Equivalent stabilizing force of members parabolically compressed by longitudinally variable axial force

The EN 1993-1-1 model of equivalent stabilizing force qd and Rd of bracings conservatively assumes that the braced member is compressed with a force constant along its length. This assumption is incorrect since the axial force distribution varies along the length of the braced member. As a result, the braced member generates equivalent stabilizing forces different from equivalent force qd and Rd acc. to EN 1993-1-1. This paper presents parametric studies of the equivalent stabilizing forces of the braced, compression top chord of roof trusses. The girder’s top chord is compressed parabolically by a variable axial force. The values of the axial compressive forces is: Nsupp in the support zone of truss and Nspan in the central zone of truss. Parametric analyses of the equivalent stabilizing force and the stress of the purlins and the bracings depending on axial forces Nsupp and Nspan in the braced member were carried out. The investigated problem is illustrated with exemplary calculations of the equivalent force in trusses.


Introduction
According to EN 1993-1-1 [1], an evaluation of the loading capacity of frame systems and bracings should take into account the forces due to random initial bow imperfections with amplitude e0 (Fig. 1a). In calculations the forces are replaced with equivalent stabilizing force qd1 and Rd1 (Fig. 1c) (2) where: NEd,maxthe axial force in the braced member, e0the maximum amplitude of the braced member imperfection, Lthe span of the braced member. Uniformly distributed imperfect span equivalent stabilizing force qd1 = const. acc. to (1) and support reactions Rd1 acc. to (2) were determined assuming that the braced member is compressed by axial force N1(x) = NEd,max, constant along its length (Fig. 1b). Frame columns satisfy this assumption since they are compressed by a longitudinally invariable force [2][3][4].
In the EN 1993-1-1 [1] computational model also (1) and (2) are used to analyse the stress of purlins and bracings caused by the actions of the bow curved laterally stiffened top flanges of roof girders. In this standard model, even though axial force N2(x) varies along the braced member (e.g. as in Fig. 2b), it was assumed that the latter is compressed by axial force N1(x) constant along its length (Fig. 1b).
It should be noted that in the case of the equivalent stabilizing force of the braced top flanges of roof girders the assumption N1(x) = const. is incorrect since it does not correspond to the actual longitudinally variable distribution of the axial force in the braced member (Fig.  1d). The axial force usually changes parabolically (e.g. as in Fig. 1e, 2c) or stepwise parabolically (as in Fig.  2b), and also its sign can change (compression and tension occur).
In the case of distribution of axial force as in Fig. 1e, the equivalent stabilizing force qd2(x) is also variable over the length of memberit changes longitudinally (parabolically) and changes sign (Fig. 1f). The equivalent stabilizing force qd2(x) differs fundamentally from the qd1(x) (compare Fig. 1c and Fig. 1f) which results in significant differences in the assessment of purlin safety and bracings in comparison to calculated according to EN 1993-1-1 [1].
As demonstrated in [5][6][7][8], this seemingly safe assumption about constant axial force N1(x) = const. can lead to the underrating of the stress of purlins and bracings. Figure 2a shows the loading diagrams of the transverse systems of steel industrial sheds in which the roof girder is pin jointed with the columns. Due to the overhead crane load (V, H) and wind loads (wp, wn) axial force Nsupp is transferred to the braced top flange of the roof girder. This force and the parabolically longitudinally variable axial force in the top flange generated by the roof girder's equivalent stabilizing force (p) add up. In the considered case, axial force N3(x) changes stepwise parabolically along the length of the braced top flange of the roof girder, as shown in Fig. 2b. In the support zone compressive force N3(0) = Nsupp, whereas in the central zone N3(0.5L) = Nspan = NEd,max. The consequence of the initial bow imperfection e0 of the braced upper chord of truss, in the roof plane, is the twist of the girder's principal plane (because the upper chord is curvilinear, and the lower chord is, for example, rectilinear). Hence not only the bow imperfection e0 of compression flange, but also an imperfection consisting in a twist of the truss's principal plane by angle 0(x) occurs. As a result of the action of vertical loads PEd in the upper joints on the truss twisted by angle 0(x) additional equivalent horizontal forces Hi(x) arise [5]. The forces are transmitted to the purlins and the transverse bracing whereby their stress increases. The bracing calculation model adopted in EN 1993-1-1 [1] does not take into account equivalent horizontal forces Hi(x) generated by the twist of the girder.
The aim of this paper is to identify the equivalent stabilizing forces of the braced top chord of the roof truss. The study takes into account the initial bow imperfection e0 of the braced top chord and the imperfection consisting in a twist of the roof girder's principal plane by angle 0(x). Moreover, compressive axial force N3(x) in the top chord of the truss is assumed to be longitudinally parabolically variable and to have a distribution as shown in Fig. 2b. The values of the axial forces in the support zone (Nsupp = αNEd,max) and in the central zone (Nspan = NEd,max) of the braced top flange of the girder depend on the load parameters (p, wp, wn, V, H) and the stiffness parameters of the transverse system (moment of inertia Ig and span L of the roof girder, moment of inertia Ic and height h of the column). As part of this study parametric analyses of the equivalent stabilizing forces and the stress of the purlins and the bracings depending on axial forces Nsupp and Nspan in the braced member were carried out.  Nspan = NEd,max and axial force in the support zone is Nsupp = αNEd,max.
The longitudinal variation of axial force N3(s) of the braced member was defined using a parametric parabolic function (Fig. 2c) described by the relation where: NEd,maxthe value of the extreme axial force at the midspan of the braced member, a dimensionless coefficient of the axial force at respectively the left and right end of the braced member, assuming values from the ‹0,1› interval (a "+" valuecompression), s = x/Lthe relative location (Fig. 2c) of the considered cross section along the length of the braced member, assuming values from the ‹0,1› interval. Using the adopted function of the variation of axial force N3(s) acc. to (3) one can analyse the equivalent stabilizing forces of braced members compressed by force N1(x) constant along the length (acc. to EC 1993-1-1 [1]) and by longitudinally variable axial forces N2(x) and N3(x) whose distributions are shown in Figs 1e, 2c.
The initial geometric bow imperfection of the braced member was assumed to have the form of a parabola [1], [4] where: e0the maximum amplitude of imperfection of the braced member (acc. to [1], e0 = L/500). General formulas for span equivalent stabilizing force q(s) and support reactions Rd of a member loaded by longitudinally variable axial force N(s) with any distribution NEd(0)  NEd (0.5)  NEd (1) were derived in [8].
If axial force N3(s) varies longitudinally as in Fig   Schemes 2-11 concern the parabolically variable distributions of axial force N3(s) in the braced members, shown in Fig. 2c. They take into account the variation in the values of the compressive forces at the support: N3(0) = Nsupp (variable parameter ) and the constant value of the compressive force at the midspan of the braced member: N3(0.5) = Nspan = NEd,max. Scheme 11 ( = 0) corresponds to the distribution of axial force N3(s) in the braced top flange of a pin-supported roof girder. The latter is uniformly loaded and the axial force at the support in the braced top flange is N3(0) = Nsupp = 0 (Fig. 1d, e).
The considered schemes of the longitudinal variation of axial force N3(s) in the braced members as a function of parameter  are shown in Fig. 3a. The distributions of span equivalent stabilizing force qd3(s) resulting from the stress of the braced member consistently with the considered schemes of axial force N3(s) are shown in Fig. 3b. Figure 3c shows support reactions Rd3 of the compressed members versus , corresponding to the considered diagrams of axial forces N3(s).
An analysis of the parametric calculations indicates that the parabolic distribution of span equivalent stabilizing force qd3(s) corresponds to the parabolic distribution of axial force N3(s) in the braced member (compare Fig. 3a and 3b). Span equivalent stabilizing force qd3(s) and support reactions Rd3 in the analysed diagrams of the longitudinal stress of the brace members are a linear function of dimensionless axial force parameter .
The equivalent stabilizing forces of the braced member consists of span force qd3(s) (Fig. 3b) and support reactions Rd3 (Fig. 3c). Equivalent stabilizing force qd3(s) and support reactions Rd3 together form a self-equilibrating system. Whereas in the case of the equivalent force of the member acc. to scheme 11 ( = 0, the braced flange of the pin-supported roof girder -Figs 2 a, c) there is no support reaction (Rd3 = 0, see Fig. 3c). Then span equivalent stabilizing force qd3(s) is selfequilibrating along the length of the braced member.
Span equivalent stabilizing force qd3(s) (Fig. 3b) is variable (nonuniform) along the length of the braced member and when  < 0.65, it is also longitudinally sign-variable.
A comparison of equivalent stabilizing forces qd1 and qd3(s) clearly indicates that their estimates acc. to EN 1993-1-1 [1] differ fundamentally from the ones acc. to the adopted model. The differences are both qualitative and quantitative. This has a significant bearing on the stress of both the purlins and the bracings.

Parametric analysis of stress of purlins and bracings due to initial bow imperfection e0 of braced top chord of truss
A scheme of the analysed bracing is shown in Fig. 4a The axial force at the midspan of the girder's braced top chord is Nspan = NEd,max = 163.64 kN, while the axial force at its support is Nsupp = αNEd,max (Fig. 4c).
The diagonals of the bracing are assumed to be thin beams which do not transfer compressive forces (tie rods with diameter 20mm).

Fig. 4. Schemes of: a) roof bracing, b) truss, c) distribution of axial force in braced chord of truss
The axial forces in the bracing's purlins and diagonals caused by the equivalent stabilizing force generated by one stabilized top flange of the girder were analysed. Eleven schemes of the variation of force N3(s) in the braced top flange of the girder were considered. Parameters α of the variability of axial forces N3(s) are given in Table 1. The values of the axial forces in the purlins (Fd,i) and in the bracing's diagonals (Sd,i) in schemes 1-11 of the stress in the girder's braced flange are presented in Table 2. The top part of Table 2 shows parameter α for schemes 1-11. In Table 2 the numbers of the roof structure members are given in column 1 (the numbering of the members is shown in Fig. 4a) while the numbers of the columns are given in row 4.
The following conclusions can be drawn from the parametric analyses: 1. If the real parabolic distribution of the axial force in the braced member is adopted in the computational model, the equivalent stabilizing force and the axial forces in the bracing's purlins and members differ both qualitatively and quantitatively from the ones yielded by the model recommended by EN 1993-1-1 [1]. 2. In comparison with the evaluation of the stress acc.
to the model recommended by EN 1993-1-1 [1] (column 2 in Table 2, equivalent stabilizing force qd1 as in scheme 1 at  = 1), if the longitudinally parabolically variable axial force in the braced member of the uniformly loaded pin-supported girder (column 12 in Table 2, equivalent stabilizing force qd3 for scheme 11 at  = 0) is taken into account in the analysis, this follows: • an 84% reduction in the axial force in the eaves purlin (in element 1), • a 9.2% increase in the axial force in the beforeeaves purlin (in element 2), • in the model acc. to scheme 1 (qd1) the strongest axial force occurs in the before-eaves purlin (in element 1, S1 = 1.200 kN), • in the model acc. to sheme 11 (qd3) the strongest axial force occurs in the before-eaves purlin (in element 2, S2 = 0.238 kN), • the axial force in the purlin at the midspan of the bracing (in element 7) is the same for both the computational models, • an 8.7% (element 6) 75.7% (element 4) reduction in the axial forces in the other purlins (in elements 3-6), • a 0.9% (element 13) 84% (element 8) reduction in the axial forces in all the diagonals of the bracing, • when equivalent stabilizing force qd1 is assumed, the maximum axial force reliable for dimensioning the bracing occurs in eaves diagonal 8 (S8 = 1.265 kN), • assuming qd3, the maximum axial force reliable for dimensioning the bracing occurs in diagonal 10 (S10 = 0.531 kN), • the axial force in diagonal S10 (generated by equivalent force qd3) is by 58% weaker than the axial force in diagonal S8 (generated by equivalent force qd1). 3. For all the shemes 2-11 as the support axial force decreases Ssupp ( < 1), the axial forces in all the purlins decrease (in comparison with the ones calculated acc. to [1]). In the central zone this decrease is only slight. Also the axial force values reliable for dimensioning purlins decrease (relative to the ones calculated acc. to [1]. As a result, the axial forces in the diagonals decrease (relative to the ones calculated acc. to [1]) and the axial force values reliable for dimensioning the bracing also decrease.  1.200 -1.099 -0.998 -0.898 -0.797 -0.696 -0.595 -0.494 -0.393 -0 In the calculations acc. to EN 1993-1-1 [1] braced top chords are treated as initial bow curved members "isolated" from the truss (not connected via lattice work with the bottom chords). This computational model does not reflect the behaviour and stress of a real truss, i.e. the twist of its principal plane by angle 0 due to initial bow imperfection e0 of the top chord (Fig. 5) and also the stiffness parameters of its members and joints. Figure 5a shows initial bow imperfection e0 of the truss's top chord in the plane of the roof. Since the top chord is curvilinear while to bottom chord is rectilinear (Fig. 5c) the bow imperfection results in the twist of the principal plane of the truss. Hence besides the initial bow imperfection y(s) of top chord, there is an imperfection consisting in the twist of the truss's principal plane by angle 0(s). As a result of the action of vertical loads PEd,i in top joints i on the truss twisted by angle 0(si) equivalent horizontal forces Hi arise. They are transferred to the purlins and to the bracing (Fig. 5b), increasing their stress.
Assuming that vertical forces PEd are identical and that only the top chord is bow curved (Fig. 5c) acc. to (4), equivalent horizontal forces Hi,m in joint i originating the external loads (e.g. wind load W), obtained from the 1st order analysis (if the 2nd order theory is used one can assume δq,H,w = 0), hithe construction depth of the truss in joint i. The distribution of equivalent horizontal forces Hi due to the twist of the principal plane of the truss varies along the length of the latter. In the considered case, the distribution is parabolic consistently with the adopted bow imperfection of the axis of the braced member (y(s)). The strongest equivalent horizontal force Hi occur at the midspan of the truss when y(0.5) = e0. 5 Analysis of total equivalent stabilizing forces due to initial bow imperfection e0 of braced flange of girder and twist of its principal plane by angle 0 Initial bow imperfection e0 of the braced flange and the twist of the principal plane of the girder by angle 0 together generate equivalent forces transferred to the purlins and the bracing. Therefore when evaluating the equivalent stabilizing forces one should add up the force due to the braced flange bow imperfection (qd3 and Rd3) and to the twist of the girder (Hi).
In order to quantitatively evaluate the total imperfect effect due to initial bow imperfection e0 and to the twist of the principal plane of the truss by angle 0 on the stress of the purlins calculations were done for the truss presented in Fig. 4b. The truss loaded only with PEd = 10 kN and NEd,s = 0, was analysed.
The axial force in the central zone of the span of the braced top chord of truss is Nspan = NEd,max = 163.64 kN, and axial force in its support zone is Nsupp = 49.42 kN; factor α = 0.151. The results of the calculations are presented in Table 3. The latter contains the values of the axial forces in the purlins due to the equivalent stabilizing forces generated by one braced truss. The forces in the purlins due to respectively equivalent force qd3 (for α = 0.151) and the equivalent horizontal forces Hi (determined only for initial bow imperfection of top chord - Fig. 5c) are specified in columns 3 and 4 in Table 3. Notation of axial forces: "+"compression In comparison with equivalent stabilizing force qd1 (column 1 in Table 3) acc. to EN 1993-1-1 [1], the total equivalent forces: qd3 + Hi (column 5 in Table 3) causes: • an increase in the axial forces in the midspan purlins of the braced truss chord, i.e. by 11% in purlin 4, 60.1% in purlin 5, 89.4% in purlin 6 and 100% in purlin 7 (the middle one); • a reduction in the axial forces in the support zone of the braced top chord of the truss, i.e. by 71.2% in purlin 1 (the eaves purlin), 53.7% in purlin 2 and 58.3% in purlin 3.
Additional calculations of the bracings loaded with forces qd3 + Hi showed a reduction in the forces in the diagonals in the support zone and an increase in the forces in the diagonals in the central zone of the bracing (from 18.4% to 48.7%) in comparison to EN 1993-1-1 [1] model.

Conclusions
The analyses have clearly shown that the computational model of equivalent stabilizing force qd1 and Rd1 acc. to EN 1993-1-1 [1] does not reflect the behaviour and stress of the real structure. It incorrectly estimates the axial forces in the purlins and the bracing members, which may lead to a wrong assessment of their reliability. This is due to the fact that the model does not take into account either the real longitudinally variable (and often sign-variable) [5][6][7][8] parabolic distribution of the axial force in the bow curved (e0) braced flange of the girder or the twist of the girder's principal plane by angle 0.
If the distributions of the axial force in the braced members are longitudinally variable parabolic (as in schemes 2-11, sections 2 and 3; α < 1.0), their span equivalent stabilizing force qd3 are nonuniform and signvariable at α < 0.65. They can be much higher than equivalent force qd1 acc. to [1]. For example, in the case of scheme 11 ( = 0) the equivalent stabilizing force in the support zone of the braced member: qd3(0) = qd3(1) = 16q (where q = NEd,maxe0L -2 ) is twice as high as the equivalent force at midspan: qd3(0.5) = 8q (Fig. 3b).
If the actual longitudinally variable parabolic distribution of the axial force in the braced member is assumed, this results in major changes, in comparison with the model acc. to EC 1993-1-1 [1], in the schemes and parameters of the equivalent stabilizing forces. A comparison of equivalent stabilizing forces qd1 and qd3(s) and support reactions Rd1 and Rd3 clearly shows fundamental differences between the estimates of the equivalent forces. As a result, the purlins and the bracings are stressed differently. As compared with the equivalent forces calculated acc. to EC 1993-1-1 [1], equivalent forces qd3(s) and Rd3 can cause not only an increase in the axial forces, but also a different distribution of the latter in the principal purlins. The parametric analysis of the stress of the purlins and the bracing diagonals carried out in sect. 3 clearly shows that by disregarding the real distribution of the axial force in the braced member one will incorrectly assess of their reliability.
The "imperfect" model in EN 1993-1-1 [1] is limited to an analysis of the effects of initial bow imperfection e0 of the braced top flange of the girder. However, the consequence of the top flange bow imperfection is the twist of girder's principal plane by angle 0, which generates equivalent horizontal forces Hi. The latter are transferred to the purlins and to the bracing, increasing their stress. The computational model in EN 1993-1-1 [1] does not take into account equivalent horizontal forces Hi caused by the twist of the truss.
When determining the total equivalent stabilizing force (resulting of e0 i 0) one can use the nonlinear geometric analysis of the 3D computational beam model, taking into account the real stiffnesses of the members and joints of the considered structure. The proposed methods of evaluating equivalent stabilizing forces which take into account the real distribution of the axial force in the braced member with arch initial bow imperfection e0 and the twist of girder's principal plane by angle 0 enable a more precise analysis of the stress of purlins and bracings. The quantitative and qualitative differences between the proposed computational models and the evaluation acc. to [1] are considerable. Therefore one should consider introducing (after additional numerical analyses) proper corrections pertaining to the investigated problem in the revised EN 1993-1-1 [1].