Load Combinations
Load combinations define how individual load cases are combined with partial safety factors to produce design scenarios. Armatura supports both manual combination definition and automatic generation of standard Eurocode combinations.
Principle: linear superposition
Because the solver performs linear elastic analysis, results from individual load cases can be combined by weighted addition (linear superposition). This is a fundamental property of linear systems — it means each load case needs to be solved only once, and any number of combinations can be evaluated without re-solving the stiffness matrix.
The LinearSuperposition utility performs this operation:
Result_combination = γ ₁ × Result_case1 + γ₂ × Result_case2 + ...
This applies to displacements, reactions, and member forces. It is valid only for linear elastic analysis — if second-order effects or material nonlinearity are significant, each combination would need to be solved independently.
For background on why superposition works for linear systems, see [MAS] §3.1.
Eurocode load combination framework
The Eurocode system for load combinations is defined in EN 1990 (Basis of Structural Design). The basic concepts are:
Load types and partial factors
Each load case has a type that determines its partial safety factor:
| Load type | γ (ULS) | γ (SLS) | Reference |
|---|---|---|---|
| Dead (permanent) | 1.35 | 1.0 | EN 1990, Table A1.2(A) |
| Live (imposed) | 1.50 | 1.0 | EN 1990, Table A1.2(A) |
| Wind | 1.50 | 1.0 | EN 1990, Table A1.2(A) |
| Snow | 1.50 | 1.0 | EN 1990, Table A1.2(A) |
| Seismic | 1.00 | — | EN 1998-1 |
These are the recommended values from EN 1990. National Annexes may specify different values — consult the relevant National Annex for your jurisdiction.
Combination types
Ultimate Limit State (ULS) — checks against structural failure (yielding, buckling, fracture). Uses factored loads with γ > 1.0.
Serviceability Limit State (SLS) — checks against functional impairment (deflection, vibration). Uses unfactored or partially factored loads.
The general ULS combination expression from EN 1990, Eq. 6.10 is:
Σ γG,j × Gk,j + γQ,1 × Qk,1 + Σ γQ,i × ψ0,i × Qk,i
Where ψ₀ is the combination factor for accompanying variable actions. For simplicity, Armatura's standard generator uses ψ₀ = 0.6 for the secondary variable action in two-variable combinations.
Standard combinations generated
LoadRegistry.GenerateStandardCombinations() produces:
| Combination | Type | Expression |
|---|---|---|
| 1.35D | ULS | 1.35 × Dead |
| 1.35D + 1.5L | ULS | 1.35 × Dead + 1.5 × Live |
| 1.35D + 1.5W | ULS | 1.35 × Dead + 1.5 × Wind |
| 1.35D + 1.5S | ULS | 1.35 × Dead + 1.5 × Snow |
| 1.35D + 1.5L + 0.9W | ULS | 1.35 × Dead + 1.5 × Live + 0.9 × Wind |
| D + L | SLS | 1.0 × Dead + 1.0 × Live |
| D + W | SLS | 1.0 × Dead + 1.0 × Wind |
These are common combinations for building-type structures. They are meant as reasonable defaults — for real design work, the engineer should define combinations appropriate for their specific structure and the governing National Annex.
Users can also define fully custom combinations:
var combo = new LoadCombination("Custom ULS", CombinationType.Ultimate);
combo.AddFactor(deadCase, 1.35);
combo.AddFactor(liveCase, 1.50);
combo.AddFactor(windCase, 0.75);
model.AddLoadCombination(combo);
Envelopes
An envelope extracts the maximum and minimum values of a result quantity across all load cases and combinations. Each envelope value tracks which scenario governed, providing traceability:
var envelope = results.GetDisplacementEnvelope(nodeId);
// envelope.Dy.Min → maximum downward displacement
// envelope.Dy.MinScenario → which load case or combination caused it
Envelopes are computed lazily (on first access) and cached. They are useful for identifying governing design scenarios without manually iterating through all combinations.
Combination factors ψ
The ψ-factors from EN 1990, Table A1.1 reduce accompanying variable actions to account for the low probability of all loads acting at their maximum simultaneously. For context:
| Action | ψ₀ | ψ₁ | ψ₂ |
|---|---|---|---|
| Imposed (office) | 0.7 | 0.5 | 0.3 |
| Imposed (storage) | 1.0 | 0.9 | 0.8 |
| Snow (H ≤ 1000m) | 0.5 | 0.2 | 0.0 |
| Wind | 0.6 | 0.2 | 0.0 |
These values vary by National Annex. The table above shows the recommended values from EN 1990. If you need exact ψ-factors for your jurisdiction, consult the applicable National Annex directly.
Practical notes
- Linear superposition is exact for linear elastic analysis. If the αcr check indicates that second-order effects are significant (αcr < 10), the superposed results are unconservative and the structure should be re-analysed with second-order effects included.
- The standard combinations are generated from whichever load cases exist in the model. If you only have dead and wind cases, the generator will not produce combinations involving live load.
- Envelope results are invalidated whenever the model changes (same invalidation mechanism as the analysis results).
References
- EN 1990:2002, §6.4 and Tables A1.1–A1.2: load combination rules and partial factors
- EN 1993-1-1:2005, §2.4: design values and partial factors for steel
- Kassimali (2015), §3.1: superposition principle for linear systems