Ultimate shear strength of intact and cracked stiffened panels

Fang Wang a,n, Jeom Kee Paik b, Bong Ju Kim b, Weicheng Cui a,c, Tasawar Hayat c,d,B. Ahmad c

a Hadal Science and Technology Research Center, Shanghai Ocean University, Shanghai 201306, Chinab Department of Naval Architecture and Ocean Engineering, Pusan National University, 30 Jangjeon-Dong, Geumjeong-Gu, Busan 609-735, Republic of Koreac Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabiad Department of Mathematics, Quaid-i-Azam University, 45320, Islamabad 44000, Pakistan

a r t i c l e i n f o

Article history:Received 19 October 2013Accepted 1 December 2014Available online 18 December 2014

Keywords:Stiffened panelCrackUltimate shear strength

a b s t r a c t

The present paper focuses on the ultimate shear strength analysis of intact and cracked stiffened panels.Several potential parameters influencing the ultimate shear strength of intact panels are discussed,including the patterns and amplitudes of initial deflection, the slenderness and aspect ratios of theplates, and the boundary conditions defined by the torsional stiffness of support members. An empiricalformula for the ultimate shear strength of intact stiffened panels is proposed based on parametricnonlinear finite element analyses in ANSYS. Furthermore, the ultimate shear strength characteristics ofcracked stiffened panels are investigated in LS-DYNA with the implicit method. Three types of cracks areconsidered, namely vertical crack, horizontal crack and angular crack. A simplified method is putforward to calculate the equivalent crack length. And the formula for the ultimate shear strength ofcracked stiffened panels is derived on the basis of the formula for intact stiffened panels.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Ship and offshore structures now tend to be designed based onthe ultimate strength. Significant progress has been made on theultimate strength characteristics of various types of structuralcomponents under typical load conditions as axial compressionand edge shear [1]. Comparing to axial compression, less attentionhas been given to the shear loading [2,3]. Though several empiricalformulae have been developed and applied in industry, most ofthem are based on the elastic buckling strength of the individualplate [e.g. 1,2]. In order to accurately assess the limit-state-basedcapacity of ship structures, systematic study on the ultimate shearstrength of stiffened panels should be carried out further.

Moreover, failure due to shear loads will arise particularly fordamaged structures [2]. Fatigue cracks are among the most typicaldamage or defect forms, which would be initiated in the concentra-tion area of the structure. The importance on assessing residualultimate strength of cracked ship structures under monotonicallyincreasing extreme load has been highlighted [4,5]. Some studieshave been paid on buckling characteristics of the cracked platesunder shear [6], while the study on post-buckling or large-deflectionregime is still not enough.

The aim of this paper is therefore to investigate the ultimateshear strength of intact and cracked stiffened panels. Since thedeflection pattern of the plates under edge shear in the post-buckling or large deflection regime is quite complex, the analyticalapproach may not be straightforward to use [1]. In this case,nonlinear FEA softwares, ANSYS [7] and LS-DYNA-implicit [8] willbe used for carrying out analyses on intact and cracked stiffenedpanels respectively, while cracked panel models used in LS-DYNAwith initial deflection are preprocessed by ANSYS. Based on thecomputed FEA results, an empirical formula for estimating theultimate shear strength of stiffened panels will be derived. And itwill provide the basis for the residual shear strength assessment ofcracked panels. Factors for shear strength reduction due to typicalcracks will be put forward at the same time.

2. Ultimate shear strength of intact stiffened panels

The one-bay SPM (stiffened panels model) shown in Fig. 1 willbe adopted for calculation in the present study. The plate dimen-sion is a� b� tp; the stiffener web dimension is hw � tw; and thestiffer flange dimension is bf � tf . The Young's modulus is E; thematerial yield stress is σY; and the Poisson ratio is ν. The shearmodulus can be expressed by G¼ E=2ð1þνÞ; and the shear yieldstress can be expressed by τY ¼ σY=

ffiffiffi3

p. The plate slenderness ratio

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/tws

Thin-Walled Structures

http://dx.doi.org/10.1016/j.tws.2014.12.0010263-8231/& 2014 Elsevier Ltd. All rights reserved.

n Corresponding author. Tel.: þ86 13812061166; fax: þ86 21 61900308.E-mail address: wf0224@aliyun.com (F. Wang).

Thin-Walled Structures 88 (2015) 48–57

representing the degree of plate thickness is defined as β¼ ðb=tÞffiffiffiffiffiffiffiffiffiffiffiσY=E

p.

Uniform shear load will be applied on the edges of the SPM.The simply supported condition is assumed at the panel edges,and all edges will remain straight by proper displacement con-straint equations during loading. Elastic perfectly plastic materialmodel is adopted. Strain-hardening effect will not be included innumerical model. The ultimate shear strength of stiffened panelsmay be affected by the patterns and amplitudes of initial deflec-tion, the slenderness and aspect ratios of the plates surroundedby longitudinal and transverse support members, and also theboundary conditions defined by the tortional stiffness of supportmembers. Parametric studies will be conducted to investigate theeffects of those potential influential factors.

2.1. Effect of initial deflection pattern

Initial deflections have to be considered for ultimate limit stateassessment of stiffened panels. The geometric configuration offabrication-related initial deflections in stiffened plate structures isquite complex [1]. For practical design purposes, the measure-ments of initial deflection for plate elements in steel-platedstructures are usually idealized to a multi-wave shape predomi-nant in the long direction and one half wave in the short direction.And the initial deflections into FE models are generally based onbuckling mode under compression. More precisely, the initialdeflections of stiffened panels can be divided into the local andglobal modes, which include the local buckling modes of platesand stiffeners, together with the vertical and horizontal compo-nents of global buckling mode. For more complicated structuressuch as cracked panels, it is more inconvenient to consider the realinitial deflection patterns. In view of this, initial deflections arefurther simplified to the buckling mode by eigen analysis underthe considered loading conditions. Generally speaking, the buck-ling mode under compression is more close to the real initialdeflection pattern, while it is quite different from the bucklingmode under edge shear. And the buckling mode of stiffened panelsmay also be varied if support members are included during eigenanalysis. In this part, the effect of initial deflection patterns will bediscussed first to provide the basis for further discussion. Fourtypes of patterns will be considered, namely,

a) Compression-based initial deflection pattern considering localbuckling modes of plates and stiffener webs respectively, andalso the global failure modes of the stiffened panel;

b) Compression-based initial deflection pattern from eigen analysis;c) Shear-based initial deflection pattern from eigen analysis with

support members included in the model (but the supportmembers will be removed during ultimate strength analysisin view of the same boundary conditions for comparison);

d) Shear-based initial deflection pattern from eigen analysis with-out support members included in the model.

The FE models of SPM are illustrated in Fig. 2 with four types ofinitial deflection patterns listed above. Initial deflection with theamplitude of wopl ¼ b=200 is assumed for the local mode of pattern(a) and the same amplitude is considered in other three patterns.A series of ANSYS ultimate strength analyses were undertakenwith varying the initial deflection patterns and plate thickness.Fig. 3 shows the average shear stress (τav) versus strain (γav)curves of the SPM. It is noted that the initial deflection patternswill slightly change the modeling results. Relatively larger effect isfound when plate thickness is 10mm and 8mm, but it can beignored for thick plates and very thin plates. And the shear-basedinitial deflection patterns will induce relatively lower strengthin FEA.

2.2. Effect of slenderness ratio

Based on the analysis above, the effect of plate slendernessratio can be obtained at the same time. Fig. 4 gives the curves ofnormalized ultimate shear strength (τu=τY) of SPM versus plateslenderness ratio with the four types of initial deflection patternsconsidered. A significant decreasing tendency can be observedwith increasing slenderness ratio. Slight difference due to differentinitial deflection patterns can be further noticed, but the differ-ences are within 6%.

2.3. Effect of initial deflection amplitude

Another potential parameter which may affect the ultimateshear strength of SPM is the initial deflection amplitude. Calcula-tions are performed here on the SPM withtp ¼ 10 mmconsideringtwo different initial deflection patterns. The initial deflectionamplitude is changed from 0:01β2t to 0:3β2t. The average shearstress versus strain curves are shown in Fig. 5 and the strengthchanging tendency can be further clarified by Fig. 6. Where, thevalue of wopl=ðβ2tÞ is adopted to represent the level of initialdeflection. It can be seen from the results that a proximate extentof 10% should be considered when the initial deflection amplitudechanges from very small level to severe level.

2.4. Effect of aspect ratio

It has been pointed out that the ultimate shear strengthdepends weakly on the plate aspect ratio, especially for relativelythick plates [1]. However, for the completeness of the parametricstudy on SPM in the present paper, the effect of the aspect ratiocombined with plate slenderness ratio will be further clarifiedhere by a series of FE analysis results shown in Fig. 7. The aspectratio will be changed from 1.0 to 5.0. From the results, it can be

Fig. 1. One-bay SPM.

F. Wang et al. / Thin-Walled Structures 88 (2015) 48–57 49

observed that the effect of the aspect ratio will be more obviouswith the increase of plate slenderness ratio. And the difference ofthe strength between SPM with a=b¼ 1:0and a=b¼ 5:0 reaches10% when β¼ 3:32.

2.5. Effect of torsional rigidity of support members

2.5.1. Effect of stiffener's rigidityIt is well investigated that the different height of the stiffener

may induce different collapse mode of SPM under axial compres-sion [1]. The effect of the stiffener's rigidity should also be studiedreasonably for the SPM under edge shear, which will be repre-sented according to the rotational restrained parameters of thelongitudinal (x) and transverse (y) support members put forwardin [9], namely,

ζL ¼ CLGJLbD

; ζS ¼ CSGJSbD

ð1Þ

Where, the constants

JL ¼hwxt3wxþbfxt3fx

6; JS ¼

hwyt3wyþbfyt3fy6

CL ¼ JLJPLwhen JL

JPLr1

n1 when

JLJPL

41;CS ¼ JSJPS

when JLJPSr1

n

1 whenJLJPS

41

Where, JPL ¼ bt3=3, JPS ¼ at3=3

Fig. 8 (1) compares the FEA results of the ultimate shear strengthof SPM when web height is 350 mm and 150 mm respectively. It isobserved that the effect of the web height on shear strength can beignored. And Fig. 8 (2) illustrates the effect of the rotational restrainedparameter of stiffener, which shows that the ultimate shear strengthcould be increased slightly with more rigid stiffener. However, itseffect is not significant.

2.5.2. Effect of tortional rigidity of support membersThe adopted one-bay SPM with simply-supported boundary

conditions has not taken into account the rotational constraints bylongitudinal girders and transverse frames. In a continuous shipstructures, the rotation along the SPM edges depends on the torsionalrigidity of the support members. The support members will beincluded into the ultimate strength analysis in this part and theirrotational restraint parameters will be changed with thickness. Fig. 9illustrates the normalized ultimate shear strength of SPM versusrotational restrained parameter of transverse frame. It can be seenthat whenζSincreases from zero to 1.0, the ultimate shear strength ofSPM will almost reach to a constant value. The strength from simply-supported boundary conditions is approximately 9.5% of the constantvalue when the rigidity of support members is large enough.

Fig. 2. SPM with four types of initial deflection patterns (Scale factor¼20).

F. Wang et al. / Thin-Walled Structures 88 (2015) 48–5750

Fig. 3. Average shear stress versus strain curves of the SPM obtained by FEA in ANSYS with varying plate thickness and initial deflection patterns ((a)–(d) in the figuresindicate the four initial deflection patterns listed in the text).

F. Wang et al. / Thin-Walled Structures 88 (2015) 48–57 51

2.6. Formula for ultimate shear strength of intact stiffened panels

Based on the analysis above, the following formula for theultimate shear strength of intact stiffened panels considering maininfluential parameters,

τuτY

¼1:0 for βo1:5

f 1f 2 U e

ðβ=f3 Þ þ f 4for βZ1:5 :

(ð2Þ

Where

f 1 ¼ τu0τY�1

3wopl

β2t

�

f 2 ¼ 0:00804þ0:07582Uab

f 3 ¼ 3:05128þ0:44105Uab

f 4 ¼ 0:95173�0:11177Uab

Here, τu0represents the ultimate shear strength of intact stiffenedpanel without initial deflection.

3. Ultimate shear strength of cracked stiffened panels

Defects of cracks are inherent in structural components; they areeither present during production or developed under service load[10]. Ultimate strength can be weakened by the cracking defectssignificantly. The ductile cracks with large-scale yielding zone areusually regarded as new free surface for residual strength assessmentunder monotonic extreme loading. However, cracks may also growunder some cases of monotonically increasing extreme loads, whichwill further decrease the strength of the structure. Therefore, thepossibility of crack propagation should be checked carefully duringultimate strength analysis. Regardless of the microscopic parametersgoverning the ductile fracture behavior, critical plastic strain isusually adopted as the criterion for ductile crack propagation used

Fig. 4. The effect of the plate slenderness ratio on the ultimate shear strengthof SPM.

Fig. 5. Average shear stress versus strain curves of the SPM (tp ¼ 10 mm) obtained by FEA in ANSYS with varying initial deflection amplitudes considering two initialdeflection patterns: compression-based initial deflection pattern (with global and local modes); Shear-based initial deflection pattern (without support members includedduring Eigen analysis).

Fig. 6. The effect of initial deflection amplitude factor on the ultimate shearstrength of SPM.

F. Wang et al. / Thin-Walled Structures 88 (2015) 48–5752

in FEA. During the calculation on the ultimate strength of crackedstiffened panels, the increase of maximum plastic strain in thevicinity of crack tips will be monitored and compared with thefailure plastic strain defined by material fracture strain and meshsize. The failure plastic strain for uniform shell elements recom-mended in [11] will be adopted here.

Generally speaking, three types of cracks need to be consid-ered, namely vertical crack, horizontal crack and angular crack.Fig. 10 shows a schematic of a stiffened panel with the three typesof cracks [5]. The ultimate strength characteristics should beanalyzed carefully with different crack locations and orientations.

Fig. 11 shows the FEA models for the stiffened panel with threetypes of cracks under edge shear, of which the edge shear loadingorientation will be considered to induce the angular crack with openmode. And shear-based initial deflection pattern without supportmembers supported will be adopted. It will be obtain after eigenanalysis for cracked SPM in ANSYS. However, LS-DYNA with implicitmethod will be adopted for ultimate strength analysis on the modelsinputted from ANSYS. Possible contact within crack edges or between

crack edges and panel surfaces could be consideredmore convenientlyin LS-DYNA. In order to make comparison between the ultimatestrength of intact and cracked panels, a brief comparison between the

Fig. 7. The effect of aspect ratio (a=bÞon the ultimate shear strength of SPM.

Fig. 8. Effect of stiffener's rigidity: (1) the normalized ultimate shear strength of SPM versus plate slenderness ratio when web height is 350 mm and 150 mm respectivelyand (2) the normalized ultimate shear strength of SPM versus rotational restrained parameter of stiffener.

Fig. 9. The normalized ultimate shear strength of SPM versus rotational restrainedparameter of support members.

Fig. 10. A schematic of a stiffened steel panel with three types of crack orientationsand under axial loads or edge shear [5].

F. Wang et al. / Thin-Walled Structures 88 (2015) 48–57 53

shear strength of SPMwith an angular crack obtained from ANSYS andLS-DYNA-implicit is performed, as shown in Fig. 12. The results agreewell. And all the notations in Fig. 12 will be used for the sameparameters in the following analysis.

3.1. The ultimate shear strength of a cracked individual plateelement

In a stiffened panel, multiple cracks may be formed in differentplate elements. It is impossible to consider all of the conditionswith different cracks distribution. In order to further simplify theproblem, each cracked plate will be considered individually first toprovide the basis for the strength of the stiffened panel.

A series of finite element analysis will be performed on asimply-supported plate with an angular center crack by changingcrack length. Fig. 13 gives the average shear stress versus straincurves. The decrease of shear strength can be observed and it isnoted that the maximum plastic strain in the vicinity of crack tipsat the same load level will increase with crack length. But thecrack will not propagate easily even in post-collapse period.

In order to consider more common cases with random crackorientation, the crack will be normalized in the following way byassuming that the angular cracks will be equivalent to vertical

Fig. 11. FEA models for the stiffened panel with three types of cracks under edge shear: (a) SPM with a vertical crack; (b) SPM with a horizontal crack; (c) SPM with anangular crack; and (d) Mesh pattern in the vicinity of crack tip.

Fig. 12. Results comparison between the ANSYS and DYNA implicit method.

F. Wang et al. / Thin-Walled Structures 88 (2015) 48–5754

cracks or horizontal cracks under different conditions,

cF ¼cU sin ðβpÞ=b when tanðβpÞZb=a

cU cos ðβpÞ=a when tanðβpÞob=a

(ð3Þ

Fig. 14 gives the normalized ultimate shear strength of the simply-supported plate with an angular center crack versus normalized cracklength. The following equation can be used for expressing the shearstrength decrease of stiffened panels due to the crack,

τucτu

¼ 1:00491�0:13362UcF�0:28721Uc2F ð4Þ

It is assumed here tha2t the strength of the stiffened panel withthe vertical cracks is the average value of cracked plate elements;the number of plate elements in a stiffened panel is denoted byn.But the strength of the stiffened panel with the horizontal cracks isthat of a plate element with a longer parallel crack. Then the shear

strength of the cracked panel can be expressed by,

τucτY

¼ τuτY

U f crack ð5Þ

Where

f crack ¼∑n

i ¼ 1ð1:0049�0:13362

cbsin βp�0:28721ðc

bsin βpÞ2 Þ for tan βp4b=a

1:0049�0:13362ca cos βp�0:28721ðca cos βpÞ2 for tan βprb=a

8><>:

3.2. The ultimate shear strength of cracked stiffened panels

The assumption and formula put forward above will be verifiedin this part for cracked stiffened panels. A series of ultimatestrength analysis is undertaken first on a stiffened panel with ahorizontal crack under edge shear with varying crack length.Fig. 15 shows the average shear stress versus strain curves. AndFig. 16 gives the normalized ultimate shear strength versusnormalized crack length. It can be seen that Eq. (4) can besuccessfully used for the ultimate shear strength prediction ofthe cracked panel with a horizontal crack.

Fig. 13. Average shear stress versus strain curves of a simply-supported plate withan angular center crack obtained by FEA in LS-DYNA with varying crack length; allplate edges will remain straight during deformation; shear loading orientation foropen mode crack is considered.

Fig. 14. The normalized ultimate shear strength of the simply-supported plate withan angular center crack versus normalized crack length.

Fig. 15. Average shear stress versus strain curves of a stiffened panel with ahorizontal crack obtained by FEA in LS-DYNA with varying crack length.

Fig. 16. The normalized ultimate shear strength of a stiffened panel with ahorizontal crack versus normalized crack length.

F. Wang et al. / Thin-Walled Structures 88 (2015) 48–57 55

Fig. 17. The FEA results for ultimate shear strength of cracked panels with different crack forms and initial deflection amplitudes: (1) SPM with a full crack in one plateelement; (2) SPM with an angular crack (tp ¼ 19 mm); (3) SPM with an angular crack (tp ¼ 10 mm); (4) average shear stress versus strain curves of SPM with an angular crackwith varying initial deflection amplitude (tp ¼ 10 mm); (5) average shear stress versus strain curves of SPM with a vertical crack with varying initial deflection amplitude(tp ¼ 19 mm); (6) average shear stress versus strain curves of SPM with a vertical crack with varying initial deflection amplitude (tp ¼ 10 mm); (7) the effect of initialdeflection amplitude factor on the ultimate shear strength of SPM with a vertical crack; and (8) the effect of initial deflection amplitude factor on the ultimate shear strengthof SPM with an angular crack.

F. Wang et al. / Thin-Walled Structures 88 (2015) 48–5756

Some other cases of cracked panels are calculated for validation ofthe proposed formula by varying crack forms and initial deflectionamplitudes. The FEA results are shown in Fig. 17. And the correlationcurve between the results from FEA and Eq. (5) is given in Fig. 18. It isnoticed that the proposed method correlate well with FEA results andcan accordingly be used to predict the ultimate shear strength ofcracked panels reasonably.

4. Concluding remarks

The aim of the present paper is to investigate the ultimate shearstrength of the intact and cracks stiffened panels. Nonlinear finiteelement analyses in ANSYS and LS-DYNA-implicit are carried out forthis purpose. Several potential parameters influencing the ultimateshear strength of intact panels are discussed. Three types of cracks areconsidered, namely vertical crack, horizontal crack and angular crack.And the possibility of crack propagation is checked with shear load

increasing. Based on the parametric analyses, an empirical formula forthe ultimate shear strength of intact stiffened panels is proposed. Asimplified method is put forward to calculate the equivalent cracklength. And the formula for the ultimate shear strength of crackedstiffened panels is derived on the basis of the formula for intactstiffened panels. The formula for the ultimate shear strength ofcracked panels is verified by FEA results. However, with regard tothe complexity of crack problem, further studies should be made onthe ultimate shear strength of stiffened panels with randomlydistributed cracks.

Acknowledgments

The present paper was undertaken at the Ship and OffshoreStructural Mechanics Laboratory, Pusan National University, that isa National Research Laboratory funded by the Korea Science andEngineering Foundation (Grant no. ROA-2006-000-10239-0). Partof the study was also supported by the Lloyd's Register EducationTrust (LRET) through the LRET Research Centre of Excellence atPusan National University.

References

[1] Paik JK, Thayamballi AK. Ultimate limit state design of steel-plated structures.Chichester, UK: Wiley; 2003.

[2] Zhang S, Kumar P, Rutherford SE. Ultimate shear strength of plates andstiffened panels. Ship Offshore Struct 2008;3(2):105–12.

[3] Ambur DR, Jaunky N, Hilburger MW. Progressive failure studies of stiffenedpanels subjected to shear loading. Compos Struct 2004;65:129–42.

[4] Paik JK, Thayamballi AK. Ultimate strength of aging ships. J Eng Marit Environ2002(M1):57–77 (216).

[5] Paik JK, Satish Kumar YV, Lee JM. Ultimate strength of cracked plate elementsunder axial compression or tension. Thin-Walled Struct 2005;43(2):237–72.

[6] Alinia MM, Hosseinzadeh ASS, Habashi HR. Numerical modelling for bucklinganalysis of cracked shear panels. Thin-Walled Struct 2007;45(12):1058–67.

[7] ANSYS 10.0 Users Manual, Swanston Analysis Systems (SAS) Inc., USA; 2005.[8] LS-DYNA User's Manual, Version 971, Livermore Software Technology Cor-

poration; 2006.[9] Paik JK, Thayamballi Ak. Buckling strength of steel plating with elastically

restrained edges. Thin-Walled Struct 2000;37:27–45.[10] Zacharopoulos DA. Arrestment of cracks in plane extension by local reinforce-

ments. Theor Appl Fract Mech 1999;32:177–88.[11] Simonsen BC, Törnqvist R. Experimental and numerical modelling of ductile

crack propagation in large-scale shell structures. Mar Struct 2004;17(1):1–27.

Fig. 18. Comparison between the results from FEA and Eq. (5).

F. Wang et al. / Thin-Walled Structures 88 (2015) 48–57 57