Combination of Variable Loads in Structural Design

Abstract

This study delves into the intricacies of variable load combination factors (ψ) within structural codes under fundamental design scenarios, with Eurocodes serving as the primary reference. Currently, variable loads are combined by adding one load, the leading load with its full value, and the other load, the accompanying load, with a reduced value multiplied by a combination factor ψ. This approach employs an independent load combination methodology, utilizing hypothetical reference materials. In contrast, this paper advocates for a shift towards dependent load combination, anchored in the use of actual reference materials. Specifically, it is proposed that imposed loads be combined without the combination factor, i.e., ψ = 1. Given that combination factors are in approximate unity or pertain to infrequent load cases, this research recommends the elimination of ψ from codes altogether. This recommendation stems from the recognition that the current combination factor calculation excels in cases with approximately equal loads with a significant reliability gain, while more frequent unequal loads introduce a minor reliability gain and harmful unsafe errors. Despite the overall minor safety advantage of about 2–3% being negligible considering unavoidable safe errors of about 7% in codes, this simplification significantly reduces code complexity, enhances user-friendliness, and substantially decreases the workload associated with design processes.

1. Introduction

The Eurocodes [1,2], fundamental standards governing structural design, are currently undergoing a significant update process to usher in their second generation.

1.1. Background

Drawing from decades of practical structural design experience and extensive research in structural probability theory, the author identifies potential avenues for improvement within these pivotal standards. The combination of variable loads and combination factors is a crucial part of reliability and safety factor calculation. Variable load combination has been a challenging issue for the author, resulting in various unsuccessful articles over the past two decades.

This article stems from four primary motivations: the pursuit of simplification, international consensus, and two key observations within the current Eurocodes:

The primary objectives of the Eurocodes’ update include enhancing user-friendliness and cultivating exemplary levels of international consensus [3,4]. However, within the draft update version [2], the author notes a lack of improvement in user-friendliness, and three options are given for load combination, indicating missing international consensus.

Within multi-storey buildings, permanent loads are combined dependently, while variable loads are combined independently—an arrangement deemed unconventional.

The author contends that permanent loads and variable loads should be combined dependently. Although variable loads are seemingly combined independently, it is crucial to note that certain loads, such as snow loads, behave akin to permanent loads due to their sustained nature, thus warranting dependent combination.

The author asserts that the insights and suggestions provided in this article correct prevalent errors in the current theory and codes, resulting in improved user-friendliness, increased international consensus, and reduced complexity and workload.

1.2. Novelty

This article presents a new model for the combination of variable loads and calculation of combination factors in fundamental design scenarios involving snow, wind, and imposed loads, featuring six discoveries:

• Variable loads are traditionally perceived as independent when considered as single load pairs and are normally combined independently. However, this article reveals that certain scenarios necessitate dependent load combinations for accurate analysis.
• A new procedure is provided for calculating combination factors for one load in proportion α and the other 1 − α, including the basic scenarios where the loads have the same magnitude, i.e., the load proportion is 0.5 and the combination factor is the lowest.
• The current combination factors in codes are based on load proportions where loads are equal, i.e., a load proportion of 0.5. However, it is imperative to incorporate values derived from various load proportion scenarios in setting combination factors for codes. This approach yields higher values compared to those obtained solely from a load proportion of 0.5. To facilitate practical application, approximate code values are provided.
• The primary objective of employing combination factors is to mitigate excessive safety margins inherent in the combination of two variable loads. The existing model for variable load modelling, relying on a single combination factor, is deemed inadequate. While it offers a notable advantage when dealing with rare cases with loads of similar magnitudes, its effectiveness diminishes in scenarios involving frequent cases with loads of varying magnitudes, including a harmful unsafe error.
• There is no basis for setting a target reliability limit for the combination of serviceability limit state (SLS) loads. Any limit results in a combination factor arbitrarily decreasing or increasing the combination load. However, the SLS loads are mean loads which are added up deterministically and dependently without constraints regardless of whether the loads are independent or dependent. A combination factor may be applied due to intermittence or to convert the loads to simultaneous.
• This article conducts a comparative analysis of the gains introduced by combination factors with errors and approximations inherent in structural codes. It concludes that the gains associated with combination factors are negligible regarding the unavoidable errors and approximations present in the codes, highlighting the insignificance of combination factors in structural analysis.

By introducing these innovations, this article aims to enhance the precision and user-friendliness of structural design methodologies, particularly in scenarios involving variable loads, thereby contributing to advancements in the field of structural engineering.

1.3. Notation, Limitations

The notation outlined in the Eurocodes [1,2] is applicable. This article addresses the combination factor calculation as they appear in the design equations of common variable loads—including snow, wind, and imposed load—in fundamental design scenarios typical of standard construction practices focusing on the combination of the ultimate limit state (ULS) loads and solely on basic linear elastic cases excluding considerations of nonlinearity, plasticity, and fatigue [5].

Later in this article, the notation ψ is used for the combination factor. It corresponds to Eurocodes’ notation ψ₀. The Eurocodes include other combination factors in the SLS: ψ₁ = frequent factor and ψ₂ = quasi-permanent factor caused by load intermittency. Loads are not typified here as frequent or quasi-permanent. Therefore, combination factors ψ₁ and ψ₂ are not needed in the codes in this load model when normal loads are combined. The notation ψ in this article applies mainly to the combination factor of the codes and sometimes the value calculated for load proportion of 0.5. When confusion may arise, explicit notations are used: ψ_0.5 and ψ_code.

The terms “reliability” and “safety” are used here in relation to the application of a safety factor. For example, when a factor increases safety by 1%, it means reducing the safety factor corresponding to the target reliability by 1%.

1.4. Current Theory

References [1,2,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24] address the current combination of variable loads in structural design. In this approach, variable loads are combined by adding the leading load with its full value and the accompanying load with a reduced value, multiplied by a combination factor ψ. The calculation procedure for these combination factors employs an independent load combination methodology using a fictitious ideal material, V_M = 0, and a semi-probabilistic reliability model. The current method bases the combination factors solely on a load proportion of 0.5, without considering other load proportions. Discussion is lacking on how combination factors impact the reliability of the overall structural design. The current approach to combining SLS loads involves using mean loads for cases with a single variable load and extreme loads for cases with multiple variable loads. This inconsistency is not addressed.

Eurocodes [1] include equations for variable load combination factors ψ₀.

The general equation is

$${\psi }_0={\displaystyle \frac{{F_{\mathrm{s}}}^{-1}{\{\varphi (0.4\left({-\varphi }^{-1}\right(\varphi \left(-0.7\mathsf{\beta }\right))/N_1))}^{N_1}\}}{{F_{\mathrm{s}}}^{-1}{\{\varphi (0.7\mathsf{\beta })}^{N_1}\}}},$$

(1)

The approximation for large N₁ is

$${\psi }_0={\displaystyle \frac{{F_{\mathrm{s}}}^{-1}\left[\mathrm{e}x\mathrm{p}{-N}_1(\varphi (-0.4\left({-\varphi }^{-1}\right(\varphi \left(-0.7\mathsf{\beta }\right))/N_1)\left)\right)\right]}{{F_s}^{-1}(\varphi \left(0.7\mathsf{\beta }\right))}},$$

(2)

The approximation for a normal distribution is

$${\psi }_0={\displaystyle \frac{1+(0.28\mathsf{\beta }-0.7\mathrm{l}\mathrm{n}{N}_1)V}{1+0.7\mathsf{\beta }V}},$$

(3)

The approximation for the Gumbel distribution is

$${\psi }_0={\displaystyle \frac{1-0.78V(0.58+\ln \left(-\mathrm{l}\mathrm{n}\varphi \left(0.28\mathsf{\beta }\right)\right)+\mathrm{l}\mathrm{n}{N}_1)}{1-0.78V(0.58+\ln \left(-\mathrm{l}\mathrm{n}\varphi \left(0.7\mathsf{\beta }\right)\right)}},$$

(4)

where F(.) is the probability distribution function of the extreme value of the accompanying action in the reference period T; Φ(.) is the standard Normal distribution function; T₁ is the greater of the basic periods for the actions to be considered; N₁ is the ratio T₁/T, approximated to the nearest integer; β is the reliability index; V is the coefficient of variation of the accompanying action for the reference period.

Assuming a Gumbel distribution, an exceedance probability of 0.02, i.e., β = 2.05, V_Q = 0.4, i.e., μ_Q = 0.491, σ_Q = 0.196, T = 1, T₁ = 1, i.e., N₁ = 1, Equation (1) provides ψ₀ = 0.72; Equation (2), ψ₀ = 0.57; Equation (3), ψ₀ = 0.78; and Equation (4), ψ₀ = 0.72.

The current draft Eurocodes [2] include equations for variable load combination factors ψ₀:

Q₁ is dominating:

$${\mathsf{\psi }}_{0,2}={\displaystyle \frac{F_{Q_{2,max,{\tau }_1}}^{-1}\{\varphi \left({\alpha }_{Q_{2,\mathrm{m}\mathrm{a}x,{\tau }_1}}{\beta }_{\mathrm{t}}\right)\}}{F_{Q_{2,max,T}}^{-1}\{\varphi \left({\alpha }_{Q_{2,\mathrm{m}\mathrm{a}x,T}}{\beta }_{\mathrm{t}}\right)\}}}, with F_{Q_{2,\mathrm{m}\mathrm{a}x,{\tau }_1} }\left(q\right)=F_{Q_{2,\mathrm{m}\mathrm{a}x,T}}^{\left({\tau }_1/T\right)}\left(q\right)$$

(5)

Q₂ is dominating:

$${\mathsf{\psi }}_{0,1}={\displaystyle \frac{F_{Q_{1,max,{\tau }_1}}^{-1}\{\varphi \left({\alpha }_{Q_{2,\mathrm{m}\mathrm{a}x,{\tau }_1}}{\beta }_{\mathrm{t}}\right)\}}{F_{Q_{1,max,T}}^{-1}\{\varphi \left({\alpha }_{Q_{2,\mathrm{m}\mathrm{a}x,T}}{\beta }_{\mathrm{t}}\right)\}}}, with F_{Q_{1,\mathrm{m}\mathrm{a}x,{\tau }_1} }\left(q\right)=F_{Q_{1,\mathrm{m}\mathrm{a}x,T}}^{\left({\tau }_1/T\right)}\left(q\right)$$

(6)

where $F_{Q_{i,\mathrm{m}\mathrm{a}x,T}}$ is the extreme value distribution function for Q_i, where i = 1, 2 for reference period T; ${\tau }_i$ is the basic period, where the load intensity is assumed to be constant (i = 1, 2); $F_{Q_{i,\mathrm{m}\mathrm{a}x,{\tau }_1}}\left(q\right)$ is the extreme value distribution function for Q₂ for reference period ${\tau }_1$; $F_{Q_{1,{\tau }_1}}\left(q\right)$ is the distribution of an arbitrary realization Q₁; Ф is the cumulative standard normal distribution function; ${\alpha }_{Q_{i,\mathrm{m}\mathrm{a}x,T}}$ is the FORM sensitivity factor of Q_i dominating (i = 1, 2), ≈0.7; ${\alpha }_{Q_{i,\mathrm{m}\mathrm{a}x,{\tau }_1}}$ is the FORM sensitivity factor of Q_i dominating (i = 1, 2), ≈0.24;${\beta }_{\mathrm{t}}$ is the target reliability for the reference period T, usually 1 year.

Assuming V_Q = 0.4, i.e., μ_Q = 0.491, σ_Q = 0.196, T = 1, ${\tau }_1$ = 1, β_t = 4.7, ψ₀ = 0.45; if ${\tau }_1$ = 1/6, i.e., the annual load duration is two months, ψ₀ = 0.28.

Equations (1)–(6) apply to ideal material and the combination factor when the loads have the same magnitude, i.e., the load proportion is 0.5. These equations have limited significance as they provide the combination factor for a load proportion of 0.5 and for an ideal material with V_M = 0 only.

Table A1.1 of the Eurocodes [1,2] comprises combination factors for snow. The factors are different in the north, i.e., Finland, Iceland, Norway, and Sweden, vs. the south, and at different altitudes, ψ₀ = 0.7, 0.5, which is due to snow load duration. This issue is addressed later in this article.

2. Materials and Methods

2.1. Constant Occurrence Probability over Time

Here, the occurrence probability of the variable load is assumed to be constant over time. This assumption is appropriate in the combination factor calculation of snow and imposed load and approximate in the combination factor calculation of wind load.

In this model, the snow load is the constant mid-winter load in each combination. The possible intermittency of another load plays no role in the combination, and the assumption of constant occurrence probability is justified.

Combination of imposed loads with each other is made dependently regardless of the intermittency of these loads.

When the wind load is combined independently with another load, like the imposed load, intermittency is meaningful. However, in this load combination model, the intermittency is disregarded and constant occurrence probability is assumed over time. The combination factor ψ for wind obtained in this model is about the same as that given in the current Eurocodes.

The primary outcome and recommendation of this article is ψ = 1, though only imposed loads combined with each other are combined clearly without a combination factor. Other combination factors are almost unity, or the load combination is rare. Therefore, ψ = ψ₀ = ψ₁ = ψ₂ = 1 is recommended here for all normal loads. Multiple arguments are given to justify this suggestion.

2.2. The Target Reliability

The ULS one-year reliability index of the Eurocodes is β₁ = 4.7 [1,2], meaning the service time failure probability is P_f,50 = 1/15,400. However, β₁ = 4.2, i.e., P_f,50 = 1/1499 is assumed here as it is proposed in [7], and studies by the author indicate that the current safety factors of the Eurocodes approximately correspond to this reliability when the load combination rule (8.12) is applied; rules (8.13,a,b) and (8.14,a,b) result in lower reliability [25]. These reliabilities apply to structures with linear elastic behaviour. When the actual structure has yielding capacity and plastic behaviour, a different reliability index and analysis procedure can be applied [5,22,23,24].

The combination factors in the SLS are calculated for β₁ = 2.9 reliability [1,2]. However, new findings indicate that the reliability index β₁ = 2.9 should not be used as a constraint for combining SLS loads.

2.3. The Reference Calculation

The Eurocodes [1,2], especially the section on ‘combination of actions’ regarding variable loads, are used as a reference. However, the Eurocodes’ safety factors and parameters do not correspond with the target reliability, with the reliability of variable loads being particularly below the target [25]. Therefore, the safety factors are calculated here based on a reliability model which matches the target reliability in the reference calculation, i.e., the calculation does not include errors in the safety factor and parameter settings regarding the target reliability.

In codes, the combination factors, though different, are the same for all materials [26,27,28]. Here, timber is selected as the reference material as it has a mean variability of V_M = 0.2, compared to V_M = 0.1 for steel and V_M = 0.3 for concrete. A check is performed to confirm that the combination factors based on timber are comparable to those obtained based on ideal material, V_M = 0, steel, and concrete, too.

In the SLS, the calculations are based on the ideal material, V_M = 0.

In codes, the combination factors are the same regardless of the amount of permanent load in the combination. In this study, the combination factors are first calculated without the permanent load. A check is then made to confirm the calculation outcome remains as long as the permanent load is added. To simplify calculations, the reliability model is designed such that the material factor remains constant for both permanent and variable loads. This means that the material factor for timber is not affected by the amount of permanent load in the load combination.

The load safety factors can be fixed arbitrarily and setting the permanent load factor the same as the variable load factor, i.e., γ_G = γ_Q is feasible as it simplifies calculation. In the author’s opinion, γ_G = γ_Q = 1 would be the best option. However, γ_G = γ_Q = 1.2 produces a similar outcome and is selected here as it corresponds to the common partial safety factor design model.

The basic calculation is that variable loads 1 and 2 are combined in proportions α and 1 − α, α = ${\mu }_1/({\mu }_1+{\mu }_2$) where ${\mu }_1\mathrm{a}\mathrm{n}\mathrm{d} {\mu }_2$ are the means of the loads. A combination factor is fixed in one load and an equation for the material factor of timber γ_M is set for one-year loads. The condition for this design is that the safety factor for the service time combination load must be equal to the safety factor for a single load where the combination factor ψ is solved. A check is then made to confirm the obtained combination factor is the same or approximately the same as a permanent load is added. The same check is carried out when the calculation is made for ideal material, steel, and concrete. With the loads being often similar it is irrelevant to ascertain load the combination factor is fixed. If the loads are not similar, the combination factor is fixed to both loads and the bigger factor is selected.

2.4. The Load Proportion α

Each load proportion results in different combination factors. To obtain a proper design, the calculation should be made considering the actual load proportion α in each design case. However, codes simplify the problem and use the lowest combination factor ψ_0.5 for a load proportion of 0.5 only. This means that unsafe design cases may be included in the final design. The combination factor is fixed to the smaller load. Therefore, it is sufficient to study load proportions α = 0–0.5. Setting the combination factor to the codes is an optimization process and evaluation between safe and unsafe design cases. The suitable combination factor of the codes ψ_code is about (2ψ_0.5 + 1)/3. In this context, determining the precise numerical value of the combination factor according to the codes is of secondary importance, as the outcome is generally that the combination factors will be either unity or close to unity, and thus of little impact on the overall design. Thus, the recommendation is that the combination factors in codes are always unity, ψ = 1, and in the SLS, sometimes ψ = 0.

The combination factors are approximately calculated considering mainly the factors at load proportion α = 0.5 only. This approximation leads to minor errors in the combination factors for asymmetric variable loads. Specifically, it involves applying snow load, which is only active in winter, in combination with another variable load that is active year-round.

2.5. The Design Point

In the reliability calculation, a design point is needed where all pertinent distributions are fixed, i.e., 0.5 fractile of the permanent load, 0.98 fractile of the variable load, and 0.05 fractile of the material property. It can be selected arbitrarily. Unity is selected here.

2.6. The Basic Distributions

The permanent load plays a minor role in calculating combination factors. However, it is necessary to consider the permanent load when determining its impact on the combination factors. Normal distribution and V_G = 0.1 are assumed. These assumptions result in the material safety factor of timber γ_M = 1.21 in the permanent load, γ_G =1.2, P_f,50 = 1/1499.

The variable load distribution is assumed here to be the Gumbel distribution. The cumulative distributions of the loads 1 and 2 to be combined are $F_1(x; {\mu }_1,{\sigma }_1)$ and $F_2\left(x; {\mu }_2,{\sigma }_2\right).\mathrm{T}\mathrm{h}\mathrm{e}$ density distributions are $f_1(x; {\mu }_1,{\sigma }_1)$ and $f_2\left(x; {\mu }_2,{\sigma }_2\right)$ correspondingly. The coefficient of variation V_Q = 0.4 is assumed as such loads are the most common. Thus, the results given later apply to V_Q = 0.4 and μ_Q = μ₁ = μ₂; σ_Q = σ₁ = σ₂. The combination factors for loads V_Q < 0.4 are higher than the ones presented here for loads V_Q = 0.4. This fact is disregarded here, and therefore the presented ψ values are too low. The variable load parameters are defined in a way that the material safety factor for timber is for the 50-year target reliability P_f,50 = 1/1499 and γ_Q = 1.2, γ_M = 1.21, i.e., similar to the permanent load. The outcome is μ_Q = 0.372, σ_Q = 0.149, meaning that the characteristic load is 32% higher than the current one, and the one-year return load is 400 years. If the active time of the load is other than one year the distribution is exponentiated to the active time and the distribution is ${F_{\mathrm{Q}}(x; {\mu }_{\mathrm{Q}},{\sigma }_{\mathrm{Q}})}^t$ where t is the active time of the load in years.

Timber is used here as the reference material and the combination factors are calculated for this material [28]. The material resistance distribution is assumed log-normal. The parameters for timber are γ_M = 1.21, V_M = 0.2, μ_M = 1.141, σ_M = 0.282. A check is made to confirm that the combination factor for timber is approximately the same for other materials [26,27], and for the ideal material V_M = 0, too.

2.7. Uncertainty

The uncertainty is omitted in calculations here. Thus, the target reliability β₁ = 4.2 applies the reliability including the uncertainty.

2.8. Independent vs. Dependent Load Combination

Load combinations in structural design can be approached independently or dependently, with no intermediate method available. The dominant approach is that variable loads are independently and stochastically combined [1,2,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]. However, this load combination method is inconsistent with structural probability theory. In the ULS, permanent loads and variable loads are combined independently with a load reduction, whereas in the SLS, they are combined dependently and deterministically without a load reduction for single variable loads, and with a reduction for multiple variable loads. Imposed loads on multi-storey buildings are combined independently, while corresponding permanent loads are combined dependently. Nonetheless, most loads are combined dependently. The article in [29] presents five arguments supporting that permanent and variable loads should be combined dependently. Another article [30] addresses the issue further, summarizing current load combination procedures and explaining that the 50-year variable load impacts every structure and permanent load within the design domain. The independent combination of the 50-year variable load and the permanent load yields virtually identical results to those obtained through dependent combination methods. If we recognize that structures are designed for 50-year loads, the independent–dependent problem almost vanishes. However, the current calculation with a 5-year variable load results in approximately 10% lower load. The independent combination is virtually always unsafe, except for wind loads; the dependent combination is always safe, usually fully correct, and sometimes excessively safe.

Two arguments are further presented here for the dependent combination: permanent and variable loads are not independent in the combination, and the variable load simultaneously impacts all permanent loads within the failure domain.

• The current practice of combining loads independently in structural design is based on the combination of a random permanent load with a random variable load. This approach, however, does not capture the non-random nature of actual load combinations in real-world scenarios. In practical design, one combination load in design is applied to multiple actual combinations. For example, the design of one roof girder applies to multiple roof girders and various actual load combinations. Every actual variable load, including the maximum value in the variable load domain, has a simultaneous impact on multiple permanent loads. This includes a randomly selected permanent load, its adjacent loads, and typically the maximum load in the permanent load domain. For design purposes, the highest actual loads in the design domain are simultaneous and must be added up without any reduction. This indicates that the highest variable loads and the highest permanent loads coincide and should thus be considered together. The possibility of high variable loads coinciding with low permanent loads or vice versa cannot be utilized because the design must account for the highest combination load.
• In the Eurocodes, the highest variable load in each design is the service time load, specifically the 50-year load with an occurrence probability P_f,50 = 1/15,000. The permanent load failure domain encompasses an area that includes structures with 15,000 failure options, typically much smaller than 1 km². Within this domain, only one failure is permitted in 50 years. It is normal for each variable load to have a uniform and simultaneous impact on all permanent loads within the domain. As a result, the maximum loads, the next highest loads, and so on, occur concurrently. Consequently, the combination load distribution must be constructed as if the loads were fully correlated to accurately reflect their simultaneous nature.

In the current Eurocodes [1,2], permanent and variable loads are always combined independently in the ULS safety factor calculation, resulting in a design that is approximately 10% less safe regarding the dependent calculation. Additionally, combination rules (8.13a,b) and (8.14a,b) contribute further to this discrepancy, leading to an additional 10% reduction in safety.

Permanent loads and variable loads, as well as most variable loads, are combined with each other dependently. However, some variable loads like wind and imposed loads are independently combined with each other.

In this study, snow load is combined with another variable load approximately considering only the mid-winter period when the snow load is constant each year, making this combination dependent. Snow loads at the beginning and end of winter are disregarded. This assumption obviously results in lower bound combination loads and combination factors. An alternative calculation is also presented here, assuming the loads are combined independently for the entire winter period. The dependent combination results in higher combination factors if the mid-winter period with constant snow load constitutes half of the snow load period. Consequently, the dependent combination is more critical. Additionally, the independent combination is clearly inaccurate as the snow load remains constant for a prolonged period in mid-winter. Combining snow load with another variable load is analogous to combining permanent load and variable load.

In multi-storey buildings, the combinations of imposed loads on different floors should be considered dependent, since these loads are essentially partial components of the overall imposed load on the entire building. When the imposed load on the entire building aligns with the target reliability, it constitutes the maximum design load, which is then uniformly or almost uniformly distributed across all floors. This means that all floors simultaneously bear a high load, rendering load reduction inapplicable. The current practice of treating these combinations as independent leads to inconsistencies. For example, consider a building with a ground floor supporting two apartments, one on the first floor and another on the second floor. If load reduction is applied to the imposed loads of these apartments, the combination results in contradictions. If both apartments were on the first floor, no load reduction would be applied, yet the overall load on the ground floor remains the same in both scenarios. Furthermore, applying load reduction independently to imposed loads on various floors results in unrealistically low combination loads in multi-storey buildings. While load reduction can be applied to imposed loads, it should be based on the imposed area rather than the number of floors.

2.9. The Basic Equations

The author revealed the safety factor calculation procedure in detail earlier [25,29,30,31], and thus, it is explained only concisely here. Five equations are needed in the combination factor calculation. Safety factors, combination factors, and reliabilities can be obtained from these equations and assumed parameters without approximations, i.e., the used model is a fully probabilistic model without approximations.

When a load L with the cumulative distribution $F_{\mathrm{L}}(x;{\mu }_{\mathrm{L}},{\sigma }_{\mathrm{L}})$ and safety factor ${\gamma }_{\mathrm{L}}$ affects a material with the resistance distribution $f_{\mathrm{M}}(x; {\mu }_{\mathrm{M}},{\sigma }_{\mathrm{M}})$, and safety factor ${\gamma }_{\mathrm{M}}$, the equation to calculate the safety factors γ_L and γ_M or the failure probability P_f is

$${\int }_0^{\infty }{F}_{\mathrm{L}}\left(x; {\displaystyle \frac{{\mu }_{\mathrm{L}}}{{\gamma }_{\mathrm{L}}}},{\displaystyle \frac{{\sigma }_{\mathrm{L}}}{{\gamma }_{\mathrm{L}}}}\right)f_{\mathrm{M}}\left(x; {\mu }_{\mathrm{M}}{\gamma }_{\mathrm{M}}, {\sigma }_{\mathrm{M}}{\gamma }_{\mathrm{M}}\right)\mathrm{dx}=1-P_{\mathrm{f}}.$$

(7)

When one load L₁ with the cumulative distribution $F_1(x;{\mu }_1,{\sigma }_1)$ and safety factor ${\gamma }_1$ and another load L₂ with the density distribution $f_2(x; {\mu }_2,{\sigma }_2)$ and safety factor ${\gamma }_2$ is combined in load proportions α and 1 − α, the equation to calculate the independent combination load distribution $F_{\mathrm{i}}$ is

$$F_{\mathrm{i}}(x;\alpha )={\int }_{-\infty }^{\infty }{F}_1\left(x-r; {\displaystyle \frac{{\mu }_1\alpha }{{\gamma }_1}},{\displaystyle \frac{{\sigma }_1\alpha }{{\gamma }_1}}\right)f_2\left(\mathrm{r}; {\displaystyle \frac{{\mu }_2(1-\alpha )}{{\gamma }_2}}, {\displaystyle \frac{{\sigma }_2(1-\alpha )}{{\gamma }_2}}\right)\mathrm{dr}.$$

(8)

The dependent distribution $F_{\mathrm{d}}$ is calculated as follows:

$$F_{\mathrm{d}}\left(x;\mathrm{}\alpha \right)=\mathrm{}\left|{\displaystyle \genfrac{}{}{0pt}{}{y\leftarrow \mathrm{r}\mathrm{o}\mathrm{o}\mathrm{t}\left[F_1\left[x-r;\mathrm{}\frac{{\mu }_1\alpha }{{\gamma }_1},\mathrm{}\frac{{\sigma }_1\alpha }{{\gamma }_1} \right]-F_2\left[r;\mathrm{}\frac{{\mu }_2\left(1-\alpha \right)}{{\gamma }_2},\mathrm{}\frac{{\sigma }_2\left(1-\alpha \right)}{{\gamma }_2}\mathrm{}\right],\mathrm{}r\right]}{F_{\mathrm{d}}\leftarrow F_2\left[y;\mathrm{}\frac{{\mu }_2(1-\alpha )}{{\gamma }_2},\mathrm{}\frac{{\sigma }_2(1-\alpha )}{{\gamma }_2}\mathrm{}\right]}}\right..$$

(9)

When two variable loads 1 and 2 with the parameters set at unity are combined, the equation to calculate the independent combination factor ψ using the ideal material as a reference is

$${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{F}_1\left(1-\mathrm{}r;\mathrm{}{\displaystyle \frac{{\mu }_1\alpha }{{\gamma }_1\psi }},{\displaystyle \frac{{\sigma }_1\alpha }{{\gamma }_1\psi }}\right)f_2\left(r;\mathrm{}{\displaystyle \frac{{\mu }_2\left(1-\alpha \right)}{{\gamma }_2}},\mathrm{}{\displaystyle \frac{{\sigma }_2\left(1-\alpha \right)}{{\gamma }_2}}\right)\mathrm{d}\mathrm{r}\mathrm{}=1-P_f.$$

(10)

The equation to calculate the dependent combination factor is correspondingly

$$\left|{\displaystyle \genfrac{}{}{0pt}{}{y\leftarrow \mathrm{r}\mathrm{o}\mathrm{o}\mathrm{t}\left[F_1\left[1-r;\mathrm{}\frac{{\mu }_1\alpha }{{\gamma }_1\psi },\mathrm{}\frac{{\sigma }_1\alpha }{{\gamma }_1\psi } \right]-F_2\left[r;\mathrm{}\frac{{\mu }_2(1-\alpha )}{{\gamma }_2},\mathrm{}\frac{{\sigma }_2(1-\alpha )}{{\gamma }_2}\mathrm{}\right],r\right]}{F_{\mathrm{d}}\leftarrow F_2\left[y;\mathrm{}\frac{{\mu }_2(1-\alpha )}{{\gamma }_2},\mathrm{}\frac{{\sigma }_2(1-\alpha )}{{\gamma }_2}\mathrm{}\right]}}\right.=1-P_f.$$

(11)

3. Results

The combination factors ψ are given next for common variable loads, snow, wind, and imposed loads.

3.1. Independent Loads, Wind–Imposed Load

When the calculation is made for the ideal material and two independent loads V_Q,1 = 0.4 and V_Q,2 = 0.4, like wind and imposed load and for reliability β₁ = 2.05, i.e., for an exceedance probability of 0.02, ψ = 0.72 is obtained, which aligns with the values given by Equations (1) and (4).

Assuming the code parameters match the target reliability, when loads V_Q,1 = 0.4 and V_Q,2 = 0.4 are combined for β₀ = 4.2 with ideal material, steel, timber, and concrete, the combination factors for ψ_0.5 and corresponding code factors ψ_code are given in

Table 1. Independent combination factors for α = 0.5, ψ_0.5 and corresponding code factors ψ_code.

Material/ ψ	Ideal VM = 0	Steel VM = 0.1	Timber VM = 0.2	Concrete VM = 0.3
ψ0.5	0.58	0.59	0.61	0.69
ψcode	0.70	0.75	0.75	0.80

.

The corresponding value for loads V_Q,1 = 0.2 and V_Q,2 = 0.2 and timber is ψ_0.5 = 0.70, corresponding to a code value ψ_code = 0.80. This value applies to ideal code parameters. Current Eurocodes have one variable load safety factor, only γ_Q = 1.5, defined for loads V_Q = 0.4. Therefore, designs for loads V_Q = 0.2 are excessively safe and therefore, a somewhat smaller combination factor may be applied for loads V_Q = 0.2.

The factor ψ = 0.61 for timber is obtained by solving

$${\int }_0^{\propto }{[{\int }_0^{\propto }{f}_1\left(x-r; {\displaystyle \frac{{\mu }_1\alpha }{{\gamma }_1}},{\displaystyle \frac{{\sigma }_1\alpha }{{\gamma }_1}}\right)F_2\left(r; {\displaystyle \frac{{\mu }_2(1-\alpha )}{{\gamma }_2\psi }}, {\displaystyle \frac{{\sigma }_2(1-\alpha )}{{\gamma }_2\psi }}\right)dr]}^t{f}_{\mathrm{M}}\left(x; {\mu }_{\mathrm{M}}{\gamma }_{\mathrm{M}}, {\sigma }_{\mathrm{M}}{\gamma }_{\mathrm{M}}\right)\mathrm{dx}=1-P_{\mathrm{f}}.$$

(12)

where μ_M = 1.141, σ_M = 0.282, γ_M = 1.21, P_f = 1/1499, γ₁ = γ₂ = 1.2, α = 0.5, μ₁ = μ₂ = 0.372, σ₁ = σ₂ = 0.149, t = 50, and 10 is a feasible approximation for ∝.

If the calculation is made for the current parameters of the Eurocodes and for ideal material, factor ψ_0.5 = 0.81 is obtained.

The values given above approximately apply if the combination includes a permanent load, too.

The current value in the Eurocodes is ψ₀ = 0.60, which matches well with the calculated ψ_0.5 values given in Table 1.

3.2. Imposed Loads Combined with Each Other

As explained earlier, the imposed loads are combined dependently regardless of the intermittency of the loads and without a combination factor, ψ = 1, as these loads are correlated and partial loads of the whole imposed load of the house.

3.3. Snow Load

Snow load is combined dependently with another variable load, since the combination is performed using a constant snow load value for each year. Possible intermittency of the other load has no role in this combination.

If the calculation is made for ideal material, unrealistic values are obtained, e.g., the combination factor for one-week snow is ψ_d,0.5 = 0.82 in the dependent combination, and in the independent combination, it is ψ_i,0.5 = 0.53. Combination factors are given in

Table 2. The dependent combination factors for snow for timber ψ_d,0.5, highlighted code factors ψ_d,code, the corresponding factors calculated independently ψ_i,0.5, and code factors ψ_i,code.

Time/ ψ	One Day	One Week	One Month	Two Months	Four Months	One Year
ψd,0.5	0.51	0.67	0.79	0.86	0.92	1.00
ψd,code	0.70	0.80	0.85	0.90	0.95	1.00
ψi,0.5	0.30	0.39	0.47	0.50	0.54	0.61
ψi,code	0.55	0.60	0.65	0.65	0.70	0.75

for timber. The combination factors for the dependent values, α = 0.5 and ψ_d,0.5, and highlighted approximate rounded code values ψ_d,code are given for mid-winter constant snow durations of one day–one year. Values ψ_{d, 0.5} are the lower bound factors for a half winter period. For comparison, the corresponding incorrect independent values are ψ_I,0.5 and ψ_i,code.

We may assume that the constant snow load in the south lasts approximately one week. The current code value applies to α = 0.5, ψ_0.5,code = 0.50, meaning ψ_code = 0.70. It is a little lower than obtained in this calculation: ψ_d,code = 0.80. Correspondingly, the updated current one-month code value assumed in the north is 0.8, which is slightly lower than 0.85 obtained in this calculation.

3.4. More than Two Variable Loads

The analysis indicates that the combination factor for two variable loads is nearly unity. This suggests that the combination factors for additional loads are similarly close to unity. Given the low load proportion of these additional loads, approximating the combination factor as unity is reasonable.

3.5. Combination Factors in the SLS

The target reliability index of the Eurocodes in the SLS is β₁ = 2.9, i.e., P_f,50 = 1/11. The characteristic load for the variable load is the same in the SLS and in the ULS, i.e., it is the 50-year return load.

As explained above, the imposed loads are combined dependently without a combination factor, ψ = 1 as the loads are correlated.

Two independent loads like imposed load and wind combined in the SLS with each other have an unrealistically high combination factor ψ = 1.1. This high value means that the combination load is higher than the arithmetic sum of the partial loads. The high value is attributed to a mismatch between the characteristic variable load and the reliability index β₁ = 2.9. This contradiction diminishes if the characteristic load is changed for a longer return time [25] and the robust upper tail of the Gumbel distribution is cut off. If the calculation is made for ideal parameter settings for β₁ = 2.9 and for the Gumbel distribution, the parameters are μ_Q = 0.360 and σ_Q = 0.144, i.e., the return time of the characteristic load is 536 years and the combination factor is ψ = 0.65, meaning that the code value is about ψ = 0.80.

The combination factors for snow in the SLS are unrealistically high if the current load parameters are used and about the same as in the ULS if ideal parameters are used.

The calculations above indicate that the current approach to load combination in the SLS is questionable, as the fixed β₁ value and the characteristic load of the variable load arbitrarily affect the outcome.

A new perspective suggests that SLS loads should be combined without using a constraint, like β, to guide the combination. In the ULS, whether there is one variable load or multiple variable loads, all loads are multiplied by a safety factor defined by β, making the combination load always represent the extreme load. Conversely, in the SLS, when one variable load is combined with the permanent load, the mean loads are added up, and no combination factor or constraint is applied. However, when the permanent load is combined with multiple variable loads, combination factors apply, meaning that the combination load is the extreme load. The use of combination factors imposes a constraint on the load combination through the fixed β₁ value, which is flawed. Instead, SLS loads are mean loads which are combined by adding up the loads as such regardless of whether the loads are independent or dependent or the combination is dependent or independent. No constraint, such as a combination factor, should be applied when combining loads in the SLS that act full-time. If one load acts year-round while the other acts part-time, like snow, a combination factor is applied due to conversion of the distributions to the same active time, with one-month, one-week, and one-day ψ_0.5 values being 0.6, 0.4, and 0.1, respectively, obtained from dependent combinations. If both loads act part-time or intermittence is considered, a further reduction may be needed. Setting these factors at unity is also desirable. Design criteria such as the deflection limit of a girder in the SLS are indeterminate. These criteria can be adjusted in cases where the assumption of ψ = 1 would result in an overly robust outcome. In any case, SLS design is secondary to ULS, and higher errors are tolerable.

Combining variable loads in the SLS is a complex issue that is not addressed in depth here. However, it is evident that the current combination factors are too low for the same reasons discussed regarding the ULS combination factors.

3.6. Combination of Intermittent, Frequent, and Quasi-Permanent Loads

In this load combination model, the imposed loads are combined dependently at ψ = 1 regardless of the load intermittency. All loads on the floors share similarities, exhibit correlation, and constitute partial loads that contribute to the overall loading of the entire house. Load reduction is feasible based on the exposed area, not on floors.

Snow loads are combined dependently, too. The load intermittency has no effect on the combination as the load combination is calculated for the constant load in mid-winter.

The load intermittency is meaningful in the independent load combination like the combination of imposed load and wind. Though the intermittency is disregarded here, virtually the same combination factor is obtained as given in the current Eurocodes in the ULS: ψ_0.5 = 0.61 vs. ψ₀ = 0.6. Frequent and quasi-permanent load combinations are applied in the SLS. The current values of the Eurocodes are low: ψ₁ = 0.2 and ψ₂ = 0. These combinations are rare in the design domain. Therefore, in the author’s opinion, the approximation ψ = ψ₁ = ψ₂ = 1 is feasible to simplify codes and to lessen the workload. However, loads acting for very short periods evidently require a mitigating arrangement in the SLS, such as being combined only with sustained loads like snow, i.e., sometimes setting ψ = 0.

4. Discussion

The calculations here are based on dependent or independent load combinations depending on the nature of loads. However, the conclusions remain valid even when the dominant approach of independent combination is applied:

• Although the combination of imposed loads is assumed to be independent, removing the combination factor is justified to avoid obvious contradictions. Such removal simplifies codes and does not affect the design outcome, as the load reduction can still be applied to the exposed area.
• In this article, wind load is independently combined in the ULS.
• Snow load is combined dependently here. If combined independently, slightly lower combination factors are obtained. In any case, the current factors in codes should be increased, as they are ψ_0.5 values. However, the overall effect is negligible.

These points illustrate that the approach to combining loads should be re-evaluated, particularly in light of the contradictions, potential simplifications, and cultivating international consensus [4] that can be achieved with improved safety.

Current combination factors in structural codes are the lowest values obtained from a load proportion of 0.5. However, the code values should be higher since other load proportions must also be considered. If only the ψ_0.5 values were used in codes, the outcome would be correct at a load proportion of 0.5 but unsafe by up to about 5% in every other case. This is illustrated in Figure 1, which includes a left load figure and a right error figure. The following can be seen in the left load figure:

• The solid line represents the combination factor ψ as a function of load proportion α = 0–50% in the ultimate case of a wind–imposed load combination, V_Q = 0.4, where the combination factor is lowest, ψ_0.5 = 0.61 and ψ_code = 0.75.
• The dashed line shows the total load assuming no permanent load is included in the combination, and the correct combination factor is applied for each design case, i.e., for each load proportion α.
• The dash-dotted line shows the total load when the combination factor used in the codes is ψ_code = 0.75 (instead of the lowest value ψ_0.5 = 0.61).

The following can be seen in the right error figure:

• The dashed line indicates the error in the total load if the combination factor obtained from α = 0.5, ψ_0.5 = 0.61, is used in the codes.
• The solid line shows the error when ψ_code = 0.75 is used in the codes.

Figure 1. The solid line in the left load figure shows the combination factor ψ as a function of load proportion α; the dashed line shows the total load; the dash-dotted line shows the total load when ψ = 0.75. The solid line in the right error figure shows the error in the total load if factor ψ = 0.75 is selected for the codes; the dashed line shows the error when ψ = 0.61 is selected for the codes.

Figure 1. The solid line in the left load figure shows the combination factor ψ as a function of load proportion α; the dashed line shows the total load; the dash-dotted line shows the total load when ψ = 0.75. The solid line in the right error figure shows the error in the total load if factor ψ = 0.75 is selected for the codes; the dashed line shows the error when ψ = 0.61 is selected for the codes.

Figure 1 shows the following:

• The combination factor in codes ψ_code should be higher than the lowest factor ψ_0.5 at a load proportion of 0.5; an approximate relationship is ψ_code ≈ (2ψ_0.5 + 1)/3.
• The current model for variable load calculation with only one parameter approximately includes up to about 7% safe and 2% unsafe errors.
• Combination factors are effective in mitigating excess safety in variable load combinations, when the loads are of similar magnitude, typically within the range of load proportions α ≈ 0.3–0.5. In these scenarios, the calculation results in a reliability gain of 6–12% for the variable loads. When considering that the combination includes 20–50% permanent load, the overall reliability gain ranges from 4 to 10% (the upper value is theoretical, light floor with equal wind and imposed load). However, load cases with proportions α ≈ 0.3–0.5 are rare.
• Most variable load pairs involve loads of different magnitudes, typically with load proportions α < ≈0.3. In these cases, the reliability advantage ranges from 0 to 7% for the variable loads. Without applying combination factors, the potential reliability gains indicated by the light grey area above the dash-dotted line cannot be realized. The dash-dotted line dropping below the dashed line indicates a margin of 2% unsafety, represented by the dark grey area. Unsafe errors are more detrimental than safe ones. The light grey area is approximately three times larger than the dark grey area, implying that if code writers consider unsafe errors three times more harmful than safe ones, the combination factor calculation becomes almost unfeasible due to the unsafe errors it introduces.
• Assuming loads α = 0–0.5 have equal occurrence and disregarding the unsafe error at α < ≈0.3, the overall reliability gain obtained by combination factors is 3–5%.

Estimation of the overall gain due to combination factors is difficult. The unsafe error in the combination factor model, though disregarded here, should somehow be considered. The prevalence of variable load cases is unknown; the high prevalence of unequal loads is apparent. The above calculation applies to a rare case of the highest combination factor gain ψ_code = 0.75 for wind–imposed load and loads V_Q = 0.4 (loads V_Q < 0.4 result in higher factors). Other combination factors are higher (ψ_code = 0.8–1) with lower gain. The combination factors are meaningful in structures with low permanent load, i.e., in timber and steel structures, and the meaning is negligible in concrete structures. The combination factors are meaningless in cases of different load magnitudes, including a little unsafe error. Virtually, the combination factors are meaningful in one case only, i.e., in the ULS design of light roofs with about equal snow and wind. However, it can be assumed that the overall reliability gain obtained from the combination factors is about 2–3% in the ULS with an ultimate highest gain of 8% in rare cases (light roof with equal wind and snow). A gain of similar magnitude likely applies in the SLS as well. Thus, the meaning of the combination factor is negligible.

Besides reliability, there are points which must be considered in the evaluation of the combination factors. Next, six points support the removal of the combination factors from codes:

• Removing combination factors would simplify the structural codes, making them easier to understand and apply.
• The elimination of combination factors lessens the workload for engineers, streamlining the design process.
• When combination factors are included, the outcomes are approximate, encompassing both safe and unsafe cases.
• In most cases, the impact of combination factors is negligible, with clear benefit gained virtually in one load case only, such as a steel or timber roof loaded with approximately equal snow and wind.
• In the current Eurocodes, the reliability in cases dominated by variable loads is significantly below the target. Removing combination factors reduces this discrepancy and improves reliability.
• If combination factors are removed from codes, the resulting error is always safe and negligible compared to other unavoidable errors and approximations inherent in structural design.

These issues are further discussed next.

4.1. Codes Become Simpler and the Design Work Lessens

When codes are simplified, it can impact reliability and the costs associated with design outcomes, either negatively or positively. A pertinent question is what harmful consequence is justified if one load case is removed from the codes. The current load combination of the Eurocodes provides an example. In most European countries, combination rule (8.12) is applied, which includes only one permanent load safety factor and omits at least one load case in the entire design domain in the ULS regarding rules (8.13a,b) and (8.14a,b). Rule (8.12) results in approximately 10% excess safety and about 5% excess material usage based on current understanding. Code writers in most European countries prefer simplicity and consider one extra load case harmful despite a 10% excess reliability.

Implementing combination factors for variable loads results in an overall reliability gain of 2–3% but introduces many additional load cases in virtually all design tasks. The current Eurocodes are deemed complicated, and the multiple load cases induced by combination factors are a major reason for this complexity. The author estimates that removing the combination factors would decrease the overall numerical calculation work by at least 25% and reduce the overall design work by at least 15%.

Simplification and user-friendliness are major targets in the development of the Eurocodes [4]. Considering the example above, the proposed amendment of removing combination factors is realistic.

The proposal eliminates one abstraction—combination factor—from the codes, which is an essential step towards code simplification.

The author previously suggested some other simplifications for the Eurocodes [25].

• The current characteristic load for variable loads, based on a 50-year return period, leads to unequal load factors γ_G ≠ γ_Q. A suitable alternative is to set the load factors the same (γ_G = γ_Q) and preferably at unity (γ_G = γ_Q = 1), while keeping the characteristic load for permanent loads unchanged at the 0.5 fractile. The characteristic load for variable loads should be increased by 15–30, i.e., adjusted to a 300–400-year return period to match the characteristic load of the permanent load and account for the actual variability of the variable loads.
• Additionally, removing the material safety factor, i.e., setting γ_M = 1, can be achieved by changing the characteristic load, the 0.05 fractile, which is currently the same for all materials, to a much lower value specific to each material that fulfils the target reliability in the design equation.
• Using a truncated Gumbel distribution in reliability calculations can also be beneficial, as the basic Gumbel distribution results in unrealistically high safety factors for materials of low variation.
• Currently, all variable loads have the same characteristic load, i.e., the 50-year return load, which results in excess safety for loads with low variability. This can be largely avoided by setting characteristic variable loads tailored to each type of load. This amendment would not increase code complexity or design work but would require code writers to adjust the characteristic values for each variable load.
• The rounding rule for material safety factors introduces about a 2% excess error, which can be avoided by removing the material safety factors altogether.

In the author’s opinion, structural codes should include no load safety factors, γ_G = 1, γ_Q = 1, no material safety factors, γ_M = 1, and no combination factors, either ψ = 1 or ψ = 0. Such a change would necessitate modifications to the load tables for variable loads and alterations to the characteristic load values for material properties. Additionally, some design equations especially for concrete that currently use factored parameters would need to be revised to use unfactored parameters.

4.2. Removing Combination Factors Improves the Reliability of the Current Eurocodes

In the current Eurocodes, the permanent load and the variable load are combined independently, which can lead to a reduction in the variable load reliability. Consequently, the reliability of variable loads is often calculated for 5-year loads [17,18,19,20], and in some cases for 1-year loads, but with reduced reliability [16]. Both calculations result in too-low variable load reliability regarding the target.

Table 3. Material factors γ_M corresponding to reliabilities β₀ = 4.2 and β₀ = 4.7 in permanent and variable loads, γ_G = 1.35, γ_Q = 1.5.

	Permanent Load			Variable Load
Material/ β, γM	Steel VM = 0.1	Timber VM = 0.2	Concrete VM = 0.3	Steel VM = 0.1	Timber VM = 0.2	Concrete VM = 0.3
γM,code	1.0	1.3	1.5	1.0	1.3	1.5
β0 = 4.2	0.97	1.08	1.23	1.27	1.28	1.37
β0 = 4.7	1.05	1.23	1.48	1.51	1.55	1.72

includes the material safety factors γ_M for steel, V_M = 0.1; timber, V_M = 0.2; and concrete V_M = 0.3 at reliabilities β₀ = 4.2 and β₀ = 4.7, γ_G = 1.35, γ_Q = 1.5. The current material factors γ_M,code for steel, timber, and concrete are 1.0, 1.3, and 1.5. The calculated material factors match well with the code factors for the permanent load for β₀ = 4.7; however, the code factors for the variable load should be much higher.

Table 3 includes material safety factors for both permanent and variable loads. Material factors for mixed loads can be obtained approximately using linear interpolation. The material factors for steel are considerably higher than the code factors. This contradiction is significantly mitigated when the Gumbel distribution is truncated at about a 300-year return load, which is approximately 30% higher than the current characteristic load [31]. When the combination factors are removed from the current Eurocodes, the error in the variable load reliability lessens in the ULS.

4.3. The Overall Effect Is Negligible

The codes inherently contain unavoidable errors and approximations, as achieving perfect alignment of code parameters with ideal reliability targets is unattainable. Another article [16] proposes optimal parameters for the Eurocodes, incorporating approximately 20% safe and unsafe errors relative to the target reliability. The author suggests an alternative code model with zero unsafe error regarding the target and a safe error of only 7% when considering only a permanent load and one variable load [25]. When multiple variable loads are considered, the safe error is approximately doubled according to the calculations above, resulting in a mean safe error of about 7%.

The current independent load combination results in significantly higher reliability for permanent loads compared to variable loads and leads to approximately 10% unsafe design when considering the dependent combination.

The Eurocodes include three combination rules (8.12), (8.13a,b), and (8.14a,b). Rule (8.12) results in about 10% excess safety and 5% excess material compared to other rules.

The characteristic load for variable loads is the same for all variable loads. This approximation makes about 15% excess safety for loads V_Q = 0.2 compared to loads V_Q = 0.4. In the whole design domain, the safe effect is about 2–3%.

The rounding error of safety factors introduces about a 2% safe and unsafe error.

The 2–3% impact of combination factors on overall reliability is insignificant in the context of the larger approximations present in structural codes.

As proposed earlier in [25], setting code parameters to demand the target reliability in all significant design cases supports the removal of combination factors, given their inclusion of unsafe design cases in the calculation process.

4.4. Other Variable Loads

The analysis above applies to loads with a uniform occurrence probability over time. Combining other loads such as construction loads, temperature, and ice requires data and assumptions about their occurrence probability. However, loads caused by ice and temperature are analogous to snow loads, and therefore, the same combination factor can apparently be applied. The combination factor for construction loads may be set at unity ψ = 1 due to simplification of the codes.

4.5. Some Proposals

If the combination factors are removed from codes, the reliability is higher in load cases of multiple variable loads regarding single variable loads. It is feasible to mitigate this deficiency by decreasing the safety factor for single variable loads to about 1%. This amendment lessens the inevitable safe errors associated with load cases featuring multiple variable loads. Further, in the SLS design, loads acting for very short periods apparently need a mitigating arrangement, e.g., such loads are combined with sustained variable loads only, and the design criteria should probably be adjusted.

5. Implementation

The proposal can be implemented in the Eurocodes [1,2] by deleting the combination factors in Table A1.1, i.e., setting the values at unity, ψ = 1; in the SLS, it is apparently feasible to set some factors at zero, ψ = 0.

In the ULS, the load combination rule (8.12) is correct and should always be used. The current equation for the combination action calculation is

$${\sum }_{j\ge 1}{\gamma }_{G,j}{G}_{k,j}“+”{\gamma }_{P}P“+”{\gamma }_{Q,1}{Q}_{k,1}“+”{\sum }_{i>1}{\gamma }_{Q,i}{\psi }_i{Q}_{k,i}.$$

(13)

It is changed by removing ψ, i.e., it becomes

$${\sum }_{j\ge 1}{\gamma }_{G,j}{G}_{k,j}“+”{\gamma }_{P}P“+”{\gamma }_{Q,1}{Q}_{k,1}“+”{\sum }_{i>1}{\gamma }_{Q,i}{Q}_{k,i}$$

(14)

Corresponding changes are made in the SLS. The amended equations, other equations, and the design process remain.

The updated codes are safe regarding codes with combination factors.

The findings of this study should be verified in actual design practice by conducting comparative designs using both the current Eurocodes and the proposed amended version. The comparison should be made using the combination factors given here. The current factors are based on ψ_0.5 values, which exaggerate the reliability gain. In this comparison, a penalty should be applied when the calculation results in an unsafe outcome. This task is relatively straightforward when using an automated design software programme. Key metrics should be evaluated:

• Safety Margins: Determine if the amended codes maintain or improve the safety margins compared to the current codes.
• Design Efficiency: Evaluate any changes in material usage and overall design efficiency.
• Complexity and Workload: Measure the reduction in complexity and workload achieved by the proposed amendments.

The study should end with a document of findings to provide a justification for or against the proposed amendments including benefits and drawbacks.

If the removal of combination factors is executed, the author estimates a significant decrease in complexity and workload, with a slight increase in material use. However, if other proposed amendments [25] are also considered, a significant decrease in material use can be achieved along with decreased complexity and workload.

6. Conclusions

This research recommends the removal of combination factors from codes as these factors are unity, ψ = 1, can be approximated at unity, or apply to rare load cases only:

• Imposed loads are combined with each other without a combination factor, ψ = 1.
• The combination factor for a one-week constant snow load during mid-winter is ψ = 0.8, and the one-month factor is ψ = 0.85.
• The combination factor of wind is ψ = 0.75. Such a combination is rare as it applies to wind–imposed loads only.

These factors apply to loads V_Q = 0.4; loads V_Q < 0.4 result in higher factors.

The combination factors are meaningful in light structures only and merely in cases of about two equal variable loads and virtually in one load case only: timber or steel roofs with about equal snow and wind.

The existing model for variable load modelling, relying on a single combination factor, is deemed inadequate. While it offers a notable advantage when dealing with rare cases with loads of similar magnitudes, its effectiveness diminishes in scenarios involving frequent cases with loads of varying magnitudes, including a harmful unsafe error.

Despite offering a modest overall reliability advantage of around 2–3%, given the unavoidable errors and approximations of approximately 7% that are inherent in structural codes due to the difficulty in perfectly accounting for all parameters in reliability models, the effect of combination factors is negligible.

Removing the combination factors from the codes would be a simple task. This can be achieved by removing the combination factors from the relevant equations, leaving the rest of the equations and the design process unchanged. The resulting design without combination factors maintains safety levels comparable to designs incorporating combination factors.

Regarding the nature of the loads, the combination is calculated here either dependently or independently. However, the conclusions remain valid even if the dominant approach of independent combination is applied.

In light of their limited reliability advantage and the significant complexity and additional work they introduce to the design process, the removal of combination factors from structural codes is warranted. This study strongly recommends their complete removal to streamline structural design processes and mitigate unnecessary complexities.