Thrust

An arch carries a load down to its two feet, but there is a catch: the feet are pushed outward as well as down. Think of two people leaning against each other with their feet apart. The pair stays up only if the floor stops their feet from sliding apart. An arch is the same. The sideways push at each foot is called the horizontal thrust, H.

You can find it with a short piece of balance. Take a symmetric arch of span L and rise (height) f, carrying a load w per horizontal metre. The two vertical reactions are wL ÷ 2 each. Cut the arch at the top (the crown) and look at the left half. Take moments about the crown: the vertical reaction wL ÷ 2 acts at a horizontal distance L ÷ 2, the half-load wL ÷ 2 acts at L ÷ 4, and the thrust H acts at a height f below the crown. Balancing gives

H = w · L² ÷ (8 · f)

A worked example (assumed numbers). An arch spans L = 8 m and carries w = 20 kN/m. With a rise of f = 2 m:

• Each foot carries a vertical force wL ÷ 2 = 80 kN.
• The thrust is H = 20 × 8² ÷ (8 × 2) = 80 kN.
• The resultant at each foot is √(80² + 80²) ≈ 113 kN, pointing 45° down and outward.

Now change only the rise. Halve it to f = 1 m and the thrust doubles to 160 kN. Double it to f = 4 m and the thrust halves to 40 kN. The vertical force stays 80 kN throughout. So the flatter the arch, the harder it shoves its supports sideways, and in the limit of a flat beam-like arch (f → 0) the thrust becomes enormous. That is why a very flat arch needs very strong abutments.

This is also why pointed arches helped medieval builders. A semicircle of span L has a fixed rise of L ÷ 2. A pointed arch can be made taller or shorter at will, and arches of different spans can be given the same height. A tall pointed arch pushes less sideways and sends its load more steeply down.

What stops the thrust? There are four classic answers:

• Massive abutments or walls. Roman barrel vaults sat on very thick walls, and the weight of the wall tilts the line of force down toward the ground.
• Buttresses. Extra masonry added on the outside where the thrust arrives (Gothic builders developed this into the flying buttress, in World 3).
• A tie. A tension rod or beam joining the two feet, like the string on a bow. The tie takes the thrust as a pull, so the foundations only feel the vertical load. Tie rods are common in arcades and in the arches of many old buildings.
• Rock. A stone arch in a rocky gorge can push straight into the rock.

If the supports move apart even a little (the ground settles, a wall leans), the arch lowers, the thrust goes up, and cracks open. Many ruined arches failed because their abutments gave way, not because the arch itself was too weak.