Loads, Pressure Mapping & Remote Loading

What Is It?

Load representation is the process of transferring the external forces, moments and pressure distributions that act on the real structure into equivalent nodal loads on the finite element model. The real load may be a distributed pressure over a surface, a line load along an edge, a force at a point, a moment about an axis or an inertia body force through the volume. The FE model accepts these as nodal forces and moments, as surface pressures, as edge tractions or as body force accelerations. The way the load is applied — the distribution over the nodes, the spatial extent, the point of application — determines how realistically the model represents the load path and the local stress. Load representation is an idealisation: the real load is never a perfect point force or a perfect uniform pressure, and the model must approximate it in a way that captures the engineering-relevant behaviour without introducing artificial singularities.

Why It Matters

The load representation affects both the global response and the local stress. Globally, the resultant force and moment must be correct — if the model applies the right resultant to the right location, the overall load path and the reaction forces will be correct. Locally, the distribution matters — a force applied as a point load produces a stress singularity (infinite stress at the point), while the same force applied as a distributed pressure produces a finite, realistic stress. The analyst must decide what level of load distribution fidelity is needed. For global stiffness and load path, the resultant is enough. For local stress at the load application point, the distribution must be modelled realistically — a bearing load spread over the contact arc, a bolt force spread over the washer area, a pressure distribution over the actual loaded surface. Getting the load representation right is as important as getting the mesh and the boundary conditions right — a wrong load gives a wrong result, no matter how good the mesh.

THE LOAD REPRESENTATION MUST MATCH THE LEVEL OF THE RESULT. For global stiffness and load path, the correct resultant at the correct location is sufficient. For local stress at the load application point, the distribution must be modelled realistically. A point load is acceptable for global analysis and unacceptable for local stress — it produces a singularity that has no physical meaning.

Point Loads and Singularities

A point load — a force applied to a single node — produces a stress singularity at that node. The stress at the exact point of application is theoretically infinite, and mesh refinement increases the stress without convergence. This is not a physical result — no real load is applied at a mathematical point. The singularity is an artefact of the idealisation. Point loads are acceptable when the load application point is far from the region of interest (Saint-Venant: the disturbance decays with distance) or when only the global resultant matters. They are unacceptable when the local stress at the load point is the objective. In that case, the load must be distributed — over a patch of elements, along an edge or through a bearing distribution — so that the stress is finite and mesh-convergent.

COMMON MISTAKE: Applying a point load and then refining the mesh at the load point to "converge" the stress. The stress will not converge — it will increase with refinement because the singularity is mathematical, not physical. The remedy is to distribute the load over a physically realistic area, not to refine the mesh at the singularity.

Pressure Mapping and Distributed Loads

Distributed loads — surface pressures, edge tractions, body forces — are the physically realistic way to apply load to a region. The pressure is applied to the element faces or edges, and the solver converts it to equivalent nodal forces through the shape functions. The stress under a distributed load is finite and mesh-convergent (provided the element is adequate). The distribution of the pressure should match the real distribution as closely as the engineering question requires: a uniform pressure for a uniformly loaded panel; a varying pressure for a hydrostatic or aerodynamic load; a Hertzian contact distribution for a bearing or gear tooth. When the real distribution is unknown, a conservative but realistic distribution should be chosen — spreading the load over the expected contact area rather than concentrating it. The key principle is that the load should be distributed over the area that the real load actually acts on, not over a smaller area (which over-concentrates) or a larger area (which under-concentrates).

• Uniform pressure — for uniformly loaded surfaces; simplest and most common
• Varying pressure (hydrostatic, aerodynamic) — for loads that vary over the surface
• Bearing load (cosine distribution) — for cylindrical contact; spreads over the contact arc
• Hertzian contact pressure — for concentrated line or point contact; elliptical distribution
• Body force (acceleration × density) — for inertia loads; distributed through the volume
• Edge traction — for line loads along an edge; force per unit length

Remote Loading and RBE

Remote loading applies a force and moment at a point that is not on the mesh — typically the centre of a bearing, the centre of a bolt pattern or the centre of gravity of an attached component — and transfers it to the mesh through a kinematic coupling. The remote point is connected to the loaded surface by a rigid beam (RBE2) or a distributed coupling (RBE3), and the force and moment at the remote point are transferred as a distribution of nodal forces on the surface. Remote loading is the standard way to apply a bearing load, a bolt pattern load or an equipment CG load without modelling the bearing, the bolts or the equipment in detail. The coupling determines the distribution: an RBE2 (rigid) makes the surface rigid — it over-stiffens; an RBE3 (distributing) makes the surface follow the remote point without adding stiffness — it is usually preferred. The remote loading approach avoids the singularity of a point load on the mesh and gives a realistic distribution over the loaded surface.

Remote load transfer (RBE3 distributing coupling):

  Applied at remote point:  F_remote, M_remote
  Transferred to surface nodes:  {F_i}

  The coupling distributes the remote load as nodal forces
  such that the resultant is preserved:

    Σ F_i = F_remote
    Σ (r_i × F_i) = M_remote

  where r_i is the position of node i relative to the remote point.

  RBE2 (rigid): surface nodes are rigidly tied to remote point
    → adds stiffness, over-stiffens the loaded region
  RBE3 (distributing): load is distributed, no stiffness added
    → preferred for load application

Load Mapping Approaches

Conservative vs Realistic Load Mapping

When the real load distribution is uncertain, the analyst must choose between a conservative mapping (which over-concentrates the load and gives higher local stress) and a realistic mapping (which spreads the load over the expected area and gives lower, more representative stress). The conservative approach is safer for margin calculation but can be excessively conservative — it may predict failure where the real structure is adequate. The realistic approach gives a more representative result but relies on an assumption about the distribution that may be wrong. The best practice is to run both: a realistic case for the nominal result and a conservative case for the bounding margin. The difference between the two tells the analyst how sensitive the result is to the load distribution — a large difference means the distribution matters and should be characterised more carefully; a small difference means the result is robust to the assumption.

Bearing Loads and Contact Arcs

A bearing load — a pin in a hole, a shaft in a housing, a bolt in a clearance hole — is typically applied as a cosine pressure distribution over a contact arc (commonly 180 degrees for a close-fit pin, less for a clearance fit). The cosine distribution peaks at the centre of the arc and falls to zero at the edges, matching the pressure distribution of a close-contact bearing. The contact arc and the distribution shape depend on the fit and the clearance — a tight fit has a wide arc and a gentle distribution; a loose fit has a narrow arc and a concentrated distribution. Applying the bearing load as a uniform pressure over half the hole is a crude approximation; applying it as a cosine over the correct arc is more realistic. The local stress at the hole edge is sensitive to the arc and the distribution, so for fatigue-critical hole stress, the bearing distribution should be modelled carefully.

Bearing load — cosine pressure distribution over contact arc:

         Pin load F (downward)
              │
              ▼
         ┌────────┐
         │   ●    │  ← pin centre
         │  / \   │
        ╱│ /   \  │╲
       ╱ ││  p   ││ ╲  contact arc (e.g. 180°)
      │  ││ (θ)  ││  │
      │  ││      ││  │
       ╲ ││ \   / ││ ╱
        ╲││  \ /  ││╱
         ││   ●   ││
         └────────┘

  p(θ) = p_max · cos(θ)   for |θ| ≤ α/2
  p(θ) = 0                otherwise

  α = contact arc angle
  Resultant: F = ∫ p(θ) · r · dθ  (integrated over the arc)

Idealisation Considerations

IDEALISATION CONSIDERATION: The load distribution is often as uncertain as the boundary conditions. When the real distribution is unknown, run a realistic case and a conservative case and envelope the results. The sensitivity of the result to the distribution tells you whether the assumption matters. For local stress at load application points, never use a point load — distribute the load over the physically realistic area.

Key Takeaways

• Point loads produce stress singularities — acceptable for global analysis, unacceptable for local stress
• Distributed pressures give finite, mesh-convergent stress — match the real distribution as closely as needed
• Remote loading (RBE3) transfers a force and moment from a remote point to the mesh without adding stiffness
• Bearing loads use a cosine distribution over the contact arc — the arc and fit determine the concentration
• When the distribution is uncertain, run realistic and conservative cases and envelope the results