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Influence of imperfections on the strength of unstiffened, fabricated, tubular beam-columns
J. Construct. Steel Research 25 (1993) 43-61
Influence of Imperfections on the Strength of Unstiffened, Fabricated, Tubular Beam-Columns S. Z. Hu Defence Research Establishment-Atlantic, PO Box 1012, Dartmouth, Nova Scotia, Canada B2Y 3Z7 &
H. G. L. Prion Department of Civil Engineering, University of British Columbia, Vancouver, BC, Canada V6T 1Z4 &
P. C. Birkemoe Department of Civil Engineering, University of Toronto, Toronto, Ontario, Canada MSS IA4
ABSTRACT Fabricated tubular steel members are used in offshore structures as primary and secondary members, typical applications in fixed structures incorporate large diameter tubes in truss-type assemblies. These unstiffened tubes are formed by rolling plate into cylindrical components which are then seam and oirth welded to create the circular tubular shape. The members produced in this fashion have characteristic imperfections which are different from those found in smaller diameter seamed or seamless hollow structural and pipe sections. These imperfections and their influence on the structural behaviour of beam-columns are the subject of the study and results presented here. Earlier experimental studies at the University of Toronto on the influence of fabrication parameters are the foundation of the results reported here. Extensive measurements of as-fabricated geometry provided information on actual shape and these measured geometries are used to establish the analytical models usin0 a finite element program, A N S YS; this program permits the modellin 0 of non-linear material
43 © 1993 Canadian Crown Copyright
44
S. Z. Hu, H. G. L. Prion & P. C. Birkemoe and geometric effects. The ultimate and post-ultimate local buckling behaviour was studied in the analyses to the same extent as observed in experiments. The load-displacement behaviour of tubular members depends significantly on the tube geometry and material properties. The discretization of the tube into elements is shown to require careful consideration so that potential deformation modes are not missed. Bilinear kinematic hardening material based on coupon tests of the 'as fabricated' steel, in combination with geometric modelling techniques yielded a behaviour response with a close resemblance to the experimental observations and measurements. The relative importance of various parameters is discussed and direct comparisons of experimental and analytical results with current code resistance formulations are made. The specific effects of residual strains are not addressed in this paper.
INTRODUCTION Large diameter tubular members, as used in fixed steel offshore structures, are usually not available from tube manufacturers but rather are fabricated from steel plate. Design codes have, in general, relatively stringent tolerance requirements L2 and the fabrication of these large members within these limits is challenging. The high cost of replacement, repair, and production losses, combined with the inability to perform reliable assessments of the safety related to the presence of specific imperfections, has resulted in conservative limits. In response to these concerns, a large scale experimental study on fabricated steel tubes was conducted at the University of Toronto 3-6 and was followed up by a finite element study 7 with the aim of modelling fabrication imperfections and studying their influence on the behaviour of such members. This paper summarizes the first phase of the analytical study which focused on the effect of geometric imperfections. The fabrication of large diameter tubular members is conducted by cold rolling the steel plate into cylindrical shapes which are subsequently welded along their longitudinal joints to form cans, the length of which is limited by the rolling width of the steel plate. Several cans are then welded together, end to end, to form members of the desired length. Despite stringent quality control, imperfections are introduced during these fabrication processes which affect the strength of the members. To minimize and distribute local imperfections, the seam welds on mating cans typically are staggered from 45 ° to 180° about the circumference; the resulting junctions of longitudinal and circumferential welds remain locations of stress concentrations and potential structural distress. Fabrication tolerances are stipulated in specifications L2 to limit the reduction of strength caused by such imperfections. Generally, three types of geometric imper-
Influence of imperfections on beam-column strength
45
fections are defined in various specifications, namely out-of-straightness, out-of-roundness and mismatch at welds. Another factor considered was that the stress-strain characteristics of the steel are changed in the cold-rolling process. The cold working during the forming process changes the stress-strain characteristics beyond the proportional limit to a non-linear relationship with gradually declining stiffness and strain hardening; the resulting mechanical characteristics are those of cold formed steel.
OBJECTIVE The objective of the study reported here was to develop a reliable analytical method to assess the effects of fabrication imperfections on the strength and behaviour of fabricated steel tubes, using generally available software tools. An extensive testing program on large scale tubular members at the University of Toronto 3-6 provided the experimental data base which enabled direct comparisons and evaluation of the numerical procedures. The tested tubes had a cross-sectional slenderness, D/t, which falls into a category for which local buckling could be expected at stress levels at or beyond the onset of first yielding in compression. Several specimens were tested as stub-columns with the aim of determining the effect of imperfections on the squash load, whereas others were subjected to axial and bending loads, some of which also included the effect of overall column buckling. As part of the experimental program, complete records of the geometric profiles of test specimens were taken. The analytical study reported here forms part of a comprehensive numerical investigation of fabricated steel tubes. 7'a The objective of the first phase was to incorporate specimen profile data and material properties to model the behaviour of fabricated tubes under flexural and axial loading. The numerical investigations were carried out using the commercial finite element program ANSY. 9,~° An incremental load--displacement analysis, including large displacements and material non-linearities, was employed to establish the pre- and post-ultimate load-deformation characteristics. The bifurcation of equilibrium was not considered in the analytical work since it was considered unlikely for tubes with significant imperfections and it was not observed in the experiments. During the fabricating process, imperfections are created that can be categorized as being either in the longitudinal direction, affecting the cross-section of the cylinder, or localized at the abutting ends of cans.
46
s, z . Hu, H. G. h Prion & P. C. Birkemoe
Generally, three types of geometric imperfections are defined in various specifications, namely out-of-straightness, out-of-roundness and mismatch at welds. These parameters were used to categorize various effects in the results which follow.
DESCRIPTION OF SPECIMENS The experimental study was done on tubes with a nominal inside diameter of 430 ram, wall thickness ranging from 4-5 mm to 9 mm and lengths of 1.5 m and 10.0m. Because of the global complexity of the problem, this phase of the analytical study was limited to short columns. The purpose of the 1-5 m short column in the experimental program was to determine the cross-sectional strength and behaviour of a fabricated column subjected to concentric and eccentric loads without the interactive effects of local and column buckling. The test setup of an eccentrically loaded short column is shown in Fig. 1. The steel plate used to produce the experimental specimens was sheared or flame cut to the required dimensions and cold formed in a rolling press to form cylindrical tubes. The members were fabricated using automatic
Uniaxial Rouktiom~Head
LongitudinalLVDT Seam Weld
Girth Weld
Aluminum She~
e m 50ram
Fig. 1. Experimentalset-up for eccentricallyloaded short column.4
Influence of imperfections on beam-column strength
47
and manual welding, staggering the longitudinal welds at 90 ° to follow typical practice. The axial deformation of the specimens was measured on four locations over a gauge length of 1200 ram, as shown in Fig. 1. The average value of these four readings is regarded as the axial shortening of the specimen in the load--displacement comparisons which follow. Four specimens with different wall thickness, material properties and loading eccentricities were chosen for comparison with analytical results. The geometric and material properties of these specimens are summarized in Table 1. Test results showed that the ultimate load of most specimens was reached with concurrent elastic-plastic buckling distortions of the cross-section in the vicinity of the girth weld.
TABLE 1 Geometric and Material Properties of the Models
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
Spec. No.
t (nun)
r (ram)
e (ram)
D/t
No.
Map
Girth Weld
~y MPa
E, MPa
1 2 3 4 5 6
i i • e e e
C S1 C Sl S1 $2
266 266 266 254* 266 266
4200 4200 4200 O* 4200 4200
ST7
4.5
223.2
0
100
ST6
6.5
222.6
25
70
1 2
i e
C C
302 302
3200 3200
STI2
4.5
223'2
50
100
1
e
C
266
4200
ST13
7.3
218.7
0
60
1
e
C
450
0
(1) Specimen designation, Ref. [4]; (2) wall thickness; (3) average measured mid-wall radius; (4) load eccentricity; for e = 0, loading plattens were fixed against rotation; (5) diameter to thickness ratio; (6) finite element model number;, (7) model parameters: T and 'e' represent girth weld reinforcement included or excluded in the mapping. 'C' indicates that the girth region is modelled by constraint equations; 'SI' and '$2' represent shell element modelling: 'SI' has a thickness of 6 mm with yield strength of 560 MPa, and '$2' has thickness of 4.5 mm with yield strength of 266 MPa; (8) material properties: ey and E, are taken from tensile coupon results of the formed member material except '*' indicates the property is from a coupon test of virgin plate. Elastic modulus and Poisson's ratio equal to 205 000 MPa and 0-3 respectively were used.
48
S. Z. Hu, H. G. L. Prion & P. C. Birkemoe
FINITE ELEMENT MODEL
Tubular geometry For each specimen, either 50, 72 or 74 profiling data sets were recorded, at 30 mm or 20 mm spacing along the length; each set contains 72 radiusversus-angle readings for a full circumferential profile. To reduce the large number of profile coordinate points, a simple linear interpolation scheme was chosen to generate the finite element nodal points at the required positions. In consideration of the wave front size and realistic computational time, 48 nodal points around the circumference, which resulted in an interval of approximately 30 mm, were generated. Twenty-six finite element nodal points were selected in the longitudinal direction for each can, also at 30 mm intervals. The nodal points for the top and bottom can of the specimen were generated individually, starting at the tube edge and progressing towards the girth weld location, where the real and modelled mismatch occurs. The peak magnitude of the measured mismatch was on the order of ~ of the mean diameter. In the experimental program, it was found that the initiation of local buckling occurred at or close to the girth weld location. In the close proximity of the girth weld, 180mm to either side, a fine mesh of 30 mm x 30 mm elements was used in recognition of the anticipated local buckling behaviour. Towards both ends of the tube, where no significant deformation occurred during the test, larger elements were used. The four-node plastic quadrilateral shell element, 11'12 a member of the family of degenerate isoparametric shell elements, was used to model the tube. Triangular elements, formed by suppressing the last node of the quadrilateral element, were used in the transition zone between the coarse and fine meshes. The development of the finite element model from the test profile measurements for specimen ST-12 is illustrated in Fig. 2. The mismatch at the girth weld jointing the two cans constituted a discontinuity in thr. structure which locally influenced the initiation of local buckling and 'ailure. Two types of modelling were used to represent this mismatch reg! ,n as shown in Fig. 3. First, the girth weld was treated as part of the tut- , modelled with a narrow band of shell elements as an extension of the be wall. The thickness and material properties of these shell elements w ,', in some cases, chosen to be the same as the tube itself; in other cases the actual weldment properties and thickness were used. Secondly, the assumption was made that the nodal points on either side of the cans' juncture were rigidly compatible; constraint equations were used to provide this compatibility.
Influence of imperfections on beam-column strength
Di.un'eteModel
Can Assembly
Fig. 2. Modelling of measured can geometry for finite element analysis.
J
Shell Elements Model "S"
Model "C"
Fig. 3. Modelling of mismatch and weld at junction of cans.
49
50
S. Z. Hu, H. G. L. Prion & P. C. Birkemoe
The profiling data set which represented coordinates on the ridge of the girth weld reinforcement was included in some of the models, but it was found to create a large misrepresented mismatch. It was included in the parametric study to demonstrate the importance of this aspect.
Material modelling The classic bilinear kinematically hardening material was used in modelling the material. The material characteristics were described in the form of a bilinear stress-strain curve, starting at the origin, and with positive stress and strain values; the initial slope of the curve was taken as the elastic modulus of the steel (205 000 MPa). After reaching the specified yield stress, the curve continued along a second slope which was defined by the strain-hardening modulus. Two types of stress-strain curve were used in the analysis: (1) elastic-perfectly plastic material representing virgin plate test coupons, (2) elastic-plastic kinematically hardening material, representing the steel properties after cold rolling fabrication. The yield stress and strain hardening modulus were taken from tensile coupon tests from the fabricated tubes.
Boundary conditions Elastic shell elements with increased thickness were used at the ends of the tube to create a stiff loading fixture similar to that used experimentally for the eccentrically loaded short beam-columns (ST12 and ST6). Specified displacements parallel to the longitudinal axis were imposed at eccentric loading points on the top. The corresponding points on the bottom can were allowed to rotate. The concentrically loaded short columns, ST7 and ST13, were tested with ends fixed against rotation. In this case, a constant displacement boundary condition, with rotational and lateral displacements restricted, was employed in the model.
Model parameters For clarity, every analysis was given a specific designation, consisting of the specimen number as given in the experimental reports, followed by a single digit number. Six analyses were performed on specimen ST7, two analyses on specimen ST6 and one each on specimens ST12 and ST13. A summary of the model parameters is tabulated in Table 1. The first two analyses of the specimen ST7 (ST7-1 and ST7-2) were performed on the model with the inclusion of the girth weld measurements in the mapping process. The cans of model ST7-1 were joined by the
Influence of imperfections on beam-column strength
51
constraint equations, whereas the cans of model ST7-2 were joined by a narrow band of thicker shell elements. The elastic-perfectly plastic stressstrain relation with a yield stress of 560 MPa, from weld coupon tests, was used in these elements. The other analyses (ST7-3, ST7-4, ST7-5 and ST7-6) were performed with the exclusion of the girth weld measurements. The model ST7-C was otherwise identical to ST7-1, for which the girth welds were modelled by constraint equations. The girth weld was modelled by shell elements for ST7-4, ST7-5 and ST7-6. These shell elements had a thickness of 6 mm with elastic-perfectly plastic material properties of yield stress 560 MPa for ST7-4 and ST7-5. For specimen ST7-6, a thickness of 4.5 mm with kinematically hardening material with a yield stress of 266 MPa and strain hardening modulus of 4200 MPa were assigned to these elements. The main body of ST7-4, which differed from other models, was composed of an elastic-perfectly plastic material with a yield stress of 254 MPa. Two analyses were performed on specimen ST6. The first run, ST6-1, was performed on the model with the inclusion of the girth weld measurements in the mapping process. The second run, ST6-2, utilized the same finite element model, but with the exclusion of the girth weld measurements. The mismatch at the junction of the cans of both analyses was modelled by constraint equations. Specimens ST12 and ST13 were analysed without the inclusion of the girth weld measurement and were named ST12-1 and ST13-1. The junction between the cans was modelled by constraint equations. Material properties were obtained from coupon results unless otherwise indicated.
RESULTS O F T H E N U M E R I C A L ANALYSIS 1. ST7-1: In the experimental study of specimen ST7, five alternating buckles formed around the circumference at the girth weld. Although the inclusion of girth weld measurement in the mapping process created a relatively large mismatch between cans, the final deformed shape, Fig. 4, of ST7-1 compared well with those observed in the experiments. The comparison of the load-displacement curve with that obtained from the test is shown in Fig. 5(a); the similarity in slope in the post-ultimate region is evident. 2. ST7-2: Thicker shell elements with higher material strength were used in the modelling of the girth weld instead of using constraint equations. A completely different deformation pattern was obtained and no sudden buckling was observed (Fig. 4). The load-displacement curve, as shown in Fig. 5(b), indicates a very flat plateau following the elastic portion. No obvious unloading behaviour was observed.
52
S. Z. Hu. H. G. L. Prion & P. C. Birkemoe
Top
South
B m g i mi O i m O Wi
'0 0
O
m
Fig. 4. Deformed geometry of finite element models for ST7.
3. ST7-3: The girth weld measurements were excluded in the mapping process and the only difference from ST7-1 was the size of the mismatch. A comparison of the load-displacement curve with the test result is shown in Fig. 5(a). The unloading part is stiffer than observed in the test. Instead of an alternating diamond buckling pattern, the pattern as shown in Fig. 4 was obtained analytically. 4. ST7-4: Thicker shell elements with high yield stress material (Fy = 560 MPa) were used to model the girth weld. The elastic-perfectly plastic material was used in the modelling of the main body of the tube. The load--displacement behaviour showed a linear elastic slope followed by a sudden load drop as unloading occurred (Fig. 5(c)). No buckling deformation occurred at the junction of the cans, but slight bulging occurred in the bottom can, 180 m m below the girth weld.
53
Influence of imperfections on beam-column strength 2000.
200O
1500
"o
1"7"2
3
°°°
jT:ST
lS00
-5
1000
5OO
50C
0
0
15
10
5
O~mmt
0
&L {mrn)
. . . . . . . . . . . . . .
5 10 Axild Di=p~emont AL (ram)
(a)
15
(b)
-
1500 ¸
TEST
t
~
~
l ST73 ST7-6
•o 100(] o
....
-~ ....
15
Ib ....
Axial ~
AL (ram)
(c)
0
. . . . . . . . . . . . . .
5 10 Axial Disp~a~m~n~ AL |ram}
15
(d)
150(
lO= 50(;
C~
0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
5 10 Axial D~,plocsment &L (ram)
15
(e)
Fig. 5. Load-displacement behaviour of concentrically-loaded specimen ST7 compared with finite element models. 5. ST7-5: The difference between ST7-5 and ST7-4 was that a kinematically hardening material was used in ST7-5 to model the main body of the tube. A completely different behaviour was observed in both the load--displacement curve and the deformed shape. The load-displacement curve of ST7-5 has a flat plateau following the elastic portion as shown in Fig. 4. No obvious sudden unloading was observed. The display of the deformed shape shows no local change in shape at the girth weld junction.
S. Z. Hu, H. G. L. Prion & P. C. Birkemoe
54
6. ST7-6: The thickness and material properties of shell elements representing the girth weld were changed in this analysis. The thickness of the shell elements at the girth weld was changed from 6 mm (in ST7-5) to 4.5 mm (equal to the wall thickness of the main body). The material properties were also changed from the higher yield stress of 560 MPa to 266 MPa (equal to the yield stress of the material of the main body). A substantial change in the behaviour, both in the load-displacement curve and the deformed shape, occurred. The load started to drop along with the occurrence of large deformation as shown in Fig. 5(e). Both the loaddisplacement response and the deformed shape are very similar to the results of ST7-3 as shown in Fig. 5(d) and Fig. 4. 7. ST6-1: The girth weld measurements were included in the interpolating scheme of model ST6-1. An asymmetric pair of buckles formed on the concave side of the tube at the junction of the cans. The deformed shape, as well as the load-displacement response shown in Fig. 6, show good agreement with the test results. 8. ST6-2: The girth weld measurements were again excluded in the mapping process. A similar deformed shape was obtained as in the case of ST6-1, the ultimate load was slightly higher and the load--displacement curve varied slightly. The deformed shape and the load--displacement curve, Fig. 6, also agreed well with the test results. 9. ST12-1: A symmetric alternating set of buckles, with the initial buckles in the region of maximum compression, was formed as shown in Fig. 7, and were similar to the test observations. The load--displacement curve shows that the finite element model exhibited a stiffer post-ultimate response than the test specimen. 10. ST13-1: In comparison with other specimens that were analysed, ST13-1 had a much higher yield stress. The load-displacement curve had a
•
~'l
Jelll 2
oI
§
I
I
I
2 4 8 AXIAL ~ E N T
1
8
I
10 12 AL (mm)
Fig. 6. Behaviour of eccentrically-loaded specimen ST6 compared with finite element models,
Influence of imperfections on beam-column strenoth
I_
e
55
I..-¥~t
1200f~
ST12.1
0~ / II4 / 18 lb , AXIALDISPLACEMENT AL(mm) Fig. 7. Behaviour of eccentrically-loadedspecimen ST12 compared with finite element model.
5O0O 4000
i o/" °
200O
1000 S 10 15 AXIALDISPLACEMENT AL(ram) Fig. & Behaviour of concentrically-loadedspecimen ST12 compared with finite element model. linear elastic slope followed by a sudden drop as unloading occurred. In the comparison (Fig. 8), the curves for the test specimen and the finite element model have very similar slopes in the elastic and in the unloading regions.
COMPARISON
WITH PLASTIC INTERACTION EQUATIONS
The Canadian code for fixed offshore structures 2 takes account of local buckling under compressive stress by substituting the yield stress with a reduced value which depends on the diameter-to-thickness (D/t) ratio and the specified minimum yield stress, Fy, of the steel. These local buckling provisions were based on recommendations by Plantema ~3 and Sherman. ~4 The axial load resistance using these relationships is calculated as:
(1)
56
S. Z. Hu, H. G. L. Prion & P. C. Birkemoe
where F¢ is the reduced compressive stress as recommended by Plantema I 3 F¢ = f y
Fc = F, [0.75 + 6200/(Fy (O/t)] Fc = Fy [66 O00/(D/t)]
for F~(D/t) ~ 25 000 for 25 000 < Fy(D/t) <<.80 000 for 80 000 < Fy(D/t) ~ 360Fy
The bending resistance is given as
Mr=ckMu
(2)
where M, is the reduced moment resistance as recommended by Sherman ~4 M u .~- M p
M, = Mp [ff775 + 3200/(Fy(D/t)] Mu = Mp [51 O00/(D/t)]
for Fy(D/t) <<.14 200 for 14 200 < Fy(D/t) <<.621300 for 62 000 < Fy(D/t) <<.360Fy
To determine the cross-sectional strength of a member at any location, an interaction formula that does not take account of the column stability is used. This strength interaction relationship, based on the theoretical derivation of the plastic resistance of a cylindrical section, is
Mt E,cl ~
~=cos
(3)
where Mt is the total moment at the section. In Table 2, the results of finite element models ST6-2, ST12-1, ST7-6 and ST13-1 are compared with the test results and the calculated results of the cross-sectional strength according to eqns (1), (2) and (3). Because of the small second-order moment for the short members, Mt in this equation is replaced by the product of the axial load C and its eccentricity, e. Local buckling is accounted for in the evaluation of Cr and M, in which the resistance factor is set equal to 1"0. From Table 2 it can be seen that a good correlation exists between the experimental results, the analytical results, and the empirical design equations for the level of imperfection and material strengths examined.
DISCUSSION OF THE RESULTS A finite element model has been shown to provide a good approximation of the behaviour of an imperfect tubular member. In the analytical study, it
Influence of imperfections on beam-column strenoth
57
TABLE 2 Comparison with Plastic Interaction Equations 2
Spec. No.
Fc (MPa)
ST6 STI2 ST7 ST13
304 262 262 441
C, M, C eqn (1) eqn (2) eqn (3) (kN) (kN m) (kN) 2764 1654 1654 4424
364 214 N/A N/A
2465 1323 1654 4424
C.u test (kN)
Cuu FE (kN)
Test C
FF C
FE test
2596* 1360 1743" 4424
2425 1330 1630 4395
1'05 1"03 1"05 1"0
0"98 1"0 0"99 0"99
0"93 0"98 0"94 0"99
Cr and M, are obtained from eqns (1) and (2) respectively for, O= 1.0. C is calculated using Mt = C x e. Asterisk indicates that the test results of ST6 and ST7 are higher than the calculated squash load using the tensile coupon properties.
was found that the load-displacement behaviour of tubular members very much depends on the geometry of the tube and is sensitive to the material properties, the initial imperfections and residual stresses, Discretization of a tube into elements thus requires careful consideration since some of the possible deformed modes may be missed in the analysis. The only geometric differences between models ST7-1 and ST7-3, and models ST7-2 and ST7-5 were the size and shape of the mismatch, yet completely different load-displacement responses and deformed shapes were observed. These results indicate the importance of the mismatch modelling on the behaviour of the tube. A comparison of the buckling deformations indicates that the finite element mesh size may not have been fine enough to describe the distortion at the girth weld of STT. Consequently the mesh size may be the cause of the discrepancy in the buckling mode from the observed experimental deformed shape. Classic linear buckling analyses of a concentrically loaded circular tube Is indicate that variations in the buckling loads for buckling modes near the observed mode are small. Thus, with random imperfections present, differing mode shapes may be expected although the loads may show good agreement. The results of analyses ST7-3 and ST7-6 show very similar buckling modes and very similar load-displacement curves. The use of constraint equations to model the girth weld (ST7-3) gave a reasonable method to approximate the behaviour of the central region. In the case where the girth weld was modelled by shell elements, however, the element thickness and material properties were found to have a significant influence on the behaviour of the tube. The results of the analyses of ST7-4 and ST7-5 illustrate how the strain hardening modulus caused a significant difference between the load-
58
S. Z. Hu, H. G. L. Prion & P. C. Birkemoe
displacement response of the tested specimens and the finite element method. In addition, the deformed shapes varied depending on the value of the strain hardening modulus. The method of using shell elements with higher yield strength and greater thickness to model the junction of the cans did not give satisfactory results in the analyses of ST7-2, ST7-4 and ST7-5. These elements likely stiffened the centre and led to a change in deformed shape from one mode to another. Consequently, buckling modes observed in the experiments were prevented. On the other hand, it was found that by modelling the joint with shell elements of thickness and material properties identical to the main body of the tube (ST7-6), good agreement with the experiments was obtained.
S U M M A R Y AND C O N C L U S I O N S The conclusions deal with the topics of finite element modelling and with observations of the modelled and experimental behaviours of tubular members. The finite element method has been shown to be a valid analytical approach for the study of behaviour of imperfect steel tubular members. It has been clearly demonstrated that the element availability and sophistication of state-of-the-art finite element programs requires careful development and verification of models. Several observations summarizing the findings suggest topics for further study and refinement.
Finite element modelling and member behaviour (1) The numerical modelling of fabricated tubes with non-linear shell elements, as provided in the commercial finite element program ANSYS, successfully simulated the local buckling behaviour of eccentrically loaded tubular members with initial imperfections (ST6 and ST12). The analytically determined buckled shape did not consistently correlate with experimental observations. (2) The ultimate loads obtained in the analyses were quite close to the experimentally obtained values (ST12 and ST13); others (ST6 and ST7) underestimated the experimental values as shown in Table 2. It should be kept in mind that the experimentally obtained ultimate loads for ST6 and ST7 were higher than the calculated squash load based upon coupon properties. The quantitative comparisons of the analytically obtained ultimate load with the resistance as stipulated in the Canadian Standards for Fixed Offshore Structures, 2 on the other hand, showed very good agreement.
Influence of imperfections on beam-column strenoth
59
(3) The geometric modelling of the girth weld was found to be a very important factor in the finite element study. The best correlation with experimental results was achieved by using constraint equations or by using connecting shell elements with the same thickness and material properties as the main body of the tube. The use of thicker elements and/or stronger material properties typically caused a change in the buckling mode and consequently altered the load-displacement response. (4) The material properties of the finite elements played a very important role in the solution. The elastic-perfectly plastic stress-strain relationship, obtained from tension coupon tests of the virgin plate, gave unsatisfactory results. A bilinear, kinematically hardening, material relationship, based on tests of tension coupons cut from the tube after cold rolling, resulted in load--displacement responses from the analytical models which closely resemble the experimental observations. (5) The mismatch at abutting ends was found to be the principal parameter to initiate local buckling and resulted in a reduction of the cross-sectional strength for the tubes studied. The influences of outof-straightness and out-of-roundness on the sectional ultimate strength were found to be negligible. The deformed shape which develops during the post-ultimate behaviour depends very strongly on the size and shape of the mismatch and likely affects the post-ultimate load-displacement response. (6) The fluctuations in the load-displacement response of the finite element model in the vicinity of the ultimate load can be avoided by using smaller load increments. It should be realized, however, that such a refinement also implies a significant increase in computing time and cost. Further research
The mismatch tolerances as specified in current codes need to be reexamined. These are typically given as a maximum value, expressed as a percentage of the wall thickness, without reference to the extent to which they may occur around the circumference. For the analysed specimens, for example, the specifications were violated over only a small portion of the circumference. It is felt that not only the magnitude, but also the extent of a large mismatch with respect to the circumference, plays an important role in the formation of local buckles. Further studies are needed to determine the effect of the shape of the mismatch, combined with its extent. As more results become available, the tolerances and the interpretation of the magnitude, extent and location of mismatch can be defined more specifically and realistically in specifications for offshore structures.
60
s. z. Hu. H. G. L. Prion & P. C. Birkemoe
Measurements of imperfections for longer cylinders (5 m and 10m) included in the experimental program were also reported in detail. A finite element model similar to the one discussed in this paper should be developed to investigate the behaviour of long specimens which typically fail as a result of the interaction of local and overall buckling. With the current techniques, however, a significant increase in computational costs would result. Residual stresses are very much a part of the total behaviour of any buckling and yielding scenario and they have not been discussed here. The effects of welding residual stresses on the conditions which cause local buckling and on the strength of fabricated tubes must be added to the modelling for a complete evaluation of cross-sectional member behaviour.
ACKNOWLEDGEMENTS The authors wish to acknowldge the support provided through various grants, and fellowships by the Natural Sciences and Engineering Research Council of Canada and the University of Toronto.
REFERENCES I. API Specification for Fabricated Structural Steel Pipe, Spec 2B, 3rd edn, American Petroleum Institute, Washington, DC, Nov., 1977. 2. Canadian Standards Association, Code for the Design, Construction and Installation of Fixed Offshore Structures, Preliminary Standard $473, Dec. 1989. 3. Sato, J. A., The compression behaviour of unstiffened fabricated tubes. MASc thesis, Department of Civil Engineering, University of Toronto, 1985. 4. Prion, H. G. L. & Birkemoe, P. C., Experimental behavior of unstiffened fabricated tubular steel beam-columns. Department of Civil Engineering, University of Toronto, Report No. 88-03, 1991. 5. Birkemoe, P. C., Prion, H. G. L. & Sato, J. A., Compression behaviour of unstifl'ened fabricated steel tubes. Pre-print; ASCE Annual Convention and Structures Congress, Houston, Texas, Oct., 1983. 6. Prion, H. G. L. & Birkemoe, P. C., Beam-column behaviour of fabricated steel tubular members. J. Struct. Enono. ASCE, 118(5) (1992) 1213-32. 7. Hu, S. Z., An analytical investigation of the compressive behaviour of fabricated steel tubes. Ph D Dissertation, Department of Civil Engineering, University of Toronto, 1991. 8. Hu, S. Z., Birkemoe, P. C. & Prion, H. G. L., Analysis of a fabricated cylindrical member of ANSYS. Proceedinos, Supercomputino Symposium '89, Ontario Centre for Large Scale Computation, Toronto, 1989, pp. 411-21. 9. A N S Y S User's Manual 4"4. Swanson Analysis System, Inc. Houston, 1989.
Influence of imperfections on beam-column strength
61
10. ANSYS Theoretical Manual 4"4. Swanson Analysis System, Inc., Houston, 1989. 11. The Four Noded Plastic Shell Element (STIF43). Swanson Analysis System, Inc., Houston, 1989. 12. Bathe, K. J. & Dvorkin, E. N., A formulation of general shell elements--the use of mixed interpolation of tensorial components. Int. J. Num. Meth. En#ng, 22 (1986) 697-722. 13. Plantema, F. J., Collapsing stresses of circular cylinders and round tubes. Report S.280, Nat. Luchtvaart laboratorium, Amsterdam, The Netherlands, 1946. 14. Sherman, D. R., Bending equations for circular tubes. Proceedings: SSRC Annual Technical Session, Cleveland, OH, April, 1985, pp. 251-62. 15. Timoshenko, S. P. & Gere, J. M., Theory of Elastic Stability, 2nd edn. McGraw-Hill, New York, 1961.
Influence of Imperfections on the Strength of Unstiffened, Fabricated, Tubular Beam-Columns S. Z. Hu Defence Research Establishment-Atlantic, PO Box 1012, Dartmouth, Nova Scotia, Canada B2Y 3Z7 &
H. G. L. Prion Department of Civil Engineering, University of British Columbia, Vancouver, BC, Canada V6T 1Z4 &
P. C. Birkemoe Department of Civil Engineering, University of Toronto, Toronto, Ontario, Canada MSS IA4
ABSTRACT Fabricated tubular steel members are used in offshore structures as primary and secondary members, typical applications in fixed structures incorporate large diameter tubes in truss-type assemblies. These unstiffened tubes are formed by rolling plate into cylindrical components which are then seam and oirth welded to create the circular tubular shape. The members produced in this fashion have characteristic imperfections which are different from those found in smaller diameter seamed or seamless hollow structural and pipe sections. These imperfections and their influence on the structural behaviour of beam-columns are the subject of the study and results presented here. Earlier experimental studies at the University of Toronto on the influence of fabrication parameters are the foundation of the results reported here. Extensive measurements of as-fabricated geometry provided information on actual shape and these measured geometries are used to establish the analytical models usin0 a finite element program, A N S YS; this program permits the modellin 0 of non-linear material
43 © 1993 Canadian Crown Copyright
44
S. Z. Hu, H. G. L. Prion & P. C. Birkemoe and geometric effects. The ultimate and post-ultimate local buckling behaviour was studied in the analyses to the same extent as observed in experiments. The load-displacement behaviour of tubular members depends significantly on the tube geometry and material properties. The discretization of the tube into elements is shown to require careful consideration so that potential deformation modes are not missed. Bilinear kinematic hardening material based on coupon tests of the 'as fabricated' steel, in combination with geometric modelling techniques yielded a behaviour response with a close resemblance to the experimental observations and measurements. The relative importance of various parameters is discussed and direct comparisons of experimental and analytical results with current code resistance formulations are made. The specific effects of residual strains are not addressed in this paper.
INTRODUCTION Large diameter tubular members, as used in fixed steel offshore structures, are usually not available from tube manufacturers but rather are fabricated from steel plate. Design codes have, in general, relatively stringent tolerance requirements L2 and the fabrication of these large members within these limits is challenging. The high cost of replacement, repair, and production losses, combined with the inability to perform reliable assessments of the safety related to the presence of specific imperfections, has resulted in conservative limits. In response to these concerns, a large scale experimental study on fabricated steel tubes was conducted at the University of Toronto 3-6 and was followed up by a finite element study 7 with the aim of modelling fabrication imperfections and studying their influence on the behaviour of such members. This paper summarizes the first phase of the analytical study which focused on the effect of geometric imperfections. The fabrication of large diameter tubular members is conducted by cold rolling the steel plate into cylindrical shapes which are subsequently welded along their longitudinal joints to form cans, the length of which is limited by the rolling width of the steel plate. Several cans are then welded together, end to end, to form members of the desired length. Despite stringent quality control, imperfections are introduced during these fabrication processes which affect the strength of the members. To minimize and distribute local imperfections, the seam welds on mating cans typically are staggered from 45 ° to 180° about the circumference; the resulting junctions of longitudinal and circumferential welds remain locations of stress concentrations and potential structural distress. Fabrication tolerances are stipulated in specifications L2 to limit the reduction of strength caused by such imperfections. Generally, three types of geometric imper-
Influence of imperfections on beam-column strength
45
fections are defined in various specifications, namely out-of-straightness, out-of-roundness and mismatch at welds. Another factor considered was that the stress-strain characteristics of the steel are changed in the cold-rolling process. The cold working during the forming process changes the stress-strain characteristics beyond the proportional limit to a non-linear relationship with gradually declining stiffness and strain hardening; the resulting mechanical characteristics are those of cold formed steel.
OBJECTIVE The objective of the study reported here was to develop a reliable analytical method to assess the effects of fabrication imperfections on the strength and behaviour of fabricated steel tubes, using generally available software tools. An extensive testing program on large scale tubular members at the University of Toronto 3-6 provided the experimental data base which enabled direct comparisons and evaluation of the numerical procedures. The tested tubes had a cross-sectional slenderness, D/t, which falls into a category for which local buckling could be expected at stress levels at or beyond the onset of first yielding in compression. Several specimens were tested as stub-columns with the aim of determining the effect of imperfections on the squash load, whereas others were subjected to axial and bending loads, some of which also included the effect of overall column buckling. As part of the experimental program, complete records of the geometric profiles of test specimens were taken. The analytical study reported here forms part of a comprehensive numerical investigation of fabricated steel tubes. 7'a The objective of the first phase was to incorporate specimen profile data and material properties to model the behaviour of fabricated tubes under flexural and axial loading. The numerical investigations were carried out using the commercial finite element program ANSY. 9,~° An incremental load--displacement analysis, including large displacements and material non-linearities, was employed to establish the pre- and post-ultimate load-deformation characteristics. The bifurcation of equilibrium was not considered in the analytical work since it was considered unlikely for tubes with significant imperfections and it was not observed in the experiments. During the fabricating process, imperfections are created that can be categorized as being either in the longitudinal direction, affecting the cross-section of the cylinder, or localized at the abutting ends of cans.
46
s, z . Hu, H. G. h Prion & P. C. Birkemoe
Generally, three types of geometric imperfections are defined in various specifications, namely out-of-straightness, out-of-roundness and mismatch at welds. These parameters were used to categorize various effects in the results which follow.
DESCRIPTION OF SPECIMENS The experimental study was done on tubes with a nominal inside diameter of 430 ram, wall thickness ranging from 4-5 mm to 9 mm and lengths of 1.5 m and 10.0m. Because of the global complexity of the problem, this phase of the analytical study was limited to short columns. The purpose of the 1-5 m short column in the experimental program was to determine the cross-sectional strength and behaviour of a fabricated column subjected to concentric and eccentric loads without the interactive effects of local and column buckling. The test setup of an eccentrically loaded short column is shown in Fig. 1. The steel plate used to produce the experimental specimens was sheared or flame cut to the required dimensions and cold formed in a rolling press to form cylindrical tubes. The members were fabricated using automatic
Uniaxial Rouktiom~Head
LongitudinalLVDT Seam Weld
Girth Weld
Aluminum She~
e m 50ram
Fig. 1. Experimentalset-up for eccentricallyloaded short column.4
Influence of imperfections on beam-column strength
47
and manual welding, staggering the longitudinal welds at 90 ° to follow typical practice. The axial deformation of the specimens was measured on four locations over a gauge length of 1200 ram, as shown in Fig. 1. The average value of these four readings is regarded as the axial shortening of the specimen in the load--displacement comparisons which follow. Four specimens with different wall thickness, material properties and loading eccentricities were chosen for comparison with analytical results. The geometric and material properties of these specimens are summarized in Table 1. Test results showed that the ultimate load of most specimens was reached with concurrent elastic-plastic buckling distortions of the cross-section in the vicinity of the girth weld.
TABLE 1 Geometric and Material Properties of the Models
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
Spec. No.
t (nun)
r (ram)
e (ram)
D/t
No.
Map
Girth Weld
~y MPa
E, MPa
1 2 3 4 5 6
i i • e e e
C S1 C Sl S1 $2
266 266 266 254* 266 266
4200 4200 4200 O* 4200 4200
ST7
4.5
223.2
0
100
ST6
6.5
222.6
25
70
1 2
i e
C C
302 302
3200 3200
STI2
4.5
223'2
50
100
1
e
C
266
4200
ST13
7.3
218.7
0
60
1
e
C
450
0
(1) Specimen designation, Ref. [4]; (2) wall thickness; (3) average measured mid-wall radius; (4) load eccentricity; for e = 0, loading plattens were fixed against rotation; (5) diameter to thickness ratio; (6) finite element model number;, (7) model parameters: T and 'e' represent girth weld reinforcement included or excluded in the mapping. 'C' indicates that the girth region is modelled by constraint equations; 'SI' and '$2' represent shell element modelling: 'SI' has a thickness of 6 mm with yield strength of 560 MPa, and '$2' has thickness of 4.5 mm with yield strength of 266 MPa; (8) material properties: ey and E, are taken from tensile coupon results of the formed member material except '*' indicates the property is from a coupon test of virgin plate. Elastic modulus and Poisson's ratio equal to 205 000 MPa and 0-3 respectively were used.
48
S. Z. Hu, H. G. L. Prion & P. C. Birkemoe
FINITE ELEMENT MODEL
Tubular geometry For each specimen, either 50, 72 or 74 profiling data sets were recorded, at 30 mm or 20 mm spacing along the length; each set contains 72 radiusversus-angle readings for a full circumferential profile. To reduce the large number of profile coordinate points, a simple linear interpolation scheme was chosen to generate the finite element nodal points at the required positions. In consideration of the wave front size and realistic computational time, 48 nodal points around the circumference, which resulted in an interval of approximately 30 mm, were generated. Twenty-six finite element nodal points were selected in the longitudinal direction for each can, also at 30 mm intervals. The nodal points for the top and bottom can of the specimen were generated individually, starting at the tube edge and progressing towards the girth weld location, where the real and modelled mismatch occurs. The peak magnitude of the measured mismatch was on the order of ~ of the mean diameter. In the experimental program, it was found that the initiation of local buckling occurred at or close to the girth weld location. In the close proximity of the girth weld, 180mm to either side, a fine mesh of 30 mm x 30 mm elements was used in recognition of the anticipated local buckling behaviour. Towards both ends of the tube, where no significant deformation occurred during the test, larger elements were used. The four-node plastic quadrilateral shell element, 11'12 a member of the family of degenerate isoparametric shell elements, was used to model the tube. Triangular elements, formed by suppressing the last node of the quadrilateral element, were used in the transition zone between the coarse and fine meshes. The development of the finite element model from the test profile measurements for specimen ST-12 is illustrated in Fig. 2. The mismatch at the girth weld jointing the two cans constituted a discontinuity in thr. structure which locally influenced the initiation of local buckling and 'ailure. Two types of modelling were used to represent this mismatch reg! ,n as shown in Fig. 3. First, the girth weld was treated as part of the tut- , modelled with a narrow band of shell elements as an extension of the be wall. The thickness and material properties of these shell elements w ,', in some cases, chosen to be the same as the tube itself; in other cases the actual weldment properties and thickness were used. Secondly, the assumption was made that the nodal points on either side of the cans' juncture were rigidly compatible; constraint equations were used to provide this compatibility.
Influence of imperfections on beam-column strength
Di.un'eteModel
Can Assembly
Fig. 2. Modelling of measured can geometry for finite element analysis.
J
Shell Elements Model "S"
Model "C"
Fig. 3. Modelling of mismatch and weld at junction of cans.
49
50
S. Z. Hu, H. G. L. Prion & P. C. Birkemoe
The profiling data set which represented coordinates on the ridge of the girth weld reinforcement was included in some of the models, but it was found to create a large misrepresented mismatch. It was included in the parametric study to demonstrate the importance of this aspect.
Material modelling The classic bilinear kinematically hardening material was used in modelling the material. The material characteristics were described in the form of a bilinear stress-strain curve, starting at the origin, and with positive stress and strain values; the initial slope of the curve was taken as the elastic modulus of the steel (205 000 MPa). After reaching the specified yield stress, the curve continued along a second slope which was defined by the strain-hardening modulus. Two types of stress-strain curve were used in the analysis: (1) elastic-perfectly plastic material representing virgin plate test coupons, (2) elastic-plastic kinematically hardening material, representing the steel properties after cold rolling fabrication. The yield stress and strain hardening modulus were taken from tensile coupon tests from the fabricated tubes.
Boundary conditions Elastic shell elements with increased thickness were used at the ends of the tube to create a stiff loading fixture similar to that used experimentally for the eccentrically loaded short beam-columns (ST12 and ST6). Specified displacements parallel to the longitudinal axis were imposed at eccentric loading points on the top. The corresponding points on the bottom can were allowed to rotate. The concentrically loaded short columns, ST7 and ST13, were tested with ends fixed against rotation. In this case, a constant displacement boundary condition, with rotational and lateral displacements restricted, was employed in the model.
Model parameters For clarity, every analysis was given a specific designation, consisting of the specimen number as given in the experimental reports, followed by a single digit number. Six analyses were performed on specimen ST7, two analyses on specimen ST6 and one each on specimens ST12 and ST13. A summary of the model parameters is tabulated in Table 1. The first two analyses of the specimen ST7 (ST7-1 and ST7-2) were performed on the model with the inclusion of the girth weld measurements in the mapping process. The cans of model ST7-1 were joined by the
Influence of imperfections on beam-column strength
51
constraint equations, whereas the cans of model ST7-2 were joined by a narrow band of thicker shell elements. The elastic-perfectly plastic stressstrain relation with a yield stress of 560 MPa, from weld coupon tests, was used in these elements. The other analyses (ST7-3, ST7-4, ST7-5 and ST7-6) were performed with the exclusion of the girth weld measurements. The model ST7-C was otherwise identical to ST7-1, for which the girth welds were modelled by constraint equations. The girth weld was modelled by shell elements for ST7-4, ST7-5 and ST7-6. These shell elements had a thickness of 6 mm with elastic-perfectly plastic material properties of yield stress 560 MPa for ST7-4 and ST7-5. For specimen ST7-6, a thickness of 4.5 mm with kinematically hardening material with a yield stress of 266 MPa and strain hardening modulus of 4200 MPa were assigned to these elements. The main body of ST7-4, which differed from other models, was composed of an elastic-perfectly plastic material with a yield stress of 254 MPa. Two analyses were performed on specimen ST6. The first run, ST6-1, was performed on the model with the inclusion of the girth weld measurements in the mapping process. The second run, ST6-2, utilized the same finite element model, but with the exclusion of the girth weld measurements. The mismatch at the junction of the cans of both analyses was modelled by constraint equations. Specimens ST12 and ST13 were analysed without the inclusion of the girth weld measurement and were named ST12-1 and ST13-1. The junction between the cans was modelled by constraint equations. Material properties were obtained from coupon results unless otherwise indicated.
RESULTS O F T H E N U M E R I C A L ANALYSIS 1. ST7-1: In the experimental study of specimen ST7, five alternating buckles formed around the circumference at the girth weld. Although the inclusion of girth weld measurement in the mapping process created a relatively large mismatch between cans, the final deformed shape, Fig. 4, of ST7-1 compared well with those observed in the experiments. The comparison of the load-displacement curve with that obtained from the test is shown in Fig. 5(a); the similarity in slope in the post-ultimate region is evident. 2. ST7-2: Thicker shell elements with higher material strength were used in the modelling of the girth weld instead of using constraint equations. A completely different deformation pattern was obtained and no sudden buckling was observed (Fig. 4). The load-displacement curve, as shown in Fig. 5(b), indicates a very flat plateau following the elastic portion. No obvious unloading behaviour was observed.
52
S. Z. Hu. H. G. L. Prion & P. C. Birkemoe
Top
South
B m g i mi O i m O Wi
'0 0
O
m
Fig. 4. Deformed geometry of finite element models for ST7.
3. ST7-3: The girth weld measurements were excluded in the mapping process and the only difference from ST7-1 was the size of the mismatch. A comparison of the load-displacement curve with the test result is shown in Fig. 5(a). The unloading part is stiffer than observed in the test. Instead of an alternating diamond buckling pattern, the pattern as shown in Fig. 4 was obtained analytically. 4. ST7-4: Thicker shell elements with high yield stress material (Fy = 560 MPa) were used to model the girth weld. The elastic-perfectly plastic material was used in the modelling of the main body of the tube. The load--displacement behaviour showed a linear elastic slope followed by a sudden load drop as unloading occurred (Fig. 5(c)). No buckling deformation occurred at the junction of the cans, but slight bulging occurred in the bottom can, 180 m m below the girth weld.
53
Influence of imperfections on beam-column strength 2000.
200O
1500
"o
1"7"2
3
°°°
jT:ST
lS00
-5
1000
5OO
50C
0
0
15
10
5
O~mmt
0
&L {mrn)
. . . . . . . . . . . . . .
5 10 Axild Di=p~emont AL (ram)
(a)
15
(b)
-
1500 ¸
TEST
t
~
~
l ST73 ST7-6
•o 100(] o
....
-~ ....
15
Ib ....
Axial ~
AL (ram)
(c)
0
. . . . . . . . . . . . . .
5 10 Axial Disp~a~m~n~ AL |ram}
15
(d)
150(
lO= 50(;
C~
0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
5 10 Axial D~,plocsment &L (ram)
15
(e)
Fig. 5. Load-displacement behaviour of concentrically-loaded specimen ST7 compared with finite element models. 5. ST7-5: The difference between ST7-5 and ST7-4 was that a kinematically hardening material was used in ST7-5 to model the main body of the tube. A completely different behaviour was observed in both the load--displacement curve and the deformed shape. The load-displacement curve of ST7-5 has a flat plateau following the elastic portion as shown in Fig. 4. No obvious sudden unloading was observed. The display of the deformed shape shows no local change in shape at the girth weld junction.
S. Z. Hu, H. G. L. Prion & P. C. Birkemoe
54
6. ST7-6: The thickness and material properties of shell elements representing the girth weld were changed in this analysis. The thickness of the shell elements at the girth weld was changed from 6 mm (in ST7-5) to 4.5 mm (equal to the wall thickness of the main body). The material properties were also changed from the higher yield stress of 560 MPa to 266 MPa (equal to the yield stress of the material of the main body). A substantial change in the behaviour, both in the load-displacement curve and the deformed shape, occurred. The load started to drop along with the occurrence of large deformation as shown in Fig. 5(e). Both the loaddisplacement response and the deformed shape are very similar to the results of ST7-3 as shown in Fig. 5(d) and Fig. 4. 7. ST6-1: The girth weld measurements were included in the interpolating scheme of model ST6-1. An asymmetric pair of buckles formed on the concave side of the tube at the junction of the cans. The deformed shape, as well as the load-displacement response shown in Fig. 6, show good agreement with the test results. 8. ST6-2: The girth weld measurements were again excluded in the mapping process. A similar deformed shape was obtained as in the case of ST6-1, the ultimate load was slightly higher and the load--displacement curve varied slightly. The deformed shape and the load--displacement curve, Fig. 6, also agreed well with the test results. 9. ST12-1: A symmetric alternating set of buckles, with the initial buckles in the region of maximum compression, was formed as shown in Fig. 7, and were similar to the test observations. The load--displacement curve shows that the finite element model exhibited a stiffer post-ultimate response than the test specimen. 10. ST13-1: In comparison with other specimens that were analysed, ST13-1 had a much higher yield stress. The load-displacement curve had a
•
~'l
Jelll 2
oI
§
I
I
I
2 4 8 AXIAL ~ E N T
1
8
I
10 12 AL (mm)
Fig. 6. Behaviour of eccentrically-loaded specimen ST6 compared with finite element models,
Influence of imperfections on beam-column strenoth
I_
e
55
I..-¥~t
1200f~
ST12.1
0~ / II4 / 18 lb , AXIALDISPLACEMENT AL(mm) Fig. 7. Behaviour of eccentrically-loadedspecimen ST12 compared with finite element model.
5O0O 4000
i o/" °
200O
1000 S 10 15 AXIALDISPLACEMENT AL(ram) Fig. & Behaviour of concentrically-loadedspecimen ST12 compared with finite element model. linear elastic slope followed by a sudden drop as unloading occurred. In the comparison (Fig. 8), the curves for the test specimen and the finite element model have very similar slopes in the elastic and in the unloading regions.
COMPARISON
WITH PLASTIC INTERACTION EQUATIONS
The Canadian code for fixed offshore structures 2 takes account of local buckling under compressive stress by substituting the yield stress with a reduced value which depends on the diameter-to-thickness (D/t) ratio and the specified minimum yield stress, Fy, of the steel. These local buckling provisions were based on recommendations by Plantema ~3 and Sherman. ~4 The axial load resistance using these relationships is calculated as:
(1)
56
S. Z. Hu, H. G. L. Prion & P. C. Birkemoe
where F¢ is the reduced compressive stress as recommended by Plantema I 3 F¢ = f y
Fc = F, [0.75 + 6200/(Fy (O/t)] Fc = Fy [66 O00/(D/t)]
for F~(D/t) ~ 25 000 for 25 000 < Fy(D/t) <<.80 000 for 80 000 < Fy(D/t) ~ 360Fy
The bending resistance is given as
Mr=ckMu
(2)
where M, is the reduced moment resistance as recommended by Sherman ~4 M u .~- M p
M, = Mp [ff775 + 3200/(Fy(D/t)] Mu = Mp [51 O00/(D/t)]
for Fy(D/t) <<.14 200 for 14 200 < Fy(D/t) <<.621300 for 62 000 < Fy(D/t) <<.360Fy
To determine the cross-sectional strength of a member at any location, an interaction formula that does not take account of the column stability is used. This strength interaction relationship, based on the theoretical derivation of the plastic resistance of a cylindrical section, is
Mt E,cl ~
~=cos
(3)
where Mt is the total moment at the section. In Table 2, the results of finite element models ST6-2, ST12-1, ST7-6 and ST13-1 are compared with the test results and the calculated results of the cross-sectional strength according to eqns (1), (2) and (3). Because of the small second-order moment for the short members, Mt in this equation is replaced by the product of the axial load C and its eccentricity, e. Local buckling is accounted for in the evaluation of Cr and M, in which the resistance factor is set equal to 1"0. From Table 2 it can be seen that a good correlation exists between the experimental results, the analytical results, and the empirical design equations for the level of imperfection and material strengths examined.
DISCUSSION OF THE RESULTS A finite element model has been shown to provide a good approximation of the behaviour of an imperfect tubular member. In the analytical study, it
Influence of imperfections on beam-column strenoth
57
TABLE 2 Comparison with Plastic Interaction Equations 2
Spec. No.
Fc (MPa)
ST6 STI2 ST7 ST13
304 262 262 441
C, M, C eqn (1) eqn (2) eqn (3) (kN) (kN m) (kN) 2764 1654 1654 4424
364 214 N/A N/A
2465 1323 1654 4424
C.u test (kN)
Cuu FE (kN)
Test C
FF C
FE test
2596* 1360 1743" 4424
2425 1330 1630 4395
1'05 1"03 1"05 1"0
0"98 1"0 0"99 0"99
0"93 0"98 0"94 0"99
Cr and M, are obtained from eqns (1) and (2) respectively for, O= 1.0. C is calculated using Mt = C x e. Asterisk indicates that the test results of ST6 and ST7 are higher than the calculated squash load using the tensile coupon properties.
was found that the load-displacement behaviour of tubular members very much depends on the geometry of the tube and is sensitive to the material properties, the initial imperfections and residual stresses, Discretization of a tube into elements thus requires careful consideration since some of the possible deformed modes may be missed in the analysis. The only geometric differences between models ST7-1 and ST7-3, and models ST7-2 and ST7-5 were the size and shape of the mismatch, yet completely different load-displacement responses and deformed shapes were observed. These results indicate the importance of the mismatch modelling on the behaviour of the tube. A comparison of the buckling deformations indicates that the finite element mesh size may not have been fine enough to describe the distortion at the girth weld of STT. Consequently the mesh size may be the cause of the discrepancy in the buckling mode from the observed experimental deformed shape. Classic linear buckling analyses of a concentrically loaded circular tube Is indicate that variations in the buckling loads for buckling modes near the observed mode are small. Thus, with random imperfections present, differing mode shapes may be expected although the loads may show good agreement. The results of analyses ST7-3 and ST7-6 show very similar buckling modes and very similar load-displacement curves. The use of constraint equations to model the girth weld (ST7-3) gave a reasonable method to approximate the behaviour of the central region. In the case where the girth weld was modelled by shell elements, however, the element thickness and material properties were found to have a significant influence on the behaviour of the tube. The results of the analyses of ST7-4 and ST7-5 illustrate how the strain hardening modulus caused a significant difference between the load-
58
S. Z. Hu, H. G. L. Prion & P. C. Birkemoe
displacement response of the tested specimens and the finite element method. In addition, the deformed shapes varied depending on the value of the strain hardening modulus. The method of using shell elements with higher yield strength and greater thickness to model the junction of the cans did not give satisfactory results in the analyses of ST7-2, ST7-4 and ST7-5. These elements likely stiffened the centre and led to a change in deformed shape from one mode to another. Consequently, buckling modes observed in the experiments were prevented. On the other hand, it was found that by modelling the joint with shell elements of thickness and material properties identical to the main body of the tube (ST7-6), good agreement with the experiments was obtained.
S U M M A R Y AND C O N C L U S I O N S The conclusions deal with the topics of finite element modelling and with observations of the modelled and experimental behaviours of tubular members. The finite element method has been shown to be a valid analytical approach for the study of behaviour of imperfect steel tubular members. It has been clearly demonstrated that the element availability and sophistication of state-of-the-art finite element programs requires careful development and verification of models. Several observations summarizing the findings suggest topics for further study and refinement.
Finite element modelling and member behaviour (1) The numerical modelling of fabricated tubes with non-linear shell elements, as provided in the commercial finite element program ANSYS, successfully simulated the local buckling behaviour of eccentrically loaded tubular members with initial imperfections (ST6 and ST12). The analytically determined buckled shape did not consistently correlate with experimental observations. (2) The ultimate loads obtained in the analyses were quite close to the experimentally obtained values (ST12 and ST13); others (ST6 and ST7) underestimated the experimental values as shown in Table 2. It should be kept in mind that the experimentally obtained ultimate loads for ST6 and ST7 were higher than the calculated squash load based upon coupon properties. The quantitative comparisons of the analytically obtained ultimate load with the resistance as stipulated in the Canadian Standards for Fixed Offshore Structures, 2 on the other hand, showed very good agreement.
Influence of imperfections on beam-column strenoth
59
(3) The geometric modelling of the girth weld was found to be a very important factor in the finite element study. The best correlation with experimental results was achieved by using constraint equations or by using connecting shell elements with the same thickness and material properties as the main body of the tube. The use of thicker elements and/or stronger material properties typically caused a change in the buckling mode and consequently altered the load-displacement response. (4) The material properties of the finite elements played a very important role in the solution. The elastic-perfectly plastic stress-strain relationship, obtained from tension coupon tests of the virgin plate, gave unsatisfactory results. A bilinear, kinematically hardening, material relationship, based on tests of tension coupons cut from the tube after cold rolling, resulted in load--displacement responses from the analytical models which closely resemble the experimental observations. (5) The mismatch at abutting ends was found to be the principal parameter to initiate local buckling and resulted in a reduction of the cross-sectional strength for the tubes studied. The influences of outof-straightness and out-of-roundness on the sectional ultimate strength were found to be negligible. The deformed shape which develops during the post-ultimate behaviour depends very strongly on the size and shape of the mismatch and likely affects the post-ultimate load-displacement response. (6) The fluctuations in the load-displacement response of the finite element model in the vicinity of the ultimate load can be avoided by using smaller load increments. It should be realized, however, that such a refinement also implies a significant increase in computing time and cost. Further research
The mismatch tolerances as specified in current codes need to be reexamined. These are typically given as a maximum value, expressed as a percentage of the wall thickness, without reference to the extent to which they may occur around the circumference. For the analysed specimens, for example, the specifications were violated over only a small portion of the circumference. It is felt that not only the magnitude, but also the extent of a large mismatch with respect to the circumference, plays an important role in the formation of local buckles. Further studies are needed to determine the effect of the shape of the mismatch, combined with its extent. As more results become available, the tolerances and the interpretation of the magnitude, extent and location of mismatch can be defined more specifically and realistically in specifications for offshore structures.
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Measurements of imperfections for longer cylinders (5 m and 10m) included in the experimental program were also reported in detail. A finite element model similar to the one discussed in this paper should be developed to investigate the behaviour of long specimens which typically fail as a result of the interaction of local and overall buckling. With the current techniques, however, a significant increase in computational costs would result. Residual stresses are very much a part of the total behaviour of any buckling and yielding scenario and they have not been discussed here. The effects of welding residual stresses on the conditions which cause local buckling and on the strength of fabricated tubes must be added to the modelling for a complete evaluation of cross-sectional member behaviour.
ACKNOWLEDGEMENTS The authors wish to acknowldge the support provided through various grants, and fellowships by the Natural Sciences and Engineering Research Council of Canada and the University of Toronto.
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