:.1f} N*m")
print(f"difference {abs(left - right):.1e} N*m, effectively zero")

The 45 kg child sits 1.07 m out, both torques come to 470.4 N m, and the difference prints as exactly zero. Notice that g dropped out of the balance entirely: both sides multiply by the same 9.8, so the balance condition m1 times d1 equals m2 times d2 never needed it.

Tension, compression, and bending

Loads travel through members in three ways. Tension pulls a member apart from both ends: the cable of a crane hook is in tension, and rope is excellent at it, because a rope cannot be pushed. Compression squeezes a member end to end: the legs of a table are in compression, and stone, brick, and concrete handle it well, which is why ancient columns still stand. A member in pure tension or pure compression is the simple case, and well-designed trusses keep every bar in one of these two states.

Bending is the complicated one. Load a horizontal member in the middle and its top surface squeezes together in compression while its bottom surface stretches in tension, with a nearly unstressed neutral layer between them. Material near the top and bottom faces does most of the work, which is why an I-beam concentrates its steel there and fills the middle with a thin web. Concrete makes the point brutally clear: it is strong in compression and weak in tension, so a concrete slab gets steel reinforcing bars cast into its bottom, exactly where the tension lives.

Hollow shapes earn their place the same way. A round tube keeps its material far from the neutral center, where bending stress is highest, so scaffolding poles and bicycle frames are tubes rather than solid rods: a fraction of the material for nearly the same stiffness. The same logic makes a flat ruler easy to bend flatwise and almost impossible to bend edgewise. Shape, not just material, resists bending.

Why triangles hold and squares fold

Pin four rigid sticks together at their corners to form a square and lean on it: the square folds into a parallelogram without any stick bending. Now pin three sticks into a triangle and lean on that. Nothing happens, and the reason is geometric: three bar lengths determine exactly one triangle shape. A pin-jointed triangle cannot change shape unless a bar bends or a joint breaks. That single property is the entire secret of the truss.

A truss is a frame built from triangles, usually out of slender bars. Look at a roof truss or a railway bridge and count the triangles: each diagonal converts what would be a folding square into two rigid ones. The diagonals are not decoration. They are the difference between a structure and a mechanism, which is why every frame that must resist sideways load, from bicycle frames to crane booms, is full of them.

In a loaded truss some bars stretch and carry tension while others squeeze and carry compression, and which is which depends on the geometry and the load. Designers care, because the two jobs suit different members: thin rods make fine tension ties, while compression members can bow sideways and fail early, so they need more girth. A bridge truss is a visible ledger of these alternating pulls and pushes.

Every load also needs a continuous path to the ground, called the load path. Snow on a roof presses on the rafters, the rafters load the truss, the truss pushes down through the walls, and the walls deliver everything into the foundation. Break the path anywhere and everything above it comes down. Storm damage often starts when wind pressure reverses the load path and lifts the roof off walls that were never fastened to hold it down; the connections, not the beams, are frequently the weak link.

The path is easy to follow on furniture. A chair seat passes a sitter's weight into the legs, the legs into the floor, and the floor into the building's foundation. A cracked leg does not merely look bad; it interrupts the path, and the chair fails at that link. Good design makes every link in the chain stronger than the loads it will ever see, and inspection hunts for exactly the links that rust, crack, or work loose.

Statics treats every member as strong enough and moves on. Whether a 10 mm rod actually survives 10,000 N, or stretches permanently, or snaps, is a question about the material inside the shape. The next lesson, Stress and Materials, brings the material in: stress, strain, stiffness, and the yield point where springing back stops being guaranteed.