Finite element Investigation of Sandwich Panels Subjected to Local Buckling Dr. Hayder H. Alkhudery

Finite element Investigation of Sandwich Panels Subjected to Local Buckling Dr. Hayder H. Alkhudery and Prof. Kuldeep V. Abstract: The sandwich panels, one of the attractive engineering structures mixed between two different materials to achieve high ultimate strength associated with light weight structures.

Usually, the sandwich panels comprise of foam core and thinner high strength steel faces. This report discusses currently design formulae of local buckling behaviour of sandwich panels with profiled faces using finite element method. Multiple wave finite element models adopted to investigate and examine the adequacy of currently approach for design. This report presents the details of examining the FEA model including geometry, dimensions, load pattern and boundary conditions. The FEA model gives well agreement using experimental programme of Pokharel and Mahendran (2003). However, it appears the currently design formulae are conservative for the plate elements with low b/t ratios while over conservative for high b/t ratios (slenderness plate). A unified design formula of local buckling behaviour is developed.

1-Introduction: In the recent years, we have observed an increase in practical application of sandwich panels in civil engineering construction. High bending stiffness coupled with small weight and very good thermal and damping properties make sandwich panels attractive structures for designers. Ease of transport and assembly in all conditions are additional advantages. These aspects have also generated a growth in computational and experimental research.

Sandwich panels are composite structural elements, consisting of two thin, stiff, strong faces separated by a relatively thick layer of low-density and low stiffness material. The faces are commonly made of steel, aluminium, hardboard or gypsum and the core material may be polyurethane, polyisocyanurate, expanded polystyrene, extruded polystyrene, phenolic resin, or mineral wool. The sandwich panels most often used in civil engineering consist of two steel flat or profiled faces and a relatively soft core (Fig.(1)). The faces carry normal stresses, while the three principal roles of the core are to carry shear stress, to protect the compressed faces against 1

buckling and to provide thermal insulation. Because sandwich panels have flexible cores, their behaviour is therefore more complex than that for plain plates.

Figure (1): Types of sandwich panels.

Few standards are available for the design and testing of sandwich panels. One of the important documents is the European Recommendations for Sandwich Panels which is based on rational analysis and test results.

It is important to understand the structural behaviour of sandwich panels, since they display numerous failure modes. The different failure modes are related to external loads, overall buckling, local buckling, wrinkling and overstressing of faces. Local buckling of flat plate elements is the critical failure mode for fully profiled sandwich panels, whereas flat and lightly profiled panels undergo a flexural wrinkling type failure (Fig.(2)).

2

a) Local Buckling of flat plate element

b) Flexural wrinkling of lightly profiled panels.

Figure (2): Failure modes of sandwich panels.

2- Motivation: The currently used design formulae of sandwich panels have been suggested by many researchers depending on assumptions which sometimes do not represent actual structural behaviour of sandwich panels. Most of these formulae are complicated and follow complicated procedure to determine the ultimate strength of panels.

In the present work, a unified design formula is presented based on finite element analysis investigations. Essentially, the ultimate strength of sandwich panel is related to b/t ratio and material properties of faces and core. The Waves-Length model (W-L) used for FEA was verified through comparison with experimental results.

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3- Literature Review: 3-1 General: Davies and Hakmi (1990) [5] studied the local buckling behaviour of a compressed plate element supported by relatively weak isotropic core medium. When the sandwich panel is subjected to uniform compression the authors represented the panel as a simply supported plate resting on half space linear elastic foundation. The critical buckling stress was determined using the principle of minimum strain energy method following Timoshenko and Gereâ€&#x;s method (1961). For simply supported plates without foundation they added the strain energy contribution for the core assuming it to be the elastic foundation with exponentially decaying displacement.

Parameter representing the decay in the displacement of the core was found by minimizing the strain energy. This leads to a formula for the buckling coefficient of sandwich pane which is used to achieve the ultimate strength of plate using the effective width concept (i.e.Winter formula). The authors found that the results did not quite agree with the experimental results.

Improved correlation was obtained by arbitrarily reducing the value of an intermediate parameter.

Davies and Hakmi (1991) [6] reviewed the analytical methods for the evaluation of buckling stress of sandwich panels for plane and profiled faces. They developed analytical formula for wrinkling stress based on considering the core as an elastic half space, by treating the width of the plate as wider flat face (i.e. width of the plate increases to infinity), this formula is modified for lightly profiled faces by introducing the effect of flexural rigidity of profiled face.

Also the authors introduced practical design reduction factors for both cases due to imperfection and material non-linearity. For the profiled sandwich panels authors adopted design formula similar to Winter formula based on experimental mean values and numerical post-buckling analysis.

Davies (1993) [7] has reviewed the state of the art with regard to the design of sandwich panels as load bearing structural members. Mainly two cases of the sandwich panel analysis are investigated. Firstly, global analysis was carried out by considering the sandwich panels as an ordinary beam with the principle of antiplane core which means constant shear stress distribution across core depth and no deformation of the core in a perpendicular direction of faces. Secondly, 4

local buckling analysis of compressed face elements was considered when sandwich panels are classified into three cases, flat faces at which the wrinkling failure is governing the failure mode, profiled faces which failed with local buckling failure of plate element, and lightly profiled faces as an intermediate case which my fail with interaction mode between the two previous cases depending on the profiled depth and spacing part between folds. Author has introduced a design formula for each discussed sandwich panel type with practical reduction factor determined using tests results of products or using a value which will be generally conservative.

Mahendran and Jeevaharan (1999) [8] investigated the validity of Europe Design Recommendations for local buckling behaviour of sandwich panels made of high steel strength (i.e. Australian panels). They carried out laboratory experiments and finite element analysis on steel plate elements of varying yield stresses and thickness with and without supported polystyrene foam core. Through qualifying the FEA with experimental results, they introduced modified design stress buckling formula based on FEA results.

Mahendran and McAndrew (1999 and 2001) [9 and 10] studied the wrinkling failure of sandwich panels of flat steel faces with presence of joints between the polystyrene foam slabs in the transverse direction(i.e., a “gap” between the foam slabs and a “step” due to the difference in the heights of foam slabs. They investigated that effect through a series of full scale tests and finite element analysis. Three types of foam joints are conducted half-width butt, full-length butt, and full-width butt glued; also two grade foam cores were used. On average, authors indicated from experimental results about 20% reduction to wrinkling stresses due in presence of joints. In FEA, they used a half-wave buckled model to simulate a sandwich panel without joints which agreed well with the theoretical results, the model was then used to investigate the effects of foam and steel face thicknesses. Both experimental and FEA with half-wave model results showed that the face thickness and foam thickness 75mm or more do not affect the wrinkling stress of flat face of sandwich panels. Also, they used a larger-length model to simulate a panel with joints which were adequate to obtain accurate results. This model is used to investigate the effect of presence of gap and step imperfections. They found a significant reduction in wrinkling stress occurred even with very small gap size, and this initial reduction was gradual with increasing the gap size up to 10mm, while the step imperfections did not cause any reduction in wrinkling capacity.

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Mahendran and McAndrew (2003) [11] extended previous work to study the effects of lightly profiled faces and transverse joints on the flexural wrinkling stress of lightly profiled sandwich panels. They carried out a series of tests using two types of profiled face namely ribbed and satinlined with different face thickness. Also, two foam grades were used. In numerical investigation using FEA method, the authors adopted previous models (i.e. half-wave and largerlength model) to simulate the sandwich panels. They indicated similar significant reduction in wrinkling stress for lightly profiled face as in the previous study of flat face in the presence of transverse joints with gaps and step imperfection. In the experimental results, they did not observe the expected increase in the wrinkling stress due to the lightly profiled face. Authors indicated that the failure mechanism of satinlined profiled is similar to flat face, while in the ribbed profiled, an interaction between wrinkling of whole face and local buckling of flat element plate appears which is behind the significant difference between the FEA and theoretical results.

Pokharel and Mahendran (2002, 2004) [4, 1] carried out extensive experimental test for simply supported steel plate element supported by foam core subjected to uniaxial load with different steel grade (250 and 550) and different plate dimension to cover large variation of (b/t) ratio. Authors conducted three models in FEA to simulate the local buckling behaviour of sandwich panels. They indicated good agreement to use the FEA with full-length and half-length models to simulate the experimental tested cases. Then they use third FEA model, half-wave model to simulate the practical local buckling of element plate supported by foam core, then they use half-wave model with large range of (b/t) ratio to introduce improved design formula of local buckling of sandwich panel. But, it should be noticed that these two papers were gave two results of the same plate element using the same half-wave model.

Pokharel and Mahendran (200) [3] used the previous experimental test for simply supported steel plate element supported by foam core subjected to uniaxial load with different steel grade (250 and 550) and different plate dimension to cover large variation of (b/t) ratio, to modify the design formula of local buckling of sandwich panels by modifying the buckling coefficient formula.

The experimental programme of Pokharel and Mahendran [2] has been adopted as a source of experimental data in this paper. In this programme of laboratory tests were conducted 6

on plate elements supported by polystyrene foam as used in the profiled sandwich panels. A large range of b/t ratios (between50 to 500), both the thickness and width of the plates were varied. Also two different grades of steel were used, one mild steel with a minimum yield stress of 250 MPa and the other high strength steel with a minimum yield stress of 550 MPa. A single type of foam core was used with average tested mechanical properties Ec=3.8 MPa and Gc=1.76 MPa. The widths „b‟ of the plates were; 50, 80, 100, 120, 150, 180, and 200 mm. The lengths of the plates were three times the width „b‟ plus 10 mm for clamping. Also a constant foam thickness of 100 mm was used in all the tests. The steel plate was glued to the foam core by using a suitable adhesive. The specimens were tested after 48 hours to ensure the adhesive was set and the steel face and the foam core were joined properly. The details of the test programme and specimens are given in Table (1), and a complete schematic diagram of the test rig is shown in Fig. (4).

3-2 Literature Review of Linear Elastic Buckling: As the thickness of the core is large compared with the thickness of the face plate, it is customary to consider only one plate face together with the core material. Thus, in elastic buckling analysis of sandwich panel, thin steel faces supported by a thick foam core can be considered as plates on elastic foundation as shown in Fig.(3). A simply supported rectangular plate is subjected to an applied stress p along the two transverse edges. The elastic critical buckling stress σcrt of this plate can be given by [5]:

Figure (3): Plate on elastic foundation [1].

7

(

)

*

( )

+

√

(1)

√

(2)

( ) (actual)

(3)

( ) (simplified)

(4)

where: Ef : Modulus of elasticity of the plate.

Ec : Modulus of elasticity of the core.

Gc : Shear Modulus the core. νf : poison‟s ratio of the plate. νc : poison‟s ratio of the core. b : width of the plate. tf : thickness of the plate K : buckling coefficient.

R: dimensionless stiffness coefficient. n : term number buckling mode.

The value of ϕ is a ratio of half-wave buckle length „a‟ to the width of the plate „b‟ (ϕ= a/b), the critical value of half-wave buckle length decreases with increase in buckling coefficient K and thus raises the critical buckling stress σcrt . The first critical buckling mode relates to the

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minimum value of buckling coefficient K which can be minimized with respect to ϕ (

).

Then Eq. (2) can be written as: (

)

(5)

√

This equation can be solved for ϕ using a suitable numerical method and, hence, K can be evaluated. However, this theoretical method of determining the buckling coefficient K has not been adopted in the design procedure because it is seem as complicated.

3-3 Approximation for buckling coefficient: For practical purposes, explicit mathematical formulae are required, and the following expressions have been proposed for the solutions of Eq.(5): 1. By Hassinen [13] based on the elastic half-space assumption: with

√

(6)

2. By Hassinen [13] based on the simplified elastic half-space model: with

√

(7)

3. Davies and Hakmi [5] based on simplified foundation model: √

(8)

3-4 Review of Design Formula for Local Buckling: 9

It is well-known that the critical buckling stress itself does not provide any satisfactory basis for design, but it can be used as a useful parameter. Typically the b/t ratios are large in cold-formed steel, and therefore local buckling becomes a major design criterion for compression members. Local buckling causes a loss of stiffness and redistribution of stresses.

However, in the plated structures, considerable post-buckling strength exists that enables additional loads to be supported. Much of the load after elastic buckling is carried by the regions of the plate near the edges. Thus only a fraction of the plate width is considered effective in resisting the applied compression load. Based on this, a simplified assumption has been developed that the maximum edge stress acts uniformly over two strips of plate and the central region is unstressed. This assumption leads to the development of effective width principles for the design of cold-formed steel members subject to local buckling effects. The original effective width formula for the plate elements was developed by Winter (1947) based on many tests and studies of post-buckling strength on light-gauge cold-formed steel plates and sections. This design formula is given by [5]:

*

+ for λ > 0.673

ρ=1.0

for λ ≤ 0.673

(9)

( )√ where: fy : yield stress of face.

In the case of plate element without foam support the buckling coefficient K is constant for a particular type of boundary conditions (e.g. for simply supported conditions K=4.0), while in the supported foam steel plate elements the K value varies with changes in b/t ratios and mechanical properties of core foam and steel faces. However, the effective width approach for plain elements has been extended to the profiled faces of sandwich panels by modified values of the buckling coefficient K [5].

Davies and Hakmi [5] conducted a series of tests on thin-walled steel beams in which the compression flange was stiffened by foam to investigate the possibility of extending the effective 10

width formula to sandwich panels. They found that Eq. (8) is unsafe when compared with the test results for increasing values of b/t ratios. They proposed following equation for K by replacing R in Eq. (8) by 0.6R. √

(10)

Mahendran and Jeevaharan [8] proposed Eq.(11) for K based on a series of tests and finite element analyses on foam supported steel plate elements to investigate the local buckling behaviour. √

(11)

Currently European Recommendations for Sandwich Panels, Part I: Design (CIB 2000) [13] is based on an empirical reduction factor recommended initially by Davies and Hakmi [5], with simplified formula to determine the dimensionless parameter R. √

√

with

( )

(12)

Pokharel and Mahendran [1,4] proposed an improved effective width design formula Eq.(13) and Eq.(14) based on a the same series of tests and finite element analyses on foam supported steel plate elements.

*

+

for λ > 0.673 for λ ≤ 0.673

=1.0

(13)

( )√ ( ) *

+ ( )√

(14)

( )√

11

4- Experimental Data Used in this Study [2]: In order to verify or to calibrate the finite element method as an appropriate analysis method to investigate local buckling and post-buckling behaviour of sandwich panels subjected to uniaxial load, the details of the Pokharel test programme and specimens are given in Table (1), and a complete schematic diagram of the test rig is shown in Fig. (4).

Table 1: Test Programme and Specimens [2] Test

G250 Steel Plates Thickness Measured Measured (mm) b/t Ratio fy Ef fy Ef Spec. bmt Spec. Bmt (MPa) (GPa) (MPa) (GPa) 50 0.95 0.95 637 226 52.6 1 0.93 326 216 1 50 0.8 0.8 656 230 62.5 0.8 0.73 345 217 2 50 0.6 0.6 682 235 83.3 0.6 0.54 360 218 3 50 0.42 0.42 726 239 119 0.4 0.39 368 220 4 80 0.95 0.95 637 226 84.2 1 0.93 326 216 5 80 0.8 0.8 656 230 100 0.8 0.73 345 217 6 80 0.6 0.6 682 235 133.3 0.6 0.54 360 218 7 80 0.42 0.42 726 239 190.5 0.4 0.39 368 220 8 100 0.95 0.95 637 226 105.3 1 0.93 326 216 9 100 0.8 0.8 656 230 125 0.8 0.73 345 217 10 100 0.6 0.6 682 235 166.7 0.6 0.54 360 218 11 100 0.42 0.42 726 239 238.1 0.4 0.39 368 220 12 120 0.95 0.95 637 226 126.3 1 0.93 326 216 13 120 0.8 0.8 656 230 150 0.8 0.73 345 217 14 120 0.6 0.6 682 235 200 0.6 0.54 360 218 15 150 0.95 0.95 637 226 157.9 1 0.93 326 216 16 150 0.8 0.8 656 230 187.5 0.8 0.73 345 217 17 150 0.6 0.6 682 235 250 0.6 0.54 360 218 18 150 0.42 0.42 726 239 357.1 0.4 0.39 368 220 19 180 0.6 0.6 682 235 300 0.6 0.54 360 218 20 180 0.42 0.42 726 239 428.6 0.4 0.39 368 220 21 200 0.95 0.95 637 226 210.5 1 0.93 326 216 22 200 0.8 0.8 656 230 250 0.8 0.73 345 217 23 200 0.6 0.6 682 235 333.3 0.6 0.54 360 218 24 200 0.42 0.42 726 239 476.2 0.4 0.39 368 220 25 Note: fy – measured yield stress of steel face, Ef – measured Young‟s modulus of steel face b/t ratio – plate width b/bmt, Spec. – specified thickness bmt – estimated base metal thickness based on measured total coated thickness Plate Width b (mm)

G550 Steel Plates

Thickness (mm)

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b/t Ratio 53.8 68.5 92.6 128.2 86 109.6 148.1 205.1 107.5 137 185.2 256.4 129 164.4 222.2 161.3 205.5 277.8 384.6 333.3 461.5 215.1 274 370.4 512.8

Figure (4): Schematic Diagram of Test Rig [2].

5- Finite Element Model: Linear elastic buckling analysis is used to obtain critical eigenvalue-buckling (i.e. first buckling mode). Elastic buckling analysis is also used to obtain the buckling shape to represent the geometric imperfection distribution shape required for non-linear analysis. Ultimate strength of the foam supported steel plate elements was determined from a non-linear analysis.

The finite element programme used to investigate local buckling behaviour of sandwich panels subjected to uniaxial load is MARC 2010 with PATRAN 2010 as pre-and post-processer.

The sandwich panel was modelled using four-noded quadrilateral shell elements (CQUAD4) for the steel face plate and eight-noded solid brick elements (CHEXA) for the foam core.

When using the finite element method, various types of numerical models can be used.

The full-scale model may be the easiest model (LĂ—b), but the disadvantage associated with this model is that either poor level of accuracy is obtained when using smaller number of elements or a large computational time needed when using a fine mesh. In order to reduce difficulties, a halflength model (L/2Ă—b/2) can be used which represents a quarter size of the full panel with appropriate boundary conditions using symmetry as shown in Figure (5). This quarter size model was used to validate the FEA method against experimental results for local buckling behaviour of sandwich panels.

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Another reduced model used in the numerical analysis of sandwich panels is the Half Multiple wave model (2.5ɑ×b/2). The length of this model is variable and depended on the b/t ratio and mechanical properties of the core and the faces. This model contains 2.5 waves as shown in figure (6). This model is used to review the current design rules and develop new design rules for local buckling effects. A detailed description of these finite element models is presented later.

Half-Length Model L/2 x b/2

p

p

b

Face

y Core

z

Figure (5): Principle of Half-Length Model

L/2=2.5 ɑ ɑ

Half Multiple Wave Model

ɑ

ɑ

ɑ

ɑ

b/2

p

p

b

y z

Face

Core

Figure (6): Principle of Half-Five-Waves Buckle Model 14

x

6-1 Full-Length model: Figure (7) shows the model geometry size (b×3b×(tc + tf)), the compressive line load along one edge face, and the appropriate boundary conditions ( for the edge faces only as in experimental test). The numbers shown in the figure represent the prevented degrees of freedom along the edge face, where (1, 2, and 3 are X-axis, Y-axis, and Z-axis translations, and 4, 5, and 6 are the X-axis, Y-axis, and Z-axis rotations).

2356 Steel Face 346

1 “for middle node” only

356

tc + tf

Applied Line load

1 “for middle node” Z only Y X

Foam Core

b Figure (7): Full-Length Model

6-2 Half-Length model: The steel plates supported by foam tested in the experiments were modeled and analyzed using half-length experimental model. The width of each model was b/2 (half the plate width), length 3b/2 (half the length of the specimen), and thickness sum of the foam and steel thicknesses (tc + tf). Figure (8) shows the model geometry, load applied, and the appropriate boundary conditions including that of symmetry.

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246 “face and core“ 346 “face only”

Applied Line load

Steel Face

356 “face only”

tc + tf

156 “face and core“

Z

Foam Core

Y X

b/2

Figure (8): Half-Length Model

7- Geometric Imperfections: It is important that appropriate geometric imperfections are introduced in a finite element model while undertaking non-linear analysis to simulate the true shape and structural behaviour of the specimens, because it is unlikely to find any cold-formed steel member with original perfect geometry. Actual members always deviate from their perfect shape.

The presence of geometric imperfections seriously affects the strength of compressed plate elements and ignoring them in the numerical analysis could result in unrealistic strength predictions. It is generally considered that the most detrimental type of imperfection is one which has the same shape as the critical local buckling mode.

However, in practice, maximum imperfection magnitudes are specified in terms of either the thickness or the width of the plates. Experimental observations showed that the actual magnitudes of geometric imperfections in sandwich panels are quite small. The maximum imperfection adopted in this paper is (0.1tf) as adopted by Pokharel and Mahendran [1].

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8- Numerical Convergence Study: As the mesh density increases, the accuracy of a finite element model also generally increases and converges to a numerically „correct‟ solution. The accuracy of the model can then be compared with the experimental results and other theoretical solutions. In order to determine the appropriate mesh density, a convergence study was conducted with gradually decreasing mesh sizes for half-length model.

Table (2) shows results of the numerical convergence study for half-length model for specimen no. 10 (Table (1)) with b/t=125. It is found that, a mesh with 10×10 mm square surface elements (for steel plate) and solid elements with 10×10×5 mm throughout the foam depth provides satisfactory results in terms of accuracy. Pokharel and Mahendran [1] also adopted the element size in their studies.

Table (2) Numerical Convergence Study for specimen no.10 Element mesh size (mm)

Number of

Elastic Buckling stress

Ultimate strength

Shell

Solid

Elements

25×25

25×25×25

60

129.63

319.35

15×15

15×15×15

240

122.43

214.40

10×10

10×10×10

825

120.57

184.04

10×10

10×10×5

1575

120.46

182.95

(MPa)

(MPa)

9- Validation of Half-Length model: To confirm the half-length model results, a full-length model was first analysed for some of the specimens using the same element size for elastic buckling and non-linear analysis. Table (3) shows the comparison between the full-length and the half-length models. It is shown that, the maximum difference was 4%. Since this comparison results between the two models agreed well, half-length models was adopted for further studies.

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Table (3) Comparison between Full-Length and Half-Length Models Plate Length Thickness width b L (mm) t (mm) (mm) 80 240 0.80 100 300 0.80 120 360 0.95 150 450 0.95

Buckling Stress (MPa) FLM HLM 143.85 149.11 119.15 120.46 116.35 116.96 99.63 100.24

Ultimate Strength (MPa) FLM HLM 224.97 222.73 184.59 182.87 174.38 173.61 141.88 143.27

10- Comparison of FEA and Experimental results: To calibrate the FEA method for investigating the local buckling behaviour of sandwich panels subjected to uniaxial compressive line load, all specimens tested by Pokharel and Mahendran are modelled using half-length finite element model. Firstly, linear elastic buckling analysis is carried out to evaluate the buckling stress and buckling shape which is then used to as an imperfection shape for non-linear analysis with (0.1tf) as a maximum imperfection. Table (4) presents the comparison of FEA and experimental buckling and ultimate stresses for G550 and G250 steel plates, respectively. Results of FEA and experimental agree well for both G550 and G250 steel plates.

10-1 Results of Elastic Buckling Stress: The mean values of the ratio of FEA and experimental buckling stresses were found to be 0.98 and 1.03 for G550 and G250, respectively. The corresponding coefficients of variation (COV) were 0.06 and 0.08 for G550 and G250, respectively. Figures (9 and 10) present the comparison of buckling stress results from FEA and experiments for G550 and G250 steel plates, respectively.

10-2 Results of Ultimate Strength: Similarly, for ultimate strength results, the mean values of the ratio of FEA and experimental ultimate strength were found to be 0.84 and 0.91 for G550 and G250, respectively.

The corresponding coefficients of variation (COV) were 0.07 and 0.08 for G550 and G250,

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respectively. Figures (9 and 10) present the comparison of buckling stress results from FEA and experiments for G550 and G250 steel plates, respectively.

All these comparisons confirm that the half-length model can be satisfactorily used to analyse the local buckling behaviour of foam-supported steel plates used in the experiments. In addition that the FEA method is power full technique can be used to model the non-linear or post buckling behaviour of sandwich panels.

Table (4): Comparison of FEA Results for Half-Length Model with Experimental Results G550 Steel Plates Test No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

b/t ratio

52.60 62.50 83.34 119.00 84.20 100.00 133.34 190.50 105.34 125.00 166.67 238.10 126.34 150.00 200.00 157.90 187.50 250.00 357.10 300.00 428.60 210.50 250.00 333.34 476.20

G250 Steel Plates

Buckling

Ultimate Strength

Stress (MPa)

(MPa)

FEA

Expt.

FEA

Expt.

337.53 264.22 189.17 133.44 177.96 149.11 118.51 100.70 138.05 120.46 103.24 91.67 116.96 106.03 92.93 100.24 93.10 85.16 79.88 79.71 76.04 83.63 81.58 76.94 73.98

336.20 293.00 238.30 141.90 170.30 155.80 112.90 97.00 139.40 132.80 102.00 87.60 122.10 119.80 91.00 104.60 96.10 83.90 79.00 78.60 77.80 88.20 79.90 71.40 80.00

400.10 358.65 305.42 264.26 259.48 222.73 190.09 163.15 203.67 182.87 153.18 129.20 173.61 156.36 129.73 143.27 129.37 109.25 93.05 96.53 84.54 113.62 103.57 90.37 80.50

434.30 428.30 305.00 264.30 279.20 275.00 223.30 186.00 232.60 203.40 181.80 184.00 205.20 203.50 169.90 158.10 172.80 133.90 119.20 122.60 118.40 136.50 117.10 118.00 100.10

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b/t ratio 53.80 68.50 92.60 128.20 86.00 109.60 148.10 205.10 107.50 137.00 185.20 256.40 129.00 164.40 222.20 161.30 205.50 277.80 384.60 333.30 461.50 215.10 274.00 370.40 512.80

Buckling

Ultimate Strength

Stress (MPa)

(MPa)

FEA

Expt.

FEA

Expt.

313.82 223.41 160.68 120.63 168.33 131.17 107.6 94.87 131.33 109.9 95.18 87.18 112.32 98.18 87.71 96.88 87.45 80.46 76.81 75.86 73.4 83.26 77.68 73.46 71.53

272.50 221.90 151.90 146.70 171.20 152.60 98.10 88.10 123.40 121.40 90.90 76.90 119.40 98.60 85.20 96.80 87.20 76.50 67.20 73.60 64.10 84.90 72.70 65.70 64.60

273.02 236.1 196.37 164.6 186.82 153.23 130.41 111.84 148.25 130.03 110.21 94.9 129.77 114.09 96.93 110.82 98.18 86.11 78.29 78.98 73.69 91.77 83.04 75.55 71.51

285.40 251.50 201.50 186.20 201.60 185.40 159.00 149.00 178.10 162.10 148.10 111.30 150.20 126.40 113.30 125.50 101.90 91.20 80.30 86.00 78.30 91.60 78.00 71.90 75.60

Figure (9): Comparison of FEA and experimental buckling stress results G550.

Figure (10): Comparison of FEA and experimental buckling stress results G250.

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Figure (11): Comparison of FEA and experimental ultimate strength results G550.

Figure (12): Comparison of FEA and experimental ultimate stress results G250 21

11- Multiple Wave Finite Element Model: In Pokharel and Mahendran experimental programme, the foam width of the tested specimens was made the same as the steel face width [3]. Only the steel plates were restrained along the four sides leaving the foam unrestrained. However, in real sandwich panels the foam continuous in the length and width direction.

In the elastic bucking analysis, the first mode of buckling occurred associated with critical buckling stress depending on mechanical properties of faces and core. In non-linear ultimate analysis, the behaviour is different due to nonlinear of stress-strain relations and some yield criterion must be taken into account.

Also, the numerical convergence of element mesh size is considered for wavy length model, and acceptable mesh size in terms of accuracy is (5×5) mm for shell element and (5×5×5) mm for solid element.

11-1 Multiple Wave Model length: According to local buckling behaviour sandwich panel, initially at elastic buckling many waves is constructed in steel face, and their numbers and wavy length depend on b/t ratio and mechanical properties of faces and core [8]. At the post buckling stage these waves are rearranged depending upon the stress redistribution, until one of these waves reaches ultimate strength and the panel fails. So in this study, the wave length is examined. The parametric study used one, two, three, and five waves as model length with different b/t ratio and constant material properties as shown in table (5). Symmetry is used to reduce the computation efforts.

Table (5) Material Properties of Faces and core for Multiple Wave Model Material Steel face Foam core

Young Modulus

Poisson‟s ratio

Shear Modulus

Yield stress

E (MPa)

υ

G (MPa)

F (MPa)

233000.00

0.30

89615.38

3.80

0.08

1.76

22

675.00 -

Table (6) shows comparison for different number of waves with different b/t ratios. The results show that, the length of the model strongly influences the ultimate strength for high b/t ratios. This is expected as the high slenderness ratio results in factor redistribution during post buckling stage. It is clear concluded that, the models with five waves (or 2.5 waves for half model) can be considered as appropriate for calculations the ultimate strength critical model length of wavy model, hence the ultimate strength become constant for different (b/t) ratio. Als, the same is shown in figure (13).

Table (6) Comparison of Multiple Wave model with different model length Buckling stress (MPa)

b/t

Ultimate strength (MPa)

ratio

1 wave

3waves

5waves

7waves

1 wave

3waves

5waves

7waves

63

271.71

272.48

272.57

272.65

345.91

345.40

346.55

345.93

125

134.09

134.63

134.80

134.41

201.712

203.55

203.46

203.35

200

109.06

109.64

109.81

109.91

136.34

144.31

144.23

144.19

250

103.54

104.23

104.26

104.36

120.99

129.80

129.35

129.74

500

96.63

96.89

96.97

97.03

96.80

103.54

104.47

104.67

Figure (13): Comparison of Multiple Wave model with different model length 23

11-2 Foam Thickness: To find a suitable foam core thickness which can be used, a parametric study was conducted for different b/t ratio with different foam core thickness. Table (7) shows comparison results of different foam core thickness of multiple wave model associated with different b/t ratios. The table shows that with a foam core thickness greater than 75mm there is only a marginal effect on elastic buckling stress and ultimate strength. This agrees with the use of 100mm as foam core thickness in the experimental programme conducted by Pokharel and Mahendran [3].

Table (7) Comparison of Multiple Wave model with different foam core thickness b/t ratio

Buckling stress (MPa)

Ultimate strength (MPa)

Foam core thickness (mm)

Foam core thickness (mm)

50

75

100

50

75

100

63

272.37

272.56

272.57

346.52

346.55

346.55

125

134.78

134.80

134.81

203.45

203.45

203.29

200

109.31

109.79

109.81

143.94

144.21

144.53

250

104.13

104.25

104.26

129.17

129.33

129.35

500

96.74

96.96

96.97

103.99

104.44

104.47

11-3 Double Faces: Daviesâ€&#x;s principle [5] to simulate each face in sandwich panel as plate on elastic foundation leads to omitting the composite action between two faces. To investigate validity of this principle in multiple wave models, a parametric study was carried out for different b/t ratio with the full cross-section of two faces together with foam core. Table (7) shows the results for different b/t ratios. The table shows that, the multiple wave models with a full cross section give a marginal effect on elastic buckling stress and ultimate strength. This comparison confirmed Daviesâ€&#x;s principle that is the effect of presence double faces in wavy model is marginal effect.

24

Table (8) Comparison single and double faces of multiple wave model

b/t ratio

Ultimate strength

Buckling stress (MPa)

(MPa)

Single

Full

Single

Full

Face

Section

Face

Section

63

272.57

272.51

346.55

346.54

125

134.80

134.78

203.29

203.42

200

109.81

109.80

144.23

144.18

250

104.26

104.26

129.35

129.29

500

96.97

96.97

104.47

104.43

12- Application of Multiple Wave Model to Design: The results presented above show that multiple wave model can be used as acceptable finite element model to simulate and investigate the behaviour of sandwich panels. Figure (14) shows the geometry, boundary conditions and pattern load of multiple wave models.

The design formula of local buckling of sandwich panel can be modelled as a relation between beff/b and b/t ratios, also this relation is affected by mechanical properties of face and foam core, to investigate this relation without any effects of other factors, constant magnitude of mechanical properties for faces and foam core are considered as shown in table (5) (which is average values of Pokharel and Mahendran experimental programme).

Table (9) show the results of parametric study conducted using multiple wave models. As in the case of the half-length model, the multiple wave model models was analysed first using elastic buckling analysis, followed by non-linear analysis. The first buckling mode from the elastic buckling analysis was used to input the geometric imperfection for the non-linear analysis with a magnitude of 0.1t. Figure (15) shows the typical buckled shape and stress distribution of multiple wave models.

25

246 “face and core“

Steel Face

346 “face and core“

tc + tf Applied Line load 156 “face and core“

356 “face and core“ Z

Foam Core

Y X

b/2 Figure (14): Multiple Wave Model undeformed shape maximum stress

deformed shape

Z Y X Z Y X

a-buckled shape

b- stress distribution

Figure (15): Typical buckled shape and stress distribution multiple wave model

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Table (9): Comparison Results of Multiple Wave model for steel face Grade 550.

No.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

b

tf

b/t

(mm) (mm) ratio 50 50 50 50 50 100 100 100 150 100 200 150 200 250 150 200 250 300 200 300

0.8 0.7 0.6 0.5 0.4 0.7 0.6 0.5 0.7 0.4 0.7 0.5 0.6 0.7 0.4 0.5 0.6 0.7 0.4 0.5

63 71 83 100 125 143 167 200 214 250 286 300 333 357 375 400 417 429 500 600

Wave length (mm) Theory 41.8 39.4 36.3 32.4 27.6 49.7 43.8 37.4 52.8 30.6 54 38.7 46.7 54.7 31.3 39.2 47.2 55.1 31.6 39.7

FEA 42.4 40.0 36.8 32.8 28.4 50.8 44.8 38.4 54.4 31.6 55.6 40 48 56.4 32.4 40.4 48.4 56.8 32.8 40.8

L/2 (mm) 106 100 92 82 71 127 112 96 136 79 139 100 120 141 81 101 121 142 82 102

Buckling stress

Ultimate

(MPa)

strength

Theory

FEA

278.70 233.64 195.70 164.67 140.17 130.23 121.74 114.64 112.58 108.89 106.51 105.79 104.45 103.72 103.25 102.71 102.40 102.20 101.29 100.51

272.57 227.65 189.82 159.00 134.80 125.09 116.74 109.81 107.79 104.26 101.88 101.25 99.94 99.17 98.85 98.29 97.96 97.70 96.97 96.18

(MPa) 346.55 309.75 273.76 238.42 203.29 177.26 160.21 144.23 137.61 129.35 120.57 119.16 114.46 111.83 111.15 109.15 107.82 106.94 104.47 101.80

Also, table (9) shows the wave length a and the critical buckling stress were obtained from FEA method which compared with the theoretical results obtained from Eq.(1), (2), (4) and (5). As seen from these results from FEA agreed well with the theoretical results.

13- Effective Width Comparison: To determine the beff/b ratio of steel plate elements supported by foam core, the ultimate stresses given in tables (9) divided by fy as in the following formula:

(15)

27

These effective width ratios obtained from the FEA results, those evaluated from Eq.(9) using K values predicted by Davies and Hakmi (1990) and ECCS(2000) [17], and those evaluated from Eq.(13 and 14) using proposed by Pokharel (2002 and 2005, respectively) are plotted against the b/t ratios in Figure (16) for G550 steel plates.

Figure (16): Effective width of the steel element stiffened by form for G550 steel

It can be observed from Figure (16) that the effective widths ratio evaluated from Eq.(9) using K values predicted by different buckling formulae agreed reasonably well with the FEA results for low b/t ratios (<100). However, for higher b/t ratios, all the formulae predicted very high effective width values compared with the FEA results. The FEA results also indicated that, none of the current design formulae of ECCS 2000 [17] could estimate reasonable values of effective width beff for slender plates with high b/t ratios (> 100). In addition, FEA results indicated that, the proposed design formula of Pokharel and Mahendran (2002 and 2005) which are based on half wave buckle model using ABAQUS programme gives reasonable values of effective width beff for slender plates with b/t ratios (< 300), and underestimation for high 28

slenderness b/t ratios (> 300). In fact these formulae of Pokharel and Mahendran based on half wave buckle model which is neglected the effect of stress redistribution process which become more effect for high slenderness b/t ratios.

14- Effects of Yield Stress: Usually for building construction, the sandwich panels are made steel for the faces, and polystyrene for foam core. Generally modulus of elasticity and Poissonâ&#x20AC;&#x;s ratio of these two mechanical are constant. The ultimate strength of the sandwich panel subjected to local buckling depends on the post-buckling behaviour in which the yield stress of steel faces becomes more effective. To study the effect of yield stress of steel face, a parametric study of sandwich with different steel grade was carried out. Figure (17) shows the FEA results of element steel plate supported by foam core with different yield stress of steel face using multiple wave model. It seems that the ultimate strength of sandwich panel for low b/t ration is affected by the yield stress of steel face, while the effects become marginal for higher b/t ratio or slenderness plate.

Figure (17): Ultimate strength of sandwich panel with different yield stress of steel faces 29

Similarly, a parametric study was carried out to investigate the validity of elastic assumption behaviour of the foam core. Table (10) shows the FEA results of specimen no.4 from table (9) with elastic perfectly plastic behaviour assumption of the foam core for different yield stress. It is obvious that, the effect of yield stress begins when the yield stress of the foam core is less than fy/3000. This means the elastic behaviour assumption of the foam core is more realistic.

Table (10): Comparison Results of Different Yield Stress of the Foam core Yield stress (fc)

Ultimate strength

(MPa)

(MPa)

fy/10

238.20

fy/100

238.20

fy/500

238.20

fy/1000

238.20

fy/3000

238.20

fy/4000

230.00

15- New Design Formula: The local buckling behaviour of steel plate elements supported by foam core with simply supported longitudinal edges is investigated in the previous articles which is an essential preliminary step towards the development of design formula for profiled sandwich panels.

Based on FEA results using multiple wave model and using best fit techniques, the basic relation of beff/b and b/t ratios can be approximated as type of power formula as following:

(

⁄ )

(⁄ )

(16)

where: μ, γ, and δ constants related to mechanical properties of steel face and foam core

30

To determine the unified values of μ, γ, and δ constants, all mechanical properties are assumed to be constant except the yield stress of steel faces due to it‟s effects on the ultimate strength of sandwich panel. Figure (18) shows the best fitting results process to the data shown if figure (17) and the new design formula of effective width can written as following:

(

⁄ )

(⁄ )

(17)

Figure (18): Formula of ultimate strength of sandwich panel

16- Conclusions: The finite element method represents one of the powerful methods to study the local buckling behaviour of fully profiled sandwich panel. A well agreement of finite element results comparison of experimental test results conducted by Pokharel and Mahendran (2003). The 31

multiple wave finite element models checked in terms of geometry, dimensions and boundary conditions to insure a good practical simulation of sandwich panels. The multiple wave finite element models used to review the existing design rules by which generally can be conclude that presented design formula gave acceptable results for low b/t ratio whereas inadequacy agreement to higher slenderness ratio or high b/t ratio. A new design rule is developed with a good agreement.

References: 1- Narayan Pokharel, Mahen Mahendran,(2004),„‟Finite Element Analysis and Design of Sandwich Panels subject to Local Buckling Effects‟‟, Original Research Article ThinWalled Structures, Volume 42, Issue 4, April 2004, Pages 589-611 2- Narayan Pokharel, Mahen Mahendran, (2003), „Experimental investigation and design of sandwich panels subject to local buckling effects‟‟, Journal of Constructional Steel Research 59 (2003) 1533-1552.

3- Narayan Pokharel, Mahen Mahendran, (2005), „‟An investigation of lightly profiled sandwich panels subject to local buckling and flexural wrinkling effects‟‟, Journal of Constructional Steel Research 61 (2005) 984 –1006.

4- Pokharel, N., and Mahendran, M., (2002), „‟Numerical Modeling of sandwich panels Subject to local buckling‟‟, Proc. of the 17th Australasian Conference on the Mechanical of Structures and Materials, Gold Coast, Queensland, Australia.

5- Davies, J.M. and Hakmi, M.R. (1990), “Local Buckling of Profiled Sandwich Plates”, Proc. IABSE Symposium, Mixed Structures including New Materials, Brussels, September, pp. 533-538.

6- Davies, J.M., Hakmi, M.R. and Hassinen, P. (1991), “Face Buckling Stress in Sandwich Panels”, Nordic Conference Steel Colloquium, pp. 99-110.

7- Davies, J.M. (1993), “Sandwich Panels”, Thin-Walled Structures, Vol. 16, pp. 179-198.

8- Mahendran, M. and Jeevaharan, M. (1999), “Local Buckling Behaviour of Steel Plate Elements Supported by a Plastic Foam Material”, Structural Engineering and Mechanics, Vol. 7, No. 5, pp. 433-445.

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9- McAndrew, D. and Mahendran, M. (1999) “Flexural Wrinkling Failure of Sandwich Panels with Foam Joints”, Proc. 4th International Conference on Steel and Aluminium Structures, Helsinki, Finland, pp.301-308.

10- Mahendran, M. and McAndrew, D. (2001), “Effects of Foam Joints on the Flexural Wrinkling Strength of Sandwich Panels”, Steel Structures, Vol. 1, pp. 105-112.

11- M. Mahendran and D. McAndrew, (2003) ‟‟Flexural Wrinkling Strength of Lightly Profiled Sandwich Panels with Transverse Joints in the Foam Core‟‟, Advance in structural Engineering 6(4): pp.325-337.

12- Hassinen, P. (1997), “Evaluation of the Compression Strength of Profiled Metal Sheet Faces of sandwich Panels”, Thin-Walled Structures Vol. 27, No. 1, pp, 31-41.

13- Hassinen, P. (1995), “Compression Failure Modes of Thin Profiled Metal Sheet Faces of sandwich Panels”, Sandwich Construction 3-Proceedings of the Third International Conference, Southampton, pp. 205-214.

14- Davies, J.M. and Heselius, L. (1993), “Design Recommendations for Sandwich Panels”, Journal of Building Research and Information, Vol. 21,No.3, pp.157-161.

15- Davies, J.M. (1987b), “Axially Loaded Sandwich Panels”, Journal of Structural Engineering, Vol. 113, No. 11, pp. 2212-2230.

16- Allen, H.G. (1969), Analysis and Design of Structural Sandwich Panels, Pergamon Press, New York, U.S.A. 17- International Council for Building Research, Studies and Documentation (CIB) (2000), “European Recommendations for Sandwich Panels Part 1: Design”, CIB Publication.

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