Action-Reaction at Joints — Newton's Third Law in Structures

Why joints come in pairs Concept

A joint is just the place where two (or more) members meet — a pin, a bolt, a hinge, a ball-and-socket, even a rope tied to a hook. Newton's third law says that whatever force one member exerts on the joint, the joint exerts the exact opposite back on that member.

That means every internal force in a structure shows up twice, once on each side of the connection:

Member A pushes/pulls on the joint with force F
The joint pushes/pulls on member A with force −F
If a second member B is at the same joint, the same logic applies to it, independently

at any joint: F_{on joint from A} = − F_{on A from joint}

This is the whole reason free-body diagrams work: cut a structure at a joint, and you replace the missing material with the reaction force it was exerting — equal and opposite to what your isolated piece was exerting on it.

Two members pinned together. Whatever force member A exerts on the pin, the pin returns the exact opposite onto member A.

Worked example 1: two-bar pin joint Walkthrough

Two truss bars meet at a pin. Bar 1 is in tension and pulls on the pin with F₁ = (60, 0) N (pulling the pin toward bar 1, i.e. to the left). By Newton's third law, the pin pulls back on bar 1 with the opposite force.

F_{pin on bar 1} = − F₁ = (−60, 0) N

This reaction is what keeps bar 1 in equilibrium — it's the force the rest of the truss exerts on that bar through the pin, exactly cancelling the bar's own pull in the free-body sense.

Bar 1 pulls the pin left (tension); the pin pulls bar 1 right with equal magnitude.

Worked example 2: hinge between beam and wall Walkthrough

A beam is hinged to a wall at O. Under load, the beam pushes on the hinge with F_beam = (−25, −40, 0) N (down and into the wall).

F_{beam on hinge} = (−25, −40, 0) N

F_{hinge on beam} = −F_{beam on hinge} = (25, 40, 0) N

The wall doesn't know or care what's pushing on it — it simply supplies whatever reaction is needed at that hinge. The beam's free-body diagram shows this reaction as a separate, independent force at O, equal and opposite to what the beam itself transmits into the hinge.

The beam loads the hinge down-and-left; the hinge supplies an equal up-and-right reaction back into the beam.

Worked example 3: three members at one joint Walkthrough

At a truss joint, three bars meet. Two of the forces the bars exert on the pin are known: F₁ = (30, 40, 0) N and F₂ = (−10, 20, 0) N. For the pin to be in equilibrium, the third bar's force on the pin must balance both of them.

F_{3 on pin} = −(F₁ + F₂)
F_{3 on pin} = −[(30−10), (40+20), 0]
F_{3 on pin} = (−20, −60, 0) N

And by the third law, the pin pushes back on bar 3 with the opposite of that: (20, 60, 0) N. Action-reaction applies pairwise to every single member at the joint, not just to the joint as a whole.

Three bars at one pin. The unknown bar force is whatever makes the vector sum at the pin zero.

Build the reaction pair Interactive

Enter the force one member exerts on a joint (or the other member it touches). We'll give you the equal-and-opposite reaction.

Force on joint (N)

F_x F_y F_z

Reaction back on member (computed)

F_x F_y F_z

F_{on joint} =

F_{reaction on member} = − F_{on joint} =

Same magnitude, same line of action, opposite sense — that's all the third law asks for. It never depends on what the member is made of or how big it is.

Quick checks Self-test

Five short ones. Two come with a diagram of the actual connection.

Check 1

A rope pulls on a hook with F = (0, −80, 0) N. What force does the hook exert on the rope?

x y

Check 2 — pinned bar

Bar pulls on the pin (tension) with F = (45, 0) N. What does the pin exert on the bar?

x y

Check 3 — hinge

A beam pushes on its wall hinge with F = (−20, −35, 0) N. What does the hinge push back on the beam with?

x y

Check 4

A block rests on the ground; the ground pushes up on the block with N = (0, 90, 0) N. What force does the block exert on the ground?

x y

Check 5

Member A exerts F = (12, −18, 6) N on a ball-and-socket joint. Find the force the joint exerts on member A.

x y z