International Journal of Civil Structural Research Vol.1 ... · Non-Linear Behaviour Of Thin-Walled...

NON-LINEAR BEHAVIOUR OF THIN-WALLED

HOLLOW CIRCULAR STEEL TUBES

K.Chithira 1 and K. Baskar2*

1Research Scholar, 2Associate Professor National Institute of Technology,Tiruchirappalli.

*Communicating Author, e-mail: [email protected] Phone: 0091-431-2503161, Fax: 0091-431-2500133

ABSTRACT

This paper deals with a numerical investigation on Thin-walled Hollow

Circular steel tubular columns under Axial load Condition. A finite element

model is developed using the general purpose finite element software ANSYS

and is used to carry out complete non-linear analysis. The proposed FE model

is validated through the available experimental results reported by other

researchers. In total, sixty six specimens with different D/t and L/D ratios have

been considered in this present study. Analyses were carried out until collapse

of specimen and the complete non-linear behaviour was studied. The

numerically predicted values have been compared with Perry-Robertson and

modified Perry-Robertson predictions and were found that those two equations

predicted the ultimate load carrying capacity of column members to a higher

accuracy for columns having a L/D ratio less than 30 and after a large difference

was found between the numerically predicted values and the other two

predictions. Based on the numerical results a new design equation has been

proposed herein in view of predicting the ultimate load carrying capacity of

hollow circular steel tubular columns which were having larger L/D ratios until

75. The proposed design equation has been verified through available

experimental results reported by other researchers and found to be more reliable.

International Journal of Civil Structural

Environmental And Infrastructure Engineering

Research Vol.1, Issue.1 (2011) 86-117

© TJPRC Pvt. Ltd.,

K.Chithira and K. Baskar

63

This paper presents the complete procedure involved in the numerical analysis

and the proposed design equations.

Keywords: Hollow Steel Tube;Circular Hollow Section;CFT; FEM;Buckling

NOMENCLATURE

CFT Concrete Filled Tube

CHS Circular Hollow Section

VHS Very High Strength Tubes

SSHS Structural Steel Hollow Section

IS Indian Standard Code of Practice

MPa Mega Pascal (N/mm2)

GPa Giga Pascal (N/mm2)

ECCS European Convention for Constructional Steelwork

AIJ Architectural Institute of Japan

L/D Length to Depth ratio

E Young’s Modulus, (N/mm2)

DF Ductility Factor

ε Yield stress ratio ( 250/fy)1/2

fy Yield stress

fc Compressive strength

fu Ultimate strength

fcr Euler crippling stress (π↑2E / (λ)↑2)

FEA Finite Element Analysis

Imperfection factor or Perry factor 0 2

y

h

rδ

λ Non-dimensional slenderness ratio

y

cr

f

f

Non-Linear Behaviour Of Thin-Walled Hollow Circular Steel Tubes

64

1. INTRODUCTION

Hollow circular steel tubular sections have been used extensively as

structural members in the modern onshore, offshore construction and

industries.These tubular members are used now as beam-columns to offer larger

Initial bow

Slenderness ratio

Radius of gyration

Parameter dependent on the shape of the cross section

y

0.001h

r

Crushing/ Squash load

Eigen buckling load

Euler Critical load

Yield load by Finite Element Model

Ultimate load by Finite Element Model

Load predicted by the Perry Robertson

Load predicted by the Modified Perry Robertson

Strength or resistance reduction factor

Modified Slenderness ratio y

E0.2

fπ


65

strength and ductility and to offer economical geometric properties. The load

carrying capacities of such members depend on many parameters including L/D

ratios varies from 5 to 75 and D/t 28.2 to 84.7. The steel tubes with larger D/t

ratio provide economical design but, prone to local buckling failure.The

buckling capacity of the steel tubular column and the post buckling behaviour of

column is mainly depending on the geometrical properties of the column.

In many applications these sections are filled with concrete in order to gain

advantages of both steel and concrete infill.In such cases, the erection of beam

column joint is carried out before making the concrete infill. The steel tube in

CFT columns should be able to resist the minimum load coming from the

structural systems. In steel and concrete composite columns, the Young’s

modulus of the steel is about 10 times higher than that of concrete. Hence, while

loading, the steel tube is stressed high and it would lead the steel portion to

buckle outwards and to undergo local buckling failure.In all the situation it is

very much essential to study the behaviour of unfilled steel tubular sections.

Rasmussen (2001) summarised the researches undertaken at University of

Sydney during the 1990’s on stainless steel tubular members and connections.

Tests were performed on square, rectangular and hollow circular column and

beam sections, as well as welded X- and K-joints in square and circular hollow

sections.

An attempt was made by Elchalakani et al.(2001) to establish more accurate

slenderness limits for cold-formed circular hollow sections.A design curve was

developed and recommended for the design of cold-formed CHS under pure

bending.

Frode and Torgeir (2002) presented an analytical model for the

determination of local post-buckling and suck-in deformations in aluminium

alloy rectangular hollow section formed in stretch bending.


66

Martin Pircher et al.(2002) studied the influence of the fabrication process

on the buckling behaviour of thin walled steel box sectionsand suggested that,

when residual stresses were ignored and the buckling analysis was based on

purely geometric imperfections, amplitude of these imperfections had a strong

influence on the response of the box column.

Hancock and Zhao (2003)studied the behaviour of cold-formed stub

columns, long columns and beams. The design rules have been proposed for

localised failure at loading points for the columns subjected to concentrated

force and combined bending and concentrated force.

Kiymaz (2005) performed a series of tests consisting of various cross-

section geometries on structural stainless steel circular hollow sections (CHS)

subjected to bending.

Zhu and Young(2006)investigated the performance of aluminum alloy

circular hollow sections undercombined axial compression and bending.

Ling et al. (2006) presented an investigation on block shear tear-out (TO)

failure for gusset-plate welded connections in both Very High Strength (VHS)

tubes and Structural Steel Hollow section (SSHS).

Experimental and numerical investigations of cold-formed stainless steel

square and rectangular hollow sections subjected to concentrated bearing load

were presented by Feng Zhou and Young (2007).

Ernest and Young ( 2007) presented a finite element model to assess the

structural performance of stainless steel tubular columns at elevated

temperatures.

Poonaya et al. (2007) proposed a theoretical model to predict the collapse

mechanism of thin-walled circular tube subjected to pure bending.


67

Mamaghani et al. (2008) dealt with the elasto-plastic analysis and ductility

evaluation ofsteel tubular columns subjected to cyclic loading. The results of

finiteelement analyses on cyclic elasto-plastic behaviour of steel tubular columns

were presented.

From the literature review it is noted that the studies on CHS with larger D/t

and L/D ratios arelimited. In view of understanding the behaviour of CHS

sections with larger D/t and L/D ratio, further numerical study carried out has

been reported in this paper.

2. PARAMETERS CONSIDERED

The main objective of the present study is to predict the non-linear

behaviour of hollow Steel tubular column with larger D/t ratio and larger L/D

ratio which are prone to local and global buckling mode of failure. The D/t and

L/D ratios are taken by considering various code specifications as follows.

As per IS 800: (2007) , the limiting values D/t of circular hollow tube

including welded tube subjected to axial compression.

D/t ≤ 88

ε = √250/

As per “The Technical General Secretariat of The ECCS for composite

construction (1981)”, the diameter of the tube should be greater than 100 mm.

D ≥ 100 mm

According to Euro code 4 – Part 1.1(2004), the D/t ratio should be less than

the following prediction for composite structures.

D/ t ≤ ( )

y

90*235

f

Using the above equation, the D/t ratio is limited to 68.22 for fy = 310 MPa.


68

The sections are chosen from the Indian Standard code of Steel Tubes for

Structural purposes-specification IS: 1161-1998(1998) to confirm the practical

availability. The sections given in IS 1161 – 1998 are having the D/t ratios varies

from 6.7 to 51.4.In practical case, it is noted that D/t varies from 30 to

50.Considering all, the D/t ratio ranging from 28.2 to 51.4 are considered in the

present study.

As per The Technical General Secretariat of The ECCS for composite

construction, the length-to-diameter ratio (L/D) should be less than or equal to

45 and as per SRC standard of Architectural Institute of Japan (AIJ), the length-

to-diameter ratio (L/D) should be less than or equal to 50.Ten different L/D

ratios ranging from 5 to 75 are selected for the finite element analysis to study

the local and global buckling behaviour of steel tube. L/D ratio less than 12 is

classified as short column and L/D ratio greater than 12 is classified as long

column(Shosuke Morino and Keigo Tsuda). For short column, three different

L/D ratios with 9 different D/t ratios, in total 27 specimens are considered. Long

columns with L/D of 15, 25 and 35 are analysed with 9 different D/t ratios, in

total 27 specimens and for column with L/D ratios 45, 55, 65 and 75,three

different D/t ratios such as 28.2, 39.1 and 89.7, in total 12 specimens are

subjected to investigation. Therefore, 27 short specimens and 39 long specimens

(total 66) are considered in the present analyses.Thediameter, thickness and

length of the specimens have been listed in Table 1.


69

Table 1. Geometrical Properties of Sections Considered

Sl.No Specimen

ID

Outside

Diameter

D (mm)

Wall

Thickness

t (mm)

Length of

Specimen

L (mm)

D/t ratio L/D ratio

1. S.1.1 101.6 3.6 508 28.2 5

2. S.1.2 101.6 3.6 812.8 28.2 8

3. S.1.3 101.6 3.6 1117.6 28.2 11

4. L.1.1 101.6 3.6 1524 28.2 15

5. L.1.2 101.6 3.6 2540 28.2 25

6. L.1.3 101.6 3.6 3556 28.2 35

7. L.1.4 101.6 3.6 4572 28.2 45

8. L.1.5 101.6 3.6 5588 28.2 55

9. L.1.6 101.6 3.6 6604 28.2 65

10. L.1.7 101.6 3.6 7620 28.2 75

11.

12. S.2.1 139.7 4.5 698.5 31.0 5

13. S.2.2 139.7 4.5 1117.6 31.0 8

14. S.2.3 139.7 4.5 1536.7 31.0 11

15. L.2.1 139.7 4.5 2095.5 31.0 15

16. L.2.2 139.7 4.5 3492.5 31.0 25

17. L.2.3 139.7 4.5 4889.5 31.0 35

18. S.3.1 152.4 4.5 762 33.9 5

19. S.3.2 152.4 4.5 1219.2 33.9 8

20. S.3.3 152.4 4.5 1676.4 33.9 11

21. L.3.1 152.4 4.5 2286 33.9 15

22. L.3.2 152.4 4.5 3810 33.9 25

23. L.3.3 152.4 4.5 5334 33.9 35

24. S.4.1 165.1 4.5 825.5 36.7 5

25. S.4.2 165.1 4.5 1320.8 36.7 8

26. S.4.3 165.1 4.5 1816.1 36.7 11

27. L.4.1 165.1 4.5 2476.5 36.7 15

28. L.4.2 165.1 4.5 4127.5 36.7 25


70

29. L.4.3 165.1 4.5 5778.5 36.7 35

30.

31. S.5.1 193.7 4.8 968.5 40.4 5

32. S.5.2 193.7 4.8 1549.6 40.4 8

33. S.5.3 193.7 4.8 2130.7 40.4 11

34. L.5.1 193.7 4.8 2905.5 40.4 15

35. L.5.2 193.7 4.8 4842.5 40.4 25

36. L.5.3 193.7 4.8 6779.5 40.4 35

37. S.6.1 219.1 4.8 1095.5 45.6 5

38. S.6.2 219.1 4.8 1752.8 45.6 8

39. S.6.3 219.1 4.8 2410.1 45.6 11

40. L.6.1 219.1 4.8 3286.5 45.6 15

41. L.6.2 219.1 4.8 5477.5 45.6 25

42. L.6.3 219.1 4.8 7668.5 45.6 35

43. S.7.1 323.9 6.3 1619.5 51.4 5

44. S.7.2 323.9 6.3 2591.2 51.4 8

45. S.7.3 323.9 6.3 3562.9 51.4 11

46. L.7.1 323.9 6.3 4858.5 51.4 15

47. L.7.2 323.9 6.3 8097.5 51.4 25

48. L.7.3 323.9 6.3 11336.5 51.4 35

49. S.8.1 101.6 2.6 508 39.1 5

50. S.8.2 101.6 2.6 812.8 39.1 8

51. S.8.3 101.6 2.6 1117.6 39.1 11

52. L.8.1 101.6 2.6 1524 39.1 15

53. L.8.2 101.6 2.6 2540 39.1 25

54. L.8.3 101.6 2.6 3556 39.1 35

55. L.8.4 101.6 2.6 4572 39.1 45

56. L.8.5 101.6 2.6 5588 39.1 55

57. L.8.6 101.6 2.6 6604 39.1 65


71

3. FINITE ELEMENT SIMULATION

An experimental investigation on sixty six number of Circular Hollow Steel

tubular column will involve huge fabrication cost and will require sophisticated

testing facility. Considering the various factors involved in experimental study

and also keeping in mind of the authors limitations, a numerical experimentation

has been proposed in this study. The general purpose Finite Element software

‘ANSYS’ has been used to develop a FE model and to carryout the numerical

experimentation through a complete Non-linear analysis.

Shell elements are generally used to model thin-walled structures. ANSYS

includes general-purpose shell elements as well as elements that are specifically

formulated to analyse ‘thick’ and ‘thin’ shell problems. SHELL281 is suitable

for analyzing thin to moderately-thick shell structures. Therefore, Shell281

element shown in Figure1. is used in this study to model the steel tube in view of

predicting the local and global buckling effect of steel tube. It is an 8-node

element with six degrees of freedom at each node: translations in the x, y, and z

axes, and rotations about the x, y, and z-axes. SHELL281 is well-suited for

linear, large rotation, and/or large strain non-linear applications. Change in shell

thickness is accounted for in non-linear analyses. The element accounts for

follower (load stiffness) effects of distributed pressures.

Figure 1. Typical Details of the ANSYS Element SHELL 281


72

3.1 Material Modelling

Development of an appropriate FE model requires a more accurate material

model as same as used in the structure the correct representation.Inaccurate or

inappropriate modelling of the basic material behaviour would overshadow the

performance of even the most refined FE models.In this present study, the

properties of structural steel tube given in IS 1161:1998 is employed. The

‘E’value is taken as 200 GPa, the yield (fy) and ultimate (fu) strength of materials

are taken as 310 MPa and 450 MPa respectively. Typical stress-strain curve of

the considered material is shown in Figure2.

Figure 2. Typical stress-strain behaviour of steel (Ref: IS 1161-1998)

3.2 Geometric Modelling

Sixty six hollow steel tubes under axial loading have been modelled by

using the SHELL 281element and material non-linearity as discussed in the

section 3.1.

3.3 Boundary Conditions

One end of the hollow steel tube columns is fixed against all translations

and another end of the column is fixed against the translations in horizontal

plane as shown in Figure3. It is allowed to displace in vertical direction and free


73

to rotate in all three directions at the loaded edges.In comparison with nominally

pinned conditions, rigid connections cause plastic hinges to form at the member

ends, in addition to that at mid-length, leading to improved inelastic

performance.

Figure 3. Typical View of the FE Model with Boundary Conditions

3.4 Analysis Technique

In the present study, Eigen buckling analysis has been carried out to obtain

the Eigen modes, which were subsequently used to represent initial geometric

imperfections. The non-linear effects arising from geometric and material

non-linearity were included using the ‘NLGEOM’ (large deformation effect)

option as stated in ANSYS. It offeredseveral techniques to analyse this type of

problem and among the available options the ‘Arc length method’ has been

chosen because of its simplicity and widespread use in similar applications.

3.5 Validation of the Finite Element Model

Past experimental results have been taken from the journal and were used to

validate the finite element model. The finite element model has been based on

experimental investigation carried out by Mathias Johnson and Kent Gylloft


74

(2002). The non-linear finite element analysis was carried out and the following

results have been obtained.

Load-deflection behaviour has been compared between the finite element

prediction and the experimental data as shown in Figure4. The comparison

showed that the developed FE has been capable of predicting the linear part of

load-deflection behaviour to a highly accuracy level and the post failure

behaviour to an acceptable limit. Thus, the developed model was validated and

used in the present work to carry out the further parametric studies. The authors

specified only yield and ultimate point of stress strain relationship. If the exact

stress strain behaviour of the model was known, the post buckling failure could

be predicted with great accuracy.

Figure 4. Validation of FE Model with the Existing Experimental Results

3.5.1Convergences study

Selecting a suitable mesh is one of the most important aspects of FE

modelling to predict the accurate response of the problem considered. Finer

meshes are generally preferred to obtain better predictions although there is no

general guideline for such fineness, which largely depends on the type of


75

structure and analysis involved. Therefore, performing a convergence study is a

pre-requisite for finding a suitable mesh for any FE investigation.

A compromise is therefore needed between the required level of accuracy

and the computational time of a solution. Five different mesh sizes are used to

simulate the load–deformation response of steel tube columns considered in the

present research. The steel tube columns are analysed using these five meshes

and the results are shown in Figure 5. The results show that there is a small

improvement in predictions for both peak load and the corresponding

deformation with the finer mesh. There is not much difference in load –

deformation curve when the element edge length of 10 mm, 20 mm, 30 mm,

60 mm and 80 mm are used. But the element edge length of 80 mm is showing

the variation in load deformation behaviour. So the element edge length of

60mm is used to model the structure. No further refinement is attempted since

the predictions are found to be in good agreement with the test results and this

mesh size has been adopted in the subsequent FE models.

Figure 5. Results of Convergence Study


76

4. ANALYSES OF COLUMN SPECIMENS

The non-linear analysis has been carried out on the columns having D/t ratio

varying from 28.22 to 84.7 and with six different slenderness ratio ranges from 5

to 75 using finite element package ANSYS 11. Element edge length of 60mm

has been chosen from the convergence study.Boundary conditions are simulated

and developed model has been validated as discussed in the section 3.3 and 3.5.

The results are detailed below.

4.1 Results and Discussion

All the sixty six columns are subjected to non-linear analysis. The load

versus axial deformation curves are generated as shown in Figures6-8. The yield

load and the ultimate load of each column are picked up from the load deflection

curves. These predicted yield load , Ultimate load are

summarised in Table.2. The Euler Critical load and the load predicted

by the Perry Robertson and modified Perry Robertson are

calculated for all column and are summarised with the predicted values

and in Table 2.

The calculated Euler Critical values are compared with the Eigen

buckling load obtained from the FEM analysis. For L/D value of 5 and 8,

the Euler Critical load is found to be very high compared to the Eigen buckling

load. When the L/D ratio increased beyond 8, the Euler Critical loads and Eigen

buckling loads are closer to each other.

The crushing of all the column specimens are calculated as the

product of cross sectional area of member and the ultimate stress of the material

and are included in Table 2. for comparison purposes. From the comparison it

can be noted that for smaller D/t ratios the values are closer to


77

with a difference of around 30%. But when the D/t of the specimen

increases, a major difference is found between and in the

range upto maximum of 65%. Values in Table 2. shows the influence of D/t and

L/D ratios of tubular columns in the load carrying capacity.

Also, it can be noted that D/t ratio and L/D ratio of column members

influence more on the ductility of sections. The Ductility Factor is the ratio

between the displacement at plastic stage and displacement at yield point. The

Ductility Factors (DF) of individual column specimens are calculated from the

load-deflection behaviour plotted through the FEM analysis are included in

Table 2. A larger ductility factor up to a maximum of 18 is found in short

column with a L/D of 5 and D/t of 28.

Figure 6. Load versus Deformation Behaviour of Column with D/t Ratio 28.2


78




79

Table 2. Results of Hollow Steel Tubes

Sl.

No.

Specime

n ID

(kN)

(kN)

(kN)

(kN)

(kN)

(kN)

(kN)

%

Reductio

n in load

DF

1. S.1.1 339.44 353.17 333.57 333.57 20382.55 498.76 5460 29.19 18 1.02 1.02

2. S.1.2 329.38 336.27 327.15 327.81 7961.93 4776 32.58 8.75 1.00 1.01

3. S.1.3 307.29 331.89 320.41 321.95 4211.27 3981.6 33.46 3.9 0.95 0.96

4. L.1.1 307.51 322.76 310.58 313.87 2264.73 2217.6 35.29 2.8 0.98 0.99

5. L.1.2 307.00 319.25 278.52 289.90 815.30 819.12 35.99 2 1.06 1.10

6. L.1.3 291.41 291.41 231.29 255.57 415.97 421.14 41.57 1.18 1.14 1.26

7. L.1.4 252.97 252.97 177.37 208.88 251.64 255.2 49.28 1.12

5

1.21 1.43

8. L.1.5 244.80 246.26 132.73 160.39 168.45 171.36 50.63 1.12 1.53 1.84

9. L.1.6 222.14 222.17 100.76 121.44 120.61 122.8 55.46 1 1.83 2.20

10. L.1.7 180.57 192.75 78.38 93.32 90.59 92.32 61.35 1 1.93 2.30

11. S.2.1 578.00 615.19 574.01 574.01 35376.12 860.10 10648.56 28.48 10.3

1

1.01 1.01

12. S.2.2 559.00 600.79 562.99 564.12 13818.80 9839.28 30.15 6.38 0.99 0.99

13. S.2.3 548.00 585.54 551.42 554.06 7309.12 6908.64 31.92 3.53 0.99 0.99

14. L.2.1 536.00 538.67 534.55 540.18 3930.68 3848.52 37.37 1 0.99 1.00

15. L.2.2 485.00 493.60 479.56 499.05 1415.04 1423.44 42.61 1 0.97 1.01

16. L.2.3 475.00 479.40 398.58 440.22 721.96 731.82 44.26 1 1.08 1.19

17. S.3.1 613.42 650.08 628.17 628.17 38907.07 940.90 10816 30.91 8 0.98 0.98

18. S.3.2 607.36 638.65 616.14 617.37 15198.08 9867.2 32.12 5 0.98 0.99

19. S.3.3 602.67 629.24 603.53 606.39 8038.65 6545.6 33.12 2.8 0.99 1.00

20. L.3.1 594.80 628.19 585.14 591.26 4323.01 4232 33.24 2.5 1.01 1.02

21. L.3.2 592.40 617.99 525.30 546.47 1556.28 1566 34.32 2 1.08 1.13

22. L.3.3 585.16 606.56 437.18 482.51 794.02 805.2 35.53 1.5 1.21 1.34

23. S.4.1 684.58 706.82 682.32 682.32 42440.01 1021.69 10956 30.82 3 1.00 1.00

24. S.4.2 675.06 684.76 669.30 670.62 16578.13 9892 32.98 2.27 1.01 1.01

25. S.4.3 662.86 676.93 655.64 658.73 8768.60 8285.2 33.74 2.25 1.01 1.01

26. L.4.1 633.84 671.74 635.74 642.34 4715.56 4608 34.25 1.28 0.99 1.00

27. L.4.2 571.62 582.21 571.04 593.89 1697.60 1696 43.02 1 0.96 1.00

28. L.4.3 513.48 525.16 475.78 524.79 866.12 878.4 48.60 1 0.98 1.08

29. S.5.1 838.00 857.88 856.72 856.72 53510.35 1281.85 12686.64 33.07 7.45 0.98 0.98

30. S.5.2 810.00 844.76 840.41 842.06 20902.48 11348.88 34.10 3.61 0.96 0.96

31. S.5.3 795.00 796.38 823.31 827.17 11055.86 10441.20 37.87 1.64 0.96 0.97

32. L.5.1 745.00 751.05 798.41 806.66 5945.59 5819.52 41.41 1.37 0.92 0.93

33. L.5.2 699.00 699.48 717.54 746.06 2140.41 2153.42 45.43 1 0.94 0.97

34. L.5.3 655.00 655.09 598.50 659.74 1092.05 1107.19 48.89 1 0.99 1.09


80

Table 2. Results of Hollow Steel Tubes

Sl. No.

Specimen ID

(kN)

(kN)

(kN)

(kN)

(kN)

(kN)

(kN)

% Reduction

in load DF

S.6.1 978.45 1016.09 971.05 971.05 61054.89 1454.21 12770 30.13 5.75 1.01 1.01

S.6.2 953.12 995.79 952.63 954.48 23849.57 11380 31.52 3 1.00 1.00

S.6.3 860.00 981.99 933.32 937.66 12614.65 11044 32.47 1.7 0.92 0.92

L.6.1 820.00 974.3 905.22 914.50 6783.88 6640.00 33.00 1.7 0.90 0.91

L.6.2 817.50 945.43 814.10 846.17 2442.20 2458.00 34.99 1.5 0.97 1.00

L.6.3 811.63 916.04 680.00 749.01 1246.02 1263.60 37.01 1.4 1.08 1.19

S.7.1 1862.00 1954.89 1887.95 1887.95 119346.75 2828.68 22215.48 30.89 5.58 0.99 0.99

S.7.2 1794.00 1874.09 1852.12 1855.72 46619.82 19891.20 33.75 2.12 0.97 0.97

S.7.3 1736.00 1789.82 1814.55 1823.00 24658.42 19232.08 36.73 1.38 0.95 0.96

L.7.1 1754.00 1759.59 1759.88 1777.95 13260.75 12978.56 37.79 1 0.99 1.00

L.7.2 1631.00 1636.45 1582.55 1644.99 4773.87 4803.96 42.15 1 0.99 1.03

L.7.3 1548.00 1548.99 1321.58 1455.88 2435.65 2470.19 45.24 1 1.06 1.17

S.8.1 239.67 241.72 243.08 243.08 15165.98 363.89 3709.2 33.57 2 0.99 0.99

S.8.2 235.59 238.50 238.45 238.91 5924.21 3324 34.46 2 0.99 0.99

S.8.3 228.99 229.88 233.59 234.69 3133.47 2952 36.83 1.43 0.98 0.98

L.8.1 227.97 227.97 226.52 228.86 1685.11 1649.4 37.35 1.33 1.00 1.01

L.8.2 218.46 218.46 203.54 211.65 606.64 610.32 39.97 1.2 1.03 1.07

L.8.3 211.53 211.53 169.72 187.12 309.51 313.8 41.87 1 1.13 1.25

L.8.4 199.92 199.92 130.72 153.64 187.23 188.8 45.06 1 1.30 1.53

L.8.5 168.09 168.09 98.07 118.46 125.34 124 53.81 1 1.42 1.71

L.8.6 152.73 157.86 74.54 89.89 89.74 88.64 56.62 1 1.70 2.05

L.8.7 149.02 155.03 58.03 69.15 67.40 66.64 57.40 1 2.16 2.57

S.9.1 115.70 115.76 113.82 113.82 7296.87 170.32 934.8 32.04 1.5 1.02 1.02

S.9.2 114.02 114.02 111.69 111.90 2850.34 804 33.06 1.2 1.02 1.02

S.9.3 113.92 113.92 109.45 109.95 1507.62 721.2 33.12 1.1 1.04 1.04

L.9.1 108.05 112.74 106.20 107.27 810.76 698.64 33.81 1 1.01 1.02

L.9.2 102.43 102.43 95.72 99.39 291.87 293.64 39.86 1 1.03 1.07

L.9.3 92.61 92.61 80.30 88.24 148.92 150.96 45.63 1 1.05 1.15

L.9.4 91.08 91.08 62.26 72.96 90.08 91.52 46.53 1 1.25 1.46

L.9.5 84.95 84.95 46.90 56.60 60.30 61.36 50.12 1 1.50 1.81

L.9.6 67.82 67.82 35.72 43.11 43.18 43.92 60.18 1 1.57 1.90

L.9.7 60.05 60.05 27.84 33.22 32.43 33.04 64.74 1 1.81 2.16

4.2 Load Versus Deflection Behaviour

The non-linear Load-Axial deformation behaviour of short and slender

columns having D/t ratio 28.22 and L/D Varies from 5 to 75 are shown in

Figure 6. The Comparison clearly indicates that the ultimate load carrying

capacity decreases while increasing the slenderness ratio from 5 to 75. The


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ultimate load of column having slenderness ratio 5 is 353.17 kN and it is reduced

to 192.75 kN when the slenderness ratio is increased to 75. When compared to

the theoretical crushing values, the strength obtained from numerical analysis

decreases from 29.19% to 61.35 % with increasing of slenderness ratio from 5 to

75.

From the comparison of load deformation behaviour of all the hollow steel

tubes, it is noted that the short columns showed a linear behaviour upto yield

load and after showed a non- linear behaviour, a sudden drop in the load

carrying capacity is found with larger deformation. But, the slender columns

having L/D ratio varies from 15 to 35 behave in a different manner. It undergoes

large deformation after reaching the peak load.

4.3 Effect of D/t ratio on Ultimate Load

Figures 6-8 show the load deformation behaviour for column having D/t

ratios of 28.2,39.1 and 84.7 respectively.

Table 2. Indicates that the ultimate load carrying capacity for smaller D/t

ratios is higher than that of larger D/tratios. The maximum percentage reduction

in ultimate load carrying capacity is 32% when D/t ratio increases from 28.2 to

39.1, whereas the maximum percentage reduction in ultimate load carrying

capacity is 69% when D/t ratio increases from 28.2 to 84.7.

The Figure 9 shows the ultimate load prediction with different slenderness

ratios for different D/t ratios of 28.2, 39.1 and 84.67. The reduction rate in

ultimate load is 45% when the slenderness ratio increases from 5 to 75 in case of

D/t ratio 28.2, whereas the reduction rate is 35% and 48% for the columns

having D/t ratios 39.1 and 84.67. Hence, the ultimate load carrying capacity

decreases with increase in D/t ratios (decrease in thickness), however the

reduction rate is not much affected by the variation in thickness


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Figure 9 Ultimate load vs. L/D Ratio

4.4 Effect of L/D ratio on Ductility Behaviour

Figure 10. shows the ductility factor with respect to the different L/D ratios

for various D/t ratios ranging from 28.2 to 45.6. The ductility factors of different

columns are calculated from the load-deformation curve and it is tabulated in

Table 2. From Table 2, it can be seen that the ductility factor is varying from 18

to 3.9 for short columns with increase in slenderness ratio from 5 to 11 and 2.8

to 1 for long columns with increase in slenderness ratio from 15 to 75.

The comparative study clearly shows that the short columns having less

slenderness ratio offers more ductility ratio. The slender columns take the load

up to yield point and immediately after that load carrying capacity reduces by

undergoing large deformation. This is mainly due to global buckling failure of

column at starting and then by local buckling failure of column.


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Figure 10. Ductility Factor vs L/D Ratios for Hollow Steel Tube

4.5 Buckling Behaviour of Steel Tube

The buckling modes observed during the analysis have been reported in

Figures 11 and 12. Results are compared to study the buckling failure modes

and section capacities. It is noticed that the column buckles in both inward and

outward direction. But inward buckling is predominant among both cases

because the column is made of hollow section and further it is noted that short

columns with L/D ratios varying from 5 to 11 go for inward local buckling

failure whereas the column with larger slenderness ratio go for initial global

buckling failure and then deform locally.


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Figure 11. Deformed Shape of Column S.1.3 at Ultimate Load

Figure 12. Deformed Shape of Column L.8.1 at Ultimate Load


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4.6 Development of Mathematical Model

Figure 13. Strength Curve in a Non-dimensional Form

In view of comparing the numerically predicted values with Perry-

Robertson prediction and Modified Perry-Robertson prediction[16] , a plot

between normalised stress ( ) and non-dimensional slenderness form λ =

is plotted as shown in Figure 13.

Perry-Robertson formula can be written as

(1)

Where


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= = Euler crippling stress

= Yield stress

= Compressive Strength

= Imperfection factor or Perry factor

Based on many steel column tests, Robertson concluded that the initial bow

could be taken as length of column/1000 and hence is given by

= = = =

= diameter of the column

= Length of the column

where , a parameter dependent on the shape of the cross section,

and . Robertson also evaluated the mean values

of for column designs. The design method based on Eq. (1) and the

value of mentioned earlier is termed as Perry-Robertson approach and has

been adopted by the British code.

It has been found that very stocky (stub) columns resisted loads in excess of

their squash load due to the effect of strain hardening. But the values

predicted by Eq.(1) will result in column strength lower than , even for very

low slenderness ratios. Hence, the British code BS-5950(1990) empirically


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modified the slenderness to to achieve a yield plateau in the design

curve. The value of was arrived at to fit the observed test data as

= (2)

Thus for calculating the elastic critical loads for , the following

Perry-Robertson modified formula was used

= for (3)

is given in Equation (2)

The axially loaded hollow steel tube yield strength predicted by

Equation (1)&(3) are compared with the analytical results are tabulated in

Table 2.

From this curve it can be easily noted that the Perry-Robertson and modified

Perry-Robertson equation predicts the ultimate load for a column upto an L/D

ratio of around 20. The / and / shown

in Table 2. also reveals that upto L/D ratio of 35, the above two equations

perform well.For L/D ratio greater than 35, deviation is found between Perry

Robertson theory and numerical results.Hence, the design compressive strength

of a hollow steel tube has been proposed, which is based on the analytical

results.

4.7 Proposed Mathematical Model

Sixty six hollow structural steel tubes are analysed under axial load Table 2.

presents the axial load capacity for these columns predicted by analytical


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and experimentally by Perry-Robertson . Furthermore, Figure

13 illustrates strength curve in a non-dimensional form.

From the graph it is noted that the equation for the lower boundary of the

present study is

= + 0.944 (4)

Here refers to and y refers to .

By substituting the and values in the equation (4), the new design

equation is proposed for calculating the compressive strength of axially loaded

thin-walled hollow steel tubes.

= -0.216 + 0.945 (5)

= (-0.216 * ) + ( 0.945 * )

=[ -0.216λ + 0.945] * (6)

Where λ = = Non-dimensional slenderness ratio

= Compressive strength of hollow steel tube

=Euler critical stress

= Yield strength of hollow steel tube


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From the Table 2. the ratio between the load predicted by FEM

and the value obtained by modified Perry Robertson equation, it can be justified

that the proposed numerical model and the applied analysis technique is capable

of predicting the yield load to the desired a accuracy with an standard deviation

of 50% upto L/D = 35 and 30% for L/D greater than 35.

Furthermore, Figure 14. illustrates the proposed model capacity of hollow

steel tubes are compared with the experimental results by O’Shea et.al. As it can

be seen from the Figure14, the coefficient of correlation between the proposed

model values and the experimental results are 0.97. And it is clear that the

experimental values are in excellent agreement with the proposed model, thus

conforming the validity of proposed model.

Figure 14. Comparison of Predicted and Experimental Capacity


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5. CONCLUSIONS

In the current numerical investigation, sixty six Thin-walled Hollow

Circular Steel tubular columns are analysed under axial compression. The

results reveal that the load carrying capacity and ductility of the columns are

found to decrease with increase in slenderness ratio. The short columns fail due

to inward local buckling, whereas the slender columns initially fail due to global

buckling and then deform due to locally.

The column strengths predicted from the parametric study using Finite

Element Model are compared with the design strength calculated using Perry

Robertson theory.It is shown that upto L/D= 35, FEM results and Perry

Robertson are performed well. The new design equation based on the analytical

results has been proposed for calculating the compressive strength of hollow

steel tubes having L/D ratio upto 75. From the comparison made between the

existing experimental results and the proposed model predictions;it is concluded

that the proposed model is capable of predicting the capacity of hollow circular

steel tubes to an acceptable accuracy.

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