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J. Inst. Eng. India Ser. A (June 2022) 103(2):601–625 https://doi.org/10.1007/s40030-021-00610-4 ORIGINAL CONTRIBUTION Scaling Effect on the Behaviour and Design of Prestressed Stayed Steel Columns 1 2 1 • • Peter Hyman Adelaja I. Osofero Srinivas Sriramula Received: 20 August 2021 / Accepted: 14 December 2021 / Published online: 26 January 2022 The Author(s) 2022 Abstract This study presents the results of a small-scale was found that small-scale structures are capable of rep- experimental campaign on prestressed stayed steel col- resenting the behaviour of large-scale prestressed stayed umns, subsequent numerical model validation and design steel columns. Furthermore, it was shown that the highest guideline development. The majority of previous experi- efﬁciency in terms of the weight of materials is found close mental studies have focused on large-scale systems, which to the transition point between symmetric and antisym- are expensive and can be difﬁcult to perform due to the metric buckling behaviour. Various scale systems were required specialised experimental set-up, whereas small- modelled numerically, and the results compared with scale experiments are less restrictive with both space and existing guidelines, which resulted in low levels of accu- experimental set-up requirements. Also, existing design racy. Therefore, existing design guidelines were updated guidelines were developed from a single system scale, so using the validated numerical model and shown to yield have not been shown to be applicable to changes in geo- more accurate results for the L/400 and L/200 imperfection metric scale. Thus, the scaling effect on prestressed stayed levels. steel columns was investigated to promote the use of small- scale experiments in the study of large-scale prestressed Keywords Prestressed stayed steel columns stayed steel column systems and update design guidelines Interactive post-buckling Design guidelines for change in geometric scale. A total of 17 prestressed stayed steel columns and a control column with no cross- arms were tested. These tests investigated the symmetric Introduction and antisymmetric buckling behaviour as well as the interactive post-buckling phenomenon. These tests were A major issue with the use of slender steel columns is the designed to investigate the scaling effect on the behaviour reduction in load-carrying capacity due to buckling insta- of the system and to determine the optimal prestress level bility. A system to improve the load-carrying capacity of of prestressed stayed steel columns close to the transition steel columns that inhibits the instability through the point. A numerical model was also validated by the addition of cross-arms and prestressed stays is commonly experimental results to perform a full geometric scaling known as a prestressed stayed steel column. These systems, comparison study and update existing design guidelines. It shown in Fig. 1, typically have load-carrying capacities several times that of unstayed columns as the prestressing force provides restraint against lateral instability. Examples of the usage of prestressed stayed columns in the con- & Adelaja I. Osofero struction industry include at Chiswick Park in London aiosofero@abdn.ac.uk where they are used to support an overhanging shading structure and Algarve stadium in Portugal, where they are School of Engineering, University of Aberdeen, Aberdeen, UK used to support the stadium roof with further examples highlighted by [1]. School of Engineering, University of Aberdeen, Fraser Noble Building, King’s College, Aberdeen AB24 3UE, UK 123 602 J. Inst. Eng. India Ser. A (June 2022) 103(2):601–625 prestressed stayed steel columns with symmetric critical modes have been carried out by [9] and [10]. However, no investigation has been carried out on the effect of the specimen scale on the observed behaviour of these struc- tures. Furthermore, only one previous experimental study has investigated the antisymmetric/interactive post-buck- ling behaviour of these systems [11]. Though, the sensi- tivity of the system to prestress at the transition point was not studied. An analytical study by [12] showed that the greatest efﬁciency in terms of weight of materials for prestressed stayed steel columns can be obtained close to the transition point, highlighting the importance of under- standing how these systems behave in this region. Previous numerical studies [13–16] have highlighted the prestress Fig. 1 Prestressed stayed steel column with applied axial load P, which yields the highest load-carrying capacity for pre- length of column and cross-arm L and a respectively stressed stayed steel columns with varying cross-arm lengths, though this has not been done experimentally for Prestressed stayed steel columns have two distinct systems close to the transition point. Therefore, this study buckling modes; symmetric (Fig. 2a) and antisymmetric aims to investigate the optimal prestress level for pre- (Fig. 2b) which take the form of a half sine wave and a full stressed stayed steel column conﬁgurations close to the sine wave, respectively, while an interactive post-buckling transition point through the use of small-scale experiments. shape can result from a combination of the distinct modes The ability to use small-scale experiments to accurately (Fig. 2c). Interactive post-buckling is an important phe- capture the behaviour of large-scale prestressed stayed nomenon that is primarily triggered close to the transition steel columns will also be investigated. point of symmetric and antisymmetric modes: i.e. critical The previous researchers [17] developed design loads of symmetric and antisymmetric buckling modes are guidelines to determine the load carrying capacity of sys- similar in magnitude. tems with various prestress levels. However, these design Investigations on prestressed stayed steel columns have guidelines were developed from results of a full-scale been carried out since the 1960s [2]. Several experimental experimental campaign and haven’t been shown to be studies have been carried out on full-scale prestressed applicable to stayed columns in other geometric scales. stayed steel column systems with symmetric critical modes Therefore, this study will also attempt to investigate the [3–8]. Studies investigating the behaviour of small-scale suitability of existing guidelines for the design of stayed columns at varying geometric scales. Material Testing Stay System Testing Testing was performed to obtain the material properties of the cable stay system. The cable stay system was made up of a mini rigging screw to apply the prestress, ferrules and thimble eyes to connect the cable, a load cell to measure the cable tension, eye nuts to connect to the load cell and a quick link. In the cable system, the galvanised steel wire made up approximately 50% of the total length. The tested system is shown in Fig. 3, the cable is made up of 3 mm 7 9 7 strand galvanised steel wire rope with the cross section shown in Fig. 4. The cable system was tested in a 10kN Hounsﬁeld uniaxial testing machine, in tension. This allowed key material properties such as the Young’s modulus and the Fig. 2 Buckling modes of prestressed stayed columns; a symmetric, breaking load of the cable system to be established. The b antisymmetric and c Interactive 123 J. Inst. Eng. India Ser. A (June 2022) 103(2):601–625 603 Fig. 3 Cable system used in the cable tests and column tests throughout the main tests the cable system was not stressed past 50% of its failure load. Column and Cross-Arm Testing To obtain the material properties for the column and cross- arm material, tensile coupon tests were performed according to [18] and the gauge length was obtained, using Eq. (1). pﬃﬃﬃﬃﬃ Gauge length ¼ 5:65 A ð1Þ where A is the area of the cross section. The method Fig. 4 Cross-section proﬁle of the 3 mm galvanised steel wire rope adopted by [11] was utilised for these tests, where a section of the material was removed from either side of the coupon cable system was tested according to the guidelines in [18]: specimen to ensure failure occurred within this region. The if Young’s modulus of the material is less than 150GPa length of the milled section was just longer than the gauge then the stress rate of the tensile test should be between length to allow the extensometer to grip the milled sec- 2 MPa/s and 20 MPa/s. The equivalent displacement rate tion. Half gauge lengths were lightly scribed on the milled was calculated using the approximate Young’s modulus section to ensure fracture occurred within the gauge length. from the material suppliers and chosen in between the two Also, rod adapters were placed in the ends of the specimen limits. This allowed the stiffness of the cable system to be to allow connection to the machine jaws, shown in Fig. 6, determined from the stress–strain curve obtained from the the adapters were then connected directly to the testing test. Figure 5 depicts the stress–strain behaviour of the machine and coupon with pins through either end of the cable system from the tensile tests of 34 specimens. adapters. From the stress–strain curve of the cable systems, the An Instron 4483 150kN load frame was used to perform behaviour is seen to be approximately linear until failure, the material tests, with an Instron 2620–601 dynamic clip- exhibiting brittle fracture. The average stiffness is calcu- on extensometer used to measure the extension of the lated from the slope, up to 2% strain, with the average gauge length. This extensometer is capable of measuring being 40.9 GPa and standard deviation 1.3GPa. As pre- travel up to 5 mm, giving it a maximum strain reading of dicted by [19], the Young’s modulus of the cable system is 40%, thus it was used throughout the tests up to failure. signiﬁcantly lower than that of conventional mild steel due A Squirrel 2010 data logger was used to store the load and to the construction of the wire inducing contact forces extension values throughout the test. Based on the rec- between individual strands. It should be stated that ommendations in [18], the stress rate was kept within the limits of 6–60 MPa/s for materials with Young’s modulus greater than 150GPa. Thus, the rate of separation of the cross-heads was set at 0.125 mm/min throughout the test, 0 0.5 1 1.5 2 2.5 3 3.5 Strain (%) Fig. 5 Stress–strain curves of the stay system tests Fig. 6 Column material test coupon specimen with adapters Stress (MPa) 604 J. Inst. Eng. India Ser. A (June 2022) 103(2):601–625 meaning that fracture occurred after about 40 min. A total of 30 coupon specimens were tested for the column material to obtain statistical data for a future reliability study, with 3 coupon specimens tested for the cross-arm material to obtain the mean material property values. Mean and standard deviation values of the material properties are presented in Table 1, with the test set-up shown in Fig. 7. Stress–strain curves of the column and cross-arm tests are shown in Fig. 8; these values are similar to those found from a previous study by [11]. Prestressed Stayed Steel Column Testing The main prestressed stayed steel column testing consisted of one unstayed control column and 17 prestressed stayed steel columns. Of these, ﬁve were designed to demonstrate antisymmetric buckling, six to demonstrate symmetric buckling and six were around the transition point to investigate interactive behaviour. These were scaled down Fig. 7 Set-up of the column tensile coupon test from a previous study [11], with the column length of 1 m chosen due to the maximum capacity of the testing nature of prestressed stayed steel columns was ﬁrst derived machine. This is consistent with other previous small-scale by [3] and is shown in Fig. 10. This highlights that if the experiments that have tested systems of similar scale prestress in the stays is less than T then there is no min ([9, 10]). Comparison of several system conﬁgurations with increase in critical buckling load from the Euler load. This different buckling modes and prestress levels was done to is known as zone 1. If the stay prestress is between T and min verify the scaling method. T there is a linear increase in the critical buckling load opt (zone 2). T is a theoretical value of prestress which gives opt Preparation of Specimens the maximum critical buckling load, beyond which the prestress has a negative effect on the critical buckling load The prestressed stayed steel column system included a one- (zone 3). The prestressed stayed steel columns in this study metre section of the column material with the cross-arms have been designed to have prestress values in both zones 2 welded directly onto the column face at mid-height at 90 and 3 to investigate the effect on the post-buckling intervals. Additionally, 5 mm thick ﬂat endplates were behaviour. added at either end of the column, parallel to each other. Connecting plates were also added to the endplates and Measurement of Specimen Dimensions cross-arms to accommodate the stays. The completed assembly is shown in Fig. 9. Before testing, the key dimensions of the specimens were measured, including the average outer diameter of the main Initial Prestress column, cross-arm and stays as well as initial imperfec- tions. The diameter along the supplied lengths of steel The initial prestress used for each test was decided tubes was measured to obtain a record of how the diameter according to the zone of behaviour to be studied. The zonal changes along its length. Thicknesses were also measured Table 1 Material properties of the column and cross-arm obtained through tensile testing. Where E is Young’s modulus, r is the Yield stress taken as the 0.2% proof stress, r is the ultimate stress, e is the percentage strain at the ultimate stress and e is the percentage strain at fracture u u f Specimen E (MPa) r (MPa) r (MPa) e (%) e (%) y u u f Column Mean 210,600 341 370 13.0 24.1 Stdev 9630 26.5 17.36 4.32 9.35 Cross-arm Mean 212,780 387 533 12.0 19.1 Stdev 7464 29.4 19.8 2.54 2.52 123 J. Inst. Eng. India Ser. A (June 2022) 103(2):601–625 605 400 600 0 5 10 15 20 25 0 5 10 15 20 Strain (%) Strain (%) (a) (b) Fig. 8 Typical stress–strain curves for a column and b cross-arm coupon tests Fig. 9 Assembled prestressed stayed steel column specimens Fig. 10 Theoretical zonal behaviour of prestressed stayed steel columns, where N is the maximum critical buckling load of the max column, N is the minimum critical buckling load or Euler load, min T is the minimum prestress and T is the optimum prestress min opt at the ends of the column and cross-arm for three points around the perimeter to obtain an average, this was repe- onto the paper underneath and a straight line was then ated once the tubes were cut to their ﬁnal lengths. The drawn connecting the ends of the curve. Distances between column material was supplied in 6 m lengths and had an the straight line and the curve of the column were measured average outer diameter of 15.9 mm and wall thickness of for various points along its length. This method yielded the 1.25 mm. The cross-arms were supplied in 5.8 m length approximate shape of the imperfections in the buckling and had an average outer diameter of 9.98 mm and a wall plane before prestressing was applied. The imperfection thickness of 1.06 mm. These sections were chosen to be shapes of all specimens had a roughly half sine wave shape. class 1 according to [20] to avoid local buckling of the Imperfection amplitudes ranged from L/106 to L/426, cross section before yielding. owing to the relatively large initial curvature from trans- portation and handling of the main column material. This Measurement of Initial Imperfections imperfection amplitude is in line with the speciﬁcations in British Standards for hot ﬁnished structural hollow sections Following assembly of the specimens, the initial out-of- of L/200 [21]. Key dimensions of the specimens are straightness was measured before testing by using an reported in Table 2, highlighting the imperfection approximate method. The specimens were laid ﬂat on a workbench and the curvature of the main column traced Stress (MPa) Stress (MPa) 606 J. Inst. Eng. India Ser. A (June 2022) 103(2):601–625 Table 2 Measured dimensions for the experimental test specimens, where a is the cross-arm length, w is the imperfection amplitude at mid- height and T is the initial prestress Specimen Main column diameter (mm) a (mm) w T (N) T/T Zone 0 opt Control 15.84 N/A 9.44 N/A N/A N/A 1000 9 60-C1 15.85 60 4.46 103.7 0.47 2 1000 9 60-C2 15.90 60 7.41 207.3 0.94 2/3 1000 9 60-C3 15.92 60 8.38 311.0 1.41 3 1000 9 70-C4 15.80 70 4.12 119.3 0.45 2 1000 9 70-C5 15.83 70 7.49 238.6 0.89 2/3 1000 9 70-C6 15.82 70 8.4 357.9 1.34 3 1000 9 80-E1 15.88 80 3.55 145.4 0.46 2 1000 9 80-E2 15.87 80 7.91 290.8 0.92 2 1000 9 80-E3 15.90 80 6.59 436.2 1.38 2/3 1000 9 110-E4 15.79 110 5.48 193.3 0.51 2 1000 9 110-E5 15.89 110 5.96 386.6 1.02 2 1000 9 110-E6 15.89 110 4.02 579.9 1.53 2/3 1000 9 150-B1 15.81 150 6.09 144.3 0.41 2 1000 9 150-B2 15.89 150 7.27 288.5 0.82 2 1000 9 150-B3 15.89 150 6.42 403.9 1.15 2/3 1000 9 150-B4 15.88 150 3.32 605.9 1.73 3 1000 9 150-B5 15.82 150 6.96 915.5 2.62 3 amplitude at mid-height of each specimen and the zone of column endpoint. The general test set-up can be seen in behaviour to be investigated. Fig. 11. Cross-arm lengths were chosen to investigate a wide Testing Procedure range of rotational stiffnesses such that the optimal con- ﬁguration could be found. Also, the level of initial pre- The main compression tests of the prestressed stayed steel tension was varied for each cross-arm length so that the columns were performed using an INSTRON 250kN uni- optimal level of prestress of each conﬁguration could be versal testing machine. Pinned end conditions were found. The labelling convention of the stays in the buckling achieved using knife edges attached to the column ends, plane used in the experimental campaign is shown in only allowing rotation in the plane of buckling. The knife Fig. 12. edges added 80 mm to the total column length, making the buckling length 1080 mm. Four linear displacement sen- Control Column sors were used along the length of the column to measure the displacements at the quarter points and at 35 mm on The control column exhibited symmetric failure mode and either side of mid-height. Four strain gauges were used on the Euler load of the control column was adjusted to either side of the column in the buckling plane at quarter account for the initial imperfection according to Eq. (2) height and at 25 mm on either side of the mid-height to [22]. check for yielding of the column. Load cells were also used cr N ¼ ð2Þ in each of the eight stays to measure the change in prestress w þ 1 during the tests. All data channels were recorded using the where N is the ultimate load of the column, P is the SignalExpress computer software. The mid-height rotation C cr Euler load, w is the measured imperfection and w is the was calculated from the linear displacement sensor read- mid-height deﬂection at the ultimate load. The calculated ings. End shortening results from the tests were displace- ultimate load for the control column using Eq. (2) was ment values from the Instron controller and thus are not 1.769kN which is about 3% higher than that obtained from appropriate for validation purposes as they included slack the experiment (1.723kN) (Fig. 13). in the system. Therefore, the initial part of the end-short- ening curves is not linear. In hindsight, an LVDT should have been used to measure the true displacement of the 123 J. Inst. Eng. India Ser. A (June 2022) 103(2):601–625 607 1.5 0.5 0 5 10 15 20 25 30 Mid-height displacment (mm) Fig. 13 Force versus mid-height deﬂection of the control column used to determine the buckling shape. The symmetric buckling mode was deﬁned when the tension in stays 1 and 4 or 2 and 3 were equal, whereas an antisymmetric mode occurred when the stay tension in stays 1 and 3 or 2 and 4 were equal. Interactive behaviour appears when all four stay tensions are notably different. Yield points for each test were measured using the strain gauges to conﬁrm an important assumption in previous numerical modelling that Fig. 11 General setup of prestressed stayed column tests the columns remain in the elastic region until the ultimate load is reached. Therefore, these tests will attempt to verify this assumption by showing the yield point of each column. However, there was a failure of the strain gauges in tests C2 and C6 so the yield point was not measured. Further- more, test C3 experienced a failure of one of the cable components at around the ultimate load as the thread was not fully tightened so the yield point was not observed. For the remainder of the tests, the yield point was shown to be after the ultimate load point, as seen in previous experi- mental studies [6, 11]. This veriﬁes the assumption that the columns remain in the elastic region until after the ultimate load point for numerical modelling. Symmetric Specimens C1-C3 Table 3 presents a summary of obtained results for speci- mens designed with symmetric buckling modes (C1-C3) while Figs. 14 and 15 show the force versus mid-height deﬂection and stay tension versus end shortening curves, Fig. 12 Stay labelling convention used during the experiments respectively. It can be seen that the load-carrying capacity is roughly constant across different prestress levels. This is Main Tests because the increase in load-carrying capacity due to the increase in prestress is being balanced by the effect of the A total of 17 prestressed stayed column systems were resultant increase in initial imperfection. Also, the level of tested, with varying cross-arm lengths and initial prestress tension in the stays on the concave side of the column values. These comprised: six specimens with symmetric remains roughly the same throughout the test, resulting in critical modes, six around the transition point to demon- symmetric buckling shapes. However, an interactive shape strate interactive behaviour and ﬁve with an antisymmetric occurred after the ultimate load was achieved. This was critical mode. All stays were prestressed to the levels caused by the main column reaching the yield point and the shown in Table 2, although only the in-plane stays were stay stresses being redistributed. Force (kN) 608 J. Inst. Eng. India Ser. A (June 2022) 103(2):601–625 Table 3 Summary of results for specimens C1-C3, with ultimate load N , the ratio of ultimate load and the control column N /N , end u u C shortening at ultimate load D and the ﬁrst observed buckling mode Specimen N (kN) N /N D Buckling mode u u C u 1000 9 60-C1 5.49 3.19 2.75 Symmetric 1000 9 60-C2 4.87 2.83 2.33 Symmetric 1000 9 60-C3 5.13 2.98 3.04 Symmetric First yield 1 1 0 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Mid-height displacement (mm) Mid-height displacement (mm) (a) 1000x60-C1 (b) 1000x60-C2 0 5 10 15 20 25 Mid-height displacement (mm) (c) 1000x60-C3 Fig. 14 Force versus mid-height deﬂection for specimens C1-C3 Symmetric Specimens C4-C6 reduction in the load-carrying capacity of specimen C6. The load-carrying capacity of the specimens varies with Results for specimens C4-C6 are summarised in Table 4. varying prestress levels, with the highest occurring for Figures 16 and 17 show the force versus mid-height specimen C4 at 3.65 times higher than the control column. deﬂection and stay tension versus end shortening curves, This is due to the imperfection of specimen C6 being respectively. Specimens C4-C6 had longer cross-arms than double that of specimen C4, i.e. increase in prestress C1-C3, i.e. closer to the transition point and are used to having no beneﬁt on the load-carrying capacity as it is show how the cross-arm length affects the post-buckling cancelled out by the effect of the initial imperfection. Also, behaviour. For specimen C4, the stress in the stays on the an interactive shape occurred for specimen C6 which concave side is roughly the same throughout the test, contributed to the reduction in ultimate load. suggesting symmetric buckling mode. However, small differences are seen for specimens C5 and C6, suggesting Symmetric/Interactive Specimens E1-E3 interactive behaviour in these specimens. It should be noted for specimen C5 that the quick link in the stay on the Results for specimens E1-E3 are summarised in Table 5. convex side of the specimen was not tightened during the Figures 18 and 19 show the force versus mid-height rota- test, and the results from this test are ignored. The tion and stay tension versus end shortening curves, appearance of interactive behaviour leads to a signiﬁcant respectively. These specimens were designed to have a Force (kN) Force (kN) Force (kN) J. Inst. Eng. India Ser. A (June 2022) 103(2):601–625 609 3.5 3 Stay 1 Ulmate Ulmate Stay 1 2.5 load point load point 2.5 Stay 4 1.5 1.5 Stay 4 Stays 2 and 3 0.5 0.5 Stays 2 and 3 End shortening (mm) End shortening (mm) (a) 1000x60-C1 (b) 1000x60-C2 3.5 Ulmate load point Stay 1 2.5 1.5 Stay 4 Stays 2 and 3 0.5 End shortening (mm) (c) 1000x60-C3 Fig. 15 Stay tension versus end shortening for specimens C1-C3 Table 4 Summary of results for specimens C4-C6, with ultimate load N , the ratio of ultimate load compared to the control column N /N , end u u C shortening at ultimate load D and the ﬁrst observed buckling mode Specimen N (kN) N /N D Buckling mode u u C u 1000 9 70-C4 6.29 3.65 2.39 Symmetric 1000 9 70-C5 3.08 1.79 1.21 Symmetric/Interactive 1000 9 70-C6 4.58 2.66 2.59 Symmetric/Interactive symmetric critical mode just before the transition point to specimen E3, suggesting that higher levels of prestress determine whether these systems were the optimal con- result in increased load carrying capacity for this ﬁguration. For specimens E1-E3, the tension in the stays conﬁguration. This is consistent with previous studies during loading suggests that the buckling shapes start as [6, 11, 13–16, 23] in that the load-carrying capacity of symmetric but become interactive soon after or just before prestressed stayed steel columns with symmetric critical the ultimate load. However, the stay tension plots suggest modes increases with increase in prestress, with specimen that specimen E1 remains in the symmetric mode for E3 being 4.65 times higher than the control column. longer than E2 and E3. This suggests that an increase in prestress results in the interactive shape occurring sooner. Antisymmetric/Interactive Specimens E4-E6 Also, the occurrence of mid-height rotation of the column and all four stay tensions being different highlights the Results for specimens E4-E6 are summarised in Table 6. interactive nature of the post-buckling shapes. Load-car- Figures 20 and 21 show the force versus mid-height rota- rying capacities of these specimens also appear to increase tion and stay tension versus end shortening curves, with increase in prestress, with a signiﬁcant increase for respectively. These specimens were designed to observe Stay tension (kN) Stay tension (kN) Stay tension (kN) 610 J. Inst. Eng. India Ser. A (June 2022) 103(2):601–625 3.5 2.5 First yield 1.5 First yield 0.5 0 5 10 15 20 25 30 010 20 30 40 Displacement (mm) Displacement (mm) (a) 1000x70-C4 (b) 1000x70-C5 0 5 10 15 20 Mid-height displacement (mm) (c) 1000x70-C6 Fig. 16 Force versus mid-height deﬂection for specimens C4-C6 the load carrying capacity of systems with an antisym- prestress levels in this study, with the maximum occurring metric critical mode just beyond the transition point. For for a prestress of 0.5T . opt conﬁgurations E5-E6 it can be seen that the post-buckling shape is interactive almost immediately the load is applied, Antisymmetric Specimens B1-B5 as the stay stresses are all distinctly different. Also, a mid- height rotation was recorded throughout the test. Changes Results for specimens B1-B5 are summarised in Table 7. in stay stress during loading for specimen E4 show that the Figures 22 and 23 show force versus mid-height rotation column took the symmetric shape until around the ultimate and stay tension versus end shortening curves, respectively. load point, where an interactive shape occurred. Specimen These specimens were scaled down from a previous E4 taking the symmetric shape appears to have increased experimental study by the researchers [11] to demonstrate the ultimate load signiﬁcantly, as specimens E5-E6 antisymmetric/interactive behaviour so that the effect of exhibited interactive shape quicker, resulting in a much changes in geometric scale could be investigated. Test B2 lower load carrying capacity. The results suggest that the was stopped prematurely as the Instron’s static break load-carrying capacity is negatively affected by an increase detector was still enabled, so the mid-height rotation results in prestress, as the highest load-carrying capacity is of B2 aren’t as high as the other tests. For the remainder of recorded for conﬁguration E4 at 5.15 times higher than the the tests, the static break detector was not used as the tests control. This also suggests that an increase in prestress did not include any material failure. All tests gave a sig- increases the likelihood that these systems will take an niﬁcant increase in load-carrying capacity over the control interactive shape. The reduction in load-carrying capacity column. Specimen B4 gave the highest load-carrying for higher prestress levels for prestressed stayed steel col- capacity for a prestress around 1.73T , with an increase in opt umns with an antisymmetric critical mode is similar to load-carrying capacity of 4.59 times the control column. previous work [11, 13, 16], although it occurs for lower This is also in part due to the initial imperfection of Force (kN) Force (kN) Force (kN) J. Inst. Eng. India Ser. A (June 2022) 103(2):601–625 611 3.5 1.2 Ulmate Ulmate load point load Stay 1 2.5 Stay 1 0.8 point Stay 4 0.6 1.5 0.4 Stay 4 Stay 3 Stay 2 0.2 Stays 2 and 3 0.5 End shortening (mm) End shortening (mm) (a) 1000x70-C4 (b) 1000x70-C5 Ulmate 2.5 load point Stay 1 Stay 4 1.5 Stay 3 Stay 2 0.5 End shortening (mm) (c) 1000x70-C6 Fig. 17 Stay tension versus end shortening for specimens C4-C6 Table 5 Summary of results for specimens E1-E3, with ultimate load N , the ratio of ultimate load compared to the control column N /N , end u u C shortening at ultimate load D and the ﬁrst observed buckling mode Specimen N (kN) N /N D Buckling mode u u C u 1000 9 80-E1 6.78 3.94 2.06 Symmetric/Interactive 1000 9 80-E2 6.96 4.04 3.13 Symmetric/Interactive 1000 9 80-E3 8.00 4.65 1.83 Symmetric/Interactive specimen B4 being signiﬁcantly lower than the other tests from the researches [11] in the scaling comparison specimens. Examining the stay stresses in Fig. 23 it can be section. seen that specimens B3-B5 exhibited an interactive post- buckling shape as the stay stresses are all signiﬁcantly Post-Buckling Shapes different. Specimens B1-B2 had lower levels of prestress so interactive behaviour occurred later in the tests. Also, Typical images of failure modes observed during the the mid-height rotation for all ﬁve specimens further experiments are presented in Fig. 24. Symmetric buckling highlights the interactive behaviour of these specimens. It shapes were observed for specimens C1-C6, similar to is also highlighted that the load-carrying capacity of pre- Fig. 24a. Symmetric shape was also observed for specimen stressed stayed steel columns with an antisymmetric criti- E4 even though the critical mode was antisymmetric, this is cal mode is negatively affected by an increase in prestress expected to be caused by the large half sine wave initial beyond a certain level. Obtained load carrying capacities imperfections. It has also been observed that the increase in are compared with results of the corresponding large-scale prestress leads to the interactive mode occurring sooner. Antisymmetric buckling shapes were only observed at the Stay tension (kN) Stay tension (kN) Stay tension (kN) 612 J. Inst. Eng. India Ser. A (June 2022) 103(2):601–625 8 8 7 7 First yield First yield 0 0.02 0.04 0 0.01 0.02 0.03 0.04 Midheight rotaon (rad) Rotaon (rad) (a) 1000x80-E1 (b) 1000x80-E2 First yield 0 0.02 0.04 0.06 0.08 Rotaon (rad) (c) 1000x80-E3 Fig. 18 Force versus mid-height rotation for specimens E1-E3 Interactive Conﬁgurations E1-E6 very beginning of the tests on specimens B1-B5 as they quickly turned interactive, hence the small deﬂections seen in Fig. 24b. Interactive post-buckling as shown in Fig. 24c Specimens E1-E3 were all designed to have symmetric was seen in the majority of the specimens but was partic- critical modes and the symmetric buckling shape occurred ularly pronounced for specimens E4-E6 and B1-B5 which initially but with a quick transition to an interactive post- had an antisymmetric critical mode. buckling shape. Specimen E1 remained in the symmetric mode longer than the other specimens due to lower initial prestress. It should be noted that this phenomenon also Analysis of Results occurred for specimens E4-E6 as an interactive shape occurred sooner for the specimens with higher prestress. Symmetric Conﬁgurations C1-C6 The appearance of an interactive shape occurs as all four stay tensions begin to diverge and soon have different Specimens C1-C6 which had symmetric critical modes maintained symmetric buckling shapes until after the ulti- values. Specimens E1-E3 were designed to have symmetric mate load, where interactive post-buckling occurs. The critical load just below that of the antisymmetric, thus close interactive post-buckling shape occurred sooner for speci- to the transition point on the symmetric side. Interactive men C6 as the cross-arm length of this system is closer to post-buckling is normally a phenomenon only seen in the transition point, so the symmetric and antisymmetric prestressed stayed steel columns with an antisymmetric critical loads are closer together. However, specimen C4 critical mode or with symmetric critical mode beyond the ultimate load [11]. However, specimens E1-E3 show that maintained the symmetric shape throughout the test, which was expected to be caused by the low level of initial pre- specimens with symmetric critical modes can take an interactive shape before the ultimate load is reached. stress. Interactive behaviour was seen in some of these specimens after the yield point as the stresses in the column Although, it is unclear whether this was due to the amplitude of the imperfections, unintentional eccentricities were redistributed. Force (kN) Force (kN) Force (kN) J. Inst. Eng. India Ser. A (June 2022) 103(2):601–625 613 1.6 3.5 Ulmate Stay 4 Ulmate 1.4 load point load point 1.2 2.5 1 Stay 1 Stay 4 Stay 1 0.8 1.5 0.6 Stay 2 0.4 Stay 3 Stay 3 0.2 0.5 Stay 2 End shortening (mm) End shortening (mm) (a) 1000x80-E1 (b) 1000x80-E2 2.5 Ulmate Stay 1 load point 1.5 Stay 4 Stay 2 Stay 3 0.5 End shortening (mm) (c) 1000x80-E3 Fig. 19 Stay tension versus end shortening for specimens E1-E3 Table 6 Summary of results for specimens E4-E6, with ultimate load N , the ratio of ultimate load compared to the control column N /N , end u u C shortening at ultimate load D and the ﬁrst observed buckling mode Specimen N (kN) N /N D Buckling mode u u C u 1000 9 110-E4 8.87 5.15 1.90 Antisymmetric/Interactive 1000 9 110-E5 6.83 3.96 1.94 Antisymmetric/Interactive 1000 9 110-E6 7.11 4.13 2.36 Antisymmetric/Interactive in loading or other factors. Specimens E4-E6 were designed to have an antisymmetric critical mode, although designed to have an antisymmetric critical load just below it took the symmetric shape. that of the symmetric mode. Accordingly, specimens E5- E6 took an interactive post-buckling shape almost imme- Antisymmetric Conﬁgurations B1-B5 diately and remained interactive throughout the tests. However, specimen E4 appeared to take the symmetric Specimens B1-B5 were designed to be scaled-down ver- shape until around the ultimate load point, resulting in an sions of the specimens with an antisymmetric critical mode increase in load carrying capacity. It was initially thought tested in previous work [11] to investigate the scaling that the conﬁguration yielding the greatest load-carrying effect. As in the study by the researches [11], specimens capacity would be one with symmetric critical mode just B1-B5 all took an interactive post-buckling shape almost before the transition point to avoid interactive behaviour immediately, with the highest load-carrying capacity seen and gain maximum beneﬁt from the cross-arms. However, for specimen B4. Also, it should be noted that only spec- it can be seen that the specimen yielding the greatest imen B5 had all four stays active at the ultimate load point increase in load-carrying capacity is E4 which was due to the prestress being much larger than T . Specimens opt B3 and B4 lost the prestress of one stay on the concave side Stay tension (kN) Stay tension (kN) Stay tension (kN) 614 J. Inst. Eng. India Ser. A (June 2022) 103(2):601–625 10 8 First yield First yield 0 0 0 0.02 0.04 0.06 0.08 0 0.02 0.04 0.06 Rotaon (rad) Rotaon (rad) (a) 1000x110-E4 (b) 1000x110-E5 First yield 0 0.02 0.04 0.06 Rotaon (rad) (c) 1000x110-E6 Fig. 20 Force versus mid-height rotation for specimens E4-E6 at around the ultimate load due to the initial prestress being lower than the symmetric and an initial prestress level of closer to T . However, all four stays in specimens B3-B5 0.5T . opt opt had signiﬁcantly different stress values during loading suggesting an interactive mode. A full comparison of the Scaling Comparison relative load-carrying capacities of these tests compared to previous work will be carried out subsequently. As well as investigating the behaviour of prestressed stayed steel columns with various cross-arm lengths, the experi- Efﬁciency ments were also designed to compare the results from this small-scale study to a previous full-scale experimental To assess the optimal conﬁguration from the various sys- study. The experiments were based on scaled-down ver- tems tested, the load-carrying capacity of each system is sions of the full-scale experimental study by the research- non-dimensionalised by the sum of the weight of the col- ers [11]. This allowed a comparison of the results from the umn, cross-arms and stays. This results in an overall efﬁ- small-scale tests with corresponding systems from the full- ciency indicator which can be used to determine the most scale study. The previous works [11] focused on large- efﬁcient system in terms of the weight of materials, shown scale structures in order to demonstrate interactive beha- in Table 8. It can be seen that the greatest efﬁciency is viour. However, the interactive systems were only inves- achieved for specimens E3 and E4. This highlights that tigated with a single prestress level. Therefore, a key prestressed stayed steel columns designed close to the difference between this study and the work [11] is that the transition point have optimal efﬁciency. Specimen E3 was interactive systems were tested for multiple prestress levels designed with symmetric critical mode just before the to investigate how the behaviour changes as well as transition point and a prestress level of 1.5T . Specimen forming the basis of the scaling comparison study. A opt E4 was designed with an antisymmetric critical mode just comparison of the load-carrying capacity of corresponding Force (kN) Force (kN) Force (kN) J. Inst. Eng. India Ser. A (June 2022) 103(2):601–625 615 1.4 2 Stay 4 Ulmate Ulmate 1.2 load point Stay 1 load point 1.5 Stay 1 Stay 4 0.8 0.6 Stay 3 Stay 2 0.4 0.5 Stay 3 0.2 Stay 2 End shortening (mm) End shortening (mm) (a) 1000x110-E4 (b) 1000x110-E5 1.6 Ulmate Stay 2 1.4 load point 1.2 Stay 3 0.8 0.6 Stay 4 0.4 0.2 Stay 1 End shortening (mm) (c) 1000x110-E6 Fig. 21 Stay tension versus end shortening for specimens E4-E6 Table 7 Summary of results for specimens B1-B5, with ultimate load N , the ratio of ultimate load compared to the control column N /N , end u u C shortening at ultimate load D and the ﬁrst observed buckling mode Specimen N (kN) N /N D Buckling mode u u C u 1000 9 150-B1 7.40 4.30 2.26 Antisymmetric/Interactive 1000 9 150-B2 6.86 3.98 1.69 Antisymmetric/Interactive 1000 9 150-B3 7.31 4.24 1.67 Antisymmetric/Interactive 1000 9 150-B4 7.90 4.59 2.06 Antisymmetric/Interactive 1000 9 150-B5 7.65 4.44 2.07 Antisymmetric/Interactive systems is shown in Table 9. This scaling comparison Numerical Modelling highlights that small-scale experiments can be used to investigate the behaviour of large-scale structures as the Model Development ratio of load-carrying capacities of the small-scale against the corresponding large-scale is fairly constant across all The numerical model was developed in the ABAQUS conﬁgurations, with an average value of 0.101, although commercial software. Validation of the model was per- small discrepancies between the various conﬁgurations formed, based on the small-scale experimental results, occur due to differences in the initial prestress level and before prestressed stayed columns with different geomet- imperfection. Also, similar post-buckling shapes were rical scales were investigated. The same model formulation observed between the small-scale and the corresponding as that used in previous studies was utilised large-scale systems. [13–16, 23–26]. The column and cross-arms were modelled using B32 beam elements, while the stay was modelled with a single T3D2 truss element. A convergence study was carried out to establish the optimum mesh size after Stay tension (kN) Stay tension (KN) Stay tension (kN) 616 J. Inst. Eng. India Ser. A (June 2022) 103(2):601–625 8 8 First First yield yield 0 0.02 0.04 0.06 0 0.005 0.01 0.015 0.02 Midheight rotaon (rad) Midheight rotaon (rad) (a) 1000x150-B1 (b) 1000x150-B2 8 10 First yield First yield 0 0.01 0.02 0.03 0.04 0.05 0 0.02 0.04 0.06 Midheight rotaon (rad) Midheight rotaon (rad) (c) 1000x150-B3 (d) 1000x150-B4 0 0.01 0.02 0.03 0.04 Midheight rotaon (rad) (e) 1000x150-B5 Fig. 22 Force versus mid-height rotation for specimens B1-B5 which a 2 mm element size was chosen for both the col- this experimental study, interactive behaviour was umn and cross-arm. As the stays are cables that go slack observed for all the specimens. The specimens designed under compressive loading, the ‘no compression’ option with symmetric critical modes had symmetric buckling was used to enable the stays to go slack when they lost their shapes initially but became interactive before or just after prestress. A buckling analysis was used to obtain the the ultimate load. Therefore, a combination of the ﬁrst two eigenmode shapes of the system which were subsequently buckling mode shapes was used in the models to induce an used as initial imperfections in the Riks analysis [27]. asymmetric post-buckling shape. To calculate the ampli- tude of each of the ﬁrst two modes, the equation from the Initial Imperfections literature [16] was used (Eq. 3). 2 2 l þ 4l ¼ 1 ð3Þ 1 2 To induce the post-buckling behaviour in the Riks analysis, a combination of the ﬁrst two buckling modes was used. where l is the imperfection coefﬁcient for the symmetric mode and l is the imperfection coefﬁcient for the Generally, a distinct symmetric imperfection is used to antisymmetric mode. The imperfection level of each model systems with symmetric critical modes. However, in Force (kN) Force (kN) Force (kN) Force (kN) Force (kN) J. Inst. Eng. India Ser. A (June 2022) 103(2):601–625 617 1 1.4 Ulmate Ulmate load point 1.2 load point 0.8 Stay 4 Stay 1 0.6 Stay 4 0.8 0.6 0.4 Stay 1 Stay 2 0.4 0.2 Stay 3 0.2 Stay 3 Stay 2 0 0.5 1 1.5 2 End shortening (mm) End shortening (mm) (a) 1000x150-B1 (b) 1000x150-B2 1.2 1.4 Ulmate Ulmate Stay 3 1.2 Stay 1 load point load point 0.8 0.8 Stay 4 0.6 Stay 2 0.6 0.4 0.4 Stay 1 0.2 Stay 4 Stay 2 Stay 3 0.2 End shortening (mm) End shortening (mm) (c) 1000x150-B3 (d) 1000x150-B4 1.2 Stay 1 0.8 0.6 Stay 3 0.4 Ulmate Stay 4 0.2 Stay 2 load point End shortening (mm) (e) 1000x150-B5 Fig. 23 Stay tension versus end shortening for specimens B1-B5 column was multiplied by l and l to obtain the imper- antisymmetric critical mode as interactive behaviour 1 2 fection amplitude of the symmetric and antisymmetric dominated, whereas mid-height displacement results at modes, respectively, to be used in the Riks post-buckling ultimate load were used for systems with symmetric critical analysis. mode as they remained in the symmetric mode until around the ultimate load point. A mean ratio of FE to the experi- Numerical Model Validation mental ultimate load of 0.98 and a standard deviation of 0.06 highlights the ability of the numerical model to cap- A comparison of the load-carrying capacities and mid- ture the load-carrying capacity of the experiment tests. height displacement/rotation of the experimental results Furthermore, the mean and standard deviation of the ratios and ﬁnite element (FE) model is shown in Table 10. Mid- of the mid-height rotation and mid-height displacement height rotation results at ultimate load were compared for results also show the FE model is capable of accurately systems close to the transition point and systems with an modelling the load-deformation behaviour of prestressed Stay tension (kN) Stay tension (kN) Stay tension (kN) Stay tension (kN) Stay tension (kN) 618 J. Inst. Eng. India Ser. A (June 2022) 103(2):601–625 Fig. 24 Post-buckling shapes seen during testing; a symmetric, b antisymmetric, c interactive Table 8 Efﬁciency of each prestressed stayed steel column specimen in terms of the weight of materials Conﬁguration N (kN) Weight (N) N /Weight u u B1 7.40 6.58 1125 B2 6.86 6.58 1042 B3 7.31 6.58 1111 B4 7.90 6.58 1200 B5 7.65 6.58 1163 C1 5.49 5.77 951 C2 4.87 5.77 843 C3 5.13 5.77 888 C4 6.29 5.86 1073 C5 3.08 5.86 525 C6 4.58 5.86 781 E1 6.78 5.95 1139 E2 6.96 5.95 1170 E3 8.00 5.95 1344 E4 8.87 6.22 1426 E5 6.83 6.22 1098 E6 7.11 6.22 1143 stayed columns. Force versus mid-height displacement/ behaviour of the specimens including the yield point. The rotation curves of specimens C4, E1 and B1 are shown in numerical model was validated across all tests, including Figs. 25, 26 and 27, respectively. These highlight the systems with symmetric buckling mode, antisymmetric effectiveness of the numerical model in capturing the buckling mode and those close to the transition point with 123 J. Inst. Eng. India Ser. A (June 2022) 103(2):601–625 619 Table 9 Scaling comparison of the ultimate load carrying capacities of corresponding conﬁgurations from this study and those of researches [11] Small-scale Full-scale N small-scale N full-scale N ratio: small-scale/full- T/T small- T/T full- u u u opt opt conﬁguration conﬁguration (kN) (kN) scale scale scale C2 A3 4.87 53 0.092 0.94 1.07 B1 B1 7.40 79 0.094 0.41 0.39 B2 B2 6.86 75 0.091 0.82 0.78 B3 B3 7.31 73 0.100 1.15 1.09 B4 B4 7.90 66 0.120 1.73 1.63 E3 C2 8 89 0.090 1.38 1.58 E6 C3 7.11 68 0.105 1.53 1.44 Average 0.101 Table 10 Comparison of the ultimate loads and mid-height displace- height displacement and the subscripts ‘FE’ and ‘exp’ represent the ment/rotation between the FE model and experimental values. Where FE model and experiment results respectively N is the ultimate load, h is the mid-height rotation, d is the mid- Column N /N h /h d /d u,FE u,exp FE exp FE exp C1 0.89 0.94 C2 0.97 0.92 C3 0.87 0.91 C4 0.91 1.07 C6 1.11 0.94 E1 0.97 1.05 E2 1.00 1.26 E3 0.99 1.03 E4 0.91 1.02 E5 1.00 1.11 E6 1.05 0.94 B1 0.96 0.94 B2 0.99 1.29 B3 0.99 1.06 B4 1.01 0.98 Mean 0.98 1.07 0.96 Standard deviation 0.06 0.12 0.07 4 4 Experiment Experiment FE FE Yield point Yield point 0 0.01 0.02 0.03 0.04 0.05 0 5 10 15 20 25 30 Displacement (mm) Mid-height rotaon (radian) Fig. 25 Comparison of the force versus mid-height displacement Fig. 26 Comparison of force versus mid-height rotation obtained between the FE model and experimental results for specimen C4 from the FE model and physical experiments for specimen E1 Force (kN) Force (kN) 620 J. Inst. Eng. India Ser. A (June 2022) 103(2):601–625 3 Experiment FE Yield point 0 0.01 0.02 0.03 0.04 0.05 0.06 Mid-height rotaon (radian) Fig. 27 Comparison of force versus mid-height rotation relationship obtained from the FE model and physical experiments for specimen B1 Fig. 29 Comparison of antisymmetric mode deformed shapes from FE model and experimental investigation Fig. 28 Comparison of symmetric mode deformed shapes from FE model and experimental investigation interactive modes at various prestress levels. Thus, the numerical modelling has been shown to be suitably accu- rate in capturing the behaviour of various system conﬁg- urations, validating the method. It can be deduced from Figs. 28, 29 and 30 that the numerical model is capable of capturing the deformed shapes of the experiments. The specimens with symmetric Fig. 30 Comparison of interactive mode deformed shapes from FE model and experimental investigation critical modes had symmetric shapes until around the ultimate load where an interactive shape occurred (Fig. 28). Specimens with an antisymmetric critical mode Force (kN) J. Inst. Eng. India Ser. A (June 2022) 103(2):601–625 621 1.2 scale to ensure an accurate comparison. All models used in the validation of the numerical model from Table 10 were y ≈ (L/L ) scaled according to the ratios in Table 11. The maximum 0.8 load-carrying capacity of each conﬁguration was then non- 0.6 dimensionalised by the load-carrying capacity of the cor- responding full-scale system as shown in Fig. 31. 0.4 The trend for the non-dimensional load-carrying 0.2 capacity against length ratio, L/L , highlights that the non- dimensional load-carrying capacity is roughly proportional 0 0.2 0.4 0.6 0.8 1 1.2 L/L f to the square of the length ratio. This can be compared by looking at the governing equations for the critical buckling Fig. 31 Non-dimensional load-carrying capacity vs length ratio load of prestressed stayed columns. Firstly, all dimensions including the inner d and outer diameter d of the column i o only took an antisymmetric shape at the very start of the were scaled according to the length ratio, thus the pro- tests, before an interactive shape occurred, hence the small portionality relationship between the column length and displacements seen in Fig. 29. Specimens close to the diameter in Eq. (4) is true. transition point with symmetric critical modes had sym- d L s s metric shape until they turned interactive soon after or just / ð4Þ d L f f before the ultimate load (Fig. 30). Thus, the FE model has been shown to be capable of where the subscripts ‘s’ and ‘f’ represent the small- and capturing the physical behaviour of the system, including full-scale columns, respectively. Dividing the second the load-carrying capacity, load-deformation response, moment of area of the small-scale I by the full-scale I s f yield point and deformed shapes and will therefore be in Eq. (5), results in a proportional relationship between utilised in performing a full geometrical scale comparison the second moment of area and the diameter to the power of the prestressed stayed steel columns’ behaviour. of four. 4 4 4 4 4 p d d d d Geometric Scale Comparison ðÞ o;s o;s i;s I i;s I d 4 4 s s s p d d ðÞ o;f i;f ¼ 64 ¼ ) / I 4 4 I d f f f d d o;f i;f Following validation of the numerical model, an investi- gation into the scaling effect on the behaviour of pre- ð5Þ stressed stayed columns was carried out. This was done by Lastly, dividing the critical buckling load of the small- investigating the behaviour of systems with varying geo- scale by the full-scale in Eq. (6) using the relationships metric scale up to that in the previous works [11]. All other from Eqs. (4) and (5) results in the same relationship for dimensions were scaled according to the ratio of the new the non-dimensional load-carrying capacity as obtained column length and the full-scale column length. In order to from the numerical modelling in Fig. 31. focus on the scaling effect, the slenderness of the small- 2 2 2 4D E I scale system was maintained irrespective of the geometric N s s I L L L u;s 2 s s f f 4D E I 2 f f ¼ f 2 2 scale as well as the cross-arm length ratios, relative 2 N L I L L L u;f f f f s s ð6Þ imperfection and prestress levels. Dimensions of the main N L u;s s columns for various geometric scales investigated are ) / N L u;f f presented in Table 11. The initial imperfection amplitude ratio was kept con- The trends for non-dimensional end shortening, mid- stant and the initial prestress for each system was kept to height displacement and mid-height rotation at ultimate the same proportion of T . The ratio of cross-arm to opt load highlight that there is no scaling effect on these column length was also kept constant for each column parameters as they follow a roughly linear trend. The minor Table 11 Dimensions and geometric scales of studied prestressed stayed columns, where the geometric scale is the length of each column L divided by the full-scale length L ,d and d are the outer and inner dimensions of the main column respectively f o i Geometric scale L/L 0.167 0.357 0.66 0.8 1 L (m) 0.468 1 1.848 2.24 2.8 d (mm) 7.4 15.9 29.4 35.6 44.5 d (mm) 6.3 13.4 24.8 30.0 37.5 Non-dimensional load-carrying capacity 622 J. Inst. Eng. India Ser. A (June 2022) 103(2):601–625 variation from the linear trend in these cases is due to (a) L/1000 1.2 uncertainties in the numerical model rather than any underlying physical behaviour. Following the conﬁrmation 0.8 that the behaviour of prestressed stayed columns can be scaled, existing design guidelines will be compared with FE data results from the numerical model for varying levels of 0.4 Proposed design geometrical scales to ascertain their applicability. expression 0 0.5 1 1.5 2 2.5 3 T/T opt Design Guidelines (b) L/400 1.2 Existing design guidance developed by the researchers [17] was used to calculate the maximum load-carrying capacity 0.8 for the conﬁgurations studied in the geometric scale com- parison section from Table 11. Results calculated using FE data 0.4 existing design guidelines were then compared with the Proposed design expression load-carrying capacity found through ﬁnite element mod- elling. It has been previously shown by the researches [28] 0 0.5 1 1.5 2 2.5 3 that the accuracy of the design expressions by earlier T/T opt investigators [17] for the L/400 and L/200 imperfection (c) L/200 1.2 levels is low. A similar trend was found in this section when comparing the load-carrying capacity from the design expressions and numerical model. Therefore, this 0.8 section attempts to update the design guidance for varying FE data imperfection levels using the numerical model. It should 0.4 Proposed design also be noted that the design guidance by [17] was devel- expression oped using a two-dimensional numerical model, whereas this study utilises a three-dimensional model. Several 00.5 11.5 22.5 3 T/T opt cross-arm lengths for each buckling mode were modelled to determine design curve trends across the range of cross- Fig. 32 Design curves for the symmetric critical mode with three arm lengths. Unlike the numerical model validation, a imperfection levels for cross-arm length ratio 2a/L = 0.05 distinct symmetric imperfection was used for the systems with symmetric critical modes as this has been shown to be Following the completion of the development of upda- the worst-case scenario [16]. Similarly, the worst-case ted design guidance, the expressions were then compared imperfection combination for the systems with an with existing guidance. The validated ﬁnite element antisymmetric critical mode was used, i.e. a combination of numerical model was used to determine the relative the two modes was used to induce interactive behaviour. improvement in the guidance. The experimental conﬁgu- Design curves for the symmetric critical mode for the three rations from Table 2 were scaled according to Table 11 and imperfection amplitudes are shown in Fig. 32, with the results from the numerical model for each imperfection updated guidance developed for the different zones of level are compared with existing and updated design behaviour shown in Table 12. guidance. Comparison of existing design guidance and the A similar procedure was repeated for the antisymmetric ﬁnite element model for the L/200 imperfection level is critical mode, with the design curves shown in Fig. 33, and shown in Fig. 34, while that of the updated guidance is updated design guidance in Table 13. shown in Fig. 35. A closer ﬁt between the numerical model Table 12 Updated design guidance for the normalised maximum load-carrying capacity (N /N ) for symmetric critical mode. Where the max subscripts z1, T and 3T refer to zone 1, the optimal prestress boundary between zone 2 and 3 and the upper boundary of zone 3 opt opt C C C Imperfection level (N /N ) (N /N ) (N /N ) max sym,z1 max Topt max 3Topt L/1000 11.0 (2a / L) ? 0.421 0.778 – 1.385 (2a / L) 0.944 – 1.03 (2a / L) L/400 9.41 (2a / L) ? 0.449 0.754 – 1.623 (2a / L) 0.872 – 1.376 (2a / L) L/200 8.13 (2a / L) ? 0.430 0.696 – 1.707 (2a / L) 0.786 – 1.611 (2a / L) N /N max c N /N max c N /N max c J. Inst. Eng. India Ser. A (June 2022) 103(2):601–625 623 Table 13 Updated design guidance for the normalised maximum load-carrying capacity (N /N ) for antisymmetric critical mode. Where the max subscripts z1, T and 3T refer to zone 1, the optimal prestress boundary between zone 2 and 3 and the upper boundary of zone 3 opt opt C C C Imperfection level (N /N ) (N /N ) (N /N ) max anti,z1 max Topt max 3Topt L/1000 1.502 (2a / L) ? 0.487 1.269 (2a / L) ? 0.348 0.284 (2a / L) ? 0.557 L/400 0.977 (2a / L) ? 0.473 0.785 (2a / L) ? 0.341 0.186 (2a / L) ? 0.434 L/200 0.716 (2a / L) ? 0.401 0.567 (2a / L) ? 0.284 0.040 (2a / L) ? 0.357 (a) L/1000 1.2 0.8 FE data 0.4 Proposed design expression 0 0.5 1 1.5 2 2.5 3 T/T opt 0 1020304050607080 (b) L/400 FE results (kN) 1.2 FE data Fig. 34 Comparison of FE and analytical load-carrying capacity Proposed design 0.8 results from existing design guidance, where the dashed line expression represents 100% ﬁt (imperfection level = L/200) 0.4 0 0.5 1 1.5 2 2.5 3 T/T opt (c)L/200 0.8 40 FE data Proposed design expression 0.4 10 0 1020304050607080 FE results (kN) 0 0.5 1 1.5 2 2.5 3 Fig. 35 Comparison of FE and analytical load-carrying capacity T/T opt results from updated design guidance, where the dashed line represents 100% ﬁt (imperfection level = L/200) Fig. 33 Design curves for the antisymmetric critical mode with three imperfection levels for cross-arm length ratio 2a/L = 0.3 ultimate load, with improvement in the other measures. Therefore, no signiﬁcant improvement is observed with the results and the design guidance is observed for the updated updated guidance for the L/1000 imperfection level. Con- design guidance. sequently, the updated guidance for the L/400 and L/200 A comparison of the updated guidance with existing imperfection levels are recommended for use, whereas guidance for all three imperfection levels is shown in existing guidance for the L/1000 imperfection level is still Table 14. Results obtained using the updated guidance for applicable for variation in geometric scale. the L/400 and L/200 imperfection levels show an improvement over existing guidance, with the average ratio of ultimate load from numerical modelling to design guidance being lower as well as a lower standard deviation/ COV and improved correlation. However, results obtained using the updated guidance for the L/1000 imperfection level show a higher average value of FE to design guidance N /N N /N N /N max c max c max c Analycal results (kN) Analycal results (kN) 624 J. Inst. Eng. India Ser. A (June 2022) 103(2):601–625 Table 14 Comparison of results obtained using numerical modelling model, existing guidance [17] and updated guidance respectively, Std. with updated guidance and existing guidance [17], where P , dev. is the standard deviation, COV is the coefﬁcient of variation and u,FE P and P are the ultimate loads from the numerical R is a measure of the correlation of the results u,Wadee u,updated P /P P /P P /P P /P P /P P /P u,FE u,Wadee u,FE u,updated u,FE u,Wadee u,FE u,updated u,FE u,Wadee u,FE u,updated (L/1000) (L/1000) (L/400) (L/400) (L/200) (L/200) Average 1.12 1.19 1.29 1.17 1.35 1.16 Std. dev 0.13 0.10 0.27 0.10 0.28 0.12 COV 0.11 0.09 0.21 0.08 0.21 0.10 R 0.97 0.98 0.95 0.98 0.93 0.98 It has been shown by comparing the load-carrying Conclusions capacities of corresponding conﬁgurations from this study and the full-scale experimental study by the research- The experimental campaign included tests on 17 small- ers [11] that small-scale systems can effectively model the scale prestressed stayed steel columns with varying cross- load-carrying capacity and post-buckling shape of large- arm lengths and initial prestress values as well as a control column with no cross-arms. This investigation aimed at scale systems. This comparison includes prestressed stayed steel columns with symmetric and antisymmetric critical studying systems close to the transition point for varying prestress to establish the optimal conﬁguration and to modes as well as systems close to the transition point for varying levels of prestress. The numerical model was also examine the effect of scale on the behaviour of prestressed stayed steel column systems. The scaling effect was validated against results from the experimental study and used to conduct a full geometric scaling comparison study. investigated through experimental investigations and numerical modelling. Furthermore, the applicability of It was found that small-scale systems are capable of replicating the load-deformation response of large-scale existing design guidance to other geometric scales is structures. studied. This study investigated various prestressed stayed steel Following the geometric scaling comparison study, the numerical model was then used to investigate the appli- column conﬁgurations close to the transition point between symmetric and antisymmetric modes. It was previously cability of existing design guidelines to systems with var- ious geometric scales. Existing design guidelines were suggested by the researchers [12] that the greatest efﬁ- ciency in terms of weight of materials can be found close to compared with results from the validated ﬁnite element model. It was shown that existing design guidelines had the transition point. This study found that conﬁgurations E3 and E4, which are either side of the transition point, gave low accuracy for small-scale systems. Therefore, updated design guidance was developed using the numerical model the greatest efﬁciency in terms of weight of materials, and shown to yield results closer to the numerical model conﬁrming the hypothesis by previous investigators [12]. for a range of system scales compared to existing design Furthermore, this highlights that there is no real beneﬁt in guidelines. designing prestressed stayed steel columns to have an antisymmetric critical mode signiﬁcantly higher than the Acknowledgements This work was carried out with assistance from symmetric as a lower efﬁciency is achieved. Also, it has technicians at the School of Engineering, the University of Aberdeen been shown that for systems close to the transition point, in assembling the specimens and equipment set up. with symmetric critical modes, that an increase in load- carrying capacity is obtained with prestress levels higher Funding This work was supported by the Engineering and Physical Science Research Council (EPSRC), UK Doctoral Training Partner- than T . Additionally, interactive behaviour is also pos- opt ship (grant number 1962441). sible before the ultimate load is achieved, although the exact cause is not yet clear. However, for systems close to Declarations the transition point with an antisymmetric critical mode, Conﬂict of interest The authors declare that there is no conﬂict of prestress levels lower than T have been shown to yield opt interest. the highest load-carrying capacity. These ﬁndings are applicable to prestressed stayed steel columns with a single Open Access This article is licensed under a Creative Commons cross-arm and further studies are required to verify if the Attribution 4.0 International License, which permits use, sharing, same conclusions can be drawn for more complex adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the conﬁgurations. source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this 123 J. Inst. Eng. India Ser. A (June 2022) 103(2):601–625 625 article are included in the article’s Creative Commons licence, unless 14. D. Saito, M.A. Wadee, Post-buckling behaviour of prestressed indicated otherwise in a credit line to the material. If material is not steel stayed columns. Eng. 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Journal of The Institution of Engineers (India):Series A – Springer Journals
Published: Jun 1, 2022
Keywords: civil engineering
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