Experimental study and numerical simulation on compressive buckling behavior of thin steel skins in unilateral contact with rigid constraints

Abstract

In this paper, a practical test and finite element analysis has been undertaken to further investigate the effects of contact buckling. A test rig was designed and constructed to record vertical and transverse deflections of compressively loaded steel skin plates. The boundary conditions were modeled as fully fixed. A finite element analysis was also undertaken using the software package Strand7. Results from both analyses have been examined and compared to data established from previous studies on contact buckling. Both the finite element analysis and practical results correlate well with this data. The result of the investigation has confirmed contact buckling theories and has foreshadowed the onset of the newly observed phenomenon of secondary contact buckling.

Keywords

compressive / buckling / thin steel skins / unilateral contact / rigid constraints

Introduction

Contact buckling is a phenomenon that occurs within composite members. The composite member is generally a combination of a thin steel skin and a filler material such as timber or concrete. When load is applied to the steel skin, it tends to buckle away from the filler material, thus inducing contact buckling. The term “contact” buckling is used as sections of the steel skin remain in contact with the filler material. Filler material such as concrete or timber provides no displacement or rotation at contact points, provided the material is stiff enough.

In 1958, Seide presented a paper that formed the foundational study of contact buckling. Seide analyzed a simply supported plate resting on a tensionless rigid foundation [1]. Since then, other studies have prevailed and contributed to the development and understanding of contact buckling including Chai et al. [2,3], Shahwan and Waas [4,5], and Ma et al. [6-10]. However, most of the above studies only focus on theoretical analysis.

The primary aim of this paper is to investigate contact buckling through both an experimental and finite element analysis to compare against results established in literature.

It is anticipated the various tests will allow observations to be made regarding the onset of secondary contact buckling. This will provide a foundation for further research and stimulate discussion regarding the phenomenon of secondary contact buckling.

Theoretical background

The onset of contact buckling may be calculated according to Eq. (1) [8].

(1)N_{\mathrm{cr}}=K({\pi }^{\mathrm{2}}E{t}^{\mathrm{3}})/[\mathrm{12}b(\mathrm{1}-v^2)],

where N_cr is the critical load for contact buckling; K is buckling coefficient; E and v are elastic modulus and Poisson’s ratio for steel; b and t are width and thickness of the steel skin.

Based on Eq. (1) and effective width theory, the ultimate compressive load may be calculated as [9]:

(2)P_{\max }={\sigma }_{\mathrm{av}}bt,

where σ_av = f_yρ; ρ = [1 - (0.22 / λ)]/λ; λ = √(f_y/σ_cr); σ_cr = N_cr/(bt); f_y is yielding strength of steel.

Basic parameters for the steel skins studied in the paper are shown in Table 1, where the <Equation NumberingStyle="Null" Important="0" Selected="0"><EquationNumber>(1)</EquationNumber></Equation>material properties are from the manufacturer, plate thicknesses are measured in the laboratory before tests.

Preliminary calculation based on Eqs. (1) and (2) are shown in Table 2.

Experimental study

A group of experimental tests were undertaken to investigate contact buckling. The composite member previously described was fabricated from sheet metal and timber. Throughout testing, loads and deflections were constantly monitored and used for determining key values.

Apparatus design

An apparatus based on ideas conceptualized in published literature and trial investigations was constructed. Figure 1 displays the final design used throughout the analysis. Figures 2 and 3 show the constructed rig ready for testing and after testing respectively.

An I-beam and “C” Channel were used as a base and center support to stabilize the test rig. A timber block, bolted to the I-beam formed the filler material. A thick steel sheet welded to the back of the I-beam carried half of the load provided by the hydraulic loader; the test sheet carried the other half. The thick steel sheet was unable to buckle and provided stability and load uniformity. The individual sheet metal specimens varied between 0.5 and 0.7 mm. The test sheets were connected to a length of “Unistrut” channel with a series of rollers. Each sheet contained cuts along both long edges to ensure all load was transferred through the desired test area of 600 by 150 mm. The front support frame screwed into the side of the I-beam and ensured the edges of the test sheet were fully fixed. Grease was applied to the surface of the frame in contact with the test sheet to reduce the resulting friction. Two lengths of channel were screwed into the sides of the timber block and stopped the sheet metal from being pulled inwards during testing; lateral movement was not constrained. A loading block ensured the load was equally distributed between the test sheet and thick sheet.

Throughout the analysis, axial load was continuously increased until the test sheet failed. Each sheet thickness was repeated three times.

Results and discussion

Figures 4 and 5 present the results from test 2 for the 0.5 mm thick sheet and test 3 for the 0.7 mm plates. This general trend was observed for all other tests.

Because there is no bonding between the steel skin and the timber block, the steel skin is not in perfect contact with constraints. Initial buckling begins with a noncontact buckling mode where the first transverse deflects negatively (top transducer in Fig. 4 and middle transducer in Fig. 5). The maximum negative deflections are between 0.2 and 0.3mm which are the initial gaps between steel and timber. Contact buckling happens after the initial noncontact buckling, where a new buckle with a positive deflection occurs (bottom transducer in Fig. 4 and top transducer in Fig. 5). The new positive (outwards) buckle develops quickly with an amplitude much larger than the negative buckle. In this study, we define the onset of the positive buckle as the initial contact buckling point.

Upon the recognition of contact buckling, additional jumps in deflection were observed for each plate thickness; these sudden changes in deflection, allowing more loads to be applied to the test sheet may indicate secondary contact buckling. Throughout the practical results, when the secondary mode change occurred, the deflection readings from the transducer would suddenly jump.

Test specimens were considered to have failed where the deflection continued to increase despite no recognition of an increase in applied load. As there is no obvious decreasing part in the deflection-load diagram, we consider the peak load before yielding as the failure load. The failure load in both Figs. 4 and 5 can be observed where the data deflection sets create a horizontal pattern, as the load is not increasing but the deflection continues to increase.

Table 3 details the critical contact buckling load, secondary buckling load and failure load for each individual test, and an average for each thickness. Results of contact buckling load and failure load obtained for both the 0.5 mm and 0.7 mm thick sheets agreed well with the theoretical values in Table 2 with minimal disparity.

Finite element analysis

The Finite Element Analysis (FEA) software package Strand7 was utilized to ascertain results for a plate contact buckling problem. Strand7 offers the capability to perform a nonlinear solver analysis which models the plate in the post buckling region while producing animation files in addition to numerical results. The nonlinear solver yields displacements, nodal reactions, nodal stresses/strains and element stresses/strains for each load increment.

The FEA nonlinear solver considers various geometric, material and boundary nonlinearities at a series of user defined and automatic increments. Additionally, Strand7 updates the stiffness matrix periodically (for each load iteration) to reflect the current deformed shape. This allowed bending and membrane stresses to be redistributed throughout the structure as it deforms; this is ideal for contact problems.

Strand7 further allowed provisions to model the contact problem with “Normal Contact” beam elements. “Normal Contact” elements generate compressive axial forces when the length of the beam decreases to zero; however, do not generate a tensile force. These elements also have the ability to generate lateral friction forces when in compression.

Basic parameters in the constitute model are taken from Table 1. To fulfill a buckling analysis, a geometric imperfection was considered here. According to the recommendation in Strand7 help files, an imperfection consisted of a point load around 0.5% of the axial load applied perpendicularly toward the middle of the plate with a resultant inwards deflection less than 0.001 mm.

Modeling

To avoid unstable detaching calculation in solving procedure, a small gap of 0.25 mm is assumed between steel and the rigid foundation. This value of the gap is taken from the negative buckling deflection in the tests as shown in Figs. 4 and 5. The steel skin will first buckle in a non-contact buckling mode. With the increase of the buckle, the lowest points will reach contact with the foundation and then contact buckling will happen (Fig. 6). As contact buckling results in a buckling mode change, observed by a change in the wavelength of the buckles, the initial contact buckling load is estimated as the load where the first mode change occurred. This was determined by analyzing FEA graphical representations of the deformed plate geometry. This result was further confirmed by analyzing plots of transverse and vertical displacement.

Contact buckling

Figure 7 depicts the transverse displacement against applied load at the midpoint of the 0.5 mm thick plate. From this graph it is evident that the midpoint experienced a local maximum positive transverse deflection just below 1.5 kN. The following sudden negative deflection represents the contact buckling mode change.

Graphical output produced by Strand7 further shows the observable mode change as seen in Fig. 8 which displays the deformed plate prior to, and after the contact buckling load.

From Fig. 8, it is apparent that Strand7 provided the ability to graphically display the contact buckling phenomenon. The mode change observed in Fig. 8 accurately aligned with the expected contact buckling loads calculated from the numerical output. Additionally, as conveyed in Table 5, the results closely match those calculated from theoretical formula.

It is evident that initial contact buckling of the 0.5 mm plate occurred at an axial load of approximately 1.42 kN, based on the trajectory of the data up until this point and Strand7 animations.

Similar results are observed in the numerical analysis for the 0.7 mm plate. Figure 9 depicts the transverse displacement against applied load at the midpoint of the plate. From this graph it is evident that the midpoint experienced a local maximum transverse deflection below a load of 4 kN. As reasoned previously, this indicates the onset of the contact buckling mode change.

Strand7 graphical output additionally shows the observable mode change, as seen in Fig. 10, which displays the deformed plate prior to, and after the contact buckling load. Again, based on the progression of the recorded deflections and corresponding loads, initial contact buckling of the 0.7 mm plate occurred at an axial load of approximately 3.75 kN.

Secondary contact buckling

Through the FEA nonlinear solver output, a subsequent buckling mode change occurred after the contact buckling mode change. This second mode change was observed in the deformed shape graphic produced by Strand7, and can be clearly viewed in the vertical and transverse displacement plots.

Although the nature of this buckling mode change is hard to investigate, it is possibly a form of secondary contact buckling. However, due to the limited research on this phenomenon, theoretical formulas to verify the results do not exist.

Figures 11 and 12 show the deformed geometry at two sequential load increments produced by Strand7 for 0.5 mm plate and 0.7 mm plate. It is evident that a mode change takes place between the two increments. As discussed, this mode change may be a form of secondary contact buckling.

Failure

The nonlinear solver did not specifically advise when the plate suffered failure. Therefore, to determine the failure load of the plate, results were carefully analyzed to determine the point where the structure became unstable or had visibly failed. The “Von Mises Yield Criterion” was selected for determining the plate stresses and strains due to the ductile behavior of the plates. The point where the yield stress and strain were exceeded was an indication that the plate had experienced plastic deformation. As members can withstand loads past their yielding point, the failure load had to be larger. However, any deflections past this load were non recoverable and reduced the overall stability of the plate. As noted, shortly after the yield stress was surpassed, the plate suffered severe deformation and failed.

The failure loads of the various plate thicknesses have been estimated at the point where the initiation of inconsistently large transverse deflections is noticed as seen in Figs. 13 and 14. Only graphs at each quarter point (1/4, 1/2, and 3/4) were placed in the report to compare the Strand7 results with the practical tests. The final failure loads are determined as follows: 5.69 kN for 0.5 mm plate and 9.99 kN for 0.7 mm plate.

Comparison among results from experiments, Stand7 and theoretical formulas

Table 4 compares the initial contact buckling loads, secondary buckling load and failure load calculated from Strand7, those calculated from published formula, and those from the experimental tests.

The initial contact buckling points and failure loads from three different methods agree well with each other with the percentage difference between all data sets falling well within a 13% envelope. The trend between each data set is similar, inferring that the problem was accurately modeled by both the FEA and practical tests. The result obtained is impressive verifying one another and confirming established formula. However, large errors are noticed between the secondary contact buckling results from experiments and Strand7. One possibility is that only three transducers were set up in one test, which may cause the missing of real secondary buckling points. The deflection jumping points recorded in the tests may be a higher order buckling-mode changing instead of the secondary buckling points. This would further explain why the test value is always higher than Strand7 results. Further experimental and theoretical investigation is required on the topic.

Conclusions

Through the analyses presented, the concept of contact buckling has been further developed and previously established theories have been confirmed. Experimental tests and FEA have yielded correlating results that conform to established buckling theories. A second mode change has been verified to exist with predictions resulting in the same level of accuracy when determining the contact buckling load.

However, in the experimental study, some limitations to the data collected were evident. The practical apparatus contained several design flaws, including the inability to control the gap between the steel skin and filler material. Future research may benefit from the use of video camera motion-detection technology as an alternative to the transducers used to record deflection. This would allow deflections to be recorded at numerous points on the plate leading to visible mode change animations similar to Strand7 results.

References

[1] Seide P. Compressive buckling of a long simply supported plate on an elastic foundation. Journal of the Aeronautical Sciences, 1958, 25(6): 382–384

[2] Chai H. Contact buckling and postbuckling of thin rectangular plates. Journal of the Mechanics and Physics of Solids, 2001, 49(2): 209–230

[3] Chai H, Babcock C D, Knauss W G. One dimensional modelling of failure in laminated plates by delamination buckling. International Journal of Solids and Structures, 1981, 17(11): 1069–1083

[4] Shahwan K W, Waas A M. A mechanical model for the buckling of unilaterally constrained rectangular plates. International Journal of Solids and Structures, 1994, 31(1): 75–87

[5] Shahwan K W, Waas A M. Buckling of unilaterally constrained infinite plates. Journal of Engineering Mechanics, 1998, 124(2): 127–136

[6] Ma X, Butterworth J W, Clifton C G. Elasto-plastic post-buckling analysis of plates resting on tensionless foundations. In: Proceedings of 19th Australian Conference on the Mechanics of Structures and Materials, Christchurch, New Zealand, 2006: 103–108

[7] Ma X, Butterworth J W, Clifton C G. Compressive buckling analysis of plates in unilateral contact. International Journal of Solids and Structures, 2007, 44(9): 2852–2862

[8] Ma X, Butterworth J W, Clifton C G. Initial compressive buckling of clamped plates resting on tensionless elastic or rigid foundations. Journal of Engineering Mechanics, 2008, 134(6): 514–518

[9] Ma X, Butterworth J W, Clifton C G. Practical analysis procedure for compressive local buckling of skin sheets in composite panels. International Journal of Advanced Steel Construction, 2008, 4(3): 230–242

[10] Ma X, Butterworth J W, Clifton C G. Unilateral contact buckling of lightly profiled skin sheets under compressive or shearing loads. International Journal of Solids and Structures, 2008, 45(3-4): 840–849

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