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≅ 1.0 (LC3, WSLC3 and Z12 columns, respectively). Also displayed in (a1)–(a3) are GBT modal participation diagrams, providing the contributions of each deformation mode to the column critical buckling modes. On the other hand, (b1)–(b3) shows the critical global and distortional buckling mode shapes of the three columns considered. The observation of these buckling results prompts the following remarks:The GBT modal participation diagrams depicted in (a1)–(a2) (LC and WSLC columns) show that the Pcr vs. L equilibrium path descending branch corresponds to 2 distinct buckling behaviours. The first one involves contributions from deformation modes 2 |
+ |
4 |
+ |
6, i.e., the column buckles in a mode combining major-axis flexure (2), torsion (4) and anti-symmetric distortion (6) (FTD), if 155 < |
L |
< 250 cm (LC columns) or 155 < |
L |
< 300 cm (WSLC columns). Due to the presence of mode 6, the designation “global” is no longer strictly correct - however, as for the sake of simplicity, it will continue to be used in this work (sometimes between quotation marks, when confusion may arise). The longer columns buckle in the (expected) flexural-torsional (FT) modes, combining deformation modes 2 and 4 - truly global buckling. Note that the participation of mode 6 gradually decreases with L up to 250 cm or 300 cm (LC or WSLC columns), value for which FTD buckling suddenly switches to FT buckling. Thus, distortional-“global” interaction is always associated with FTD buckling since (i1) the difference between the distortional and FT buckling loads is quite substantial, as can be attested by looking at the vertical distance between the dashed (PcrD)(a1)–(a2), and (i2) the post-critical strength associated with FT buckling is too small to allow for the above interaction. However, as reported in [], the participation of mode 6 invariably fades away as loading progresses.The Z column GBT modal participation diagram shows that, for lengths in the close vicinity of the transition between critical distortional and global buckling, the column global critical buckling mode involves not only predominant minor-axis flexure (as expected) but also a small (but clearly visible in (b3)) contribution from the symmetric distortional mode 5 - i.e., it is, in fact, a FD mode. Once again, the anticipated pure global (minor-axis flexural) buckling mode occurs only for the longer columns (L |
> 600 cm).The virtual coincidence between PcrD and PcrG (see ) implies that the column post-buckling behaviour and strength are certainly affected by strong D-G interaction.The elastic and elastic-plastic post-buckling analyses were performed by means of shell finite element models similar to those often employed by the authors in previous investigations (e.g., []) and whose accuracy is already well established. The main characteristics of these models are briefly described next:Discretisation. The column mid-surface is discretised into fine meshes of S4 shell finite elements (A nomenclature - 4-node isoparametric shell elements with the shear stiffness obtained by a full integration rule) - previous studies (e.g., []) showed that length-to-width ratios of about 1.0 lead to accurate results.Support Conditions. The columns analysed have fixed-ended end support conditions, i.e., all the end cross-section global and local displacements and rotations are prevented (simulating the end support conditions adopted in the vast majority of the cold-formed steel column experimental investigations).Loading. Equal compressive loads, corresponding to a 1 |
MPa uniform stress distribution, are applied at both column end cross-sections - then, the A value provides the value of the average stress acting on the column (in MPa).Material Model. The carbon steel material behaviour, deemed isotropic and homogeneous, is modelled as linear elastic (buckling analysis) or elastic-perfectly plastic (post-buckling analysis). In the latter case, the well-known Prandtl-Reuss model is adopted - it is based on J2-flow plasticity theory and combines von Mises's yield criterion with its associated flow rule. No strain-hardening is considered.Initial Imperfections. The columns analysed contain only initial geometrical imperfections, whose shape and amplitude are discussed in . These initial geometrical imperfections are automatically incorporated in the SFE elastic-plastic non-linear analysis, as linear combinations of the competing buckling modes, obtained from a preliminary linear buckling analysis based on the same shell finite element mesh. Both residual stresses and corner strength effects are disregarded - as reported by several authors (e.g., []), the combined influence of strain-hardening, residual stresses and corner strength effects has little impact on the column load-carrying capacity.This section addresses the elastic post-buckling behaviour of columns affected by D-G interaction and begins by investigating the influence of the initial geometrical imperfection shape. The study is restricted to columns with very close critical distortional and “global” buckling loads (RGD |
≅ 1.0 - highest D-G interaction effects). Moreover, only linear combinations of pure distortional and “global” buckling modes shapes are considered, combining arbitrarily the two normalised buckling modes and sharing the same overall amplitude - the Z columns results are presented first, followed by those concerning WSLC columns (the LC column results were previously reported in [Two different Z columns are analysed in this study - the major difference between them concerns the critical distortional half-wave number (nD): columns Z6 (RGD |
= 0.95) and Z13 (RGD |
= 0.94) exhibit an even and an odd nD value, respectively. They contain initial imperfections combining 10 or 9 D half-waves with a single “global” half-wave. In order to obtain column equilibrium paths that (i) cover the whole imperfection shape range and (ii) can be meaningfully compared, the following approach is adopted:Then, a given “combined” imperfection shape is obtained as a linear combination of the pure D and G modes, with coefficients vG.90 and vD.0 lying on the ellipse shown in (a) and covering every possible combination. Each selected initial imperfection shape lies on this ellipse and corresponds to an angle θ, measured counter-clockwise from the vD.0 axis, so that vD.90=r sinθ and vD.0=r cosθ, where r is the ellipse polar coordinate. The amplitudes of the pure distortional (θ=0° or 180° - (b)) initial imperfections are equal to 0.94t (corresponding to a 50% probability that a random imperfection amplitude is below this value []) and L/1000, respectively. A total of 24 initial imperfections shapes are considered (15° θ intervals, starting at 0°).To assess how the imperfection shape influences the post-buckling behaviour of the two Z columns affected by D-G interaction, numerical equilibrium paths concerning columns containing the various initial imperfections are presented and discussed - such paths are hereafter identified by their θ values. (c) shows several deformed configurations at advance post-buckling stages, which will be discussed below. The observation of all these post-buckling results leads to the followings conclusions:(a)–(b) are associated with deformed configurations combining a predominant “global” (minor-axis flexure) half-wave with several distortional half-waves, thus evidencing the occurrence of D-G interaction. For instance, (c1)–(c2) displays the Z6+θ=90° and Z6+θ=270° column deformed configurations at advanced post-buckling stages ((v |
+ |
v0)/t |
= 30). As expected, all these columns exhibit a very small post-critical strength and fail below the critical buckling load (PU/Pcr |
< 1.0).. Note that distinct initial imperfection amplitudes may lead to significant differences in the column post-buckling behavioural features (see, for instance, [Surprisingly, the two equilibrium paths of the columns with even nD and containing pure distortional imperfections (θ |
= 0° and θ |
= 180°) clearly differ from those exhibited by all the remaining columns. (c3)–(c4) show deformed configurations of these columns at (v |
+ |
v0)/t |
= 10 and 25. These figures provide evidence that these columns are affected by D-G interaction of a different nature, involving distortional deformations and torsional rotations - distortional-torsional (D-T) interaction. Note that this type of interaction does not stem from the closeness between the distortional and torsional buckling loads. In fact, yet unpublished GBT-based results show that torsional rotations may emerge in columns affected by D-G interaction and exhibiting a pure distortional post-buckling behaviour, even when the critical torsional (Pcr.T) and minor-axis flexural (Pcr.F) buckling loads are not very close - such situation “attracts”/favours the interaction with torsional buckling. If Pcr.T/Pcr.F is large enough, the above D-T interaction does not occur - this is the case of the Z13 column (see (b)). Note that this type of D-G interaction cannot occur in (“very long”) columns buckling in torsional modes - in such columns, the critical distortional buckling load is much higher than Pcr.T.Since all equilibrium paths exhibit limit points prior to merging into “common curves” (except those addressed in the previous item), the most detrimental initial imperfection can be easily identified: that leading to the lowest failure load. (a) shows that the θ |
= 75°; 105°; 255°; 285° initial imperfections are the most detrimental (they correspond to PU/Pcr |
= 0.825). In (b), on the other hand, this role is played by the θ=120°; 300° initial imperfections, which correspond to PU/Pcr |
= 0.838. Nevertheless, it should be mentioned that PU/Pcr is practically the same for the columns containing pure “global” initial imperfections - therefore, for the sake of simplicity, it is assumed, in the remainder of this work, that the pure “global” initial imperfection is the most detrimental one.A similar investigation was performed for WSLC columns and, once again, two columns differing in the nD nature (even or odd) were considered: WSLC6 (RGD |
= 1.00 and odd nD) and WSLC14 (RGD |
= 1.03 and even nD). (a), shows all the possible initial imperfection shapes and (b) displays the WSLC6 column two pure distortional (θ |
= 0° and θ |
= 180° - 7 half-waves) and two pure “global” (θ |
= 90° and θ |
= 270°) initial imperfections. On the other hand, show the elastic equilibrium paths P/Pcr vs. (v+v0)/t of both columns (WSLC6, WSLC14) containing θ = |
0°, 15°, …, 345° initial imperfections - (c) depict several deformed configurations at advanced post-buckling stages. The observation of all these post-buckling results prompts the following comments:(a)–(b) can be divided into three or four groups, depending on the nature of nD, but they only correspond to two distinct post-buckling behaviour mechanics: (i1) 15°<θ <165° + 195°<θ <345°, and (i2) θ=0° + 180°.The first mechanical behaviour (15°<θ <165° + 195°<θ <345°) concerns almost all the equilibrium paths and combines predominant counter-clockwise (15°<θ<165°) or clockwise (195°<θ<345°) rotations with distortional deformations. Therefore, these columns are affected by the “expected” interaction between the distortional and FTD buckling modes – (c1)–(c2) show the θ=90° and θ=270° WSLC6 column deformed configurations at (v |
+ |
v0)/t |
= 50. Moreover, note that the equilibrium paths θ=15°, …,165° and θ=195°, …,345° are symmetric with respect to the sign of the “global” initial imperfection, even if this symmetry is not apparent in Once again, the θ |
= 0° and θ |
= 180° equilibrium paths correspond to mechanically distinct post-buckling behaviours - (c3)–(c4) show the deformed configurations of the θ |
= 0° and θ |
= 180° WSLC6 columns at (v |
+ |
v0)/t |
= 25 and (v |
+ |
v0)/t |
= 15, respectively (including the mid-span deformed cross-sections). These configurations combine distortional and minor-axis flexural deformations, which evidences the presence of D-G interaction of a different nature. Similar observations were made in [], in the context of LC columns (the last authors performed) GBT analyses. In addition, lipped channel and rack-section columns undergoing L-D-G interaction were also studied []. It was concluded that minor-axis flexural deformations emerge due to the effective centroid shift due to the stress redistribution caused by the distortional deformations - this effect gradually (at advanced post-buckling stages) leads to the emergence of interaction with minor-axis flexure buckling. Note that, once again (like in the θ |
= 0° and θ |
= 180° Z columns), this type of interaction does not occur in (“very long”) columns with critical minor-axis flexural buckling loads - in fact, the distortional buckling load is much higher than the minor-axis flexural one. The most detrimental initial geometrical imperfection shape, in the sense of maximising the column strength erosion, is the pure “global” one (θ=90° or θ=270°) – the corresponding equilibrium paths lie below all the remaining ones.(a)–(b), associated with columns undergoing distortional-FTD interaction, exhibit much higher post-critical strength than those obtained for the Z columns, which are affected by distortional-F (minor-axis flexure) interaction (see Finally, it should be noted that the (vi1) WSLC and LC columns undergoing distortional-F (minor-axis flexure) interaction and (vi2) Z columns affected by distortional-T (torsional) interaction correspond clearly to singular post-bucking behaviours, very unlikely to occur in “real” columns. Therefore, the further investigation on columns affected by D-G interaction (relevance and design) focuses on the behaviours more likely to occur in practice, i.e., either distortional-FTD (LC and WSLC columns) or distortional-F (minor-axis flexure) (Z columns) interaction.In order to assess the relevance of D-G interaction in CFS columns, it is essential to gather substantial numerical failure load data concerning columns with (i) various RGD values and (ii) a wide range of critical (distortional or global) slenderness values λcr, in order to capture possible “secondary bifurcation D-G interactions” – 11 different slenderness values are considered, namely λcr |
= {0.50, 0.75, 1.00, 1.25, 1.50, 1.75, 2.00, 2.50, 3.00, 3.25, 3.50}. The methodology adopted in this investigation is very similar to that followed in the context of the L-D interaction in CFS columns [Identify the RGD range associated with “true D-G interaction” [], for which the distortional and “global” buckling loads are fairly close – this type of interaction may always occur.If possible, identify minimum Ry |
= |
Py/max(PcrD; PcrG) values that allow for the development of “secondary (distortional or global) bifurcation D-G interaction”. This type of interaction may occur for RGD values significantly different from 1.0, provided that the yield stress is sufficiently high to enable the interaction to develop. Naturally, the “secondary-global bifurcation D-G interaction” is more likely to occur, due to the higher post-critical strength reserve exhibited by columns buckling in distortional modes. However, since the LC and WSLC columns with moderate-to-long slenderness were shown to exhibit also a non-negligible “global” post-critical strength reserve, “secondary-distortional bifurcation D-G interaction” may also occur in such columns.The elastic-plastic post-buckling behaviour and strength of Z, WSLC and LC columns undergoing “true D-G interaction” is first addressed. show P/Pcr vs. (v |
+ |
v0)/t equilibrium paths of Z11, WSLC5 and LC6 columns with RGD |
= 1.0, 5 critical slenderness values (1.0, 2.0, 3.0, 3.5, ∞ – the last stands for the elastic behaviour) and containing pure “global” initial imperfections. On the other hand, (b1)–(b3) display deformed configurations and plastic strain diagrams at the onset of collapse for the above four elastic-plastic columns. The observation of these results prompts the following remarks:All Z columns exhibit combined distortional and (mostly) minor-axis flexural deformations since the early loading stages (see the deformed configuration at collapse of the λcr |
= 1.0 column depicted in (b1)), which characterises “true D-G interaction”. On the other hand, the WSLC and (mostly) LC columns also exhibit distortional and FTD deformations since the early loading stages (see item (iii3) below). However, the former are barely visible in (b2)–(b3), due to the presence of anti-symmetric distortion (mode 6) in the “global” buckling mode and the dominance of torsional deformations (this issue will be discussed again in With the exception of the stockiest one (λcr |
= 1.0), where the failure occurs soon after the onset of yielding, all the remaining Z columns remain elastic up to failure - moreover, the elastic limit point falls below the critical buckling load. The Z11 + |
λcr |
= 1.0 column exhibits a very small elastic-plastic strength reserveThe failure modes of the LC and WSLC are very similar and can be divided into three groups, according to the critical slenderness value:In the λcr |
≅ 2.0 columns, yielding starts in the areas identified in the previous item and is followed by a pronounced elastic-plastic strength reserve (see (b3) - this is because these columns exhibit also local deformations, i.e., undergo L-D-G interaction. Indeed, it was very difficult to find LC column geometries having the local critical buckling load much higher than its distortional and global counterparts (see ), i.e., to preclude the occurrence of L-D-G interaction. In order to overcome this difficulty, it was decided to analyse WSLC columns in this work (the local critical buckling load is much higher). display elastic and elastic-plastic equilibrium paths of the Z23, WSLC23 and LC23 columns with RGD |
= 0.5, i.e., exhibiting typical global post-buckling behaviours. The observation of these figures shows that, as already mentioned in , the post-critical strength is small in the Z columns. Thus, the occurrence of “secondary distortional bifurcation D-G interaction” (SDI) is (i) very unlikely in the Z columns and (ii) quite possible in the WSLC and LC columns - the illustration of this assertion is addressed next, first for the Z columns. show the failure modes of Z columns with several RGD values, all sharing the same “global” slenderness (λG |
= 3.5) - they all fail in the elastic range (like all the columns with λG |
> 2.0). Recalling that the “global” buckling mode contains a small contribution from the symmetric distortional mode 5 (see (b3)), the observation of these figures leads to the following comments:The columns with 0.50 ≤ |
RGD |
≤ 0.85 fail in “global” (FD) modes, which means that SDI does not develop - the distortional deformations appearing in the failure mode are not akin to the critical distortional buckling mode (they stem from the “global” buckling mode). Note that the RGD |
= 0.50 column collapses with no trace of distortional deformations, because its “global” buckling mode is truly global (F - minor axis flexural) - see Only the RGD |
= 0.90 Z column exhibits D-G interaction (see (f)). Thus, it may be argued that, in Z columns, D-G interaction only occurs for RGD between 0.85 and 0.90 (i.e., “true D-G interaction”).Next, a similar investigation is conducted for WSLC columns. In such columns (singly symmetric cross-sections) the possible (but unlikely) occurrence SDI interaction is very difficult to detect by observing the deformed configuration evolution. This is because (i) the “global” buckling mode contains a significant amount of anti-symmetric distortion (mode 6 − see (a2)) and (ii) the interaction can only develop in advanced post-buckling stages, when torsion (mode 4) plays a major role (see, for instance, (c1)), making the detection of symmetric distortional deformations (mode 5, which appears in the distortional buckling mode) an extremely difficult task (on the basis of SFEA results). In order to overcome this difficulty, this study focuses on mid-web transverse displacement profiles, thus (i) eliminating the contribution of the anti-symmetric distortional mode from the column structural response (this mode exhibits double curvature in the web – e.g., []), and (ii) reducing the impact of the torsion mode. shows these displacement profiles for WSLC columns with RGD |
= 0.50, 0.60, 0.70, 0.80, 0.85, 0.90 at several post-buckling stages, namely P/PcrG |
≅ 1.00, 1.25, 1.50, 1.75, 2.00 and peak load. The observation of these displacement profiles prompts the following remarks:All the RGD |
= 0.50 column displacement profiles show one dominant half-wave, due to the torsion mode contribution (see []), i.e., this column post-buckling behaviour is “global”.The displacement profiles of the (ii1) RGD |
= 0.60 column at P/PcrG |
= 1.941 (peak load) and (ii2) RGD |
= 0.70 column at P/PcrG |
= 2.237 (peak load) show minute “irregularities” in the predominantly single half-wave configuration, thus evidencing the presence of symmetric distortional deformations, i.e., SDI interaction. These “irregularities” become perceptible at progressively lower applied load levels in the remaining columns – for instance, compare the displacement profile of the RGD |
= 0.80 column at P/PcrG |
= 1.985 with that of the RGD |
= 0.85 column at P/PcrG |
= 1.750., that the global strength curve is able to handle adequately both the global and SDI failures. Therefore, is not necessary to establish any “border” between “global” and SDI failures – this assertion also applies to the LC columns.Regardless of the cross-section shape and provided that Ry is “sufficiently high”, “secondary-global bifurcation D-G interaction” (SGI) is likely to occur, since the distortional post-critical strength reserve is visibly higher than its global counterpart (the opposite of what happened in the case of SDI). However, evidence of this interaction can only be clearly observed in columns containing initial geometrical imperfections akin to the critical distortional buckling mode when they are subjected to high applied loads - indeed, only in such circumstances it is possible to observe the emergence of global deformations akin to the global buckling mode shape (regardless of its nature, which varies with the cross-section shape). Recall that, as discussed in , columns containing “pure” distortional imperfections exhibit a different/“unexpected” type of interaction, involving global deformations not akin to the critical global buckling mode – since this type of interaction is beyond the scope of this work, no border between distortional and SGI failures will be established for such columns. However, the inclusion of very small “global” initial geometrical imperfections makes it possible to capture SGI, since the “global” deformations are rapidly replaced/“swallowed” by distortional ones in the close vicinity of the critical distortional buckling load. show the equilibrium paths of the Z40 + |
λD |
= 3.5 and WSLC40 + |
λD |
= 3.5 columns with RGD |
= 2.0 and containing “global” imperfections with amplitudes L/1000 (value adopted in this work) and L/106 (value considered with the sole purpose of illustrating SGI). As for (b1)–(c2), they display several deformed configurations indicated on the equilibrium paths of (a). These post-buckling results show that:As mentioned before, the L/1000 columns exhibit both distortional and global deformations since the early loading stages (the latter due to the initial imperfection) and, thus, it is not possible to illustrate the SGI – see the two pairs of deformed configurations depicted in (b1)–(b2) and 11(b1)–(b2), respectively for the Z and WSLC columns.SGI is clearly illustrated by the results of the L/106 columns: (ii1) state I involves highly predominant distortional deformations, due to the column “intrinsic” post-buckling behaviour ((c1)) and (ii2) global deformations only emerge and develop at the advanced post-buckling stages − e.g., see the state II deformed configurations shown in The Direct Strength Method (DSM), developed by Schafer and Peköz [] based on an original idea from Hancock et al. [], may be viewed as an approach/procedure to establish methodologies for the design associated with a wide variety of limit states. This versatility explains the popularity of the method for the design of CFS members (and also other structural systems – e.g., []). Currently, the DSM column design (e.g., AISI []) covers limit states associated with local, distortional, global and local-global interactive modes (for columns with or without holes/perforations) – note that the DSM global strength curve is not cast in the “Winter-type” format exhibited by their local, distortional and local-global counterparts (instead, it is a “classical” design curve taken from specifications dealing with hot-rolled steel structures). The two strength curves considered in this work are those providing the resistance against distortional (PND) and global (PNG) failures, which can be found, e.g., in [Note that the current AISI specification [] prescribes that the global strength curve applies to columns failing in global modes, regardless of whether those modes are flexural, flexural-torsional or torsional. One relevant feature of the DSM is the ability to handle explicitly interactive failures. In this regard, two design approaches can be followed (as done before for columns affected by L-D or L-G interaction): (i) the NDG approach (PNDG), involving the replacement of Py by PNG, as first suggested by Schafer [], and (ii) the NGD approach (PNGD), which replaces Py by PND - one then obtains, respectively,PNDG=PNG,λDG≤0.5611−0.25λDG−1.2λDG−1.2PNG,λDG>0.561PNGD=0.658λGD2PND,λGD≤1.50.877λGD−2PND,λGD>1.5where λDG |
= (PNG/PcrD)0.5 is the distortional slenderness based on the global strength and, similarly, λGD |
= (PND/PcrG)0.5 is the global slenderness based on the distortional strength.Before addressing the quality of the estimates provided by the DSM global design curve for the numerical failure loads of columns undergoing D-G interaction, a brief study is conducted to assess how accurately does the currently codified DSM global strength curve predict the load-carrying capacity of columns failing in pure global modes – this is a necessary first step prior to investigating its applicability to columns affected by D-G interaction plot PU/Py against the “global” slenderness λG for two sets of Z, LC and WSLC columns (selected in ) with λG |
= 0.50, 0.75, 1.00, 1.25, 1.50, 1.75, 2.00, 2.50, 3.00, 3.25, 3.50 and containing global initial imperfections with an amplitude of L/1000 - the DSM global strength and elastic buckling (1/λG2) curves are also shown. The observation of these results makes it possible to draw the following conclusions:The elastic buckling curve is always above the current DSM global strength curve, which implies a minute post-critical strength. Although this is certainly true for columns buckling in pure flexural, the same does not hold for flexural-torsional buckling, usually associated with a non-negligible (moderate) post-critical strength.Although the current AISI specification [] prescribes the DSM global strength curve to predict the load-carrying capacity of columns failing in any global mode type, the observation of (a)–(c) suggests that this curve is more suitable to estimate the failure loads of columns collapsing in flexural modes (Z columns - see (a)) - recall that this strength curve has its roots in hot-rolled steel specifications, dealing almost exclusively with doubly symmetric members, failing in pure flexural modes. On the other hand, the failure loads of columns exhibiting flexural-torsional(-distortional) collapse modes (LC and WSLC columns - see (b)–(c)) are visibly underestimated by this design curve, particularly in the moderate-to-long slenderness range (λG |
> 2.0).The fact that the failure loads of the two sets of columns included in each figure are predicted with different quality indicates that such quality depends also on the cross-section dimensions (along the whole slenderness range) − this dependence is more pronounced for λG |
> 1.0, particularly in the LC and WSLC columns, which fail in FTD buckling modes. Just to illustrate this assertion, note that the failure-to-squash load ratios of the LC1 and LC2 columns with λG |
= 3.5 differ by about 60%.Given the content of the above item, a brief study on how the column cross-section dimensions impact its post-buckling strength and failure load is presented in the next section.In order to assess the influence of the cross-section dimensions on the ultimate strength of columns buckling and failing in FTD modes show elastic and elastic-plastic equilibrium paths P/PcrG vs. (v |
+ |
v0)/t of the above six columns and their observation makes it possible to conclude that:Ratio bw/bf plays an important role in the LC and WSLC column post-buckling behaviour and failure load, as attested by looking at (a1)–(a3): a bw/bf decrease causes a significant PU/PcrG increase.Similarly, ratio bf/bl also plays an important role in the WSLC (and LC) post-buckling behaviour and failure load, as is clearly visible in (b1)–(b3): a bf/bl increase causes a drastic PU/PcrG increase. Therefore, different combinations of bw/bf and bf/bl values may lead to a high variability in PU/PNG, given that PNG depends only on λG and Py. Fortunately, the DSM global strength curve invariably underestimates the WSLC and LC column failure loads (see, for instance, (b)–(c)) − the amount of underestimation (safety level) depends on the combined bw/bf and bf/bl values. The influence of the cross-section dimensions explains the “practically null”, “slightly moderate” or “significant” post-critical strengths reported in [] and for which the authors were unable to find an adequate justification (in the context of L-D-G interaction).This section addresses the DSM-based prediction of the failure loads of CFS LC, WSLC and Z columns affected by D-G interaction − although the vast majority of these columns exhibit D-G interactive collapses, some of them fail in virtually pure “global” or distortional modes. Due to space limitations, the results presented and discussed concern only a representative sample of the columns identified in . Nevertheless, they provide enough information to enable assessing the failure load prediction quality in columns exhibiting a wide variety of RGD values, namely RGD |
= 0.50-0.70-0.85-1.00-1.30-1.60-1.80-2.00. , dealing with Z, LC and WSLC columns, respectively, display (i) PU/Py vs. critical slenderness (λD or λG) plots, (ii) the currently codified DSM distortional (PND vs. λD) or global (PNG vs. λG) design curves, (iii) the elastic buckling curve and (iv) the two DSM-based design approaches to handle D-G interaction (PNDG and PNGD). To assess the overall performance of the various DSM design approaches in predicting the numerical failure load data, (a1)–(d3) plot (i) PU/PNG, PU/PNDG and PU/PNGD vs. λG, and (ii) PU/PND vs. λD for all the Z, LC and WSLC columns considered – these figures also show the averages, standard deviations and maximum/minimum values of the above ratios. A quick observation of all these results confirms that, as it would be logical to expect after the results presented in the previous sections, the DSM failure load prediction quality concerning the Z and LC + WSLC columns is quite distinct − indeed, the PU/Py values of the two column sets are well aligned along different “Winter-type” curves. Therefore, the results concerning Z and LC + WSLC columns are addressed separately − attention is first turned to the Z columns:The ultimate strengths of the Z columns failing in global modes (RGD |
≤ 0.85 – see ) are adequately predicted by the DSM global design curve, as clearly shown in (b1)–(b3) − recall that these columns do not experience SDI interaction. The PU/PNG statistical indicators given in (a1) confirm this assertion: mean and standard deviation equal to 0.99 and 0.03, respectively.(b4)), the PNG values gradually ceases to provide safe failure load estimates − this is because the columns are now affected by “true D-G interaction” (see ). Note that the PU/Py values concerning these columns are below the DSM global design curve, even if this fact cannot be clearly observed in (b4) (very low PNG/Py values). In order to obtain safe and accurate failure load estimates it is necessary to resort to the PNDG values, which successfully account for the ultimate strength erosion due to D-G interaction.The DSM distortional design curve (PND) only provides adequate failure load estimates for stocky columns with RDL |
> 1.00 − the number of accurate predictions grows very slowly with RGD and λD (for instance, compare (a1) and (a4)), i.e., as the column collapse mode nature becomes progressively “more distortional” (although global deformations are still visible, due to the initial geometrical imperfection considered – e.g., see (b1)). On the other hand, the PND values overestimate the failure loads of the RGD |
= 1.30–1.60-1.80-2.00 slender columns (see (a1)–(a4)), thus providing clear evidence of the occurrence of D-G interaction − in this case, “secondary-global bifurcation D-G interaction”. Once again, the DSM-based NDG approach provides good quality predictions, as attested by looking at (b1) and corresponding statistical indicators, which outperform those associated with the NGD approach − this last approach yields unsafe predictions for most slender columns (λG ≥2.0 − see Next, the observation of the LC and WSLC column results leads to the following conclusions:Generally speaking, the DSM global strength curve is able to handle adequately all the types of failure that may occur in these columns (including those of a few LC columns experiencing L-D-G interaction). It is not necessary to resort to the NDG approach, as was done for the Z columns.The accuracy of the PNG estimates of the ultimate strength of columns failing in “global” and D-G interactive modes (the latter due to SDI or “true D-G interaction”) depends solely on λG: while the failure loads of columns with λG |
< 2.0 are accurately predicted by the PNG values, those of the columns with λG |
> 2.0 become increasingly underestimated – see the grey circles in (a2)–(a3). The differences can be considerably high − for instance, they reach 250% for λG |
= 3.5. Naturally, this behaviour is related to the significant post-critical strength that was observed earlier (related to the bw/bf and bf/bl width ratios), which is not reflected in the PNG value – see, for instance, (a2)–(a3). Indeed, even the clear detrimental effect associated with the occurrence of “true D-G interaction” can be “disguised” by this strength increase.A similar situation occurs for the columns failing in D-G interactive modes stemming from SGI (RGD |
> 1.0), except for a few stocky columns, which are slightly inaccurately predicted by the PNG curve (see (a2)–(a3)). Since these columns correspond to the transition between distortional and SGI collapses, their failure loads are logically well predicted by the PND values.(d2)–(d3) make it possible to assess the quality of the estimates provided by the PND values (distortional strength curve) − there exists a large number of unsafe predictions, which are particularly severe for the most slender columns (because of the failure load erosion caused by SGI).This paper reported the available results of an ongoing SFEA investigation on the post-buckling behaviour, strength and DSM design of fixed-ended cold-formed steel columns undergoing distortional-global (D-G) interaction. Amongst the various findings reported in this work, the following deserve to be highlighted:The initial geometrical imperfection shape plays an important role in the post-buckling (elastic or elastic-plastic) behaviour of columns undergoing D-G interaction, since it alters the global buckling nature. Indeed, while (i1) pure distortional initial imperfections lead to “unexpected” distortional-torsion (Z columns) or distortional-minor-axis flexure (LC and WSLC columns) interactions, (i2) initial imperfections with any other shape (combining the distortional and “global” buckling modes) lead to the “expected” couplings between distortional and minor-axis flexure (Z columns) or distortional, major-axis flexure, torsion and anti-symmetric distortion (LC and WSLC columns).The global and D-G interactive post-buckling behaviours of Z and LC + WSLC columns are very different. While the Z columns exhibit a small post-critical strength, which precludes the occurrence of “secondary-distortional bifurcation D-G interaction”, the LC and WSLC columns exhibit much larger post-critical strengths, thus making it possible for the above interaction to occur. Nonetheless, the amount of distortional deformation appearing in the failure modes is always quite small (a similar feature was reported in [], for columns experiencing “secondary-local bifurcation L-D interaction”).The emergence and development of either “true D-G interaction” or “secondary-global bifurcation D-G interaction” are possible in all columns. The former, associated with close PcrG and PcrD values, is characterized by the presence of both distortional and (predominantly) “global” deformations since the early loading stages. The latter, associated with PcrG/PcrD ratios well above 1.0, is characterized by the emergence of deformations akin to the “global” buckling mode at fairly late loading stages (provided that the yield stress is “high enough” to allow it, of course).The DSM global strength curve provides excellent estimates of the ultimate strength of Z columns failing in flexural modes. However, that strength curve clearly underestimates the ultimate strength of LC and WSLC columns failing in flexural-torsional modes, particularly in the moderate-to-large slenderness range (due to the non negligible post-critical strength increase mentioned in item (ii)) – the amount of underestimation depends on the combined values of the width ratios bw/bf and bf/bl.It was found that the DSM global strength curve is able to handle adequately the three types of D-G failures occurring in LC and WSLC columns. This is because the failure load erosion due to D-G interaction (which is real) is “compensated” by the DSM global design curve safety margin in predicting flexural-torsional failure loads (as already mentioned in the previous items). In the Z columns, on the other hand, it is indispensable to employ the DSM-based NDG approach to capture adequately the failure load erosion stemming from “true D-G interaction” and “secondary-global bifurcation D-G interaction” (“secondary-distortional bifurcation D-G interaction” is never relevant).Determination and analysis of crack growth resistance in plasma-sprayed thermal barrier coatingsCeramic thermal barrier coatings (TBCs) are increasingly applied to enhance the performance of advanced gas turbine engines. However, the delamination cracks initiated in these coatings limit their applications. In this research, a sandwiched four-point bend specimen is used to evaluate the crack growth resistance in plasma-sprayed TBCs. Well controlled, stable and measurable crack extension is obtained. A rising crack growth resistance curve is found. The steady state strain energy release rate is obtained to be about 170 J/m2. The delamination crack evolution behavior is in situ observed and simulated by the finite element analysis based on a crack bridging model.Crack growth resistance is an important mechanical property of plasma-sprayed thermal barrier coatings (TBCs) which usually fail due to spallation of the coatings from the substrate. Although there have been a number of research activities to establish a standard methodology for evaluating interfacial fracture toughness in TBCs, all of them seem to not be satisfactory and the reported data on crack growth resistance in terms of interfacial strain energy release rate (G) or fracture toughness in plasma-sprayed TBCs vary largely. For example, the G values of air plasma sprayed (APS) Al2O3 coatings have been reported to be in the range from 12 to 40 J/m2This paper reports an in situ measurement of interfacial crack growth resistance in APS TBCs by using sandwiched four-point bend specimen. The technique reported in this paper, compared to that in Ref. A well-polished and cleaned SUS304 stainless steel plate of 5 mm in thickness was used as the substrate material. A NiCoCrAlY alloy (Co23Cr17Al13Y0.5 and balance Ni) was used as the bond coat layer. The bond coat layer was deposited on the substrate using APS method. A partially stabilized ZrO2 with 8 wt.% Y2O3 was used as the top coat layer, and the top coat layer was deposited on the bond coat layer by APS method. The average thickness of either the bond coat or the top coat was about 150 μm.A sandwiched four-point bend (4 PB) specimen () was used for the experiment. The thickness of the substrate of the as-deposited specimen was reduced to about 3 mm. Then, a stiffener, which was made of the same material and the same thickness as the substrate, was fixed to the surface of the top coat layer using a strong epoxy base adhesive. The thickness of the adhesive layer was about 50 μm. Hence, the coating layers were sandwiched between the substrate and the stiffener.The width and length of the sandwiched 4 PB specimen were 7 and 50 mm, respectively. After preparing the sandwiched specimens, the side surfaces of the specimens were carefully polished up to 1 μm diamond paste finish. A notch was carefully cut through the stiffener and the adhesive layer, and into the top coat layer with a diamond saw such that the notch tip was laid within the top coat layer.Pre-cracks were introduced in the notched specimens by fatigue on a servo-hydraulic test machine in an ambient air, by using three-point bend fixture with a span of 40 mm. The compression-compression load ratio was 0.1 and the frequency was 20 Hz for the fatigue pre-cracking. The pre-cracks initiated from the notch tips, then propagated and kinked symmetrically towards both sides of the notch, and finally entered into the plane close and parallel to the interface between the bond coat and top coat layers, as is shown in After pre-cracking, the specimens were monotonically loaded under four-point bend at a constant loading rate of 2 N/s. The inner and outer spans of the four-point bend fixture were 20 and 40 mm, respectively. The testing was performed in an ambient air using the same servo-hydraulic test machine as for fatigue pre-cracking. The load and the crosshead displacement were continuously recorded using a digital recording system at a sampling rate of 100 ms. The crack length was directly measured from the polished surface during loading by means of a video camera and a traveling microscope. The crack length was defined as the sum of the initial crack length and the crack extension. The initial crack length was determined as the distance from the notch root to the fatigue pre-crack tip, and the crack extension was defined as the distance from the fatigue pre-crack tip to the crack front generated by the monotonic loading. In the test, the specimen was unloaded after a crack extension of about 0.2 mm, and then reloaded to get another crack extension of about 0.2 mm. The loading–unloading process was repeated until the cracks on both sides were propagated to the inner loading lines.A linear loading behavior was observed during the whole loading stage of the first cycle. No crack extension was detected after the first cycle. During each of the following loading–unloading cycles, the initial linear loading behavior was followed by a non-linear behavior, which occurred near the maximum load on each curve and was accompanied by a crack extension. With increasing the number of loading–unloading cycles the applied maximum load was incremented, the crack length increased and the slope of the linear portion in loading stage decreased. After complete unloading in each cycle, a permanent residual displacement appeared. The residual displacement was correlated with crack extension after each loading–unloading cycle.Since each sandwiched four-point bend specimen was constructed by bonding a thin coating layer into two halves of substrate material, the whole bulk of the specimen was homogeneous except for the thin coating layer in the structure with a pre-existing crack lying close to the interface. According to Suo and Hutchinson provided that the propagating interfacial cracks are located within the inner loading lines of the specimen, where P is the total load, L is the spacing between the inner and outer loading lines, b and h are the width and half height of the specimen, and E and υ are the Young’s modulus and Poisson’s ratio of the substrate material. In the present experiment, the nominal specimen sizes necessary for the calculation of strain energy release rate in Eq. were L |
= 10 mm, b |
= 7 mm, and h |
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