Three-dimensional structures

For three-dimensional structures there are six components of moments and forces in a point. If we cut the curved beam perpendicular to the beam axis the torque is equal to the product of Force by Arm: F*r₁ and the bending moment is equal to F*r₂. Internal forces are equal to the sum of corresponding projections of external forces.

The figure shows examples of bending moment and torque diagrams for three-dimensional structures. The diagrams are built by the above-mentioned rules.

There are tensile as well as compressive stresses at external surface of the arc. The bending moment changes its sign.

For a thin-walled sphere, absolute value of radial stress at the external surface is equal to external pressure. It is smaller than meridian and tangential stress components.

The first step of the analysis includes determination of reactions in the supports. There is symmetry in the structure. There are no torque or bending moments in part of the structure.
The conclusion is purely theoretical, the structure is not stable. Loss of stability causes additional forces and moments for all parts of the structure.