Analysis of the global and local imperfection of structural members and frames

Stresses of a structure are determined with a first or a second order analysis. The choice of the method is guided by the potential influence of the structure’s deformation. In general, considering their low rigidity with regard to those of buildings, scaffolding and shoring structures quickly reach buckling failure. Imperfections, such as structural defects or residual stresses, generate significant second order effects which have to be taken into account. The main challenge is to define these imperfections and to include them appropriately in the calculations. The present study suggests a new approach to define all the structure's imperfections as a unique imperfection, based on the shape of elastic critical buckling mode of the structure. This study proposes a method allowing to determine the equation of the elastic critical buckling mode from the eigenvectors of the second order analysis of the structure. Subsequently, a comparative study of bending moments of different structures calculated according to current Eurocode 3 or 9 methods or according to the new method is performed. The obtained results prove the performance of the proposed method.


Introduction
Thin structures are very prone to buckling failure mainly caused by compression loads. Structures must be analysed using a model and assumptions that reflect their behaviour with an appropriate accuracy in relation to limit states. Stresses in a structure are determined from a first or a second order analysis, according to the potential influence of its deformation (Girão Coelho et al., 2013). Usually, it can be considered that scaffolding and shoring structures quickly reach their buckling failure thresholds. Our investigation aims to highlight a calculation method that is both safe and economical. Application of a compressive force on a member with imperfections generates a bending moment, called second order moment, which leads to greater lateral deformations of this member (Elishakoff, 1978). These deformations amplify the compressive axial force and the lever arm (Shayan et al., 2014). The bending moments cause parasitic effects which limit the load capacity of the member (Frey, 2014), explaining the need to take into account accurately imperfections of members in the structure analysis, either in including imperfections in the global analysis, or, in using appropriate criteria taken into account the unmodelled imperfections (Maquoi et al., 2001& TC8, 2006. Due to the fact that imperfections of construction elements are within the normal tolerances of manufacturing lines, they are generally not visible and cannot be quantified precisely in advance. Several approaches have been conducted to study the imperfections that can reduce the structure resistance (Taheri-Behrooz & Omidi, 2018). Kala (2005) proposed a study to determine the influence of the imperfections on the resistance of members subjected to axial compression. Taheri-Behrooz et al. (2012, 2017& 2019 analysed the buckling behaviour of composite cylinders with cutout. They developed finite element analysis, with first or second order approaches, enabling to predict on the buckling behaviour of thin cylinders, the effects of initial imperfections, the effects of the size and the direction of a cutout and the effects of combined initial imperfections and cutout. Two methods are suggested by Eurocode 3 (NF EN 1993-1-1, 2005 to include the structural defects in the model of the structure. The first is the conventional method, based on the definition of a local imperfection and a global imperfection of the structure. The second method is based on the definition of a unique global and local imperfection whose appearance is similar to predominant elastic critical buckling mode of the structure. Analytical researches have been carried out in the recent years concerning the including of the imperfections in the model by means of a unique global and local imperfection. Gonçalves and Camotim (2005) proposed an approach enabling a better understanding of the concepts involved in Eurocode 3 provisions. Agüero et a l. (2015) also proposed a method to define the unique imperfection to flexural and/or torsional buckling steel structures. Štujberová, in their articles (2013a &2013b) on the unique local and global imperfection of a structure subjected to elastic buckling, defined a tool to determine the shape of the buckling mode as an initial unique imperfection. Shayan et al. (2014) studied the number and magnitudes of eigenmodes to use in the structural analyses to determi ne the initial geometric imperfections of steel frames. Hassan et al. (2018) proposed in their investigation a comparison of three methods to define geometrical imperfections of structural steel hollow sections under cyclic axial loading. The present study suggests a new approach to define imperfections as a unique imperfection, using the eigenvectors of the second order analysis of the structure. These eigenvectors enable to put in equation the elastic critical buckling mode of the structure. This alternative method changes the approach to Eurocode 3. This present article also proposes a comparative study of bending moments of different structures calculated according to cu rrent Eurocode 3 or 9 methods or according to the new proposed approach.

Local and global imperfections of structures (conventional Eurocode 3 method)
The conventional method of taking into account imperfections according to Eurocode 3 is translated by the definition of a global imperfection and local (see Figure 1). These imperfections are represented respectively by: a global initial default balance of the structure; and a local deformation in arc. In a structure, the imperfections can be introduced either by calculating the coordinates of the nodes of the structure with imperfections or by means of equivalent loads applied on the structure without imperfections. → Global initial sway imperfections  : where where m is the number of columns in a row including only those columns which carry a vertical load not less than 50% of the average value of the column in the vertical plane considered → Local imperfections of members 0 eL : e0 is the bow imperfection for simple columns. Table 1 gives the ratio e0/l conventionally defined in the European buckling curves (Maquoi & Rondal, 1978) depending on the type of analysis conducted (elastic or plastic). Table 1. Design value of the initial local bow imperfection e0/l for members Curve a0 corresponds to hot-rolled I-sections with thin flange (at most 40mm) with steel S460, and hot-rolled hollow sections with steel S460. Curve a represents quasi-perfect shapes such as: hot-rolled I-sections (h/b>1.2) with thin flange (tf < 40mm) if buckling is about major axis, hot-rolled hollow sections (with other steel than S460)… Curve b corresponds to shapes with medium imperfections such as: most welded box sections, hot-rolled I-sections (h/b>1.2) with thin flange (tf < 40mm) whose buckling is about minor axis, welded I-sections with thin flanges (tf < 40mm) buckling about major axis, L-sections… Curve c represents profiles with a lot of imperfections such as: cold-rolled hollow sections, U-sections, T-sections, solid sections… Curve d corresponds to profiles with maximum imperfections such as: hot-rolled I-sections (h/b<1.2) with very thick flange (tf > 100mm), welded I-sections with thick flange (tf > 40mm)… Table 6.2 of Eurocode 3 enables to select the buckling curve according to the type of cross-section. The main fault of these conventional methods is that they rely only on the geometry of the structure. Moreover, it is important to consider the most unfavourable form of these imperfections. Several combinations of global and local imperfections must be considered to find the worst, which is one of the disadvantages of this method.

Unique local and global imperfection of structures
In terms of structural mechanics, the elastic critical buckling mode (ηcr) of any given frame allows anticipating its local and global behaviour. Indeed, the buckling mode expresses the type of deformation that may occur to the right of each node of the structure under the effect of a load. This being assumed, the prior existence of a structural imperfection in the direction of future displacements means that the structure is subjected to the most unfavourable conditions (Baguet, 2001).
In the case where imperfections must be taken into account, a uniq ue global and local imperfection similar to the most representative elastic critical buckling mode will be defined. It is advisable to assign an amplitude to this shape that is also representative of the desired level of imperfection. This magnitude can be determined by three methods.

Alternative method of Eurocode 3
This method is based on the determination of a unique imperfection (η init) which allows the local superposition of the critical distortion of the structure that houses the critical cross -section and a deformed reference member whose geometrical characteristics and critical load are the same as those of the critical cross -section of the structure. The alternative method of Eurocode 3 is based on the following requirement: the curvature in t he critical crosssection of the structure with the initial imperfection, based on the elastic critical buckling mode ηcr, should be equal to the maximum curvature of the reference member. The factor for amplifying the maximum curvature of the elastic cri tical buckling mode is noted Cnor. This allows defining the unique global and local imperfection of the whole structure. Indeed, for each node of the structure, the unique imperfection is the product of the amplification factor and the buckling mode (η cr) at each considered node. The normalization factor is given by: maximum curvature of the reference member maximum curvature of the elastic critical buckling mode of the structure    (2) where: e " is the maximum curvature of the deformed reference member whose geometrical characteristics and critical load are the same as those of the critical cross-section of the structure; and, η " is the maximum curvature of the dominant elastic critical buckling mode of the structure. Therefore, it is necessary to determine the maximum curvature of the elastic critical buckling mode of the structure. The calculation software only gives displacements and rotations at different nodes of the structure. Knowing the displacements and rotations of the different nodes of a member, an equation is written to reconstruct the shape of elastic critical buckling mode of the structure and subsequently to determine the curvatures at all the points of the member.

Eurocode 9 method
This method is a derivative of the method developed in Eurocode 3. Its purpose is also to define a unique global and local imperfection (ηinit) which is broadly modeled on the elastic critical buckling mode and retained locally on the deformation of the reference member. After defining the reference member, it is assumed that at the maximum bending point of the structure member, the curvature of the shape of the elastic critical buckling mode must be equal to the maximum curvature of the reference member. As the reference member is modeled with the critical load and geometric characteristics of the critical cross-section of the structure, the maximum bending moment at the critical cross -section of the structure must be equal to the maximum bending moment obtained in the reference member. Thus, Eurocode 9 (NF EN 1999-1-1/A1, 2010) approach consists in amplifying the maximum second order moment of the most heavy-loaded member of the structure so that it is equal to the maximum second order moment obtaine d in the reference member. The amplification factor denoted Cnor is that which will allow us to define the unique deformation of the whole structure. Finally, the unique imperfection for each node of the structure is the product of the amplification factor by the buckling deformation mode (ηcr). The normalizing factor in the literature is given by:

Origin of similarities in the different definition of the unique imperfection
As mentioned previously, the elastic critical buckling mode gives us the type of lateral displacements that may occur at each node of the structure under the effect of compression loads. The unique imperfection is defined according to the shape of this elastic critical buckling mode amplified by a Cnor factor enabling to achieve the desired level of imperfection called ηinit: Moreover, the desired level of imperfection is based on the behaviour or appearance of a reference member whose geometrical characteristics and critical load are the same as those of the critical cross -section of the structure. The Cnor amplification factor can be defined thanks to the comparison between: the maximum curvature of the reference bar and that of the critical cross-section of the structure: Expressions of the bending moment in a member with an initial deformation In this part, we sought to express the maximum second order bending moment of a simply supported member ( see Figure 2) as a function of the maximum initial curvature The critical load of this member is defined by the following expression:

Expression of the unique global and local imperfection ηinit
Now we will attempt to find the expression of the unique imperfection given in the alternative method of Eurocode 3 and Eurocode 9. This unique global and local imperfection has the same shape as the elastic critical buckling mode. Thus, it possible to write the following ratio: As mentioned previously, the alternative method of Eurocode 3 is based on the following requirement: the curvature in the critical cross-section m of the structure with the initial imperfection, based on the elastic critical buckling mode, should be equal to the maximum curvature of the reference member.

Principle
Alternative method of Eurocode 3 or 9 consists in comparing the maximum bending moment of the most heavyloaded section of the structure to the maximum bending moment of the reference member. We proposed studying a pinned-base frame, of 4m high and 4m wide, with a cold formed tubular section 60x6mm, whose cross-sectional characteristics are: A=10.18x10 -4 m², I=37.56x10 -8 m 4 , Wel=12.52x10 -6 m 3 . The material of the structure is S320: fy=320MPa. A virtual member is defined as a simply supported member with a height equal to the buckling length of the critical cross-section of the studied structure and with an initial imperfection as same shape as the shape of the most relevant elastic critical buckling mode. The virtual member has the same behaviour as the element of the structure when it is subjected to a compressive load equal to the critical load of the critical cross -section of the structure. It was observed, that when the length of the virtual member is defined as being the length between two success ive inflection points of the shape of the elastic critical buckling mode of the structure (see Figure 3), the maximum bending point of the structure does not always correspond to the point where the derivative of the curve is equal to zero.

Figure 3. Construction of the virtual member
Indeed, in the case of a pinned-base frame the maximum bending moment, and so the maximum curvature, is obtained in the cross-section located at Z = 4.00m. However in the reference member the maximum bending moment and the maximum curvature are obtained in the mid-height cross-section, that is to say at Z = 4.65m. The calculation of the amplification factor according to Eurocode 3 and 9 approaches is performed by comparing two points that are not located at the same position. There are other cases of structures where the gap these two points is expressed even more strongly. This gap can lead to an error in the calculation of the amplification factor and so in the calculation of the initial imperfection of the structure. To overcome this problem we propose a new approach consists in establishing a mathematical equation of the shape of the elastic critical buckling mode of the structure. The sine term of the equation of the shape of the critical buckling mode of the structure allows us to mathematically reconstruct it and set its inflection points. These inflection points allow building a virtual member from which the "imaginary" maximum deformation and the "imaginary" maximum curvature of the structure are deduced. This maximum curvature (or maximum deformation) will be compared to the maximum curvature (or maximum deformation) of the reference member. The factor C nor will then be used to amplify the deformation of buckling mode (ηcr) and then to obtain the desired unique imperfection. The equation of the shape of the predominant elastic critical buckling mode ηcr of the structure is: z  is used to determine the "imaginary" maximum curvature and the "imaginary" maximum deformation of the most heavy-loaded section of the structure. As the latter is located halfway up from the buckling length, it is compared to the maximum curvature or the maximum deformation of the reference member, respectively.

Application to the pinned-base frame
The new approach is applied to the pinned-base frame, whose cross-sectional characteristics are given in part 3.1., in order to assess its accuracy. Results of other type of structures are given in part 4. The portal frame is loaded so as to obtain a critical coefficient αcr of 2.
To establish the equation of the shape of the elastic critical buckling mode, it's necessary to use two eigenvectors. The displacements and rotations of the both following nodes are used.

Table 2. Eigenvectors of the pinned-base frame
From the second order analysis of the structure, the buckling length is determined as equal to 9.312 m. From the system of Eq. (29), we determine the value of the unknowns: Table 3. Values of the unknowns Then, thanks to Eq. (31), we determine the position of the maximum curvature point. We obtain: max z 4.656m  . From this value and Eq. (30), we determine the value of the "imaginary" maximum curvature. We obtain: On the previous figure, the shape of the elastic critical buckling mode of the structure and the theoretical deformation of the virtual member, obtained from mathematical Eqs. (28)-(31), are superimposed. This overlap enables us to check the physical impact of our approach and allows to identify the critical section which may optionally be real or fictitious, that is to say outside the structure, like the example of the pinned -base structure. Moreover, it also allows comparing the reference member and virtual member. On the other hand, we determine the maximum curvature of the reference member. This maximum curvature is determined from Eq. (18), with: From this value of the maximum curvature of the reference member and the imaginary maximum curvature of the structure, we determine the factor Cnor from Eq. (5). We obtain: 2 nor C 1.864 10   . Thus, this factor is determined from the comparison of two points located in the same cross-section in the both members. Thanks to the factor Cnor and the shape of the elastic critical buckling mode, we determine the coordinates of the initial imperfectio n at each nodes of the structure from Eq. (4). The coordinates obtained are given in the Table 4.

Table 4
Determination of the coordinates of the nodes of the structure with the initial imperfection

Presentation of the comparative study
A comparative study of the different methods is carried out in order to appreciate the impact of our new method. It concerns the previous pinned-base frame as well as three other structures (Figure 6(a)-(e)): a fixed-base frame of 4m high and 4m wide; a shoring tower of 8m high and 3m wide; and a façade scaffold of 8m high and 1.5m wide.

Figure 6. Studied structures
The characteristics of the different cross-sections are given in the Table 5. Loads on the structure are applied vertically at the top of each column such that the critical coefficient is equal to 1.5, 2 or 5.

Table 5. Cross-sectional characteristics of all structures
The purpose of the study is to compare the different methods of definition the initial imperfectio ns to the new approach, using the virtual member. Five approaches will be compared: -Conventional method of Eurocode 3, a global imperfection and a local, by calculating the coordinates of the nodes of the structure. -Conventional method of Eurocode 3, a global imperfection and a local, by using equivalent loads -Alternative methods of Eurocode 3 and 9, a unique global and local imperfection, by calculating the amplification factor Cnor by means of the curvatures -Alternative methods of Eurocode 3 and 9, a unique global and local imperfection, by calculating the amplification factor Cnor by means of the bending moments -New approach by using the virtual member in the determining of the proportional factor Cnor. For that we must compare the bending moments obtained in each structure with the different cases of definition the initial imperfection. Calculations were performed using Autodesk Robot Structural Analysis.  The results obtained with conventional methods of Eurocode 3 are underestimated compared to the results obtained with the new approach. The error with conventional methods can be up to 25%. It should also be noted that the calculation point of alternative methods of Eurocode 3 and 9 is not at the maximum point of curvature of the virtual member. Thus, there is also an error with these methods up to 6%.  Table 7. Results of the fixed-base frame 4x4m

Fixed
The results obtained with conventional methods of Eurocode 3 are underestimated compared to the results obtained with the new approach, except in the case of αcr = 5, where the results are overestimated. The error with conventional methods can be up to 40%. It is important to note that only in the case of the fixed-base frame the calculation point of alternative methods of Eurocode 3 and 9 is the same that the maximum point of curvature of the virtual member. This explains the small error observed with the results obtained from alternative Eurocode 3 and 9 method with the bending moments. On the other hand, as far as alternative Eurocode 3 and 9 method with curvatures is concerned, the error can go up to 18%. Indeed, no way to calculate the curvature is indicated in Eurocode 3 or 9, so depending on the type of equation used (polynomial, sinusoidal ...), the error can be more or less important.  IPE   Table 8. Results of the pinned-base frame 5x5m in IPE100

Pinned
The results obtained with conventional methods of Eurocode 3 are underestimated compared to the results obtained with the new approach. The error with conventional methods can be up to 13%. It should also be noted, one more time, that the calculation point of alternative methods of Eurocode 3 and 9 is not at the maximum point of curvature of the virtual member. Thus, the error made with these methods can go, in these cases, up to 13%. Figure 6(d)) Figure 9. Calculation points of the shoring tower 8x3m Table 9. Results of the shoring tower 8x3m

Shoring tower 8x3m (see
In the case of the shoring tower, the results obtained with conventional methods of Eurocode 3 are overestimated compared to the results obtained with the new approach. The error with conventional methods can be up to 50%. The calculation point of alternative methods of Eurocode 3 and 9 is close to the maximum point of curvature of the virtual member, which explains the small errors observed with alternative Eurocode 3 and 9 methods.  Table 10. Results of the façade scaffold 8x0.7m The results obtained with conventional methods of Eurocode 3 overestimated compared to the results obtained with the new approach. The error with conventional methods can be up to 20%. It should also be noted, one more time, that the calculation point of alternative methods of Eurocode 3 and 9 is not at the maximum point of curvature of the virtual member. Thus, the error made with these methods can go, in the case of the façade scaffold, up to 5%.

Conclusion of the numerical examples
Through these results, it can be observed that depending on whether the first buckling mode of the structure is in fixed or sway mode, the classical Eurocode 3 method compared to methods derived from the unique imperfection is safe in the first case and underestimated in the other cases (Tables 6-10). For structures whose the first buckling mode is a sway mode, the definition of the unique imperfection provides safe results almost identical to those of the current alternative Eurocode 3 method when the buckling length is close to the actual length of the member. However for structures whose the buckling length is superior to the actual length of the member, the determination of the amplification factor according to the alternative method of Eurocode 3 and 9 may lead to errors ranging from -18% to +5% in the present cases but can lead to larger errors in other cases. During the application of each method, it can also be seen that those based on the definition of a single defect follow the mechanical behaviour of the structure more closely. They not only take into account the geometrical aspects, but also the boundary conditions of the bars, the intrinsic mechanical properties and the load level of the structure. As for the current method of applying local and global imperfection, it leads to ambiguity when taking into account local imperfection in the bars. Moreover, this method is mainly based on geometry and does not take account of the specific properties of the bars.

Conclusion
Our study focused on the development of a method that takes into account structural imperfections in the calculation of scaffolding and shoring structures, since the latter have low critical loads and critical coefficients which are very vulnerable to second order effects. The objective was to formulate a calculation method that is both safe and economical. The conventional methods of Eurocode 3, with a global and a local imperfection are based only on the geometry of the structure. It was found that the method based on a unique imperfection has the advantage to be simple and clear, easy to apply and derived from realistic mechanical reasoning. However, in Eurocode 3 and 9, the way to determine the maximum curvature of the shape of the elastic critical buckling mode is not given explicitly. The new approach proposed in this article aims to propose a method enabling the determination of this maximum curvature. Following the structure concerned this maximum curvature can be real or imaginary. In all cases, thi s point of maximum curvature in the structure is the same as the point of maximum curvature in the reference member. This method makes it possible to change the approach to Eurocode 3.