## Combinations of actions

Representative actions (Frep) are obtained by assembling suitable combinations of characteristic values (Fk), following the rules given in ENs 1990 and 1991. The representative value of a single generic action is given by:

Frep = VFt where ^ is a combination factor, less than or equal to 1.0.

The combination factor ^ is omitted for permanent actions, i.e. a representative permanent action (G,j) is equal to its characteristic value (Gkj). The total design permanent action (Gd) is then obtained from the sum of the representative values multiplied by their appropriate partial factors yg (see Section 2.13.1). Hence:

Gd = X (YG, j X Gk, j ) = X (YG,sup, jGk,sup, j ) + X (YG,inf, jGk,inf, j ) j j j where the subscripts sup and inf denote unfavourable ('superior') and favourable ('inferior') actions respectively.

In persistent and transient situations, the value of ^ is typically equal to 1.0 for the 'leading' variable action (Qk1), but is less than one (^ = < 1.0) for all 'accompanying' variable actions (Qki). The total design variable action (Qd) is then obtained from the sum of the representative values multiplied by their appropriate partial factors yq (see Section 2.13.1). Hence:

Qd = Yq ,1 X1.0 X Qk ,1 + X Yq ,1 X v0, X Qk ,i i>1

(Note that only unfavourable variable actions are considered - favourable variable actions are ignored.)

Hence the total design action Fd in persistent and transient design situations is given by:

Fd = X Yg ,jGk ,j + Yq ,1Qk ,1 +lYQ,^Qk j>1 i>1

Alternatively, EN 1990 allows Fd to be calculated as the larger of:

Fd = XYg ,jGk ,j + Yq ,1W0,1Qk ,1 +XYq ,^0,,Qk,, j i >1

Fd = ^X Yg ,sup, G ,sup,j +X Yg m, Gk M,j + YQ,1Qk ,1 +XYQ,,n,.Qk ,i j j i>1

where ^ is a reduction factor (a.k.a. 'distribution coefficient') applied to unfavourable permanent actions Gk j only.

An example may help to illustrate the use of these equations in practice. Imagine that the motorway gantry of Figure 2.7 is subject to imposed load

and wind in the vertical and horizontal directions, as shown in Figure 2.12 and summarized in the table below.

In the table, Combination 1 assumes that the imposed load is the leading variable action (and hence ^ = 1.0) and wind is accompanying (with = 0.6); Combination 2 assumes that wind is leading (^ = 1.0) and the imposed load is accompanying (^0 = 0.7). The design actions that result are given in the row labelled 'Total': Combination 1 gives a (slightly) higher vertical action, but Combination 2 a higher horizontal value. The gantry must be designed to withstand both combinations.

Combination of actions for persistent and transient design situations Action (type*) Fk (kN) yF Fd (kN)

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