Horizontal thrust, H = (W
2)n∑zy − ∑y·∑z
n∑y2 − (∑y)2 (1)
Moment at Crown, M0 = ½W∑z − H∑y
n (2)
Shear at Crown, V0 = ½W∑zx
∑x2 (3)
For symmetrical loading such as W on the left and W on the right the horizontal thrust and crown moment due to both loads are double those found by the above formulas, while the crown shear V0 is zero. For several loads unsymmetrically placed the formulas are to be applied to each in succession and the results added algebraically, the value of V0 being taken as negative for the left semi-arch and positive for the right semi-arch.
For any joint whose middle point is at a distance x from the crown
M = M0 + Hy + V0x − ∑Wz,
V = V0 − ∑W,
where ∑W is the sum of all the loads between the joint and the crown and ∑Wz is the sum of the moments of those loads with respect to the middle of the joint. The components of the resultant thrust normal and parallel to the joints are,
N = H cos θ − V sin θ,
F = H sin θ + V cos θ,