of Phigh - WIT Press · sspii: skew symmetric progressive buckling mode; caused by an increase in the size of a lobe ebi: an extreme case of sspi which leads to overall buckling

The effect of induced imperfections on the

formation of the first lobe of symmetric

progressive buckling of thin- walled square

tubes

N.S. Marshall; G.N. Nurick

Department of Mechanical Engineering, University of Cape

Town, f rh/afe #ag TfcWeWc/z 7707,

Abstract

An experimental investigation showing the effect of induced imperfections onthe symmetric progressive buckling of thin-walled square mild steel tubes ispresented. The imperfections include: a circular hole; indentations of variousshapes; and combinations of a hole positioned centrally in an indentation. In allcases identical imperfections were induced symmetrically in opposite walls ofthe tube. The results indicate that the ultimate buckling load of the tubedecreases with an increase in the severity of the imperfection. The distancebetween the plastic hinges that form the first lobe of buckling decreases withtwo opposing holes, but increases with large indentations. The change in thesize of the first lobe leads to instabilities in the symmetric buckling mode whichin turn leads to a decrease in the stroke of the tube. The results further indicatethat combining a hole with an indentation has a cumulative effect on both theultimate buckling load and the size of the first lobe. The stability of thesymmetric progressive buckling of the tube is retained while decreasing theultimate buckling load significantly for a range of indentation sizes and holediameters.

1 Introduction

It is an established practice to use the energy absorption characteristics ofthe symmetric progressive buckling of long tubular structural members inmany impact situations because a large amount of kinetic energy can be

Transactions on the Built Environment vol 32, © 1998 WIT Press, www.witpress.com, ISSN 1743-3509

156 Structures Under Shock and Impact

Pult

Decrease Pult to the levelof Phigh

10 15Displacement (mm)

Figure 1 Typical force-displacement curve of quasi-static symmetricprogressive buckling of a thin-walled square mild steel tube.(50.8x50.8x1.2mm) (Note: Plow: low peak force; Phigh: high peakforce; Pmean: mean force; Pult: ultimate buckling force)

absorbed in a restricted distance. There are two basic requirements forcontrolled and efficient energy absorption by thin-walled tubes. Firstly, tomaximise the energy absorption capacity of the structure the entire strokeshould be used to absorb energy, i.e. instabilities in the bucklingmechanism that initiate euler buckling should be avoided. Secondly, tomaintain smooth deceleration, the force exerted during energy absorptionshould be constant over the entire stroke.

An example of a force displacement curve of a thin-walled squaresection tube collapsing in the symmetric progressive buckling mode isshown in figure 1. The figure illustrates that after the maximum peak forceis reached, the force oscillates about a mean load for the duration of thestroke. The first peak force may exceed the subsequent peaks by more thana factor of 2. To reasonably approximate a constant force throughout thestroke, the maximum peak force should be decreased to the magnitude ofsubsequent peak forces.

Many research efforts have been devoted to the analysis andoptimisation of the energy absorption characteristics of thin-walled tubularstructures *"*\ Typically indentations are used to decrease the ultimate peakforce and to induce controlled symmetric progressive buckling ™. It has


Structures Under Shock and Impact 157

been reported that while the depth of the indentation can be used to scalethe ultimate buckling load , the shape of the indentation and the position

of subsequent indentations effects the collapse mode °. The introductionof holes in circular tubes has also been reported "" as a means to decreasethe ultimate buckling load and induce specific buckling modes. In thispaper the effect of holes; dents; and a combination of the two is explored.

2 Experimental Program

The specimens in this series of tests are seam-welded mild steel squaretubes. Three lengths of tube 50.8mm wide and 1.2 mm thick were used.Each length was assigned a letter by which the test specimens were

numbered.Imperfections were induced mechanically into the tubes. In all cases two

identical imperfections were induced symmetrically in opposite walls of thetube. The induced imperfections include: a hole; indentations of variousshapes; and a combination of a hole positioned centrally in a dishedindentation. Holes of 16, 22, 25, 32 and 38mm diameter were drilled intothe tubes (A2-A6,G8,G18). Cylinders of different diameters (2mm (AA1-AA3); 30mm (AB1-AB5); 50mm(ACl-AC4)) were used to induce

opposing parallel indentations. Dished indentations were induced using ahemispherical indenter with a radius of the order of 100mm (G1-G3,G5-G7). Combined imperfections were induced by first drilling a hole andsubsequently indenting the tube with a hemispherical indenter positionedsymmetrically above the hole (G9-G16).

A number of specimens without imperfections were quasi-staticallycrushed (A1,AAO,G4). These "perfect" specimens were used as standardsby which to compare the collapse characteristics of tubes with induced

imperfections.The tubes were quasi-statically crushed on an Instron Tester with a

maximum rating of 20000M (89kN). The specimens were loaded betweenparallel plates with a constant velocity of 0.2 inches/min (5.08 mm/min).

Quasi-static uniaxial tensile tests were performed from which the staticyield stress and the ultimate tensile stress of the material was found usingthe Cowper-Symonds equation " with typical constants for mild steel(D=40.4s~*;q-5). Material from each of the lengths of tubing was testedand the results of the tests are presented in table 1.



3 Results And Discussion

Table 1. Tensile Test ResultsTube

AAA,AB,AC

G

Static Yield Stress (cr*)MPa274285282

Ultimate Tensile Stress (UTS)MPa315324322

The results of the crushing tests are presented in table 2, Buckling moderefers to the mode of collapse of the tube is defined after the table.

Table 2. Quasi-static crushing tests of 50mm square tubestestno.AlA2A3A4A5A6AAOAA1AA2AA3AB1AB2AB3AB4AB5AC1AC2AC3AC4GlG2G3G4G5G6G7G8G9G10GilG12G13G14G15G16G18

hole diam.mm01622253238

162525.725.73232.53216.438.425

dent depthmm

5.08.811.62.85.47.68.711.32.86.18.812.01.33.76

6.541.5

6.341.55.941.244

tube lengthmm300300300300300300300300300300300300300300300300300300300250250250250250250250250250250250250250250250250250

PultkN746962515245783433344031313334423234365245427541455372363744333441403157

PhighkN3330283125183234

3434282429313131

22273035302630

262833343441403130

PmeankN222321212215

21202019202116232020241718

19

bucklingmodespsspisspiebitrtrspsp

sspiisspiispsp

sspiisspiispsp

sspiisspii - ebiisspii - ebiisspii - ebii

spsspii - ebiisspii - ebiisspii - ebii

sspisspiispsspisspiispsspisspiisspisspi



Note:sp: symmetric progressive buckling mode

sspi: skew symmetric progressive buckling mode; caused by a

decrease in the size of a lobesspii: skew symmetric progressive buckling mode; caused by an

increase in the size of a lobeebi: an extreme case of sspi which leads to overall buckling.ebii: an extreme case of sspi which leads to overall buckling.tr: tearing axially along the tube.

3.1 "Perfect" specimens

Specimens Al, AAO and G4, which did not have induced imperfections,collapsed in the symmetric progressive buckling mode. The ultimatebuckling load is approximately 75kN. The high peak force, or the load towhich the ultimate buckling load should ideally be reduced, is 35kN.

3.2 Simple Imperfections

3.2.1 Two holes drilled opposite one anotherThe ultimate buckling load decreases linearly as the hole diameter increasesas shown in figure 2. This would indicate that to satisfy an ultimatebuckling load of 35kN, a hole approximately 45mm in diameter is required.However, for holes greater than 32mm the edges inter link and the tube

tears instead of folds, as shown in figure 3 (A5, A6).A linear regression of the data points of the holed specimens is shown

as a solid line in figure 2. The extrapolated line intersects the ultimatebuckling load of a specimen without holes (75kN) for a hole of 10.5mm.This infers that holes smaller than 10.5mm do not decrease the ultimatebuckling load of the tubes.

The hole diameter appears to also affect the formation of the first lobe.An increase in the diameter of the holes causes a decrease in the size of thefirst lobe as illustrated in figure 3. The result of a decrease in the size ofthe first lobe is that the lobes form skew and destabilise the buckling mode.

Surko and Meng et al™ used Von Karman's postulate of effectiveplate width to calculate the compressive buckling load of square tubes. Inorder to simplify the elastic-plastic analysis of a thin plate simplysupported at its edges and loaded compressively in plane it is proposed thatthe load is carried in two strips adjacent to the edges. The width of each



10 15 20 25 30 35

Hole Diameter (mm)

Figure 2 The effect of two holes aligned opposite one another on theultimate buckling load of square tubes. (The solid line is a least squareregression through the data points of specimens with holes, the dashedline indicates the ultimate buckling load of "perfect" specimens.)

Figure 3 Quasi-statically crushed square tubes with two opposing holes.(The diameter of the holes increases from left to right. It appears that thesize of the first lobe decreases with increasing hole diameter until tearingoccurs)



strip is defined as half the effective plate width and is given by:

* *** Sf (i)

where : E is Elastic modulus; h is effective plate width; t is wall thickness;

ju is Poisson's ratio; Oy is yield stress.This concept means that for a tube of sufficient width (greater than the

effective plate width of the tube) a hole in the centre of the width will noteffect the load bearing capacity of the tube if the hole is small enough not

to encroach into the load bearing strips.Using eqn (1), for a 1.2mm thick plate with an elastic modulus of

200GPa and yield stress of SOOMPa, the effective width of the plate is58.9mm. This exceeds the width of the tube, and hence a hole of anydiameter should effect the load carrying capacity of the tube. However thetube corners may carry extra load which would account for the discrepancyin the effective plate width of the tube. The decrease in the size of the firstlobe is also consistent with the idea of effectively decreasing the width of

the tube.

3.2.2 Two opposing Parallel IndentationsStudies reducing the ultimate buckling load of the tube by inducingindentations shaped to resemble the initiation of progressive buckling lobeshave been reported **. These results show a decrease in the ultimate

buckling load with increasing severity of the predent.The present study includes inducing indentations using cylinders of

different radii. The radius of the indentation is essentially the same as theradius of the cylinder used to create the indentation.

The results of parallel indentations on the ultimate buckling load of thetubes plotted in figure 4 show that the ultimate buckling load decreaseswith increasing severity of the indentation. It appears that the radius of theindentation does not effect the magnitude of the ultimate buckling load. Theslight increase in the ultimate buckling load of the tubes with indentationsexceeding 8mm is an anomaly. These specimens demonstrate severeinstability in the buckling mode similar to that of specimen AC4 in figure5. This may be as a result of a slight deviation in the specimen preparation.

Surko reported that the shape of the first lobe of the buckled tube isdetermined by the shape of the indentation. The length of tube affected bythe indentation increases with the depth of the indentation. It is thereforeexpected that the size of the first lobe will increase as the depth of the



Ultimate Buckling Load (kN

,388688388

c

1

*0

I (T Xx

••-«: ""0 0

) 2 ^ 4 ^ 6 , 8 10 . 12 1Dent Depth at the comer (mm)

Diameteroflndenter

O Sharp

+ 2mm

A 30mm

• 50mm

X dished

4

Figure 4 The effect of opposing parallel indentations and opposing dishedindentations on the ultimate buckling load of square tubes.

Figure 5 Quasi-statically crushed square tubes with opposing parallelindentations. (The indentations were induced with a cylinder of diameter50mm, and the depth of the indentations increases from left to right. Itappears that the size of the first lobe increases with increasingindentation depth.)



indentation increases. Tubes indented using a 50 mm diameter cylinder and

crushed quasi-statically are shown in figure 5. These specimens

demonstrate the increase in the size of the first lobe with an increase in thedepth of the indentation.

3.2.3 Dished IndentationsDished indentations produced using the hemispherical indenter are deeperin the centre than at the corners of the tube. The indentation depth reportedis the depth at the corners of the tube. The results of the depth of theindentations on the ultimate buckling load are plotted in figure 4. It is clearthat the effect of dished indentations on the ultimate buckling load is lessmarked than that of parallel indentations.

The effect on the stability of the symmetric buckling mode is shown infigure 6. The trend is similar to, but more exaggerated than that ofindentations formed by cylindrical indenters. The photograph shows thatthe size of the first lobe increases with an increase in the depth of theindentation. As the first lobe develops, opposite walls of the tube meet, thuspreventing the full formation of the lobe. The effective stroke of the tube istherefore significantly decreased (by between 1 and 2 tube widths).

A , # # ^ # P% jBk

Figure 6 Quasi-statically collapsed square tubes with opposing dishedindentations. (The indentations were induced with a hemisphericalindenter of radius 100mm, and the depth of the indentations increasesfrom left to right. The size of the first lobe increases significantly withincreasing indentation depth.)



3.3 Combined imperfections

3.3.1 Dished indentations with holesThe effect of the combined imperfections with respect to depth of theindentation at the corners of the tube is plotted in figure 7. The same datais presented with respect to the diameter of the hole in figure 8. The resultsindicate that holes and indentations acting in combination have a greatereffect on the ultimate buckling load than either holes or indentations acting

independently.Specimens with 32mm holes and indentations increasing in depth from

left to right are shown in Figure 9. The first lobe in the specimen on the left(G14; 1.5mm dent depth) appears to be smaller than the natural lobe sizefor the tube. The effect of the large hole dominates the effect of a smallindentation. The tube on the right of the figure (G12; 6.3mm dent depth)has an increased first lobe size. The greater dent depth dominates the effecton the distance between the plastic hinges.

The specimen in the centre of the figure (G13; 4mm dent depth) showsregular lobe formation, where the size of the first lobe is similar to thesubsequent lobes. The influence that the 32mm holes have on decreasingthe lobe size is countered by the 4mm indentations which increase the sizeof the lobe. Hence this combination gives a first lobe of similar size to thenatural lobe size.

Similar observations are made from figure 10. The specimens all haveindentations of between 3.5mm and 4mm. The left hand specimen (G2) iswithout holes, while the other specimens have holes increasing in size fromleft to right. Again the tubes with smaller holes and larger indentations (G2and G15) have an enlarged first lobe; the tube with larger holes and smallerindentations (G16) has a smaller first lobe. The specimens with holes of 25and 32mm (G10 and G13 respectively), and indentations 3.5mm to 4mmexhibit a first lobe sufficiently close in size to the natural lobe size so as toallow stable symmetric progressive buckling. This indicates that the rangeof imperfection dimensions that achieve an adequate first lobe size forstable symmetric progressive buckling is reasonably large.




oo

S8

68

83

2

1

1

1 ") o ++ .

A X * •A ° °^ 0 ^

] 2 4 6 (Dent Depth at the comer (mm)

3

HoleDiameter(mm)

+ none

X16

A22025

A 32

038

Figure 7 The effect of the depth of dished indentations on the ultimatebuckling load of specimens with opposing combined imperfections


oS88S883£

*

1 4 *: o •1 X 0

Xa x

) 5 10 15 20 25 30 35 4Hole Diameter (mm)

Dent Depth(mm)

+ none

n 1.2-1 .5

X 3.0-4.0

A 5.9-6.5

0

Figure 8 The effect of the diameter of holes on the ultimate buckling loadof specimens with opposing combined imperfections.



Figure 9 The effect of the depth of dished indentations on the ultimatebuckling load of specimens with opposing combined imperfections (Thetubes all had opposing 32mm diameter holes and the depth of theindentations increases from left to right.)

Figure 10 The effect of the diameter of holes on the ultimate buckling loadof specimens with opposing combined imperfections (The tubes all hadindentations of depth 3.5-4mm. The diameter of the hole increases fromleft to right (specimen G2 has no hole).



4 Conclusions

The decrease in the ultimate buckling load with the introduction of simpleimperfections illustrated in figures 2, 4, 7, and 8 is consistent with theliterature ™*. The ultimate buckling load can be controlled by adjusting theseverity of the imperfection. For the tubes considered herein the ultimatebuckling load should be reduced to 35kN. The load is then comparable to

the subsequent peak loads.The photographs (figures 3, 5, and 6) show that as the severity of

simple imperfections increases, so the stability of the symmetric bucklingmode decreases. The implication is that in many cases the imperfectionseverity cannot be increased enough to decrease the ultimate buckling loadsufficiently. The stroke of the tube would be compromised and thus themaximum amount of energy that can be absorbed would decrease.

The stability of the buckling mode is compromised due to the change inthe size of the first buckling lobe. Holes effectively decrease the width ofthe tube, and thus decrease the size of the first buckling lobe. Indentations

increase the size of the first lobe.Combined imperfections have a greater effect on the decrease in the

ultimate buckling load than either holes or indentations acting

independently.The effect of holes and indentations on the change in the size of the first

lobe is cumulative. A combination of an indentation and a hole of theappropriate severity therefore results in a first lobe the same size assubsequent lobes. The stability of the buckling mode is thus not

compromised.

References

[1] Johnson, W, Soden, P.O., & Al-Hassani, S.T.S., InextensionalCollapse of Thin-Walled Tubes under Axial Compression. J. Strain

Anal., Vol. 12, No. 4, pp. 747 - 773, 1977.

[2] Abramawicz W., The Effective Crushing Distance in AxiallyCompressed Thin-walled Metal Columns. Int. J. Impact Engng. 1, pp.

309-317, 1983.

[3] Abramowicz, W. & Jones, N., Dynamic axial crushing of squaretubes, Int. J. Impact Engng, Vol 2, No 2, pp. 179-208, 1984.



[4] Abramowicz W. & Jones, N., Static and dynamic axial crushing ofcircular and square tubes. Metal forming and Impact Mechanics, ed.S.R. Reid, pp. 225-247, Pergamon Press Oxford, 1985.

[5] Abramowicz, W. & Wierzbicki, T., Axial Crushing of MulticornerSheet Metal Columns, J. Appl Mech. 56, pp. 113 - 120, 1989.

[6] Wierzbicki, T, Bhat, S.U., Abramowicz, W., & Brodkin D,Alexander Revisited - A Two Folding Elements Model of ProgressiveCrushing of Tubes, Int. J. Solids Structures Vol. 29, No. 24, pp.

3269-3288, 1992.

[7] Schriever, T. & Helling, J., Zum Einflup gezielt eingebrachtergeomertischer Imperfektionen auf das Verformungsverhalten vonLangstragerstrukturen. (Source unknown)

[8] Langseth, M., Berstad, T., Hopperstad, OS, & Clausen, AH,Energy Absorption in Axially Loaded Square Thin-Walled AluminiumExtrusions, Structures Under Shock and Impact III, ed. P.S. Bulson,CMP, Southampton, pp. 401-410, 1994.

[9] Korneck, H, An Investigation into the Response of Square BoxColumns to Axial Loading, Project No 91, University of Cape Town,1992.

[10] Surko, W, Crushing Strength of Box Columns with Partially-Damaged Plating, Int. J. Mech. Sci., Vol. 33, No. 12, pp. 1017-1028,1991.

[11] Toda, S., Buckling of Cylinders with Cutouts under AxialCompression, Exp. Mech. pp. 414 - 417, 1983.

[12] Gupta, N.K & Gupta, S.K., Effect of Annealing, Size and Cutouts onAxial Collapse Behavior of Circular Tubes, Int. J. Mech. Sci., Vol.35, No 7, pp. 597-613, 1993.

[13] Jones, N., Structural Impact, Cambridge University Press,Cambridge, 1989.

[14] Meng, Q, Al-Hassani, S.T.S., & Soden, P. D, Axial crushing ofsquare tubes, Int. J. Mech. Sci., 25, pp. 747 - 773, 1983.


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of Phigh - WIT Press · sspii: skew symmetric progressive buckling mode; caused by an increase in the size of a lobe ebi: an extreme case of sspi which leads to overall buckling