5.3 Combination Expressions

The standards describe how to combine actions. For example, EN 1990 requires the design of the ultimate and the serviceability limit states. Ultimate limit states for load bearing capacity have to be designed in four design situations, which are subject to certain combination rules:

• Permanent situations that comply with the common conditions of use of a structural system, as well as temporary situations that refer to time-limited stages of the structure (e.g. construction stage, repairs)

• As a combination rule for permanent and temporary situations (basic combination), you have to apply either

$\sum _{j\ge 1}{\gamma }_{G,j}{G}_{k,j}+{\gamma }_P{P}_k+{\gamma }_{Q,1}{Q}_{k,1}+\sum _{i>1}{\gamma }_{Q,i}{\psi }_{0,i}{Q}_{k,i }$

• or the more unfavorable combination with Equation 5.2 and Equation 5.3 for the limit states STR and GEO.

$\sum _{j\ge 1}{\gamma }_{G,j}{G}_{k,j}+{\gamma }_P{P}_k+{\gamma }_{Q,1}{\psi }_{0,1}{Q}_{k,1}+\sum _{i>1}{\gamma }_{Q,i}{\psi }_{0,i}{Q}_{k,i }$

$\sum _{j\ge 1}{\xi }_j{\gamma }_{G,j}{G}_{k,j}+{\gamma }_P{P}_k+{\gamma }_{Q,1}{Q}_{k,1}+\sum _{i>1}{\gamma }_{Q,i}{\psi }_{0,i}{Q}_{k,i }$

• Extraordinary situations referring to extraordinary actions of the structural system or its environment (e.g. fire, explosion, collision)

$\sum _{j\ge 1}{H}_{k,j}+P+A_d+({\psi }_{1,1} \text{or} {\psi }_{2,1})Q_{k,1}+\sum _{i>1}{\psi }_{2,i}{Q}_{k,i}$

• Situations in case of earthquakes

$\sum _{j\ge 1}{G}_{k,j}+P_k+A_{Ed}+\sum _{i\ge 1}{\psi }_{2,i}{Q}_{k,i }$

According to EN 1990, you have to design serviceability limit states in three design situations which are subject to the following combination rules.

• Rare situations with irreversible (lasting) effects on the structural system

$\sum _{j\ge 1}{G}_{k,j}+P_k+Q_{k,1}+\sum _{i>1}{\psi }_{0,i}{Q}_{k,i}$

• Frequent situations with reversible (non-lasting) effects on the structural system

$\sum _{j\ge 1}{G}_{k,j}+P_k+{\psi }_{1,1}{Q}_{k,1}+\sum _{i>1}{\psi }_{2,i}{Q}_{k,i }$

• Quasi-permanent situations with long-term effects on the structural system

$\sum _{j\ge 1}{G}_{k,j}+P_k+\sum _{i\ge 1}{\psi }_{2,i}{Q}_{k,i}$

When you access the dialog box or table, the combination rules of the following design situations are preset for EN 1990:

• ULS : Ultimate limit state for permanent or temporary situation
• SLS : Serviceability limit state for characteristic situation
• SLS : Serviceability limit state for frequent situation
• SLS : Serviceability limit state for quasi-permanent situation

You can create a new combination rule in another table row or in the dialog box by using the [New] button. The design situations described below are available for selection.

Combination rules marked in the dialog list can be deleted with the [Delete] button.

The short descriptions of combination rules can be changed retroactively. The list provides some suggestions for selection.

The check box allows you to control if the selected combination rule is considered when creating result combinations. In this way, it is possible to reactivate design situations or exclude them from the generation.

The standards describe the situations for which designs of structural systems must be fulfilled. These design situations determine the conditions expected during the construction and use of the building.

The following design situations for EN 1990 are available for selection in the list:

Use the [Info] button to check the combination rule of the current design situation. A dialog box opens explaining the equation with the relevant parameters.

With this option, RFEM distinguishes between favorably and unfavorably acting permanent actions during the generation. They are considered in the superpositioning along with different partial safety factors; additional combinations are generated.

The settings of this check box only affect design situations for load bearing capacity. The distinction between favorably and unfavorably acting permanent actions is done automatically for the design situation "Static equilibrium", whereas permanent actions are not differentiated for the design situation "Serviceability".

RFEM distinguishes between two load case categories: standard load cases and load cases of the type Imperfection. Due to this special treatment of imperfections, it is possible to form any possible load combination once with an imperfection and once without.

With the Differently for each combination expression check box, it is possible to separately assign imperfections for the combination expressions: According to EN 1992-1-1, for example, imperfections have to be considered for the ultimate limit state designs. Serviceability limit state designs can be performed without imperfections.

When the check box is selected, the [Settings] button or the button is enabled. Use the buttons to access the Settings dialog box with specific settings for imperfection load cases.

The Use Load Cases of Type 'Imperfection' dialog section lists all load cases that were classified as the action type "Imperfection" (see Chapter 5.1). Use the check boxes in the Use column to specifically control, which of the load cases is included in the generation of load combinations.

The Only with Load Cases and Never with Load Cases columns are shown if the imperfection load cases are Subject to specific load cases (see description below).

The Options dialog section controls how imperfection load cases are taken into account. If All imperfection load cases act as alternative, RFEM applies only one imperfection load case for each load combination.

You can reduce the number of generated load combinations by selecting the Remove co-existence of the same load combination with and without imperfection check box: The constellations without an imperfection load case are suppressed for similar combinations. It is also possible to Remove all load combinations without imperfection.

If several imperfection load cases are available, you can select the Prefer common assignment of imperfections check box to prevent separate assignment to load combinations.

With the Subject to specific load cases option, you can further reduce the number of generated load combinations. If the option is selected, the two columns Only with Load Cases and Never with Load Cases are additionally shown in the dialog section above. Click into a cell to enable the button, which opens the Select Load Cases dialog box where you can define a relation between the imperfection load case and one or more related or mutually exclusive load cases (see Figure 5.14).

To further reduce the number of created load combinations, it is possible to classify load cases as mutually exclusive or occurring only together. By selecting the Differently for each combination expression check box, you can separately define the constellations of the load cases for combination expressions.

If you select the check box, the and buttons become available, which open a dialog box with more detailed settings for assigning load cases.

First, enter a load case into the Select Load Cases column in the Individually Acting Load Cases dialog section or define it with the button in the Select Load Cases dialog box. Then, in the Not to combine with load cases column, define the load case(s) that you never want taken into account together with the load case in the load combination. In this way, it is possible, for example, to avoid the combination of snow with human load cases.

In the Simultaneously Acting Load Cases dialog section, you can analogously specify settings for load cases that you want to always have appear together in each load combination. However, these relations are only effective if the Reduce number of generated combinations due to Examine results option (see below) is not activated.

The complexity of the structural system as well as the number of actions and load cases have a significant influence on the number of generated combinations. RFEM provides three ways to reduce the number of constellations effectively. The first two procedures are only available for the generation of load combinations, not for result combinations. They are described in an example.

With this option, you can generally limit the number of load cases that occur in the load combinations. Access to the check box is available in the General tab of the Combination Expressions (see Figure 5.10). RFEM examines, which load cases provide positive or negative internal forces and deformations. Then, all positively acting and all negatively acting load cases are combined. Thus, combinations take only the load cases into account that are relevant for the maximum or minimum values.

The advantage of this method is that it is possible to considerably reduce the number of combinations, which has a positive impact on the speed of the calculation and evaluation. A disadvantage may be the fact that if there are unfavorable load arrangements and specifications for the reduction, there is a certain factor of uncertainty for finding the extreme values.

When you select the check box, the additional Reduce - Number of Load Cases tab appears. In this tab, you can specify in detail, which load cases, internal forces, and objects you want considered for the creation of governing combinations.

The load cases can be selected Manually or determined Automatically on the basis of relevance criteria. Clicking the [Determine automatically] button starts a calculation for determining the maximum and minimum internal forces, deformations, and support reactions in the load cases.

When the automatic determination is selected, specify the Results (deformations, internal forces of members and surfaces, support reactions) and Objects (nodes, surfaces, members, etc.) that you want considered for the evaluation of the load cases. The relevant objects can be selected graphically using once the All check box is disabled. You can use the Assign separately for each object check box to assign specific result types to objects for the analysis.

The number of load cases contained in a Group after calculation depends on the settings defined in the Choice of Load Cases dialog section:

• When the Maximum number of the most relevant load cases option is selected, a group provides either the specified maximum number of load cases or only positively or negatively acting load cases in a smaller number.

• It is possible to Neglect load cases, which only contribute slightly to the maximum and minimum values. The percentage refers to the internal forces, deformations, and support forces of the load cases that respectively provide the extreme values.

Imperfection load cases are not considered when the automatic creation of groups is set.

RFEM only creates the governing load combinations (this option is not available for result combinations).

When selecting the check box, the new Reduce - Examine Results tab is added to the dialog box.

With the first Reduction Method, you can automatically evaluate generated temporary result combinations. Temporary result combinations include all load cases created in the model and consider all existing relations between them. By means of the results available on each FE node, RFEM analyzes, which simultaneously acting load cases produce a maximum or minimum at the corresponding locations. The reduction method is based on the assumption that only the combinations that contain exactly these simultaneously acting load cases can be governing.

Alternatively, it is possible to use the results of a user-defined result combination for the results reduction.

In the Examining Results dialog section to the right, you can define the deformations, internal forces, stresses, or support reactions that you want taken into account for the determination of extreme values.

The Examining Results of Objects section provides options to restrict the extreme value analysis to results of selected members, surfaces, and solids. You can use to select the objects graphically.

The third way to reduce the number of generated combinations is to only classify selected actions as leading actions. This option is available for the generation of both load and result combinations.

When selecting the check box, the new Reduce - Leading Variable Actions tab is added to the dialog box.

The list of leading actions contains only variable actions.

When you remove the check mark of an entry in the Leading Actions column, the corresponding action is only superimposed as an accompanying variable action.

Entering data in this section of the Edit Load Cases and Combinations dialog box (see Figure 5.10) affects the First number of generated Load combinations or Result combinations that are created in RFEM.

Optionally, you can Generate additionally an 'Either/Or' result combination (results envelopes). This result combination superimposes the extreme values of all load or result combinations according to the following scheme:

"CO1/permanent or CO2/permanent or CO3/permanent, etc."

If several combination expressions are specified for the generation, it is possible to Generate additionally a separate 'Either/Or' result combination for each combination expression.

Use the list to decide, which method of calculation you want to apply for analyzing combinations (see Chapter 7.3.1.1). RFEM presets the nonlinear calculation according to the second order analysis (P-Delta) for load combinations.

This dialog section or table column is filled during the generation, which starts automatically when closing the dialog tab or table. The dialog field shows a short overview of the number of generated combinations.

With the data entered in the dialog box or table, RFEM creates "Action Combinations" (AC). They are described in the following chapter. You can use the entries shown in the current dialog box to estimate how combination rules affect the number of combinations.

In the example shown on the left, a total of 47 action combinations are generated for the four specified design situations.

• ULS (STR/GEO):

AC1 to AC13

• SLS - characteristic:

AC14 to AC26

• SLS - frequent:

AC27 to AC39

• SLS - quasi-permanent:

AC40 to AC47

When jumping to the next tab with the dialog button, RFEM automatically determines the action combinations. The first action combination created with the current combination expression is selected in that dialog tab.

Enter a user-defined note or select an entry from the list.

The aim of combining actions is to find the most unfavorable load arrangement for each location in the structural system. To that end, you can

• either determine all combinations that are mathematically possible
• or try to find logical relationships before combining the actions to reduce the number of possible combinations.

For example, a symmetrical two-hinged frame has the following load cases:

The load cases result in the following axial forces available in the columns:

Table 5.3 Tensile and compressive forces of columns

Load Case	Description	Acting	Axial Force Left Column	Axial Force Right Column
1	Self-weight	Permanent	Compression	Compression
2	Wind to the right	Alternative	Tension	Compression
3	Wind lifting	Alternative	Tension	Tension
4	Snow	Alternative	Compression	Compression

The following eight combinations are theoretically possible:

Table 5.3

CO1:	LC1 + LC2 + LC3 + LC4
CO2:	LC1
CO3:	LC1 + LC2
CO4:	LC1 + LC3
CO5:	LC1 + LC4
CO6:	LC1 + LC2 + LC3
CO7:	LC1 + LC3 + LC4
CO8:	LC1 + LC2 + LC4

These eight combinations can be reduced if we only want to find the arrangements with the extreme values of the columns' axial forces, for example. It is possible to create a group of load cases that provides only tensile and compressive forces for each column in consideration of the permanently acting LC1.

Table 5.4 Groups of load cases

Group	Left Column	Right Column
Tension forces	LC1, LC2, LC3	LC1, LC3
Compression forces	LC1, LC4	LC1, LC2, LC4

Thus, the result is no longer eight but just four combinations of load cases.

This reduction can be done in the Edit Load Cases and Combinations dialog box (see Figure 5.10) by

• selecting the Reduce number of load cases check box,
• selecting only the axial forces in the Examining Results dialog section of the Reduce - Number of Load Cases tab,
• and entering only the numbers of the column members in the Examining Results of Objects dialog section (see Figure 5.20).

After clicking the [Determine automatically] button, RFEM performs a short calculation. Then the table in the Considered Load Cases dialog section lists the four groups of load cases that are also shown in Table 5.4.

With this method, a linear result combination is created from the load cases. For each point, RFEM evaluates the extreme values and the involved load cases so that a max- and a min-combination of load cases is available in each case. These extreme combinations are then used for creating the load case combinations.

The load cases result in the following axial forces N in the columns:

Table 5.5 Tensile and compressive forces of columns

1	Self-weight	Permanent	Compression (−10 kN)	Compression (−10 kN)
2	Wind to the right	Alternative	Tension (5 kN)	Compression (−5 kN)
3	Wind lifting	Alternative	Tension (3 kN)	Tension (3 kN)
4	Snow	Alternative	Compression (−12 kN)	Compression (−12 kN)

RFEM creates the following temporary result combination: LC1/permanent + LC2 + LC3 + LC4

For the axial forces of the columns, the following extreme values result when superimposing:

Table 5.6 Groups of load cases

Group	Left Column	Right Column
Maximum N	−2 kN (LC1, LC2, LC3)	−7 kN (LC1, LC3)
Minimum N	−22 kN (LC1, LC4)	−27 kN (LC1, LC2, LC4)

Again, the result is no longer eight, but just four combinations of load cases.

Setting specifications in the Reduce - Examine Results tab is similar to Figure 5.20.