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HSEHealth & Safe ty

Executive

Nonlinear analysis of stainless steel corrugatedpanels under blast loading: A numerical study

Prep ared by Imperial College of Science , Technology and

Medicine. for the Health and Safe ty Executive 2003

RESEARCH REPORT 102



HSEHealth & Safe ty

Executive

Nonlinear analysis of stainless steel corrugatedpanels under blast loading: A numerical study

V. De Rosa, J. Friis and L.A. Louca

Imp erial College of Science

Technology and Medicine.

This report presents results from a study to investigate the response of 3 standard corrugation profiles

which have been subjected to pressure time histories typical of a hydrocarbon explosion. In particular,

the study has investigated the sensitivity of peak displacements, plastic strains and dissipated energy

to mesh density and loading as these are commonly used parameters to describe the performance of

structures and to assess structural integrity in commonly used failure models.

This report and the work it describes were funded by the Health and Safety Executive (HSE). Itscontents, including any opinions and/or conclusions expressed, are those of the authors alone and do

not necessarily reflect HSE policy.

HSE BOOKS



© Crown copyright 2003

First published 2003

ISBN 0 7176 2722 5

All rights reserved. No part of this publication may bereproduced, stored in a retrieval system, or transmitted inany form or by any means (electronic, mechanical,photocopying, recording or otherwise) without the priorwritten permission of the copyright owner.

Applications for reproduction should be made in writing to: Licensing Division, Her Majesty's Stationery Office, St Clements House, 2-16 Colegate, Norwich NR3 1BQ or by e-mail to [email protected]

ii



CONTENTSEXECUTIVE SUMMARY iv

1. INTRODUCTION 12. FINITE ELEMENT MODELLING 3 2.1 Geometry and Loading 3 2.2 Material Behaviour 4 2.3 Choice of Element 5 2.4 Meshing 5 2.5 Eigenfrequency and Mesh Density Study 7

2.6 Response Parameters for Assessment 10 2.7 Results 10 3. LOCAL MODEL OF WELDED CONNECTION 26 3.1 Introduction 26 3.2 Structural Model 26 3.3 Material Model 27 3.4 Finite Element Model 27 3.5 Results 29 4. CONCLUSIONS 35 5. REFERENCES 37

iii



EXECUTIVE SUMMARY This report presents results from a study to investigate the response of 3 standard corrugation

profiles which have been subjected to pressure time histories typical of a hydrocarbon

explosion. In particular, the study has investigated the sensitivity of peak displacements, plasticstrains and dissipated energy to mesh density and loading as these are commonly used parameters to describe the performance of structures and to assess structural integrity incommonly used failure models.

An initial eigenvalue analysis was conducted to establish the first ten natural frequencies of thethree panels using different mesh densities and element types. The triangular elements availablewithin the ABAQUS finite element package were not found to perform as well as therectangular elements for the same mesh density. Up to the first six modes of vibration littledifference was seen between element types. However for the higher modes the results becamesensitive to mesh density with the results becoming less reliable for coarser meshes, indicatingfiner meshes are required to pick up the higher frequency modes.

A full non-linear dynamic finite element analysis accounting for both material and geometricnonlinearity of two of the profiles was carried out over a range of pressure time histories. Theresults indicated that the peak deflection and dissipated energy, which is a measure of the

energy absorbed by plastic deformation, were relatively insensitive to mesh density. This was both with and without added viscosity to the material model to account for strain rate effects.

However, the equivalent plastic strain was found to be very sensitive to mesh density. This wasexacerbated at low duration and high peak pressure events which can lead to brittle failuremodes developing. The addition of viscosity to the material model reduced the sensitivity but itwas not sufficient to remove this effect to mesh density.

A more refined model of the local weld detail was analysed in order to provide a more accurate

description at the critical location in the global model where failure was likely to occur. Themodel provided a more accurate three dimensional pattern of the strain distribution in andaround the weld detail. Although the peak values were occurring in the same location for thetwo models, the local model indicated that strain values higher than those in the global modelcan be achieved, despite the fact that the same loading history was applied to both models.

Despite the extensive detail in the model, the sensitivity of the plastic strain values to meshdensity could not be eliminated. However the reduced ductility of the 3D local model was

apparent and has implications for assessing ductility of structural systems on results obtainedfrom 2D analysis.

iv



1. INTRODUCTION

Corrugated panels are widely used on offshore topsides as firewalls and blastwalls to provide

separation between modules and ensure the safety of personnel. Since the Piper Alpha

1

tragedyin July 1988, a great deal of experimental and numerical work has been carried out in order toimprove understanding in the nature of the loading produced in a typical module structure

during a hydrocarbon explosion. While there has also been an interest in the blast resistance of beam and plate type structures over the same period, there is very little hard data available on

their response characteristics. This is particularly so in the area of failure, where the integrity of welded connections under large displacements, inevitable under the extreme loads being produced by explosion scenarios being considered, need to be assessed in order to establish safelower bound containment pressures for typical panel structures.

A recent study2

undertaken at Imperial College funded by the HSE, SHELL, BG Technologyand BP investigated the response of typical corrugated firewalls currently in use. A number of

full scale tests were conducted at Spadeadam as part of an ongoing safety study during the early1990‘s which provided a useful database in order to validate numerical finite element models.The firewalls were supported on novel connection details consisting of flexible angles whichwere shown to contribute significantly to the dissipation of the blast energy. It was estimated

that for a blastwall of 2.5 m span as much as 30% of total dissipated energy was absorbed in thetransverse angle perpendicular to the corrugation at the time of failure. The flexible angle was

also shown to reduce the amount of localised straining at the weld locations which gives anoverall improvement in structural integrity and estimated increase in capacity of almost 50%.

At present there is no universally accepted failure model for predicting dynamic plastic failureof a welded connection. Previous work by Holmes et al 3 has used a local damage model to

predict the ductile fracture of a welded T-joint under dynamic loading. The work was carried

out on small scale specimens with finite element analyses requiring extremely fine meshes.Bammann et al

4have used an internal state variable type of criterion which requires fracture

data from notched specimens to obtain damage parameters. Nurrick et al 5 has adopted a

relatively simple rupture strain criterion based on uniaxial tests of coupon specimens. Weldintegrity assessment using a strain based failure criteria was also adopted by Plane et al 6 as part

of a finite element analysis which was validated against large scale tests. The results highlightedthe conservative nature of simple failure models using a specified rupture strain in a finiteelement analysis.

The current numerical studies being conducted has highlighted that the prediction of plasticstrains at locations of high strain gradients is strongly dependent on mesh density as the strain

fields in the vicinity of the connection is characterised by large gradients. Although a continued

increase in the fineness of the mesh in this region will provide more information on strainvariations, the maximum value will tend to infinity due to the presence of a singularity. Thisreflects simplifications made in the geometric modelling of the weld detail which is inevitablefor practical modelling purposes. However the use of a simple strain based failure criteria was

used effectively to provide a lower bound to the containment pressure as the failure process for the numerical model gave an excellent qualitative comparison with the large scale testing.

A more universal approach to determining failure has been introduced by Jones and Shen 7,8

using an energy density failure criterion for beam and frame structures. The method assumesthat rupture occurs in a rigid-plastic structure when the absorption of plastic work reaches a

critical value which at an assumed hinge location can be determined from dynamic engineering

stress-strain curves.

1



The concept of energy was tested in reference 2 by carrying out an initial parametric studywhich suggested that the initiation of weld rupture could be predicted with sufficient accuracyusing a criteria based on the total amount of strain energy dissipated in the structure. A lower

bound for the energy capacity of the firewalls was determined for a static load condition. Thiswas then used as a limiting value in the dynamic analyses which produced encouraging results.

Importantly, studies of mesh sensitivity showed that the calculated value for dissipated energyconverge very rapidly with mesh refinements. This suggests that the safety of the firewall mayeffectively be assessed for different blast scenarios using the dissipated energy as a failurecriteria assuming failure occurs in a ductile tensile tearing mode and no local buckling of the

corrugations occur.

The study presented in this report has attempted to investigate further the effect of mesh densityon local strain values at connection details for three different corrugation profiles.

An initial eigenvalue analysis was conducted to establish the fundamental frequency of the

profiles and to study the effect of mesh density and element type on the first ten modes of vibration. The models were then analysed using a number of pressure time histories to

investigate the sensitivity of local strain values and dissipated energy to mesh density. Theinfluence of including strain rate effects have also been included in order to investigate the possibility of removing some of the strain singularity indicated in reference 2. The final sectionof the report establishes a local submodel with the weld modeled in some detail in order toestablish whether the strain singularity can be removed by the addition of viscosity.

2



2. FINITE ELEMENT MODELLING

2.1 GEOMETRY AND LOADING

The three corrugated profiles investigated in this study basically consist of unstiffenedtrapezoidal cold-formed profiles, made of stainless steel Grade 2205, spanning vertically for alength of 4.000 m between supporting structures. The geometry of the profiles is shown in

Figure 1. The profiles have been referred to as:(i) a shallow-trough profile, termed X in the following, with a wave length of 260 mm,

a depth of 41 mm and a thickness of 2.5 mm;(ii) a medium-trough profile, termed Y in the following, with a wave length of 250 mm,

a depth of 80 mm and a thickness of 2.5 mm;(iii) a deep-trough profile, termed Z in the following, with a wave length of 350 mm, a

depth of 195 mm and a thickness of 5.0 mm.

The structural scheme considered in the finite element model has been obtained by applying the

mathematical boundary conditions given, on the one hand, by symmetry considerations and, onthe other hand, by some sort of engineering judgement on constructive details typically used at

interfaces with deck plates and top girders. In particular, as far as symmetry conditions areconcerned, symmetry planes have been adopted along the longitudinal planes both at mid-peak

and at mid-trough of the transverse section, so that only a half-wave of the corrugation (assumedto be extracted from a panel with an infinite width) has been modelled

CL

67 45 67

4 1

260

40.5 40.5

t=2.5mm

40 45 80 45 40

8 0

250

CL

Shallow Trough

t=2.5mm

36 72 36

1 9 5

CL

Medium Trough

103 103

350

t=5.0mm

Deep Trough

Figure 1

Investigated profiles

3



Symmetry boundary conditions have also been accounted for about the transverse plane atmidspan of the corrugation. Therefore, only one quarter of the whole single corrugation (whosecross-section is represented in Figure 1) has been modelled. As regards modelling boundary

conditions at the terminal transverse section of the corrugation, in the —global“ model analysedin the present Section, a 10 mm thick stainless steel plate has been assumed to be rigidly

connected to the corrugation, with its lower edge being hinged. A more realistic modelling,adopted for the —local“ model, will be shown in the following.

Finally, as far as loading conditions are concerned, isosceles pressure pulses having a duration

time equal to 40, 80 and 120 ms and a peak pressure ranging from 0.50 bar to 1.50 bar , with anincreasing step equal to 0.25 bar , have been applied. Obviously, response parameters

corresponding to a load with given duration time and peak pressure have been assumed to bevalid only when the adopted failure criteria are satisfied for the given structural model.

2.2 MATERIAL BEHAVIOUR

Both the corrugations and the end plates are made of stainless steel Grade 2205. The relevantstress-strain diagram (in terms of true Cauchy stress and logarithmic strain, respectively) isdepicted in Figure 2. As far as the elastic part of the material behaviour is concerned, a linear

model (defined by a Young modulus E =2.1⋅ 105 Nmm

-2and a Poisson modulus ν =0.30) has been

chosen. As soon as the uniaxial equivalent yielding stress of 435.169 Nmm-2(defining a Von

Mises œ and therefore isotropic œ yield surface) is attained, the corresponding incremental strain

is governed by the associated plastic flow law and the adoption of an isotropic hardening model.As regards the implementation of the material strain-rate dependence in the finite elementmodel, it has been achieved by means of the well-known Cowper-Sydmonds formula:

1

+

ε&D

q

1σ σ=d 0

In particular, material constants D=7.69 s-1 and q=5.13, corresponding to a 0.1% offset proof

stress, are taken as evaluated by Jones and Birch9.

0

100

200

300

400

500

600

700

800

S t r

σ

, ( M P a )

e s s ,

E=0.200E6 MPa Nominal

True

0.00 0.05 0.10 0.15 0.20 0.25

Strain, ε , ( m/m )

Figure 2

Material stress-strain behaviour

4



2.3 CHOICE OF ELEMENTS

In the analyses illustrated in the present report, ABAQUS/Explicit10,11 has been used as the

finite element model. Three-dimensional shell elements have been adopted throughout, asdeemed correct and advisable whenever the thickness of the structure is less than 1/10 of a

typical structural dimension (in the present case, distance between supports or wavelength of thehighest vibration mode of interest). In particular, the following three finite elements have beenconsidered:

(i) the S4 finite element: a four-node, three-dimensional, fully-integrated, finite-strain,

general-purpose (i.e. valid for both thick and thin shell problems) element, with sixdegrees of freedom per node (i.e. three displacement components and three rotation

components);(ii) the S4R finite element: a four-node, three-dimensional, finite-strain, general-

purpose (i.e. valid for both thick and thin shell problems) element, with six degreesof freedom per node (i.e. three displacement components and three rotation

components), using a reduced (lower order) integration law to form the elementstiffness (the only Gauss point is situated at the centre of the element), while the

mass matrix is still integrated exactly;(iii) the S3≡ S3R finite element: a three-node, three-dimensional, finite-strain, general- purpose (i.e. valid for both thick and thin shell problems) element, with six degreesof freedom per node (i.e. three displacement components and three rotationcomponents); the number of Gauss points is equal to 1 in both cases, and it issituated at the centre of the element.

For all of the elements, five section points have been specified through the shell thickness.

Stresses and strains are calculated independently at each section point through the thickness of the shell by means of a Simpsons integration rule.

2.4 MESHING

As advisable when adopting explicit methods of solution in numerical problems, a uniformmesh density has been adopted throughout the whole model, regardless of the mesh coarseness,the adopted finite element and the boundary conditions.

As far as the meshes based on four-noded finite elements are concerned, three basic mesh

densities have been adopted for the all the investigated geometries: coarse mesh, medium andfine (termed 1, 2 and 3 in the following, respectively). In all the cases, i.e. regardless of the profile geometry or the mesh density, the number of nodes in the transverse direction has been

chosen such in a way to keep a constant width-to-depth ratio (approximately equal to 1.0÷ 2.5)for the considered finite element. Besides, an extra mesh geometry, indicated by number 4 and

characterised by a width-to-depth ratio equal to 1.0, has been considered: it has been obtained by keeping the same number of nodes on the transverse section as in the mesh geometry 3,

while the number of nodes in the longitudinal direction has been increased in such a way as toobtain square finite elements. Details of the considered mesh geometries for all the threeconsidered profiles are given in Table 1, Table 2 and Table 3 below (in the —Label“ column of each Table the letter indicating the profile type precedes the number indicating the meshdensity). For shallow- and medium-trough corrugation only, the considered mesh geometries

based on four-noded elements are also illustrated in Figure 3 and Figure 4, respectively.

As far as the meshes based on the three-node finite element are concerned, they have beenconsidered for the shallow profiles only and, in particular, only for the geometries indicated

above as X1 and X2. This is due to the relatively poor results yielded by these meshes in theeigenvalue analysis which is discussed in Paragraph 2.4. For triangular meshes, no figures are

5



shown, since they have been obtained by simply dividing every rectangular finite element intwo equal parts, and keeping unchanged the spatial distribution of nodes of both the original

geometries.

.

Figure 3

FE models for shallow-trough profile

Figure 4

FE models for medium-trough profile

Table 1 Meshing for ³ shallow trough profile

LabelTransverse Direction Elts. Longitudinal Direction Elts.

Total No. of Elts. Width-to-Depth RatioNo. Average Width [mm] No. Depth [mm]

X1 7 21.0 40 50 280 0.42 X2 9 16.3 50 40 450 0.41 X3 14 10.5 80 25 1120 0.42 X4 14 10.5 200 10 2800 1.05

Table 2 Meshing for ³ medium trough profile


Total No. of Elts. Width-to-Depth Ratio

No. Average Width [mm] No. Depth [mm] Y1 9 19.1 40 50 360 0.38 Y2 12 14.3 50 40 600 0.36 Y3 17 10.1 80 25 1360 0.40

4 17 10.1 200 10 3400 1.01

Table 3 Meshing for ³ deep trough profile


Total No. of Elts. Width-to-Depth RatioNo. Average Width [mm] No. Depth [mm]

Z1 14 20.9 40 50 560 0.42 Z2 19 15.4 50 40 950 0.38 Z3 30 9.8 80 25 2400 0.39 Z4 30 9.8 200 10 6000 0.98

6



2.5 EIGENFREQUENCY AND MESH DENSITY STUDY

In order to establish the finite element and the mesh geometry to be adopted in the nonlinear

explicit analysis of the global model, an eigenfrequency analysis has been performed for different finite element models, i.e. for some of the possible combinations between the finite

elements and mesh geometries illustrated in Paragraph 2.3. Each of the considered finiteelement models has been applied to all the three profile geometries described above; besides,different boundary conditions have been applied to nodes lying along the internal edge of the 10

mm thick plate at the end of the corrugation. In particular, for these nodes, the translational

degrees of freedom are assumed to be fixed, as well as the rotational degrees of freedom about both axes perpendicular to the cross section of the profile and perpendicular to the longitudinal

plane in the mid-plane of the corrugation. On the other hand, for the rotational degree of freedom about an axis perpendicular to the longitudinal symmetry planes, two boundary

conditions have been considered: either restrained (i.e. a fully-clamped boundary condition isobtained along the edge) or free (i.e. a simply-supported boundary condition is obtained along

the edge). These two —limit“ boundary conditions have been considered in order to determine alower bound and an upper bound to the eigenfrequencies for a given finite element model and a

given profile, according to the rotational constraint that different constructional details at theedge of the end-plate will be able to provide in reality.

The eigenfrequency analysis has been performed by means of a subspace iteration eigensolver,and the obtained eigenvectors have been normalised with respect to the structure‘s mass matrix

(i.e. the eigenvectors are scaled so that the generalised mass for each vector is unity).For each finite element and mesh geometry, results are presented in Tables 4 to 9, for shallow-,medium- and deep-trough profiles, respectively. In particular, for each profile type, the former Table gives the results for the —simply-supported“ boundary condition, while the latter refers to

the —fully clamped“ boundary condition. In each Table, the shadowed column is intended tohighlight the eigenfrequencies provided by the —most accurate“ finite element model, clearly

given by the combination between the most refined mesh geometry, indicated by number 4, and

the S4 finite element.

A comparison between the eigenfrequencies provided by the several models and the ones

yielded by the —most accurate“ model leads to the following observations:i) the accuracy of the results provided by the different models is practically

independent of the boundary conditions at the end transverse section of thecorrugation. In terms of eigenfrequencies, the difference œ even for the highest ones œ is almost invariably contained within a 2% range (the only exception being foundfor the three-node-element-based models), and no prediction is possible on the possibility that either the simply supported or the fully clamped condition will

provide the stiffer response.ii) finite-element models (such as X1,S3 and X2,S3) based on three-node elements

show a relatively poor performance, if their results are compared with the onesyielded by models characterised by the same number of nodes (but with a number

of elements equal to 50%!) but based on four-node elements, either they are full- or reduced-integration (i.e. the models indicated in Table 10 and Table 11 as X1,S4;

X1,S4R; X2,S4;X2,S4R). For example, it can be seen that, while the firsteigenfrequency is correctly predicted by models based on four-node elements even

by the X1 mesh geometry, the equivalent three-node model provides an 8÷ 11%stiffer response, depending on the geometrical boundary conditions. Therefore,three-node elements have been used only for the earliest numerical experiments(regarding the shallow-trough profile).

iii) convergence towards the —exact“ solution (i.e. the one provided by the most refinedmodel) is the same regardless of the investigated geometry of the profile

(shallow/medium trough) aqnd the adopted finite element: mesh geometries denoted

7



by X3 and Y3 provide practically the same eigenfrequency values (i.e. within a0.5% range) as the corresponding most refined models X4 and Y4. Therefore, meshgeometries characterised by a width of about 10 mm and a length of about 25 mm

provide eigenfrequency values very close to the corresponding ones yielded by themost refined model accounted for in the present analysis.

iv) as far intermediate mesh geometries X2 and Y2 are concerned, in one case(medium-trough profile) solutions practically identical to the ones given by mostrefined ones are obtained; on the other hand, quite surprisingly, for eigenfrequencies from the second onward, for shallow-trough profiles, they provide

results much stiffer (up to 40%) than the —exact“ ones and, noticeably, even worsethan the ones provided by the coarsest mesh geometries.

As a consequence of the considerations illustrated above, only the S4R finite element will be

used in the nonlinear analyses illustrated in the following (both in the —global“ and in the —local“models), while the three mesh geometries denoted with numbers from 1 to 3 will be adopted for

comparison purposes on the adopted response parameters in the nonlinear field. The nonlinear analysis, both for the local and the global model, will be limited to the X and Y profiles.

Table 4

Eigenmodes for simply supported shallow-trough profileEigenfrequency Values [Hz] for Simply Supported Shallow-Trough Corrugated Profiles (L=4.000 m)

Mode No. X1,S4 X1,S4R X2,S4 X2,S4R X3,S4 X3,S4R X4,S4 X4,S4R X1,S3 X2,S3

1

2

3

4

5

6

7

8

9

10

10.4 10.3 10.3 10.3 10.3 10.3 76.7 75.9 76.6 76.1 76.6 76.3 202 200 200 199 201 200 363 359 357 355 357 356 502 497 503 499 488 486 578 573 603 597 556 554 618 612 662 655 592 591 646 638 700 692 619 617 672 661 730 720 643 640 699 685 757 745 668 664

10.3 76.5 200 357 487 555 592 618 642 666

10.3 11.2 10.7 76.3 84.6 80.0 200 222 210 355 388 370 485 514 499 554 578 568 591 616 608 617 647 639 641 678 669 665 700 699

Table 5 Eigenmodes for fully clamped shallow-trough profile

Eigenfrequency Values [Hz] for Fully Clamped Shallow-Trough Corrugated Profiles (L=4.000 m)

Mode No. X1,S4 X1,S4R X2,S4 X2,S4R X3,S4 X3,S4R X4,S4 X4,S4R X1,S3 X2,S3

1 18.9 18.7 18.9 18.8 18.9 18.8 18.9 18.8 21.0 19.8 2 100 98.9 99.8 99.2 99.8 99.5 99.7 99.4 111 104 3 236 234 234 232 234 234 234 233 258 244 4 397 393 391 389 390 389 389 388 420 402 5 522 518 527 524 505 504 505 504 529 516 6 586 582 616 611 563 562 563 562 584 576 7 622 616 669 663 596 595 596 595 620 613 8 650 641 705 697 622 620 621 621 651 643 9 675 664 734 724 646 643 645 644 683 674

10 702 689 761 749 671 668 669 669 701 700

8



Table 6 Eigenmodes for simply supported medium-trough profile

Eigenfrequency Values [Hz] for Simply Supported

Medium-Trough Corrugated Profiles (L=4.000 m)

Mode No. Y1,S4 Y1,S4R Y2,S4 Y2,S4R Y3,S4 Y3,S4R Y4,S4 Y4,S4R

1 22.7 22.5 22.7 22.6 22.7 22.6 22.7 22.6 2 151.5 149.2 151.2 150.5 151.1 150.4 150.9 150.3 3 390.2 383.4 387.7 384.8 386.8 384.1 385.8 383.5 4 708.1 694.7 695.5 688.7 693.1 686.9 690.1 685.4 5 956.1 952.7 854.8 854.7 856.2 856.2 856.2 856.2 6 959.0 956.0 857.7 857.4 859.1 859.0 859.1 859.1 7 960.2 958.5 864.2 863.3 865.6 865.2 865.5 865.4 8 965.6 964.0 871.0 867.9 872.6 871.1 872.2 871.6 9 978.7 975.3 878.5 876.4 879.2 878.2 878.9 878.4 10 990.0 982.2 891.3 887.2 892.7 890.9 892.3 891.8

Table 7 Eigenmodes for fully clamped medium-trough profileEigenfrequency Values [Hz] for Fully Clamped

Medium-Trough Corrugated Profiles (L=4.000 m)

Mode No. Y1,S4 Y1,S4R Y2,S4 Y2,S4R Y3,S4 Y3,S4R Y4,S4 Y4,S4R

1 36.6 36.0 36.6 36.4 36.6 36.4 36.6 36.4 2 192 189 192 191 191 191 191 191 3 453 447 449 448 448 447 448 446 4 784 775 765 763 764 762 763 761 5 956 956 855 855 856 856 856 856 6 959 959 858 858 860 859 859 860 7 966 963 865 865 867 866 867 867 8 970 966 876 873 877 876 876 877 9 984 980 887 885 887 886 886 886 10 993 985 895 891 896 895 895 896

Table 8 Eigenmodes for simply supported deep-trough profile

Eigenfrequency Values [Hz] for Simply Supported

Deep-Trough Corrugated Profiles (L=4.000 m)

Mode No. Z1,S4 Z1,S4R Z2,S4 Z2,S4R Z3,S4 Z3,S4R Z4,S4 Z4,S4R

1 49.7 49.5 49.7 49.5 49.6 49.5 49.6 49.5 2 200 200 258 256 198 197 197 197 3 209 209 286 286 206 206 206 206 4 215 215 293 292 212 212 212 212 5 217 217 298 298 214 213 214 213 6 224 223 301 300 220 220 220 220 7 234 233 310 309 231 230 230 230 8 248 246 322 321 244 243 243 243 9 264 262 337 335 260 259 259 259 10 284 281 350 347 279 278 278 278

9



Table 9 Eigenmodes for fully clamped deep-trough profile

Eigenfrequency Values [Hz] for Fully Clamped

Deep-Trough Corrugated Profiles (L=4.000 m)

Mode No. Z1,S4 Z1,S4R Z2,S4 Z2,S4R Z3,S4 Z3,S4R Z4,S4 Z4,S4R

1 76.1 75.9 76.3 76.2 76.0 75.9 76.0 75.9 2 204 204 273 273 201 201 201 201 3 212 211 288 288 209 208 208 208 4 218 218 296 295 215 215 215 215 5 219 218 298 298 216 216 216 216 6 229 228 305 304 226 225 226 226 7 242 240 316 315 238 238 238 238 8 258 256 331 329 254 253 254 253 9 278 275 348 346 273 272 272 272

10 302 298 369 366 295 294 294 294

2.6 RESPONSE PARAMETERS FOR ASSESSMENT

In order to evaluate the influence of the considered load characteristics (i.e. peak pressure and

duration), material model (linear elastic with isotropic hardening, either including strain-ratedependence or not) and mesh density on the structural response, three parameters have been

considered:1. the maximum equivalent plastic strain (either in the corrugation or in the end-plate),

termed PEEQ in the Tables and Figures below:

t

ε pl =⌠ ⌡

2tr (dεpl dεpl )

30

2. the maximum transverse displacement in the midspan section at the instant t=T d

corresponding to the end of the applied blast load, termed U 3max in the Tables andFigures below;

3. the total strain energy in the structure, termed ALLIE in the Tables and Figures below.Such quantity is given by the sum of the energy dissipated by rate-independent and rate-dependent (if any) plastic deformation, and the recoverable strain energy. Both thesequantities are measured at the instant t=T d corresponding to the end of the applied blast

load. Obviously, if this measure were performed at t>T d , only the measure of theresidual elastic strain energy would be affected.

2.7 RESULTS

Table 10 and Table 11 below show the obtained results for the X and Y profiles. Whenever

strain-rate dependence has been included, symbol SR has been added to the letter and thenumber identifying the investigated geometry and the adopted mesh density, respectively.

From Figures 5 to 7 it is shown the variation of the total strain energy of the structure with theadopted mesh density, material model and time duration, for a given peak pressure. It may beobserved that results yielded by the three different mesh densities are practically coincident,

either the strain-rate dependence is accounted for or not, once the load characteristics and the profile geometry are given; even the shapes of the ALLIE-T d curves are very similar. Besides, asexpected, the influence of the strain-rate dependence on this response parameter decreases as theduration of the applied load increases: typically, for non-strain-rate-sensitive models, an

10



Table 11 Response of simply supported shallow-trough profile

Model Label Td [s] P [bar] U3,max (Td) [mm] ALLIE (Td)[J] PEEQmax [%]

X1 0.040 0.50 -123.1 1.09E+03 7.31 X1,SR 0.040 0.50 -79.0 8.49E+02 7.28 X2 0.040 0.50 -123.7 1.09E+03 8.76

X2,SR 0.040 0.50 -80.0 8.56E+02 8.72 X3 0.040 0.50 -124.9 1.09E+03 12.90 X3,SR 0.040 0.50 -78.8 8.43E+02 12.26 X1 0.080 0.50 -103.2 5.04E+02 3.54

X1,SR 0.080 0.50 -77.6 3.25E+02 3.17 X2 0.080 0.50 -103.8 5.10E+02 4.30

X2,SR 0.080 0.50 -78.3 3.30E+02 3.93 X3 0.080 0.50 -104.5 5.22E+02 6.32

X3,SR 0.080 0.50 -77.3 3.30E+02 5.57 X1 0.120 0.50 -58.7 2.51E+02 2.11

X1,SR 0.120 0.50 -41.8 1.83E+02 2.00 X2 0.120 0.50 -59.7 2.54E+02 2.59

X2,SR 0.120 0.50 -41.7 1.85E+02 2.50 X3 0.120 0.50 -60.3 2.59E+02 3.92

X3,SR 0.120 0.50 -41.4 1.89E+02 3.39 X1 0.040 0.75 -231.8 2.37E+03 14.51

X1,SR 0.040 0.75 -188.6 1.83E+03 14.53 X2 0.040 0.75 -233.8 2.40E+03 16.74

X2,SR 0.040 0.75 -189.2 1.84E+03 16.73 X3 0.040 0.75 -236.8 2.44E+03 22.50

X3,SR 0.040 0.75 -186.6 1.81E+03 22.53 X1 0.080 0.75 -160.5 1.20E+03 7.88

X1,SR 0.080 0.75 -95.6 8.00E+02 6.83 X2 0.080 0.75 -162.1 1.21E+03 9.23

X2,SR 0.080 0.75 -96.7 8.06E+02 8.28 X3 0.080 0.75 -166.2 1.24E+03 13.11

X3,SR 0.080 0.75 -95.2 7.94E+02 11.37 X1 0.120 0.75 -117.2 6.61E+02 4.57

X1,SR 0.120 0.75 -89.1 4.91E+02 4.40 X2 0.120 0.75 -117.3 6.61E+02 5.44

X2,SR 0.120 0.75 -89.0 4.91E+02 5.37 X3 0.120 0.75 -118.9 6.76E+02 7.91

X3,SR 0.120 0.75 -88.2 4.95E+02 7.26 X1 0.040 1.00 -295.1 4.18E+03 23.50

X1,SR 0.040 1.00 -237.6 3.21E+03 23.13 X2 0.040 1.00 -298.2 4.24E+03 27.17

X2,SR 0.040 1.00 -238.8 3.23E+03 26.79 X3 0.040 1.00 -303.6 4.33E+03 37.38

X3,SR 0.040 1.00 -238.7 3.17E+03 36.51 X1 0.080 1.00 -203.8 2.11E+03 13.05

X1,SR 0.080 1.00 -162.6 1.38E+03 11.14 X2 0.080 1.00 -205.3 2.15E+03 15.14

X2,SR 0.080 1.00 -162.8 1.39E+03 13.01 X3 0.080 1.00 -208.0 2.21E+03 20.59 X3,SR 0.080 1.00 -159.3 1.37E+03 17.56 X1 0.120 1.00 -158.5 1.26E+03 8.26

X1,SR 0.120 1.00 -122.1 9.77E+02 8.04 X2 0.120 1.00 -159.9 1.26E+03 9.64

X2,SR 0.120 1.00 -122.5 9.77E+02 9.48 X3 0.120 1.00 -161.4 1.29E+03 13.46

X3,SR 0.120 1.00 -122.3 9.73E+02 12.62 X1 0.080 1.25 -246.6 3.22E+03 18.76 X2 0.080 1.25 -249.8 3.28E+03 21.61 X3 0.080 1.25 -256.5 3.39E+03 29.56 X1 0.120 1.25 -205.1 1.99E+03 12.37 X2 0.120 1.25 -206.3 2.00E+03 14.26 X3 0.120 1.25 -207.8 2.01E+03 19.23 X1 0.120 1.50 -239.1 2.83E+03 16.79 X2 0.120 1.50 -240.6 2.85E+03 19.13 X3 0.120 1.50 -243.8 2.88E+03 25.60

12



d p

Td [s]

I E [ J ]

Y1 Y2 Y3

ALLIE Value on the Y Profile at t=T for P =0.50 bar

0.00E+00

1.00E+02

2.00E+02

3.00E+02

4.00E+02

5.00E+02

6.00E+02

7.00E+02

0.040 0.080 0.120

A L L

Y1,SR Y2,SR Y3,SR

Figure 7 Maximum strain energy calculated for the various FE models of medium-trough profile when

subjected to pressure pulses with a constant peak pressure of 0.50 bar

d p

0

Td [s]

U

[ m m ]

X1 X2 X3

Maximum Transverse Midspan Displacement on the X Profile at t=T for P =0.50 bar

-140 -120 -100 -80 -60 -40 -20

0.040 0.080 0.120

3 m a x

X1,SR X2,SR X3,SR

Figure 8

Maximum displacement calculated for the various FE models of shallow-trough profile when


14



d p

0

Td [s]

U

]

X1 X2 X3

Maximum Transverse Midspan Displacement on the X Profile at t=T for P =1.00 bar

-350 -300 -250 -200 -150 -100 -50

0.040 0.080 0.120

3 m a x

[ m m

X1,SR X2,SR X3,SR

Figure 9 Maximum displacement calculated for the various FE models of shallow-trough profile when


d p

0

Td [s]

U

]

Y1 Y2 Y3

Maximum Transverse Midspan Displacement on the Y Profile at t=T for P =0.50 bar

-60

-50

-40

-30

-20

-10 0.040 0.080 0.120

3 m a x

[ m m

Y1,SR Y2,SR Y3,SR

Figure 10 Maximum displacement calculated for the various FE models of medium-trough profile when


15



d

Pp ]

X1 X2 X3

PEEQ Peak Value on the X Profile for T =0.040 s

0.00 0.06 0.12 0.18 0.24 0.30 0.36 0.42 0.48 0.54 0.60

0.50 0.75 1.00 1.25 [bar

P E E Q

X1,SR X2,SR X3,SR

Figure 11 Maximum plastic strain calculated for the various FE models of shallow-trough profile when

subjected to pressure pulses with a constant duration of 0.04 sec

d

0.50 Pp ]

X1 X2

X3


0.00 0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.24 0.27 0.30

0.75 1.00 1.25 [bar

P E E Q

X1,SR X2,SR X3,SR



16



d

1.00 Pp ]

X1 X2 X3


0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20

0.50 0.75 1.25 [bar

P E E Q

X1,SR X2,SR X3,SR



l p

Td [s]

X1 X2 X3

PEEQ Peak Va ue on the X Profile for P =0.50 bar

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.040 0.080 0.120

P

E E Q

X1,SR

X2,SR X3,SR



17



l p

Td [s]

X1 X2 X3


0.00 0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.24

0.040 0.080 0.120

P E E Q

X1,SR X2,SR X3,SR



l p

Td [s]

X1 X2 X3


0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.32 0.36 0.40

0.040 0.080 0.120

P E E Q

X1,SR X2,SR

X3,SR

Figure 16

Maximum plastic strain calculated for the various FE models of shallow-trough profile whensubjected to pressure pulses with a constant peak pressure of 1.00 bar

18



l p

Td [s]

X1 X2 X3


0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60

0.040 0.080 0.120

P E E Q

Figure 17

Maximum plastic strain calculated for the various FE models of shallow-trough profile when


From all the figures above, it can be clearly seen that the equivalent plastic strain is mesh-

dependent, both for strain-rate dependent and strain-rate independent material models. For agiven duration time of the pulse pressure, Figures from 8 to 10 show that, as regards the non-strain-sensitive-models, an increase of about 100% of the parameter under consideration takes place for P p=0.50 bar , when passing from the coarsest mesh density to the finest one. On theother hand, when the assumed value of the peak pressure is the maximum one roughlycompatible with the assumed duration time (i.e. 0.75 bar for T d =40 ms, 1.00 bar for T d =80 ms,1.25 bar for T d =120 ms), the increase of the PEEQ when passing from the X1 to the X3 model,

ranges, in all the cases, between 50% and 60%. In particular, considering the —extreme cases“for the profile X , it may be observed that, evaluating the average of the maximum value of the

PEEQ: over all the considered pressures: for low time durations, there is an increase of about60% when passing from the coarse mesh to the fine one, while the increase is of 25% only when

high time durations are accounted for; over all the considered duration times: for low pressures,there is an increase of about 15% when passing from the coarse mesh to the fine one, while the

increase is of about 50% when high peak pressures are accounted for. Basically, the problem of the mesh dependency is exacerbated by short duration time and high peak pressure events. Inany case, however, the observed influence of the mesh density on the equivalent plastic strain isa considerable matter of concern, especially if the following considerations are accounted for: asfar as the other two response parameters are concerned, practically no mesh-sensitivity, as

illustrated above, can be observed, meaning that even the coarsest one is suitable to providesufficiently accurate results in terms of both global (i.e. the total strain energy in the structure)and local (i.e. the maximum transverse displacement at midspan) parameters; if the attainmentof a limit value of the equivalent plastic strain is assumed as a simple criterion to assess thefailure of the blast wall under given loading conditions, mesh sensitivity effects often showthemselves to be of crucial importance, since a load condition which may be considered as

—safe“ according the coarsest mesh density, may become —unsafe“ according to the intermediate

or finest mesh; the same applies if a more refined strain-based fracture mechanics criterion is

19



adopted in order to assess the resistance of the blast wall with respect to some of the possiblefailure forms, e.g. plate tearing; the importance of acquiring a reliable evaluation of the plasticstrain state via the finite element method is even more evident if one observes that, as shown by

Figure 18, the greatest mesh-dependence effects on the values of the PEEQ reported in Figuresfrom 11 to 17 are invariably localised in the part of the corrugated profile adjacent to the

corrugation-to-end-plate welded connection, i.e. where experiments have proved most of thefailures initiate.

Figure 18 shows, for a particular loading case on the shallow-trough profile, the variation, in the

longitudinal direction, of the maximum PEEQ value obtained, for a given finite element model.Distance reported on the abscissa is therefore the distance of the centroids of the rectangular

elements, all belonging, for a given mesh density, to the same cross-section and then having thesame distance from the end section of the corrugation. As outlined above, it can be seen that

mesh-dependence effects on the considered response parameter become less important as thelongitudinal distance from the end section increases, and they are particularly clear when

passing from the intermediate to the fine mesh, rather than from the coarse to the intermediateone. Besides, whatever is the mesh density, the equivalent plastic strain tends to infinity as the

distance from the end-plate goes to zero. More specifically, it may be interestingly observed thatthe PEEQ vs. distance curve for the finest mesh does not show a monotonic trend as in the twoother cases, since a sudden decrease in the PEEQ, followed by an immediate increase, can bedetected. Since all of the maximum values of the PEEQ, at a given cross-section, reported in

Figure 18 invariably take place in the compressed flange of the profile, such a trend of the curve

seems to indicate the occurrence of a local buckling phenomenon having a half-wave lengthshorter than the characteristic dimension of the finite element adopted for the coarse and theintermediate meshes. A diagram showing, for the X3 model and the same loading condition asabove, the PEEQ values at the centroid of the finite elements in the compressed flange of the

corrugation up to a distance of 200 mm from its terminal section, is provided in Figures 19-21.In this way, a bi-dimensional view of the PEEQ distribution in the most critical region is

provided; again, the steep increase in the PEEQ values in the part of the corrugation adjacent to

the end-plate can be appreciated.

Variation of the Maximum Cross-Sectional PEEQ

with Distance from the End Section (X Profile, P=0.50 bar, Td=50 ms)

X1 X2 X3

0.00

0.02

0.04

0.06 0.08

0.10

0.12

0.14

P E E Q

X1,SR X2,SR X3,SR

0 1 0

2 0

3 0

4 0

5 0

6 0

7 0

8 0

9 0

1 0 0

1 1 0

1 2 0

1 3 0

1 4 0

1 5 0

1 6 0

1 7 0

1 8 0

1 9 0

2 0 0

Distance from the End Section [mm]

Figure 18

Longitudinal distribution of plastic equivalent strains

20



0 25

50 75

150

a t

'

D i s t a n c

e f r o m M i d

- P l a n e o

f t h e E n

d P l a t e

[ m m ]

100 125

175 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

0.09 0.10 0.11 0.12

P E E Q

e l e m e

n t s

G a u s s p o i n t s

Figure 19 Spatial distribution of plastic equivalent strains

21



0 25

50 75

100 125

150 175

a t

'

i

D i s t a n c

e f r o m M i d

- P l a n e o f t h e

E n d P

l a t e [ m m

]

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12

P E E Q

e l e m e

n t s G a u s s p o n t s


22



0 25

50 75

100 150

D i s t a n

c e f r o m

M i d - P l a

n e o f t h e E n

d P l a t e

[ m m ]

125 175 0.00

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12

P E E Q a t e l e m e

n t s ' G a u s s

p o i n t s


23



The mesh-dependency shown by the PEEQ response parameter and discussed above has beenattributed to a strain localisation phenomenon, i.e. an intense concentration of the (plastic)deformation in narrow zones. Such zones have been identified, in particular, as those parts of

the corrugation which are not only along the boundary between two adjacent flat plates, but alsovery close to the 10 mm thick end-plate. From the mechanical point of view, the concentration

of plastic strain in these zones might be explained as the combined effect of the presence of acorner between two adjacent flat plates and the considerably stiff constraint provided by the end plate to transverse displacements taking place in the direction parallel to the mid-plane of thecorrugation. In fact, while the unrestrained corrugation would —naturally“ tend to flatten, the end

plate tends to prevent this phenomenon in its close proximity, being the in-plane stiffness of athick and flat end-plate which is orders of magnitude higher than the corresponding stiffness of

a cold-formed corrugated profile. However, if on the one hand the pattern of the plastic strainobserved in Figures 19-21 can be expected, as it is clear that this is not entirely due to numerical

singularities, the strong dependence of the magnitude of the equivalent plastic strain on theadopted mesh density (as it clearly appears from Figures 11-17 ) cannot be neglected. Such a

mesh-dependence of the plastic equivalent strain is observed even though, as shown in Figure

18 for a particular loading case and geometry, the position of the Gauss points in the

longitudinal direction, which changes slightly due to the mesh density considered, is accountedfor. In Figure 18 it can be observed, once again, that for a given cross-section of thecorrugation, if two or more Gauss points œ belonging to elements of two or more differentmeshes œ fall in that cross-section or in its vicinity, the equivalent plastic strains associated withthe more refined mesh are consistently higher than the corresponding ones associated with

coarser meshes. From Figures 11-17 it may also be noticed that the mesh-sensitivity shown bythe PEEQ, for a given profile geometry and peak pressure, is almost independent of the durationtime: therefore, even if œ as a limit case œ the loading process is quasi-static, this phenomenonwill still take place. Finally, it can be easily observed from the same Figures that, where a limit

value of the PEEQ were adopted as failure criterion, it might well happen that a given profilegeometry, under a given loading history, may be deemed to be on the safe side if a relatively

coarse mesh is adopted, while it will be considered unsafe if a more refined mesh is used.

As illustrated above, the importance, from both the theoretical and the practical point of view, of the PEEQ mesh-sensitivity is apparent and therefore it is deemed to be worthy, in the present

work, of some further considerations. Actually, the phenomenon of strain localisation has beenobserved by several Authors in ductile metals and structural metallic alloys, and occurring indifferent structural contexts, ranging from necking of a metallic bar under axial tension (Bazantand Cedolin

12) to simple shearing displacement boundary conditions of a infinite planar strip

(Needleman13) and round bars subjected to large torsional strains (Tanaka and Spretnak 14). Inthese simple cases, all relevant to rate-independent, elasto-plastic material models, the strainlocalisation phenomenon has been interpreted as an instability process, and conditions have

been found at which the material constitutive relationships allow a bifurcation fromhomogeneous or smoothly varying deformation into a band (i.e. a narrow area along which

deformation is highly concentrated). From the mathematical point of view, this leads to achange of type of the governing equations (in particular, from hyperbolic to elliptic in the

dynamic case) and, from the physical point of view, gives infinite strains over a set of measurezero in dynamic problems (Lasry and Belytschko15). In a finite element model, when the critical

stress level triggering localisation is reached, obtained results are severely mesh-dependent, i.e.deformation localises in one or few elements, irrespective of their size; furthermore, plasticenergy dissipated along the band after its formation tends to zero, as the mesh is refined. Inorder to eliminate mesh sensitivity in numerical calculations, several Authors have suggested

different methods, all based on ensuring that the localisation zone remains finite. This is done,for example, introducing non-local variables in the material model (in a kinematic-typestructural theory, this means that generalised displacements and generalised strains are averagedover a finite volume œ thus leading, for example, to an average measure of the equivalent plasticstrain, as done by Lasry and Belyschko16); or using Cosserat (or —polar“) continuum theory,

24



instead of the usual Cauchy model. Finally, according to other Authors (Needleman13

, Sluys et

al.17

, Zhu18

), a relatively simple solution can be obtained by adopting the classical continuumtheory and a classical plasticity model, but including some viscous effects in the material in

order to make the solution mesh-independent. In particular, in the simple case studied by Needleman13, it is proved that when material rate dependence is accounted for, under dynamic

loading conditions wave speeds remain real and mesh size effects do not occur for this simple problem. In the present study strain rate effects were introduced into the fininte element model by the classical metal plasticity model given by the Cowper-Sydmonds law. The material parameters required for the model were taken from Jones and Birch9, and the results obtained by

the strain-rate sensitive model were presented, along with the corresponding cases for the non-rate dependent material, in Figures 5-21, and are labelled by —SR“.

Unfortunately, as far as the influence of the mesh density on the equivalent plastic strain is

concerned, it can be easily deduced from Figures 11-18 that the amount of viscosity introducedin the model (coherent with the experimental data obtained), even though yielding some

improvement, does not seem to be sufficient, given the geometry of the model, to achieve theexpected mesh independency. Actually, as predicted by Cescotto and Li

19, while in simple cases

adopting an elastic-viscoplastic material model can be effectively employed as a localisationlimiter, in more complicated situations the mesh independency can also be achieved, but it isconditional to the amount of viscosity introduced in the model. In our case this is limited by thestrain rates achieved which for typical gas explosions are low when compared to faster eventssuch as TNT explosions or hard impacts.

25



3. LOCAL MODEL OF WELDED CONNECTION

3.1 INTRODUCTION

In the present Section, a detailed finite element model of the terminal part of the shallow-trough profiles, termed X in Section 2, as well as the end plate is presented. The obtained results will bediscussed and, for some particular load histories, they will also be compared to the onesobtained, in the particular region under consideration, from the global model considered inSection 2.

A careful analysis of the local fields of stress and strain in this area has been made necessaryfrom the results obtained in the previous Section, where it clearly appeared that, evenintroducing the strain-rate sensitivity phenomena in the material model, no mitigation of the plastic strain localisation phenomenon was achieved. On the contrary, as observed by severalAuthors Niemi

20and Fayard et al.

21, at the intersection of thin shells, where hot spots

commonly appear, the stress gradient continued to be strongly sensitive to the mesh size.

Therefore, the main goal of the model presented in the following is to assess if a more realisticfinite element model of the terminal part of the structure œ together with the adoption of finite

elements having characteristic dimensions much smaller than in the global model œ might beuseful in order to attain the aforementioned purpose of mitigating the mesh-dependence of the

solution. Secondly, the response yielded by the present three-dimensional local model allows usto evaluate the relative magnitudes of the six components of the stress and strain tensors, so that

an estimate of the error made in the use of a two-dimensional local model would be considered.In fact, because of the complicated finite element modelling of the spatial model and thecumbersome analyses to be performed subsequently, a plane strain or plane stress hypothesis isfrequently assumed in the literature, especially when dealing with fatigue studies on welded

connections as in Niemi20

and Fayard et al 21.. Thirdly, the implementation of a three-dimensional finite element model of the terminal part of the corrugation and the supporting

substructure will allow, in future work, the introduction of transverse and longitudinal cracks inthe welding, simulating weld defects. This further development, together with a more accuratematerial model for the weld metal and the introduction of proper heat affected zones, shouldmake possible a sufficiently accurate simulation of the ductile fracture phenomena frequentlytaking place at the welded connections of blast-walls, as shown by wide experimental evidence

and reported by Louca and Friis2.

3.2. STRUCTURAL MODEL

The structural scheme considered in the local model, illustrated in a three-dimensional view in Figure 22, has been obtained as a sub-model of the global model presented in the previousSection for the shallow-trough profile X. In particular, the part of the corrugation considered inthe local model corresponds to the part which, in the global model, is situated at a distance, fromthe mid-plane of the end plate, less or equal to 200 mm. The terminal plate (part of a 100x75x10

RSA unequal angle profile) has still the same geometry and material as in the global model and,as far as the boundary conditions are concerned, symmetry conditions have still been imposed

not only along the longitudinal edges of the corrugation but also on the edges of the end platelying on such symmetry planes. Besides, for the end plate a pinned boundary condition has beenapplied along the mid-plane line of the edge originally perpendicular to the direction of the blastwave and on the opposite side of the corrugation itself (with respect to the incident wave). Thesame sort of —hinge“ had also been considered in the global model, and in both cases aims at

representing, with a sufficient degree of accuracy, the actions applied by the supporting

structure on the considered part. In the global model the interaction between the corrugation and

26



the plate has been modelled by means of mathematical equations imposing the same value for all of the degrees of freedom of the nodes lying on the two sides of the boundary. In the localmodel a 6 mm continuous fillet weld, on both sides of the corrugation, has been explicitly

introduced in the model. In such a way, the rigidity locally introduced by the welding is notassumed to be infinite (Fayard et al.

21), but its finite value its explicitly accounted for, and also

makes feasible œ in future works œ a more detailed material modelling of the welding itself andthe relevant heat affected zones ( HAZ s), as well as the introduction of cracks simulating welddefects.

The isosceles pressure-time history compatible with the one adopted in the global model has been applied, in the local model, on the corrugated profile.

3.3 MATERIAL MODEL

The strain-rate sensitive, isotropic hardening model (based on the Von Mises yield surface) previously described in section 2.2 for stainless steel Grade 2205 has been adopted both for the

corrugation and the end plate. As far as the welding is concerned, the same material model asfor the corrugation has been assumed. Neither HAZ s nor residual stresses were introduced inmodelling the welded connection, so that a comparison between the results provided, in theterminal part of the structure, both by the local and the global model may be carried out.

3.4 FINITE ELEMENT MODEL

The finite element model illustrated in the following consists of 16586 nodes and 13230

elements, which are both solid (continuum) elements and three-dimensional shells. In particular,the three-node shell-type finite element S3 has been adopted to model the corrugated profile

from the cross-section at 200 mm from the mid-plane of the end plate up to the upper weld toe.

As it will be explained in detail in the following part of the present Paragraph, modellingrequirements have made necessary the use of this type of finite element, notwithstanding the poor performance provided in the elastodynamic analyses, as illustrated in Section 2. Besides,

two eight-node, first order, reduced integration prismatic solid elements with hourglass control,termed C3D8R in the following, have been adopted through the thickness of both the end plateand that part of the corrugation which is situated between the welding fillets. On the other hand,in order to model the welding seams, it has been possible to use C3D8R solid elements only for the internal part, while six-node first order triangular prisms, termed C3D6 in the following,have been used to model the external surface (i.e. the surface bounded by the upper and thelower weld toe).

The model-related boundary conditions introduced in the numerical model along the

longitudinal planes of the corrugation and along the edges of the end plate, are as described inSection 3.2. More detailed considerations are instead to be provided in order to explain the

history-related boundary conditions, i.e. the time-dependent boundary conditions applied on thetransverse section of the corrugated profile situated at a distance of 200 mm from the mid-plane

of the end plate. In fact, being the model considered herein —extracted“ from the global one,described in Section 2, it was necessary to represent properly and effectively the actions applied by the part of the structure not represented in the local model on the part, which is insteadconsidered in the present Section. In ABAQUS/Standard (Hibbit et al.

22 ,), a relevant sub-

modelling option is provided, through which the displacement and/or velocity time-histories of the nodes at the boundary between the local and the global model are saved automatically whenthe analysis of the latter is performed, and are applied as time-dependent boundary conditions inthe former. Unfortunately, this option is available only for linear analyses or, in case anymechanical, geometrical and/or boundary nonlinearities are introduced, only if a static analysis

27



is performed. In the present case, time-dependent boundary conditions were introducedfollowing a similar procedure but, because of the peculiarities of the analysis performed here,this had to be done directly in the input file of the local model. In particular, while performing

the analysis of the global model for a given profile geometry and loading history, in the cases inwhich the fine mesh geometry was adopted the displacement time-histories of all the 15 nodes

situated along the cross-section at a distance from the mid-plane of the end-plate equal to 200

mm were saved. This was done for all the degrees of freedom of each node: therefore, for theinternal nodes along the aforementioned line, six time-histories were recorded; while, for thetwo nodes lying on the longitudinal symmetry planes, only three time-histories were necessary

(one for each one of the active degrees of freedom). The 84 displacement time-historiesobtained in this way, and recorded with a time step equal to 0.002 s, were then introduced as

time-dependent boundary conditions on the 15 nodes which, in the local as well as in the globalmodel, are situated along the aforementioned cross-section. A proper —smoothing“ option

(Hibbit et al 22.,) has been introduced in the finite element model in order to avoid discontinuitiesin computing the velocity and acceleration time-histories from the given piecewise-linear

displacement time-histories. However, still aiming at the same purpose, velocity time-historieshave also been assigned for the translational degree of freedom in the transverse direction

indicated by 3 in the global reference system illustrated in Figure 22.

In the current model, and contrary to what is commonly deemed to be appropriate andcomputationally suitable for nonlinear dynamic analyses performed by means of the explicitmethod, it was not possible to keep a uniform mesh geometry in the part of the corrugated

profile modelled by means of the three-dimensional shell elements S3. The necessity of adopting a graded mesh depended on the necessity of having only 15 nodes along the cross-section at 200 mm from the mid-plane of the end plate (as required by the sub-modelling procedure described above) and, at the same time, of using, in proximity of the welded

connection, shell elements having a characteristic dimension which is small enough to allow asufficiently detailed description of the welding seams. In fact, while the size of the shell

elements adopted in the X3 mesh geometry œ and then at the aforementioned cross-section œ is

as given in Table 1, eight-node hexahedra as wide as 2.4 mm (in the transverse direction) have been used. Consequently, in order to grade the mesh from the 15 nodes at the aforementionedcross-section to the 57 nodes required along the weld toes, it was necessary to adopt the three-

node three-dimensional shell finite elements S3, at least with regard to the transition zones between two adjacent zones having different mesh densities. Further to these considerations,one possibility would have been to use triangular elements in the mesh-transition zones, andquadrangular elements in the remaining parts of the corrugation. However, aiming at reducingthe number of the finite element types introduced in the model, as well as at simplifying themesh generation procedure, S3 elements were also used for constant mesh densities zones.Further details about the adopted mesh densities throughout the corrugated plate modelled by

means of shell elements are reported in Table 12 below, where the reported distance iscomputed along the longitudinal direction (indicated by 1 in the global reference system

illustrated in Figure 22) and is taken between a given cross section (i.e. a given row of nodes inthe transverse direction) and the mid-plane of the end plate.

Table 12

Meshing for local modelDistance [mm] Transverse Direction Elts. Longitudinal Direction Elts.

Total No. of Elts. Width-to-Depth RatioNo. Average Base [mm] No. Height [mm]

12.5÷47.5 56 2.6 14 2.5 784 1.05 50.0÷95.0 28 5.3 9 5.0 252 1.05 100÷200 14 10.5 10 10 140 1.05

Finally, it has to be observed that a multi-point-constraint (Hibbit et al 22

.,), termed MPC in the

following, has been used to allow for the transition from the shell element modelling to the solid

28



element modelling through the thickness of the corrugation-to-welding edge; this has beennecessary since the nodes defining the S3R elements have six degrees of freedom, while nodesdefining the C3D8R have only got the three translational degrees of freedom. In order to

overcome this difficulty, first of all, a shell-to-solid MPC has been used to constrain each shellnode on the edge to the corresponding line of nodes on the solid-mesh-side of the interface;

then, a slider MPC has been adopted in order to constrain each interior node on each line of thesolid mesh at the interface to remain on the straight line defined by the bottom and the top nodesof that line.

Figure 22

Explicit FE modelling of weld detail

3.5 RESULTS

In the present section, results obtained from the local model, according to the finite element

procedure described in the previous part of this Section, for three particular loading histories, allof them applied to the shallow-trough profile are presented. More precisely, the load casesconsidered were those for which adopting a failure criterion based on the maximum PEEQ

attained might result in a safe/unsafe condition according to the adopted mesh density. Theseinclude isosceles pressure waves characterised by a peak pressure equal to 0.50 bar , 1.00 bar

and 1.25 bar and a duration time equal to 40 ms, 80 ms and 120 ms, respectively. The obtainedresults, in terms of PEEQ, are presented in Figures 23-25, for the low pressure-short duration,intermediate pressure-intermediate duration, high pressure-long duration cases, respectively.

Besides, the former (peak pressure equal to 0.50 bar , duration time equal to 40 ms) of the

aforementioned three cases has been considered in detail, i.e. the values of the PEEQ have been plotted on each one of the three plates making up the corrugation at the Gauss point of eachelement, in order to allow a proper comparison with the corresponding Figures 19-21 obtainedfrom the global model. Actually, more precisely, in order to make the graphical representationas clear as possible, the PEEQ values reported there are not exactly the ones obtained at the

Gauss point of each triangular element, but are given by the average of the values of the plastic

29



equivalent strain found at the integration points of two adjacent triangular elements (i.e. twoelements having the hypotenuse in common. Besides, it has to be observed that the PEEQvalues of triangular elements situated in the mesh transition zones have not been reported, since

they correspond to one row of elements (in the direction indicated by 2 in Figure 22) only, andcan be therefore neglected when looking at the general pattern of this physical quantity as

provided by the local model. Finally, it should be noticed that the values of the equivalent plastic strain have been reported only up to a distance of 12.5 mm from the mid-plane of the end plate, i.e. up to the toe of the welding seams.

As far as the discussion of the obtained results is concerned, it appears clearly from Figures 26-28 that considerably high values of the equivalent plastic strains are detected in the vicinity of

the cross-section at 200 mm from the mid-plane of the end plate, where displacement andvelocity time-histories have been applied. These values of the PEEQ, extremely close to the

limit value characterising the material failure, can be certainly regarded as local effects induced by the nodal displacements applied in the aforementioned cross-section: in fact, the pattern of

the PEEQ is decreasing while decreasing the distance from the end plate, and has not beenabsolutely noticed in the results provided by the global model. Besides, some 80 mm away from

the terminal section, this effect seems to have disappeared completely, as the values of the PEEQ are the same as provided by the global analysis. On the other hand, in proximity of thewelded connection, the PEEQ values tend to rise again, approximately following the same pattern, and this similarity in the spatial distribution of this quantity can be noticed not only inthe longitudinal direction, but also in the transverse one. Once again, the peak value of the

equivalent plastic strain takes place at the cross-section adjacent to the welding, and in particular in the flange of the corrugated profile ( Figure 26 ) which is essentially compressed bythe incident blast wave. More precisely, the highest value of the PEEQ is detected at the same place both in the global and in the local model (cf. Figure 19 and Figure 26 ), but the adoption

of a finer mesh density has again lead to an increase of about 50% of the value of the PEEQdetected in the local model.

Figure 23 Plastic strains at weld detail when subjected to a pressure pulse with a peak pressure of 0.50 bar

and a duration of 40 msec

30







31



32

200

180

160

140

120

100

80

60

40

20

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

P E E Q a t e l e m e n t ' s G a u s s P o

i n t s

D i s t a n

c e f r o m

M i d - P l a

n e o f t h

e E n d P

l a t e [ m m ]

Figure 26

Spatial distribution of plastic equivalent strains



33

200

180

160

140

120

100

8060

40

20

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

P E E Q a t e l e m e n t s ' G a u s s p o i n t s

D i s t a n c

e f r o m M i d

- P l a n e o

f t h e E n

d P l a t e

[ m m ]

Figure 27

Spatial distribution of plastic equivalent strains



180

200

160 140

120 100

80 60

40 20

m e

'

s s

i

D i s t a n c

e f r o m M i

d - p l a n e

o f t h e

E n d P l a

t e [ m m

]

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

P E E Q a t E l e

n t s

G a u

p o n t s


34



4. CONCLUSIONS

This report presents results from a study to investigate the response of 3 standard corrugation profiles which have been subjected to pressure time histories typical of a hydrocarbonexplosion. In particular, the study has investigated the sensitivity of peak displacements, plasticstrains and dissipated energy to mesh density and loading as these are commonly used parameters to describe the performance of structures and to assess structural integrity in

commonly used failure models.

An initial eigenvalue analysis was conducted to establish the first ten natural frequencies of thethree panels using different mesh densities and element types. The triangular elements availablewithin the ABAQUS finite element package were not found to perform as well as therectangular elements for the same mesh density. Up to the first six modes of vibration littledifference was seen between element types. However for the higher modes the results became

sensitive to mesh density with the results becoming less reliable for coarser meshes, indicatingfiner meshes are required to pick up the higher frequency modes.

A full non-linear dynamic finite element analysis accounting for both material and geometric

nonlinearity of two of the profiles was carried out over a range of pressure time histories. Theresults indicated that the peak deflection and dissipated energy, which is a measure of the

energy absorbed by plastic deformation, were relatively insensitive to mesh density. This was both with and without added viscosity to the material model to account for strain rate effects.However, the equivalent plastic strain was found to be very sensitive to mesh density. This wasexacerbated at low duration and high peak pressure events which can lead to brittle failure

modes developing. The addition of viscosity to the material model reduced the sensitivity but itwas not sufficient to remove this effect to mesh density.

A more refined model of the local weld detail was analysed in order to provide a more accuratedescription at the critical location in the global model where failure was likely to occur. Themodel provided a more accurate three dimensional pattern of the strain distribution in andaround the weld detail. Although the peak values were occurring in the same location for the

two models, the local model indicated that strain values higher than those in the global modelcan be achieved, despite the fact that the same loading history was applied to both models.

Despite the extensive detail in the model, the sensitivity of the plastic strain values to meshdensity could not be eliminated. However the reduced ductility of the 3D local model was

apparent and has implications for assessing ductility of structural systems on results obtainedfrom 2D analysis.

In terms of the influence of the conclusions on a practical design or re-assessment a number of issues on the modelling need to be considered. It is clear that the results are sensitive to meshrefinement and when assessing the response at the ductility level blast it is important to carry

out sensitivity studies by investigating 3 different mesh densities and applying different pressuretime histories to investigate variations in strain at critical locations. The results in combination

with engineering judgement should give a good indication of a sensible cut off for defining thecontainment pressure of the wall. The idea of using 3 different meshes should also helpminimise the likely misinterpretation of results. For example certain buckling modes may not beapparent in a coarse model but should be picked up in the finer mesh.

The critical locations are likely to be the connections where interpretation of results can

sometimes be difficult as this report has indicated. However, from the comments in the previous

35



paragraph the mesh study should provide convergence of the gradient of stress and strain at thislocation, but obviously the peak value of strain will increase with increasing mesh refinement.Although not considered in this report, the determination of the maximum strain could be

obtained by using an average over a specified area of the weld location assuming thin shellelements have been used to model the detail. The use of thick shell elements did not provide a

clear conclusion as to modelling guidelines and further work is clearly needed.

The current performance standards for assessing the response, particularly at the ductility levelare limited due to the increase in the uncertainties at the level of deformations likely to be

experienced. The current limit of 5% on the local strain is commonly used in conjunction withFE studies. This limit is not unreasonable given the uncertainties in the numerical modelling and

also in experimental measurements of large strains. However defining the ductility based on asingle parameter may be misleading and future studies should investigate the concept of a

response index which uses more than a single parameter for assessing the ductility of aconnection detail. This has been applied to the assessment of earthquake connection details to

establish fracture potential of different connection configurations.

36



5. REFERENCES

1. Cullen, Lord. The Public Inquiry into the Piper Alpha Disaster, HMSO, 1990.2. Louca, L. A. and Friis, J. Modelling Failure of Welded Connections to Corrugated

Panels under Blast Loading. HSE OTO Report No. 088/2000.3. Holmes, B. S., Kirkpatrick, S. W., Simons, J. W., Giovanola, J. H. & Seaman, L.

Modeling the Process of Failure in Structures. In Structural Crashworthiness and

Failure, ed. N. Jones & T. Wierzbicki. Elsevier Applied Science Publishers, Barking,Essex, 1993, Chap. 2.

4. Bammann, D. J., Chisea, M. L., Horstemeyer, M. F. & Weingarten, L. I., Failure in

Ductile Materials Using Finite Element Methods. In Structural Crashworthiness and

Failure, ed. N. Jones & T. Wierzbicki. Elsevier Applied Science Publishers, Barking,Essex, 1993, Chap. 1.

5. Nurick, G. N., Olson, M. D., Fagnan, R. F. & Levin, A. Deformation and Tearing of Blast-Loaded Stiffened Square Plates. Int. J. Impact Engng., Vol 16, No. 2, pp. 273-291, 1995.

6. Plane, C. A., Bedrossian, A. N. and Gorf, P. K. FE Analysis and Full Scale Blast Testsof an Offshore Firewall Panel. Int. Conf. on Offshore Structural Design AgainstExtreme Loads. ERA, London, 1994.

7. Jones, N.and Shen, W. Criteria for the Inelastic Rupture of Ductile Metal BeamsSubjected to Large Dynamic Loads. Structural Crashworthiness and Failure, Chapter 3,

Ed. Jones, N and Wierzbicki, T, 1993.8. Shen, W. Q. & Jones, N., A Failure Criterion for Beams Under Dynamic Loading. Int.

J. Impact Engng, Vol. 12, pp. 101-21, 1992.9. Jones, N. and Birch, R. S., 1998. Dynamic and Static Tensile Stress on Stainless Steel

for the Steel Construction Institute, Liverpool, UK.10. Hibbit, D., Karlsson, B. I., Sorensen, P., 1998. ABAQUS Theory Manual, Ver. 5.8.

Hibbit, Karlsson and Sorensen Inc., Pawtucket, RI, USA.

11. Hibbit, D., Karlsson, B. I., Sorensen, P., 1998. ABAQUS/Explicit User‘s Manual, Ver.5.8 (Vols. 1, 2). Hibbit, Karlsson and Sorensen Inc., Pawtucket, RI, USA.

12. Bazant, Z. P., Cedolin, L., 1991. Stability of structures: elastic, inelastic, fracture, and damage theories, Oxford University Press, Oxford, UK.

13. Needleman, A., 1987. Material Rate Dependence and Mesh Sensitivity in Localisation Problems, Computer Methods in Applied Mechanics and Engineering, 67:1:69-85.

14. Tanaka, K., Spretnak, J. W., 1973. An Analysis of Plastic Instability in Pure Shear in

High Strength AISI 4340 Steel , Metal Transactions, 4:443-454.15. Lasry, D., Belytschko, T., 1988. Localisation Limiters in Transient Problems,

International Journal of Solids and Structures, 24:6:581-597.

16. Lasry, D. and Belytschko, T., 1988. Localisation Limiters in Transient Problems,

International Journal of Solids and Structures, 24:6:581-597.17. Sluys, L. J., Bolck, J. and de Borst, R., 1992. Wave propagation and Localisation in

Viscoplastic Media, Proceedings of the International Conference on ComputationalPlasticity, Barcelona, Spain.

18. Zhu, Y. Y., 1992. Contribution to the Local Approach of Fracture in Solid Mechanics,Doctoral Thesis in Applied Sciences, University of Liege, Liege, Belgium.

19. Cescotto, S. and Li, X. K., 1996. Modelling of Strain Localisation in a Large StrainContext , Structural Engineering and Mechanics, 4:6:645-654.

20. Niemi, E. (Editor), 1995. Stress Determination for Fatigue Analysis of Welded Components, The International Institute of Welding, Cambridge, UK.

21. Fayard, J.-L., Bignonnet, A., Van Dang, K., 1996. Fatigue Design Criterion for Welded Structures, Fatigue and Fracture of Engineering Materials and Structures, 19:6: 723-

729.

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22. Hibbit, D., Karlsson, B. I., Sorensen, P., 1998. ABAQUS Standard Manual, Ver. 5.8.Hibbit, Karlsson and Sorensen Inc., Pawtucket, RI, USA.

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