Pressure Resultant Forces
How pressure integrates to resultant force and moment, F = pA for simple cases, projected area, closed ends, curved surfaces, and when p times local surface area is not the correct approach.
Technical provenance
Applicable standards / specifications
- ASME BPVC Section VIII Division 1 (2025) — Rules for Construction of Pressure Vessels
- ASME BPVC Section VIII Division 2 (2025) — Alternative Rules for Construction of Pressure Vessels
- EN 13445 — Unfired pressure vessels — Relevant European pressure-vessel code family where specified by the project.
References
- ASME Boiler and Pressure Vessel Code — 2025 edition — Primary code family reference for pressure-vessel design where ASME BPVC is the governing basis.
- Moss, D. R. & Basic, M. — Pressure Vessel Design Manual — Background engineering reference for pressure-vessel load paths, stresses and design checks.
Resultant force concept
The resultant force from a uniform pressure p acting on a flat surface of area A is F = p * A, directed normal to the surface. This is the simplest case and forms the basis for understanding more complex pressure resultants. The key insight is that the resultant depends on the projected area in the direction of the force, not the actual surface area.
F = p * A where: F = resultant force [N] p = uniform pressure [Pa or N/mm^2] A = flat surface area [m^2 or mm^2]
Projected area
For a curved surface under uniform pressure, the resultant force in a given direction equals the pressure multiplied by the projected area of the surface onto a plane perpendicular to that direction. A hemispherical end cap under internal pressure has an axial resultant of p * pi * r^2, where pi * r^2 is the projected area of the hemisphere onto a plane perpendicular to the axis. The actual surface area (2 * pi * r^2) is not used.
Closed ends
For a closed-ended cylinder under internal pressure p with radius r, the pressure acting on each end cap produces an axial resultant force of F = p * pi * r^2. This force must be resisted by the longitudinal membrane stress in the cylinder wall. Equilibrium of the end cap gives: sigma_longitudinal * 2 * pi * r * t = p * pi * r^2, which gives sigma_longitudinal = p * r / (2 * t). This is the fundamental derivation of longitudinal stress in a thin-walled cylinder.
Curved surfaces
For a curved surface, the pressure force at each point acts along the local normal. The components of the resultant in orthogonal directions are found by integrating the pressure over the projected areas in those directions. For a pipe bend under internal pressure, the pressure forces on the curved wall of the bend produce a resultant that tends to open the bend. This pressure thrust at bends must be resisted by pipe supports or anchors.
When p times local surface area is wrong
A common error is to multiply the pressure by the local surface area at a point and treat the result as a force vector at that point. This is incorrect because it ignores the vector nature of the pressure force — the direction changes with the surface normal. The correct approach is to integrate the pressure over the surface, accounting for the direction at each point. In FEA, this integration is performed automatically by the solver when pressure is applied as a surface load. The engineer should verify the resultant against the analytical projected-area calculation.
For a curved surface, the resultant force equals p times the projected area, not p times the actual surface area. Using the actual surface area overestimates the resultant by the ratio of actual to projected area — a factor of 2 for a hemisphere.
Resultant moment from pressure
When the pressure distribution or the surface geometry is not symmetric, a resultant moment is produced. For example, a non-axisymmetric pressure distribution on a vessel wall creates a bending moment in addition to the membrane force. This occurs at nozzles where the local pressure field is disturbed by the opening, and at supports where the reaction creates a local moment. The resultant moment must be included in the structural assessment of the local region.
Surface integration and equilibrium
The total force generated by pressure is obtained by integrating pressure times the local surface normal. In a finite-element model this distributed loading is converted internally into equivalent nodal forces, but the analyst should still know the expected resultant. For a flat area under uniform pressure the check is simply pA. For a curved surface the force components can be checked against projected areas.
Resultants on closed and cut systems
A complete closed vessel under uniform internal pressure has zero net external resultant from pressure, because opposing surface tractions balance. If the model contains only part of that pressure boundary, however, the missing portion must be represented by equivalent end load or by modelling the closure. Many apparent reaction-force errors arise because the analyst checks a cut model as though it were a complete closed pressure boundary.
Line of action and moment
Knowing only the resultant magnitude is insufficient when the pressure field is asymmetric. The location of the centre of pressure determines the applied moment. Partial filling, sloped liquid surfaces, nonuniform pressure or one-sided pressure on doors and covers can shift the line of action materially.
Hydrostatic pressure fields
For a liquid of constant density, pressure varies linearly with depth. The resultant on a vertical flat surface therefore acts below the geometric centroid. Modelling the load as a uniform pressure equal to the mean value recovers the same total force but can still give the wrong moment unless the distribution is applied correctly.
Pressure thrust through changes in area
Reducers, elbows, bellows, caps and valves can convert pressure into net axial or lateral forces because the projected pressure areas do not cancel locally. These resultants can be transmitted into supports and connected equipment. Piping models and local pressure-vessel models should use a consistent treatment so that pressure thrust is neither omitted nor counted twice.
Reaction check
For each global load case, compare summed reactions with the analytically expected pressure force and moment. The agreement should be within numerical tolerance once body forces and other loads are included. A mismatch is a model-definition problem, not a stress-post-processing issue.
Verification point: For each global load case, compare summed reactions with the analytically expected pressure force and moment. The agreement should be within numerical tolerance once body forces and other loads are included. A mismatch is a model-definition problem, not a stress-post-processing issue.