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)