Combined Variable Loads in ULS Design: Understanding Eurocode Load Combinations and ψ Factors
- MTS DNC ENERGY CONSULTANTS LIMITED
- Jul 12
- 6 min read
When designing structures, engineers must consider that several different loads may act on a building at the same time.
A roof structure, for example, may be subjected to:
Permanent self-weight of the structure
Roof finishes
Fixed building services
Snow loading
Wind loading
Roof maintenance access loads
A common question is:
Should all variable loads be considered at their maximum value simultaneously?
The answer is generally no.
The Eurocodes recognise that different variable actions have different probabilities of occurring together. A roof maintenance worker is unlikely to be accessing a roof during a severe snow event, and the maximum snow load is unlikely to occur simultaneously with maximum wind loading.
To account for this, EN 1990 – Eurocode: Basis of Structural and Geotechnical Design introduces load combination rules and combination factors (ψ factors).
These rules allow engineers to design structures that are:
✅ Safe
✅ Economical
✅ Realistic
✅ Compliant with Eurocode requirements
🧱 Understanding Eurocode Load Notation
Before looking at load combinations, it is important to understand the notation used in structural design.
Permanent Actions – Gk
Permanent actions are loads that remain approximately constant throughout the life of the structure.
Examples include:
Structural self-weight
Floor finishes
Roof build-up
Permanent partitions
Fixed mechanical and electrical equipment
The characteristic value of permanent action is written as:
Gk
Where:
G = permanent action
k = characteristic value
Example:
A roof structure may have:
Gk = 4.0 kN/m²
This represents the unfactored permanent load.
Variable Actions – Qk
Variable actions are loads that change during the life of the structure.
Examples include:
Occupancy loads
Storage loads
Snow
Wind
Roof maintenance loads
The characteristic value is written as:
Qk
Where:
Q = variable action
k = characteristic value
Examples:
Snow loading:
Qk,snow = 0.75 kN/m²
Roof maintenance load:
Qk,maintenance = 0.40 kN/m²
⚖️ Characteristic Loads vs Design Loads
The characteristic values are not directly used for structural resistance checks.
For Ultimate Limit State (ULS), Eurocode applies partial safety factors.
The design action is obtained by multiplying the characteristic action by the relevant factor.
Typical values:
Permanent Actions
γG = 1.35
Design permanent action:
1.35 × Gk
Variable Actions
γQ = 1.50
Design variable action:
1.50 × Qk
🔄 Why Are Combination Factors Used?
Consider a roof designed for:
Self-weight
Snow
Maintenance access
If every variable action was simply added at its maximum value, the structure would often become unnecessarily heavy and expensive.
For example:
A maintenance engineer is unlikely to inspect a roof during the most severe snow conditions.
Eurocode therefore recognises that the probability of simultaneous maximum variable actions is low.
The solution is the use of:
ψ Combination Factors
These factors reduce accompanying variable actions when they act together.
📐 What Are ψ Factors?
Eurocode defines three main combination factors.
Symbol | Purpose |
ψ₀ | Combination factor used for ULS combinations |
ψ₁ | Frequent combination factor used for SLS checks |
ψ₂ | Quasi-permanent combination factor used for long-term effects |
For Ultimate Limit State load combinations, the relevant factor is:
ψ₀
🏗️ Leading and Accompanying Variable Actions
When more than one variable action exists, one variable action is considered the leading variable action.
This action is applied at its full characteristic value.
The other variable actions are considered accompanying actions and are reduced using ψ₀.
Example:
A roof has:
Snow load
Maintenance load
If snow governs:
Leading action:
1.50 × Qk,snow
Accompanying action:
1.50 × ψ₀ × Qk,maintenance
📊 Eurocode ULS Fundamental Load Combinations
For persistent and transient design situations, EN 1990 defines fundamental combinations.
For a structural element subjected to:
Permanent action Gk
Variable actions Qk,1 and Qk,2
the following combinations are considered.
Combination 1 – Equation 6.10a
All variable actions are considered as accompanying actions.
General expression:
Ed = 1.35Gk + 1.50ψ₀,1Qk,1 + 1.50ψ₀,2Qk,2
This combination represents a situation where the probability of all variable actions reaching their maximum values simultaneously is reduced.
Combination 2 – Equation 6.10b
Variable action Qk,1 is the leading action.
General expression:
Ed = 1.15Gk + 1.50Qk,1 + 1.50ψ₀,2Qk,2
Here:
Qk,1 acts at full value
Qk,2 is reduced using ψ₀
Combination 3 – Equation 6.10b
Variable action Qk,2 is the leading action.
General expression:
Ed = 1.15Gk + 1.50ψ₀,1Qk,1 + 1.50Qk,2
Here:
Qk,2 acts at full value
Qk,1 is reduced using ψ₀
The designer must identify which combination produces the most unfavourable structural effect.
🧮 Worked Example: Roof Beam With Snow and Maintenance Loads
Consider a roof beam supporting:
Permanent load:
Gk = 10 kN/m
Snow load:
Qk,1 = 3 kN/m
Roof maintenance load:
Qk,2 = 1 kN/m
Combination factors:
Snow:
ψ₀ = 0.5
Roof Category H:
ψ₀ = 0.6
Combination 1
Formula:
Ed = 1.35Gk + 1.5ψ₀,snowQk,snow + 1.5ψ₀,roofQk,roof
Calculation:
= 1.35(10)
1.5(0.5)(3)
1.5(0.6)(1)
= 13.5 + 2.25 + 0.90
Ed = 16.65 kN/m
Combination 2 – Snow Leading
Formula:
Ed = 1.15Gk + 1.5Qk,snow + 1.5ψ₀,roofQk,roof
Calculation:
= 1.15(10)
1.5(3)
1.5(0.6)(1)
= 11.5 + 4.5 + 0.9
Ed = 16.90 kN/m
Combination 3 – Roof Maintenance Leading
Formula:
Ed = 1.15Gk + 1.5ψ₀,snowQk,snow + 1.5Qk,roof
Calculation:
= 1.15(10)
1.5(0.5)(3)
1.5(1)
= 11.5 + 2.25 + 1.5
Ed = 15.25 kN/m
Governing Combination
The largest design action is:
16.90 kN/m
Therefore, in this example:
Snow loading governs the ULS design.
🏠 Special Case: Category H Roof Loads
Roof imposed loads are treated differently from normal occupancy loads.
Category H roof loads represent:
Inspection
Maintenance
Minor repair activities
They do not represent permanent occupation.
According to EN 1991-1-1, roof imposed loads are generally not considered simultaneously with:
Snow loading
Wind loading
because maintenance activities are unlikely during severe weather.
Therefore, roof design normally requires separate checks:
Case 1
Permanent load + Roof maintenance load
Case 2
Permanent load + Snow load
Case 3
Permanent load + Wind load
The applicable National Annex must always be reviewed.
📊 Typical ψ₀ Combination Factors
Typical values from EN 1990 Table A1.1 are shown below.
Variable Action | ψ₀ |
Category A – Domestic | 0.7 |
Category B – Offices | 0.7 |
Category C – Congregation | 0.7 |
Category D – Shopping | 0.7 |
Category E – Storage | 1.0 |
Category F – Vehicle traffic | 0.7 |
Category G – Vehicle traffic | 0.7 |
Category H – Roofs | 0.6 |
Snow ≤1000m altitude | 0.5 |
Snow >1000m altitude | 0.7 |
Wind | 0.6 |
Values should always be verified against the relevant National Annex.
⚠️ Common Mistakes in ULS Load Combinations
❌ Adding all variable loads at full value
Incorrect:
1.35Gk + 1.5Qk,1 + 1.5Qk,2
This assumes all variable loads reach their maximum simultaneously.
❌ Ignoring alternative leading variable actions
Each significant variable action must be considered as potentially governing.
❌ Forgetting ψ factors
Accompanying variable actions must be reduced using the correct combination factor.
❌ Ignoring the National Annex
Eurocodes provide the framework, but National Annexes define country-specific parameters.
💡 Frequently Asked Questions
What is Qk in Eurocode design?
Qk represents the characteristic value of a variable action, such as snow, wind or occupancy loading.
What is Gk?
Gk represents the characteristic value of a permanent action, such as structural self-weight.
Why are variable loads reduced when combined?
Because the probability of several independent variable loads reaching their maximum value simultaneously is low.
Which ULS combination governs?
The governing combination is the one that produces the most adverse structural effect, such as maximum bending moment, shear force or axial load.
👷 Why Correct Load Combinations Matter
Correct application of Eurocode load combinations allows engineers to achieve:
✅ Safe structures
✅ Efficient designs
✅ Reduced material waste
✅ Compliance with EN 1990
Understanding combined variable loads is essential for engineers designing steel, concrete, timber and composite structures.
Final Thoughts
Structural design is not simply the process of adding every possible load together.
Eurocode recognises that different variable actions have different probabilities of occurring simultaneously.
By correctly applying:
Characteristic actions
Partial safety factors
Combination factors
Leading and accompanying variable actions
engineers can create structures that are both safe and economical.
Combined variable load assessment is one of the fundamental principles behind modern Eurocode-based structural design.
References
EN 1990 – Eurocode: Basis of Structural and Geotechnical Design
EN 1991-1-1 – Actions on Structures: Densities, Self-weight and Imposed Loads
EN 1991-1-3 – Actions on Structures: Snow Loads
Relevant National Annex
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Disclaimer:
The content provided in this post is for informational purposes only and should not be construed as professional engineering, architectural, or surveying advice. While every effort is made to ensure the accuracy and reliability of the information presented, it is not a substitute for a thorough, site-specific inspection or the expertise of a qualified professional. For detailed guidance on foundation issues, structural integrity, or repairs, always consult with a licensed engineer, architect, or surveyor. The authors and publishers are not responsible for any damages or losses resulting from the use or reliance on this information.
