experimental analyses of collapse behaviors of braced elliptical tubes under lateral compression

Int. J. Mech. Sci. Vol. 40, No. 8, pp. 761—777, 1998( 1998 Published by Elsevier Science Ltd. All rights reserved

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EXPERIMENTAL ANALYSES OF COLLAPSE BEHAVIORS OFBRACED ELLIPTICAL TUBES UNDER LATERAL COMPRESSION

LAN WU* and JOHN F. CARNEY IIIs*Department of Civil and Environmental Engineering, Vanderbilt University, Box 18, Station B, Nashville,

TN 37235, U.S.A. sWorcester Polytechnic Institute, 100 Institute Road, Worcester, MA 01609, U.S.A.

(Received 8 March 1996; and in revised form 17 July 1997)

Abstract—A series of experiments were conducted for braced, elliptical tubes with semi-axes ratios, b/a, rangingfrom 0.5 to 2 to verify the theoretical analysis results presented in a companion paper (Wu and Carney, Int. J.Mech. Sci., 1996). In that paper, mathematical models and finite element analyses (ABAQUS (Hibbitt, Karlssonand Sorensen, Inc. ABAQUS Manual, Version 5.3)) were employed to study the initial collapse behavior ofbraced elliptical tubes. These mathematical and numerical predictions are compared with experimental resultsin this paper. ( 1998 Published by Elsevier Science Ltd. All rights reserved

Keywords: tubes, elliptical, energy dissipation, collapse load, lateral compression.

1 . INTRODUCTION

Initial collapse analyses of braced elliptical tubes subjected to lateral compression using theequivalent structure technique (EST) [3, 4] is presented in a companion paper [1]. To verify thesetheoretical results, experiments were conducted on a series of braced tubes with b/a ratios rangingfrom 0.5 to 2. All of the elliptical experimental specimens were formed from two stocks of circularpipes with 7.62 cm (3 in) diameter. The thickness of the stocks were either 0.16 cm (0.063 in) or 0.2 cm(0.077 in). The b/a ratios were calculated using the actual measurement of the tubes. However, thediscussion classifies them into four categories according to the b/a ratios. The experiments wereperformed using an Instron testing machine interfaced with a plotter. The load—deflection curve wasrecorded for each test specimen, and a series of photographs was taken of each deforming tubeduring a test. The load—deflection curves contain numbered points. These numbered points relate tothe corresponding numbers in the companion figures containing photographs of the deformedshapes. Finite element simulations were performed using ABAQUS. The membrane forces andbending moments were recorded at specific deformation increments for each specimen in thecomputer simulations. The load corresponding to the stress distribution which initiated a mecha-nism in the tube was defined as the initial collapse load. The experimental results are shown in Fig. 1along with those from mathematical (EST) and numerical analyses (ABAQUS). The initial collapseload is normalized with the initial collapse load value of an unbraced elliptical tube with the samegeometrical features, where ‘‘M

P’’ is the plastic moment of the tube wall section, and ‘‘a’’ is the

horizontal semi-radius of the tube.

2. EXPERIMENTS

2.1. Circular tubes (b/a"1)Figure 2 shows the load—deflection curve of a 7.62 cm (3 in) diameter circular tube with horizontal

bracing. Its progressive deformation is shown in Fig. 3. The initial collapse load from the ABAQUSanalysis is marked with an asterisk in Fig. 2. The initial collapse occurs as the loading curveexperiences a significant change in its slope. However, the experimental load—deflection curve ischaracterized by two rises due to the non-uniform strength of the tube caused by a weld at the top ofthe tube. After the linear rise, the load—deflection curve starts a flatter segment. The deformation ofthe tube corresponding to this segment is shown in the second photograph in Fig. 3. Early in thisdeformation stage, plastic strains concentrate at the bottom of the tube causing large bendingdeformations within that region, and a load plateau is established. However, the loading locationsplits into two at the bottom due to the flattened shape. A stable mechanism is established in the tubeat this point which contributes to its increased load-carrying capacity. Moreover, the tube wall close

761

Fig. 1. Initial collapse load parameter versus bracing angle.

Fig. 2. Load—deflection curve for a 0° braced circular tube.

to the right bracing connection functions like a column which also strengthens the load capacity ofthe tube. As the plastic strains are fully distributed over the lower-half of the tube, a plastic hinge isdeveloped at the right bracing connection due to the restraint of braces which abrogates the‘‘column effect’’ and tends to reduce the load capacity of the tube. Meanwhile, plastic strains extendto the top part of the tube which is stiffer than the lower part due to the presence of the weld at thetop of the tube. The combination of these two mechanical effects results in a stable load plateau forthe tube. When a bifurcation of the loading point occurs in the upper-half of the tube, a stablemechanism, similar to a three-hinge arch, is established (see the third photograph in Fig. 3). As

762 L. Wu and J. F. Carney III

Fig. 3. Deformation of a 0° braced circular tube.Fig. 4. Deformation of a 10° braced circular tube.

Collapse behaviors of braced elliptical tubes 763

Fig. 5. Deformation of a 22° braced circular tube.


Fig. 6. Load—deflection curves for (a) a 10° braced, (b) a 22° braced circular tube.

a consequence, the load capacity starts another increase until the point where the loading locationseventually move to the right-hand side of the plastic hinge at the side of the tube (see the fourthphotograph). The mechanism developed in the tube at this time is no longer stable, and the finalcollapse is initiated. It can be seen in Fig. 3 that the tube undergoes a substantial amount ofdeformation before its load reaches the first plateau. Therefore, it is difficult to determine experi-mentally the actual initial collapse load. To simplify the problem, the first peak load is chosen as theapproximate initial collapse load. The same approach is used to select the initial collapse load for theother circular tubes with the same collapse behaviors.

The deformations and the load curves of a 10 and a 22° braced circular tubes are shown in Figs4—6. respectively. Similar to the 0° braced case, the 10° braced tube increases in strength during theearly stages of deformation after the initial collapse. The occurrence of the plastic hinge betweenthe left bracing connections tends to weaken the tube. However, the formation of a stiff ‘‘column’’ atthe right-hand side of the tube wall increases the strength. In addition, the plastic strain is distributedfrom the bottom to the top part of the tube. Consequently, the load—deflection curve showsan upward characteristic. When the left loading locations move to the left-hand side of the plastichinge, the mechanism is no longer stable and it reduces the load-carrying capacity dramatically(see Fig. 6(a)).

Figure 5 depicts a symmetric deformation of a small-angle braced tube. The collapse response ofa braced tube is sensitive to its geometrical imperfections. A small-angle braced tube sometimescollapses in a symmetric manner when its bracing angle is close to the limiting angle, /

8—10[1]. In

this case, the tube possesses an increasing strength when the plastic strain is distributed over the topand bottom regions of the tube after the initial collapse. However, when significant plasticdeformations concentrate at the hinge section at the bottom part of the tube, a localizedfailure mechanism results. Accordingly, the load magnitude drops sharply, as shown in Fig. 6(b).When the bottom hinge contacts the tightened braces, a three-hinge truss-like structure isgenerated in the lower part of the tube. The slope of the load-deflection curve thus becomespositive.


Fig. 7. Load—deflection curves for (a) a 20° braced, (b) a 25° braced, b/a"1.5 tube.

2.2. Non-circular tubes

2.2.1. b/a"1.5. The load deflection responses and progressive deformations of elliptical tubes(b/a"1.5) with 20° and 25° bracing are shown in Figs 7—9. Considering the 25° bracing case as anexample, the second photograph in Fig. 8 shows the deformed shape of the tube corresponding tothe first decreasing segment in the load curve. The plastic strains generated over the lower part of thetube introduce significant bending deformations in this region which replace the major membranestresses generated at the beginning of the stage. This is reflected in a temporary drop in the load. Thebifurcation of the bottom loading locations then introduces a stable mechanism (see the third picturein the figure), and a ‘‘column effect’’, similar to that mentioned in the circular tube case, is activatedon both sides of the tube. The load capacity of the tube then increases again. The fourth photographpresents the climax of this stage, after which a plastic hinge is developed between the two bracingconnections in the right-hand side of the tube. However, the ‘‘column effect’’ is still active on theleft-hand side, and the plastic strains start to expand to the top-half of the tube. A stable mechanismis thus established, and a nearly constant load is applied to the tube in this stage. However, when theright loading points move outside of the right-hand side hinge in the middle of the tube, themechanism in the tube is no longer stable, and the corresponding load curve decreases after thispoint (see fifth and sixth pictures in Fig. 8). The same approach for selecting the initial collapse loadas that in circular cases is adopted here for the tubes sharing the same yield mechanisms.

2.2.2. b/a"2. Figures 10 and 11(a) describe the deformation of a 40° braced tube with a b/a ratioof 2. In this large b/a ratio regime, the character of the load—deflection curves of the small-anglebraced cases are similar to those with a ratio of b/a 1.5. However, the first unstable segment of theload curve is deeper than in the former case because the curvature at the bottom of the tube is larger.The tube wall close to the bracing connections sustains large axial stresses before the occurrence ofsignificant bending in this region. As its bottom portion deforms, a plastic region is generated. Thestable mechanism enables the ‘‘column effect’’ to function (see the third picture in Fig. 10), and theload capacity increases accordingly. However, a highly concentrated plastic deformation then occursat the upper left bracing connection which generates a local mechanism and results in a dramaticdecrease in the load. Eventually, the braces lose their tension and the final collapse occurs. This local


Fig. 8. Deformation of a 20° braced, b/a"1.5 tube.Fig. 9. Deformation of a 25° braced, b/a"1.5 tube.


Fig. 10. Deformation of a 40° braced, b/a"2 tube.


Fig. 11. Load—deflection curves for (a) a 40° braced, (b) a 45° braced tube, b/a"2.

Fig. 12. Load—deflection curve for a 30° braced, b/a"2 tube, ABAQUS simulation.

mechanism develops so rapidly that the plastic strains have not extended to the top of the tube (seethe fifth photograph). The load curve corresponding to this stage decreases dramatically. The firstpeak value of the load curve is chosen as the initial collapse load. A similar local mechanism is alsoobserved in the finite element simulation results shown in Figs 12 and 13 for a similar b/a"2 casewith a 30° bracing angle. Since membrane forces are ignored in the EST approach, the initialcollapse loads predicted by EST are higher than those obtained from experiments and ABAQUSsimulations. The differences become pronounced for the tubes with larger b/a ratios where themembrane forces are relatively large at initial collapse, as shown in Fig. 1. Figures 11(b) and 14 showthe collapse process for a 45° braced tube.

2.2.3. b/a"0.5. Figures 15 and 16(a) show the deformation process and the correspondingload—deflection curve for an 8° braced elliptical tube with a b/a ratio of 0.5. In this b/a ratio regime,there is no distinct slope change in either ABAQUS or the experimental load—deflection curve whena mechanism is first formed (see Fig. 17). This phenomena is caused by the geometry of the tube.Figure 15 shows the deformation history of an 8° braced tube. Since both the top and bottom


Fig. 13. Deformation of a 30° braced, b/a"2 tube in ABAQUS simulation.

portions of the tube possess small curvature, the tube does not undergo much deformation beforethe loading locations bifurcate. This introduces a three-hinge arch mechanism in the tube. Becausethe top portion of the tube contains a weld, the bottom plastic hinge forms first and undergoessignificant rotation which results in a drop in load-carrying capacity. When the bottom collapsemechanism finally contacts the braces, the rotation is restrained by the braces and the strength isre-established in the tube. The similar collapse mechanism occurs in a 12° braced tube shown in Figs16(b) and 18. Since the ‘‘bifurcation’’ occurs soon after the onset of loading, it is difficult to determinethe actual initial collapse load from the load—deflection curve. To illustrate the significance of thebifurcation phenomenon, tubes of the same b/a ratio were tested under a concentrated loading


Fig. 14. Deformation of a 45° braced, b/a"2 tube.Fig. 15. Deformation of a 8° braced, b/a"0.5 tube.


Fig. 16. Load—deflection curves for (a) a 8° braced, (b) a 12° braced tube, b/a"0.5.

Fig. 17. Load—deflection curves for tubes with b/a"0.5.


Fig. 18. Deformation of a 12° braced, b/a"0.5 tube.


Fig. 19. The collapse mechanism in a b/a"0.5 tube subjected to a concentrated compressive load.

Fig. 20. Load—deflection curve for an 8° braced tube, b/a"0.5.

condition, as illustrated in Fig. 19. Figures 20 and 21 show the deformation process for an 8° bracedtube. Most of the early plastic deformation occurs at the bottom-half of the tube. The three-hingecollapse mechanism is similar to that of a fixed-end beam, as shown in Fig. 19. The initial collapseload can, therefore, be calculated as

Pu"8 M

p/¸ ,

where MP

is plastic moment, and ¸ is the equivalent length of the beam. In this case, the equivalentlength was measured to be 5.8 cm (2.3 in), and M

pequals 1.94 kgm (168.6 lbs in). Therefore, P

uis

266.1 kg (586.2 lbs), which is close to the experimental result.

3. DISCUSSION AND CONCLUSIONS

The initial collapse response of a braced tube depends upon its b/a ratio and bracing angle.Figure 1 illustrates the sensitivity of the load capacity to these two variables. The load capacitychange for a specific bracing angle range becomes larger as b/a increases.

The initial collapse models established in the theoretical analyses are ideal models. In reality, thestrength of the tube is not uniform due to a weld at the top of the tube. As a consequence, the plasticstrains are first concentrated in the bottom portion of the tube as the deformation progresses.However, the tendency of reducing the load-carrying capacity caused by concentration of the straindistribution over the weak portion of the tube is counteracted by the ‘‘column effect’’ in the tubesides and strain redistribution over the stronger top portion of the tube during the later stages ofdeformation. As a result, the overall load—deflection characteristics of the experiments are similar tothe theoretical results while the initial collapse responses are sometimes different. The differencesvary with the bracing angles and b/a ratios of the tubes. For tubes with a b/a ratio of 0.5, the loadinglocation bifurcation occurs almost immediately after the loading begins in the experiments. Thisbifurcation initiates a stable mechanism which causes an increasing load curve with deformation.Because of the relatively flat shapes of this group of tubes and geometrical imperfections, it is difficultto determine the initial collapse load from the test data. Therefore, the experimental results are not


Fig. 21. Deformation of an 8° braced b/a"0.5 tube subjected to a concentrated compressive load.


presented in Fig. 1. However, the theoretical results are compared with those of finite elementanalyses in Fig. 1 and are well correlated. For circular tubes (b/a"1.0), the initial collapse loadvalues from the EST, ABAQUS, and experiments are quite similar to each other, especially forsmall-angle braced tubes. However, for large-angle braced tubes, the experimental results are lowerthan theoretical predictions. This occurs because the tubes with large bracing angles are verysensitive to geometrical imperfections. The steeper slopes of the second portions of the four curves inFig. 1 show that the load-carrying capacity of a braced tube decreases drastically as the bracingangle increases. For tubes with b/a ratios larger than 1.0, the effects of geometrical imperfectionsbecome more severe as the b/a ratio increases. The differences in load-carrying capacities betweenexperimental and EST results are proportional to the bracing angles and b/a ratios. The ABAQUSresults for tubes with bracing angles within or close to the large bracing angle range are much lowerthan the EST results. This phenomenon can be explained by the exclusion of membrane stresses inthe EST. The axial stress becomes significant as b/a increases. The same local ‘‘column effect’’ occursin ABAQUS simulations and sometimes causes local buckling in the tube.

Acknowledgement—This research was supported by the Federal Highway Administration under Highway Safety ResearchGrant No. DTFH61-92-X-00001 of Vanderbilt University.

REFERENCES

1. Wu, L. and Carney III, J. F., Initial Collapse of Braced Elliptical Tubes Under Lateral Compression. International Journalof Mechanical Sciences, 1997, 39(9), 1023—1036.

2. Hibbitt, Karlsson & Sorensen, Inc. ABAQUS Manual, Version 5.3.3. Merchant, M. On Equivalent Structures, International Journal of Mechanical Sciences, 1965, 7, 613—619.4. Reddy, T. Y., Reid, S. R., Carney, J. F. III and Veillette, J. R. Crushing Analysis of Braced Metal Rings Using the

Equivalent Structure Technique. International Journal of Mechanical Sciences, 1987, 29(9), 655—668.


experimental analyses of collapse behaviors of braced elliptical tubes under lateral compression

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