Alternative approach to the buckling phenomenon by means of a second order incremental analysis

This article addresses the problem of determining the solicitation and deformation of beams with geometric imperfection, also called real beams under a compression action. This calculation is performed by applying the Finite Transfer Method numerical procedure under first-order effects with the entire compression action applied instantaneously and applying the action gradually under second-order effects. The results obtained by this procedure for real sinusoidal or parabolic beams are presented and compared. To verify the potential of the numerical procedure, the first and second-order effects of a beam with variable section are presented. New analytical formulations of the bending moment and the transverse deformation in the beam with sinusoidal imperfection subjected to compression are also obtained, under first and second-order analysis. The maximum failure load of the beams is determined based on their initial deformation. The results of solicitation and deformation of the real beam under compression are compared, applying the analytical expressions obtained and the numerical procedure cited. The beams under study are profiles with different geometric characteristics, which shows that it is possible to obtain maximum failure load results by varying the relationships between lengths, areas and slenderness. The increase in second-order bending moments causes the failure that originates in the beam, making it clear that this approach reproduces the buckling phenomenon. The article demonstrates that through the Finite Transfer Method the calculation of first and second-order effects can be addressed in beams of any type of directrix and of constant or variable section.

to the previous one is called second-order solicitation 2 .If the compression load increases linearly, the increase in first-order solicitation is also linear.The same does not happen with the solicitation of the second-order.If the bending generated by transverse displacements is considered, as the load increases, the effects increase more rapidly 20,21 .
When checking the compressed beam using first-order analysis, there is no elastic instability.In the secondorder analysis, the bending effect produced by the transverse displacement during the application of the compression load is variable and increasing.This makes the superposition principle invalid.Therefore, an elastic instability is generated.
The imperfection of the directrix shape can be dealt with by assimilating the actual beam to a curved beam.By applying the first-order analysis on the curved beam, all kinds of effects are obtained, both compression and bending solicitations, as well as longitudinal and transverse displacements.
This article deals with the calculation of the real beam assimilated to the curved beam under second-order conditions.To do this, the load is divided into increments first.Each load increment is applied, and the solicitations and deformations are obtained.Then, the shape of the directrix is modified by adding the displacements obtained, and the next load increment is applied again.It is a successive process of iterations until exhausting the load increments to be applied.For this iterative calculation, a numerical procedure of boundary conditions has been used [22][23][24][25][26] .
By solving the differential equation of the elastica posed by Euler, a sinusoidal function 27 is obtained, from which the critical load is deduced.Under this approach, by assimilating the calculation of a compressed beam to that of a curved beam with a sinusoidal directrix, under second-order conditions, it has been verified that there is no elastic instability.
An alternative approach is presented to address this same problem, which reinterprets the theory that is usually used in the study of elastic instability due to buckling.Unlike Euler's formulation, in which the deformation generates solicitation, we start from a differential equation of the elastica with sinusoidal strain.That is, the bending moment is generated solely by the initial imperfection.This is intended to introduce the real directrix of the beam into the behaviour model.
The loads are applied incrementally, choosing increments so small that we can assume a linear behaviour in each increase in load.
The deformed configuration obtained for each load increase is added to the starting geometry for the next calculation iteration.
The second-order analysis is applied by solving a succession of first-order analysis of a beam whose geometry changes with each load increment with respect to the previous ones.
This alternative approach is specified in a final analytical expression of the deformed directrix of the beam, under the effect of second-order, once the application of the entire load after the iterations is exhausted.
The results obtained by means of this analytical expression derived from the revision of Euler's approach are compared with those obtained numerically by calculating the real beam assimilated to the curved beam under second-order conditions.
With the aim of highlighting the research presented, as a practical case, the structural behaviour of a variable section beam is analysed.

Analysis of the beam by means of the finite transfer method under second-order effects
In this section the imperfect beam is analysed under a compression load (see Fig. 1), assimilating it to a curved beam.Together with the geometric imperfection of the directrix, it is considered that the beam has no initial manufacturing stresses, that its material is homogeneous and isotropic, and that the section is constant.
To carry out the structural calculation, the numerical procedure called Finite Transfer Method 28 is used.This procedure solves a system of linear ordinary differential equations with boundary conditions and can address the wide casuistry of the structural problem of the beam [29][30][31] .Finite Transfer Method uses a repetition strategy on the discretised directrix that allows relating the solicitation and deformation values of the ends of the structural beam through an algebraic system.The dimension of said system is always constant and independent of the intervals obtained by the discretisation.It uses a fourth-order scheme to obtain a suitable numerical approximation.
To carry out the first-order analysis, a computer program that applies the Finite Transfer Method is used on the initial directrix of the beam to be calculated, with the entire compression load.To obtain second-order solicitation and deformation values, the load has been divided into equal parts (10,000 parts).Firstly, the numerical program has been executed with the first portion of load on the initial position of the directrix.With the results obtained, the new form of the directrix has been calculated and the numerical procedure has been executed on it for the second time.This process has been repeated until the application of the entire load has been completed.With this, a second-order analysis has been carried out on the real beam.
Two cases of steel beam types and different hollow circular section with the same area have been chosen.As shown in Fig. 1, these beams are supported by a hinge at the lower end I, and linear support at the upper end II.
Table 1 shows the characteristics in terms of shape and material of the two study type beams: hollow circular profile φ200.8 and φ100.19 .In the structural analysis carried out, the shear coefficients are considered null.
The load that intervenes in the calculation represents the maximum load that these beams can support without imperfections (ideal beam) and without using safety factors to determine the resistance of the material.The value of this load is P 0 = f y A = 1713 kN.
To assimilate the beam with geometric imperfection to a curved beam, two cases are analysed: sinusoidal directrix and parabolic directrix.In both examples, the maximum initial strain in f m is 4 mm.

Sinusoidal directrix beam
In this first analysis, the real beam is identified with a curved beam whose directrix has a sinusoidal shape whose equation is: Table 2 shows the solicitation and deformation values of the beam profile φ200.8 under both first-order and second-order effects.These solicitation and strain values are presented at eight points uniformly distributed on the directrix of the beam (0, 0.5, 1, 1.5, 2, 2.5, 3, 3.5 and 4 m).
The solicitation is made up of the normal N = V t and shear forces V n , and the bending moment M b = M y in the intrinsic axes.The deformation is composed of the gyration θ y , and the transverse δ x and longitudinal δ z displacements, under the general reference system.
Analysing the results, the values of normal stress and shear stress are not comparable.Therefore, it is considered sufficiently approximate to determine the normal stress and ignore the tangential stress.
Comparing in the centre the effects of second-order with those of first-order, an increase of 36.54% is observed in the bending moment and in the transverse displacement.Regarding the longitudinal displacement, the increase is only 0.16%.
Table 3 shows the solicitation and deformation of the sinusoidal beam with a hollow circular profile φ100.19.
Logically, the solicitation values under first-order effects of the two curved beams with sinusoidal directrix are the same.
In the second order analysis the relationship between the maximum shear force and the maximum normal force is 7.42%.In relation to the bending moment and the transverse displacement in the centre, an increase of 615% is observed between the second and first-order values.Regarding the longitudinal displacement, the increase is only 80%.
(1) www.nature.com/scientificreports/Under second-order effects, the relationship between the bending moment in the centre of the beams with profile φ200.8 and φ100.19 is 534%.The relationship between transverse displacements in the centre of the beams type φ200.8 and φ100.19 is 2797%.
The relationship between the first-order transverse displacement in the centre and the length of the real beam is 0.32%.Under second order analysis, this relationship is 2.26%.

Parabolic directrix beam
In this section, the calculation of the imperfect beam is identified with that of a curved beam with a parabolic directrix whose equation is: Table 4 shows, in a similar way to the values presented in Table 2, the solicitation and deformation values are shown under effects of both first-order and second-order in the beam profile φ200.8.
Comparing the second-order effects with the first-order ones in the centre, an increase of 37.59% is observed in relation to the bending moment and 36.66% in relation to the transverse displacement.In the longitudinal displacement the increase is only 0.17%.
Table 5 shows the solicitation and strain in the parabolic beam with a hollow circular profile φ100 • 19.
Under second-order effects, the relationship between the maximum shear force and the maximum normal force is 7.66%.In relation to the bending moment in the centre, an increase of 635% is observed between second and first-order values.The relationship between the bending moment in the centre of the beams with profile φ200.8 and φ100.19 is 534%.
In the centre there is a 617% increase in second-order transverse displacement compared to first order.Regarding the longitudinal displacement, the increase is only 85%.The relationship between transverse displacements in the centre of the beams type φ200.8 and φ100.19 is 2803% under the second-order analysis.
The relationship between the first-order transverse displacement in the centre and the length of the real beam is 0.33%.This relationship under the second-order analysis is 2.33%. (2) Table 2. Sinusoidal beam profile φ200.8 : first and second-order solicitations and deformations.Significant values are in bold.

Solicitations and strains
First-order Second-order www.nature.com/scientificreports/

Results comparison
Figure 2 shows the values of transverse strain of the beams with sinusoidal and parabolic directrix under the compression load, whose numerical values are written in Tables 2, 3, 4 and 5.The bending moment graphs have not been presented because this is proportional to the transverse displacement.Both first-order effects (red dashed line) and second-order effects (solid line) have been represented.The second-order transverse displacement functions associated with different proportions of load have also been graphed.The percentage of second-order effects is greater when the load portion is greater.
Figure 2 shows that under second-order effects, greater strain is experienced than under first-order effects.The beams with the lowest moment of inertia suffer greater deformation.The differences between assimilating the imperfect beam to a beam with a sinusoidal or parabolic directrix are negligible.In the second order analysis of the beam profile φ200.8 , the difference between bending moments is 0.77% and between transverse displace- ments is 2.66%.In the beam profile φ100.19 , these differences slightly increase, being these percentages 2.91% in the case of calculating bending moments and 3.13% in the case of transverse displacements.

Sinusoidal directrix beam with variable section
In this section the calculation of the real beam is identified with that of a curved piece with a sinusoidal directrix Eq. (1).The variable circular section beam is studied.This section varies linearly along the directrix, from the initial end φ100 to the final end φ78.
Table 6 shows the stress and deformation values under both first-order and second-order effects.
The solicitation values under first-order effects of the three curved beams with sinusoidal directrix are equal.
Comparing the second-order effects with the first-order effects in the span, an increase of 1670.19% is observed in relation to the bending moment and 1676.82% in relation to the transversal displacement.
Figure 3 shows that since the beam has a variable section, symmetry is not maintained with respect to the span of the solicitations, deformations and stresses.
Under second-order effects the maximum values are: for the bending moment 122 kN at 2.13 m from the initial end, for the tangential displacement 309 mm at 2.14 m from the initial end.
The maximum normal stress of failure or collapse of the beam occurs with 52% of the applied load and 2.52 m from the initial end.

Analysis of the beam through an analytical procedure
Due to imperfection, the directrix of the beam has a deviation from the straight line.According to Euler's approach, an ideal beam subjected to a compression load can be in stable or unstable equilibrium, elastically.Figure 4 shows the starting expressions of the bending moments, under different approaches for the ideal beam and for the real beam.The expressions of the tangential displacements that are obtained by Euler's approach and by the analytical procedure that is proposed are also shown.

Beam under Euler's approach
It is usual to derive the critical Euler load from the bi-hinged beam compressed by a point load P .The directrix has only transverse strain u(z) .It is considered that the section of the beam is constant and made of the same material.No other type of load takes place.The weight of the beam itself is considered negligible.
The differential expression of the deformed directrix can be written as: The analytical solution of transverse displacement is obtained by solving this differential equation.Its expression can be noted as 32 : where C 1 , C 2 are the two constants of integration.
From the application of the support conditions u(0) = 0; u(l) = 0; it can be derived that C 2 = 0 and that sin P EIl = 0 .If it were not so, the directrix would be a straight line and there would be no transverse deformation.For this last support condition to be met, the value that the load must acquire is This value is the Euler critical load.
In the real bi-articulated beam, the effect of the imperfections is considered equivalent, with sufficient approximation, to the one produced if its directrix instead of being initially straight is a sinusoid.The analytical notation of the directrix is expressed in Eq. (1).Under Euler's approach and in the real beam, the differential expression of the deformed directrix is 6 : The analytical solution of this differential equation, applying the support condition of the real bi-hinged beam is:  This function represents the transverse displacement suffered by the real beam in compression.When the load P is equal to the critical load P K , a discontinuity in the function is produced, and therefore the real beam is in a state of elastic instability.
The bending moment M(z) produced by the load at each point of the directrix of the beam is: The bending moment remains finite while P<P K , and becomes infinite, however small is f m , when P = P K .Therefore P K is the upper limit of the load that can be applied to a double-hinged beam.In the theoretical case of an ideal beam with f m = 1, there is a stable equilibrium if P<P K , but it becomes unstable with P = P K .

Beam under second-order effects
Remembering that an ideal beam is a resistant member with a straight directrix manufactured without initial stresses or heterogeneities, and without any geometric imperfection, when it is subjected to a compression load, the displacement of each point of the directrix has only a longitudinal component.In this case, there is neither transverse displacement nor bending moment.The maximum load that the ideal beam can withstand is obtained by multiplying the resistance of the material by the area of the section.
As previously commented, in the real beam the effect of the imperfections is equivalent to considering its directrix sinusoidal instead of straight.Under this approach, the differential expression of the deformed directrix can be noted as: The solution of this equation for the beam with biarticulated support is: where P K is the Euler critical load expressed in Eq. ( 5).
In this case, the entire load P has been applied to the bi-hinged beam instantaneously.This way the first order strain δ 1 (z) is obtained.
To determine the second order deformation δ(z) = δ 2 (z) , the load P must be applied gradually.To do this, it is divided into n equal parts, and the calculation is carried out applying its first increase.From this calculation, the deformation produced is deduced and a new directrix is generated.By repeating this calculation i times, the deformation is: By gradually applying the entire load, taking this division to the limit, the second order deformation and expressed as: The first-order deformation expressions Eq. ( 9) and second-order Eq. ( 11) are continuous functions.In first-order deformation, if the value of the force action is Euler's critical load, the displacement obtained coincides with the initial sinusoidal directrix of the beam.In second-order deformation, if the action is the critical load, the directrix of the beam is e-1 ≃ 1.718 times its initial position.
The second-order bending moment can be noted as: Like what is detected in the deformation equation under second-order analysis, there is no discontinuity in the bending moment formulation.
The maximum normal stress of the beam occurs in the centre section and at the furthest point from the barycentre.Its value is: Being W the resistant module.To exhaust a beam at its maximum stress, the function that relates the initial imperfection in the centre f m and the compression load P can be noted as: As it happens in the first order strain expressions Eq. ( 9) and second order Eq. ( 11), in Eq. ( 14) there is no discontinuity.

Results and relationships between second-order effects
As previously noted, the differences in both first-order and second-order effects between the sinusoidal and parabolic directrix beam are negligible.It has also been verified that the first-order effects are not close to the reality of the structural problem posed.From this point on, the formulations associated with the sinusoidal directrix beam will be used, presenting only second-order effects.
Table 7 compares the results obtained using the Finite Transfer Method numerical procedure with those obtained by applying the analytical formulas associated with Eqs.(11) and (12).
Any difference between the values obtained by the two procedures is less than 0.30%.There are practically no differences between the values of bending moments and transverse displacements obtained by numerical or analytical methods.When using the numerical procedure for the calculation, in addition to bending moments and transverse displacements, the other values of the effect are obtained: normal stresses, shear stresses, gyrations and longitudinal displacements.As previously noted, the effects that produce tangential stresses are negligible.
From the analysis of the results expressed in Table 7, it can be deduced that, at the level of structural verification, it is considered sufficiently approximate to determine the values of the second-order effect by the analytical procedure that has been developed.
Table 8 shows the values of the maximum compression load that can be supported by different profiles of the same length and area.
It is observed that the maximum load for the beams without initial imperfection is the same: P 0 = 1713 kN .The same does not happen with the critical load P K , since all the analysed profiles have different moments of inertia.In the cases studied, it is seen that the maximum load depends on both the initial imperfections and the section of the beams.As the outer diameter of the beam decreases, the maximum compression load decreases.When the initial imperfection of the beam increases, this maximum load supported by the profile also decreases.
Table 9 shows the values of the maximum compression load that can be supported by steel profiles taken from commercial standard series.
Beams of the same length (4 m) and with the profiles φ200.8 , φ200.6 , φ200.5 , φ100.6 and φ100.5 have been chosen for this analysis.These beams are also analysed under the same initial imperfections as in the previous study.The maximum load that these beams support without initial imperfection is the product of the resistance of the material f y = 355 N mm 2 by the area of the profile used.The resulting critical loads P K are different because the moments of inertia of the profiles are different.Similarly, to the case studies presented in Table 8, in these examples the maximum load also depends on both the initial imperfections and the sections of the profiles.If the area of the profile decreases and the initial imperfection of the beam increases, the maximum load decreases.
Table 10 shows the values of the maximum load that the same profile can support under different lengths.
The standard profile φ200.8 has been chosen for this analysis, with lengths of 3 m, 4 m, 5 m and 6 m.The initial imperfections considered are the same as in Tables 7 and 8.
In these cases, when using the same profile, the maximum load for ideal beams or beams without initial imperfection is the same P 0 = 1713 kN .It is observed that the critical loads P K are different because the lengths of the beams are different.If the length of the beam increases and the initial imperfection increases, the maximum load decreases.
Table 11 shows the values of the maximum load that can be supported by different profiles with the same area and the same slenderness.
The beams analysed are the profiles studied in Table 8.The slenderness chosen is associated with the 4 m long beam and profile φ100.19 , whose value is = l I A = 136 .With this value the length of each beam is determined.Again, the initial imperfections f m are without deformation, and with a deflection-span ratio of 1/1000, 1/500, 1/250 and 1/100.Also, in these cases, when using profiles with the same area, the maximum load for beams without initial imperfection is P 0 = 1713 kN .The critical load P K = 541kN is maintained in the beams studied.If the initial imperfection increases, the maximum compression load decreases.
Table 12 shows the values of the maximum load that profile beams with the same length and the same slenderness can support.In the cases studied the chosen slenderness is associated with the beam profile φ100.19 .The analysed beams have the following hollow circular profiles: φ110.34 , φ105.26 , φ100.19 , φ95.13 , φ90.7 and φ85.2 .These beams are analysed under the same initial imperfections f m as in the previous cases.
It is observed that both the critical loads and the maximum loads are different in all the study cases.Whether the initial imperfection increases or the outer diameter of the profile decreases, the maximum compression load decreases.

Conclusions
From the traditional study of the real beam with initial sinusoidal imperfection, the analytical formulations of the transverse deformation Eq. ( 11) and the bending moment Eq. ( 12) have been deduced.These expressions are associated with second-order effects and represent continuous functions at all points.Therefore, from this continuity condition, it is deduced that the collapse of the structure does not occur under any specific critical load.This collapse is due to the failure that originates in the beam due to the increase in second-order bending moments.
Under this alternative approach to the analysis of the buckling phenomenon, it is clear that to accurately determine the bending moment under the second-order analysis it is necessary to know or establish an initial deformation of the beam.For each material, the maximum failure load of the beam under compression depends on the initial imperfection, and other properties such as the length and geometric characteristics of the section.
To determine the maximum failure load or solicitation and deformation values under second-order effects, in beams and/or pieces with any initial imperfection or any type of section, constant or variable, the numerical procedure Finite Transfer Method can be applied using the repetition strategy developed in this work.

Figure 3 .
Figure 3. Bending moments, transversal deformations and stresses in the sinusoidal beam with variable section.
P f m sin πz l .

Figure 4 .
Figure 4. Structural approaches on the ideal and real beam.

Table 1 .
Formal characteristics and materials of the beams.

Table 4 .
Parabolic beam profile φ200.8 : first and second-order solicitations and deformations.Significant values are in bold.

Table 5 .
Parabolic beam profile φ100.19 : first and second-order solicitations and strains.Significant values are in bold.

Table 6 .
Variable section sinusoidal beam: first and second-order solicitations and deformations.Significant values are in bold.

Table 8 .
Maximum compression load for different profiles of equal area.

Table 9 .
Maximum compression load for different standard profiles.

Table 10 .
Maximum compression load for the same profile and different lengths.

Table 11 .
Maximum compression load for different profiles of equal area and slenderness.

Table 12 .
Maximum compression load for different profiles of equal slenderness and length.