Structural Unit Method for Nonlinear Analysis

Thin-Walled Structures 41 (2003) 329–355www.elsevier.com/locate/tws

A concise introduction to the idealizedstructural unit method for nonlinear analysis of

large plated structures and its application

J.K. Paika,∗, A.K. Thayamballib

a Department of Naval Architecture and Ocean Engineering, Pusan National University, Busan 609-735, South Korea

b Chevron Shipping Company LLC, Camino Ramon, San Ramon, CA 94583, USA

Received 10 December 2001; received in revised form 29 October 2002; accepted 4 December 2002

Abstract

The idealized structural unit method (ISUM) has now been widely recognized by researchersas an efficient and accurate methodology to perform nonlinear analysis of large plated struc-tures such as ships, offshore platforms, box girder bridges or other steel structures. This paperpresents a summary of pertinent ISUM theory and its application to nonlinear analysis of steelplated structures. Important concepts for development of various ISUM units which are neededto analyze nonlinear behavior of steel plated structures are described. Some applicationexamples are shown, wherein comparisons of ISUM analysis predictions are made withnumerical or experimental results for progressive collapse analysis of general types of steelplated structures and ship hulls, to illustrate the possible accuracy and versatility of the ISUMmethod. The use of ISUM for the analysis of internal collision/grounding mechanics of shipsis also illustrated. This paper is in part an attempt to demystify ISUM and its applications forthe benefit of a designer of steel plates structures (Paik and Thayambali, Ultimate limit statedesign of steel plated structures; 2002). 2003 Elsevier Science Ltd. All rights reserved.

Keywords: Idealized structural unit method (ISUM); Nonlinear finite element method; Steel plated struc-tures; Nonlinear behavior; Ultimate strength; Internal accident mechanics; Collision; Grounding

∗ Corresponding author. Tel.:+82-51-510-2429; fax:+82-51-512-8836.E-mail address: [email protected] (J.K. Paik).

0263-8231/03/$ - see front matter 2003 Elsevier Science Ltd. All rights reserved.doi:10.1016/S0263-8231(02)00113-1

330 J.K. Paik, A.K. Thayamballi / Thin-Walled Structures 41 (2003) 329–355

1. Features of the idealized structural unit method

Under extreme or accidental loading, steel structures can exhibit highly nonlinearresponse associated with yielding, buckling, crushing and sometimes rupture of indi-vidual structural components. Quite accurate solutions of the nonlinear structuralresponse in many such cases can be obtained by application of the conventionalfinite element method (FEM). However, a weak feature of the conventional FEM isthat it requires enormous modeling effort and computing time for nonlinear analysisof large sized structures. Reduction in that modeling effort and the associated solutiontime while also providing an acceptable level of accuracy in results is the primarybenefit of the idealized structural unit method.

The most obvious way to reduce modeling effort and computing time in FEMapplications is to reduce the number of degrees of freedom so that the number ofunknowns in the finite element stiffness matrix decreases. Modeling the object struc-ture with very large sized structural units is perhaps the best way to do that. In orderto avoid the related loss of accuracy, ISUM requires the use of special purpose finiteelements. Properly formulated structural units within such an approach can then beused to efficiently model the actual nonlinear behavior of the corresponding largeparts of structures.

Ueda and Rashed [1,2], who suggested this idea, called it the idealized structuralunit method (ISUM). Their first effort in this regard was to analyze the ultimatestrength of a transverse framed structure of a ship as an assembly of the so-calleddeep girder units.

In an almost parallel development to ISUM, Smith [3] suggested a similarapproach to predict the ultimate bending moment of a ship hull. He modeled theship hull as an assembly of plate-stiffener combinations, i.e., stiffeners with attachedplating. The load versus end-shortening relationships for these beam–column unitswere obtained using conventional nonlinear FE analysis accounting for initial imper-fections (defections and residual stresses). The behavior of the larger structure wasthen constructed. While this method is quite properly called the Smith method, itmay also be classified as a type of ISUM in the particular context for analysis of astructure made up of repetitive structural elements.

ISUM is a simplified nonlinear FEM. Unlike the conventional nonlinear FEM,ISUM idealizes a structural component making up the structure as one ISUM unitwith a few nodal points. As a typical example, Fig. 1 compares structural modelingfor the elastic–plastic large deflection analysis of a rectangular plate under a reason-ably complex load application, as required by the conventional FEM and ISUM forcomparable accuracy in results. In conventional FEM, finer meshes would normallybe used, while on the other hand the plate is modeled using a single ISUM plateunit for ISUM analysis. It is noted that in this example the entire plate is taken asthe extent of FE analysis because of the unsymmetric characteristics related to defor-mations.

Of interest, the relative computational efforts required for the nonlinear analysisof a structure (conventional FEM versus ISUM) are now further illustrated, withdefinitions as follows

331J.K. Paik, A.K. Thayamballi / Thin-Walled Structures 41 (2003) 329–355

Fig. 1. Sample comparison of FEM and ISUM models for elastic–plastic large deflection analysis of arectangular plate of a /b = 3 under a complex load application. (a) FEM model: no. of nodes=341, degreesof freedom=6 per node, no. of unknowns=1874. (b) ISUM model: no. of nodes=4, degrees of freedom=3per node, no. of unknowns=6.

� Ratio of number of unknowns: m = FEM/ISUM.� Ratio of number of iterative loading steps: n = FEM/ISUM.� Ratio of computing time: CPU = FEM/ISUM = n × mc, with c = 1 � 3.

For the plate shown in Fig. 1, it was found that m�300, n = 5 � 10 and CPUratio�5 × 103 � 104. For larger sized structures with more complex geometries, therelative discrepancy due to the number of elements between the two methods willbecome much more significant as well. By applying ISUM, therefore, the size ofnumerical computations is much reduced leading to dramatic savings of modelingand computing times, compared to those of the conventional FEM. One trade-off isof course the possible loss of generality of the method, in the sense that ISUM mustuse specially formulated finite elements that are different, and internally supply morehigh level on structural component behavior than the elements typical of conven-tional FEM.

Among other matters, this paper presents a concise review of the most updatedtheory of the ISUM units which are available today for the nonlinear analyses ofsteel plated structures subjected to extreme or accidental loading. For a more detaileddescription of the theory of ISUM units and applications presented in this paper, theinterested reader is referred to Paik and Thayamballi [4].


2. ISUM modeling technologies for steel plated structures

A steel plated structure may be modeled as an assembly of many simpler mechan-ical structural elements or idealized elements, each type of which behaves in a knownmanner under given load application, and the assembly of which is constructed tobehave in nearly the same way as the actual structure.

Steel plated structures are typically composed of several different types of struc-tural members such as support members (or girders), rectangular plates and stiffenedpanels. In ISUM modeling, such members are regarded as the ISUM units. It isimportant to realize that an identical structure may be modeled in somewhat differentways by different analysts even within the context of ISUM, but it is of coursealways the aim to model so that the idealized structure behaves in nearly the sameway as the actual structure.

Fig. 2 shows some typical examples of ISUM modeling for steel plated structures.One of the common approaches is to model the structure as an assembly of plate–

Fig. 2. Various types of idealizations for a steel plated structure. (a) A typical steel plated structure. (b)Structural idealization by an assembly of plate-stiffener combination units. (c) Structural idealization by anassembly of plate-stiffener separation units. (d) Structural idealization by an assembly of stiffened panels.


stiffener combinations as shown in Fig. 2(b). This type of modeling may for examplenot be very appropriate when the stiffeners are relatively weak or unusually strong.In such cases, other types of modeling could be more relevant. For instance, theplate–stiffener separation models may be used, where plating between stiffeners ismodeled as one ISUM plate element, while the stiffener without attached plating ismodeled as one ISUM beam–column element, as shown in Fig. 2(c).

An entire panel together with stiffeners in either one or both directions can alsobe modeled by one ISUM stiffened panel unit as shown in Fig. 2(d). While a smallstiffener with or without attached plating may be modeled as the beam–column unit,the girder or support member with deep web may need to be modeled as an assemblyof the plate unit and the beam–column unit, the former being for the web and thelatter being for the flange.

In ISUM analyses, the load and boundary conditions are applied in a similar wayto the conventional FEM. The initial imperfections (initial deformations and residualstresses) and structural degradation (e.g., effect of corrosion or cracks) as well asgeometric and material properties of the ISUM units can be prescribed as parametersof influence as well.

3. Procedure for developments of ISUM units

For the nonlinear analysis of complex steel plated structures, it is apparent thatvarious types of the ISUM units are necessary to make a complete structural model,and that these units should then be developed in advance. Fig. 3 represents the usualprocedure to develop the ISUM units. The nonlinear behavior of each type of struc-tural members is idealized and expressed in the form of a set of failure functionsdefining the necessary conditions for the failure modes (e.g., yielding, buckling,cracking) which may take place and are described by the corresponding ISUM unit.

Fig. 3. Procedure for the development of an Idealized Structural Unit.


A set of stiffness matrices representing the nonlinear relationship between thenodal force vector and the nodal displacement vector are then formulated until orafter the particular limit state is reached. The post-failure phase is of interest, forinstance, in assessing post-limit state behavior, which one may want to do, say, indetermining the ultimate strength of a complex structure with one or more failedelements. The ISUM units (elements) so developed are used within the frameworkof the nonlinear matrix displacement procedure applying the incremental method,much like in the case of conventional FEM.

Many types of the ISUM units have so far been developed for the nonlinear analy-ses of steel plated structures. For the purposes of ultimate strength analyses, the deepgirder unit [1,2], the tubular beam–column unit [5], the I-section beam–column unit[6], the rectangular plate unit [7–12] and the stiffened panel unit [5,7–9] are available.For analysis of the internal mechanics in collisions and grounding, the rectangularplate unit, the stiffened panel unit and the gap/contact unit have been developed [13].

It is important to realize that individual developers of the ISUM units may employsomewhat different approaches from each other to idealize and to formulate the actualnonlinear behavior of the structural members. Also, the features of existing ISUMunits may be advanced continually by their developers to accommodate more factorsof influence or to improve the computational accuracy. Even though the geometricfeatures of the ISUM units are identical, their characteristics or usage in terms ofstructural behavior may be different from the purpose of the analyses, e.g., for ulti-mate strength or accidental mechanics.

ISUM has been successfully applied to the nonlinear analysis of ship structuresin overload situations, e.g., [13–15,27] among others and in collision or groundingaccidents, e.g., [16–19] among others, and also for other marine structures [20,21].It is also noted that ISUM can be readily applied to the nonlinear analysis of landbased structures such as box-girder bridges or cranes. In the following sections,ISUM formulations for many types of units that are presently available to a designerfor the nonlinear analyses of steel plated structures are summarized.

4. The beam–column unit

Fig. 4 shows the ISUM beam–column unit which may be used by a designerwith or without attached plating, the former being typically called the plate–stiffenercombination model, i.e., a stiffener with attached plating. This unit has two nodalpoints, i.e., node number 1 at the left end and node number 2 at the right end.

A deflected (or buckled) beam–column member is for the purposes of analysisreplaced by an equivalent straight (undeflected) member, but with a reduced levelof axial stiffness due to existence of the lateral deflection. This idealization leads toa decrease of degrees of freedoms to be considered at each node. As a result,rotational degrees of freedom at each node are not always needed to represent thenonlinear behavior of the unit and only three translational degrees of freedom (i.e.,in the x, y and z directions) at each node are included in the formulation of theISUM unit, which accounts for possible failure modes such as yielding and flexuralor lateral-torsional buckling.


Fig. 4. (a) The ISUM beam–column unit (a) with attached plating and (b) without attached plating (�:nodal points).

In the numerical computations, the end conditions of the unit can be applied inthe same way to the conventional FEM. For instance, all of the three translationaldegrees of freedom may need to be restrained for a fixed nodal point. The magnitudeof the resulting load effects (e.g., stress, displacement) may thus be different fromthat for a simply supported end case, for instance.

It is important to realize that an ISUM unit will behave in accordance with theinsights and knowledge built in by its developer. While one may of course try toidealize the nonlinear behavior of the unit under more relevant end conditions, theformulations typically assume that the beam–column unit is simply supported at bothends. As long as the beam–column members are bounded by relatively strong supportmembers, this approximation will give practical and adequate results on the pessi-mistic side.

Since the incremental method is used in ISUM analyses, the nodal force and dis-placement vectors of the beam–column unit are given in the incremental form, as fol-lows

{�R} � {�Rx1 �Ry1 �Rz1 �Rx2 �Ry2 �Rz2}T (1a)

{�U} � {�u1 �v1 �w1 �u2 �v2 �w2}T (1b)

where {�R} is the nodal force increment vector, and {�U} the nodal displacementincrement vector.


The relationship between the nodal force increment and the nodal displacementincrement can be expressed by

{�R} � [K]{�U} (2)

where [K] is the tangent stiffness matrix which is a nonlinear function of thenodal displacements.

As the applied loads increase, the beam–column unit can buckle or yield until theultimate limit state is reached. The tangent stiffness matrix K can then vary dependingon the condition of the structure with regard to the occurrence of such failurestogether with loading details at any point in time.

Fig. 5 represents the idealized behavior of the beam–column unit for the purposeof ultimate strength analysis. It is important to realize that in contrast to plates, thisbeam–column unit would not have reserve strength once buckling occurs, and thusthe unit reaches the ultimate limit state immediately after buckling. The effect ofstrain-hardening is neglected in the present formulations of the unit.

The tangent stiffness matrix [K] of Eq. (2) or the condition of failures of thebeam–column unit under combined axial and lateral loads can be evaluated by ana-lytical approach. The tangent stiffness matrix of the failure-free unit denoted by [K]E

is given by taking into account the effects of large deflection, combined axial andlateral loads and initial imperfections. The ultimate strength or the gross yield

Fig. 5. Idealized structural behavior of the ISUM beam–column unit for analysis of ultimate strength.


capacity can be expressed in the closed form. The tangent stiffness matrix of thebeam–column unit in the post-ultimate regime denoted by [K]U is also given by aclosed-form expression.

5. The rectangular plate unit for analysis of ultimate strength

Fig. 6 shows the ISUM rectangular plate unit which has four nodal points, i.e.,one at each corner. A buckled (deflected) plate in this case is in effect replaced byan imaginary flat (undeflected) plate, but with the corresponding reduced effectivein-plane stiffness. Since the plate configuration always remains flat in the numericalmatrix computations, rotational degrees of freedom at each nodal point are not neces-sary to represent the nonlinear behavior of the ISUM rectangular plate unit. As aresult, only three translational degrees of freedom at each corner nodal point areused to formulate the nonlinear response like in the beam–column unit.

The nodal force increment vector {�R} and the nodal displacement incrementvector {�U} of the ISUM rectangular plate unit are then given by

{�R} � {�Rx1 �Ry1 �Rz1···�Rx4 �Ry4 �Rz4}T (3a)

{�U} � {�u1 �v1 �w1···�u4 �v4 �w4}T (3b)

The relationship between the nodal force increments and the nodal displacementincrements can also be given by Eq. (2). Fig. 7 represents the idealized behavior ofthe ISUM rectangular plate unit until and after the ultimate strength is reached. Inthis case, the tangent stiffness matrices and the condition of the ultimate limit statebehavior can be evaluated using the theory of plates under combined loads, takinginto account the effects of initial imperfections and other types of structural degra-dation.

The tangent stiffness matrix of the failure-free unit denoted by [K]E is calculated

Fig. 6. The ISUM rectangular plate unit (�: nodal points).


Fig. 7. Idealized structural behavior of the ISUM rectangular plate unit for analysis of ultimate strength.

by taking into account the effects of both in-plane and out-of-plane large defor-mations by applying the conventional FE approach [13]. Before the unit reaches theultimate limit state, the internal stresses at some nodal points may satisfy the yieldcondition and if so, plastic nodes are then inserted into the corresponding nodalpoints. In this case, the elastic–plastic stiffness matrix [K]P can be derived by apply-ing the plastic node method as a function of the stress–strain matrix [D]E [22]. TheISUM rectangular plate unit reaches the ultimate strength if the ultimate limit statecriterion is satisfied upon substituting the internal stress components. In the post-ultimate regime of the unit, the stress–strain matrix [D]E should be replaced by thestress–strain matrix in the post-ultimate regime denoted by [DP]U, while the stiffnessmatrix of the collapsed unit denoted by [K]U is still given by [K]E.

6. The rectangular plate unit for analysis of collision and groundingmechanics

While the collision and grounding mechanics of a plate are also nominally rep-resented by an ISUM rectangular plate unit with four nodal points, as shown in Fig.6, the characteristics of the unit for analysis of collision and grounding mechanicsare different from those for analysis of ultimate strength. The relationship betweenthe nodal force increment vector {�R} and the nodal displacement increment vector


{�U} is also given by Eq. (2) where {�R} and {�U} are defined by Eqs. (3a)and (3b).

Fig. 8 shows the idealized behavior of the ISUM rectangular plate unit for analysisof collision and grounding mechanics. As the predominantly compressive loadincreases, the unit crushes if a crushing condition is fulfilled. It is assumed that theinternal stresses of the crushed unit are unchanged even if the compressive displace-ment increases. However, the folding process ends if the compressive strain of theunit exceeds a critical value. The unit behaves as a rigid body after the foldingprocess ends. The crushed unit eventually reaches the gross yield condition as thecrushing loads increase further.

On the other hand, with increase in the predominantly tensile loads, some nodesof the unit will yield if the yield condition is satisfied and in such a case plasticnodes are inserted into the corresponding nodal points. As long as all of the fournodes do not yield, the unit may not reach the gross yield condition, but it will showan elastic–plastic load carrying behavior. If the equivalent tensile strain exceeds thefracture strain, then ductile fracture takes place. Internal stresses of any fracturedunit must be released from the analysis, with the unit losing its resistive load capacity.

The tangent stiffness matrix of the failure-free unit denoted by [K]E is given bythe conventional FE approach [13,22]. As long as energy absorption capacity isconcerned, the effect of initial deflections may be neglected. The same stress–strainmatrix is thus used until either crushing or yielding occurs. As the applied loads

Fig. 8. Idealized structural behavior of the ISUM rectangular plate unit for analysis of collision andgrounding mechanics.


increase, the unit begins to be folded if the condition of crushing is satisfied. Thestress–strain matrix of the crushed unit is assumed to become zero, while the tangentstiffness matrix is still calculated from [K]E.

The folding process of the plate elements under crushing loads may stop if thefolding length reaches a ‘critical’ value which is a function of the geometric andmaterial properties of the plate. After the folding process ends, the unit behaves asa rigid body and thus the stress–strain matrix [D]E may become infinite, e.g., witha very large magnitude of the Young modulus. However, if the internal stress compo-nents satisfy the yield condition at any node then a plastic node is inserted into theyielded node of the crushed unit and the elastic–plastic stiffness matrix is applied.

Under predominantly tensile loading, some nodes may yield if the yield conditionis satisfied and the elastic–plastic stiffness matrix is applied. The yielded unit mayeventually rupture if the strain components satisfy the condition of ductile fracture.The fractured unit releases its internal stresses and takes a zero value in the stress–strain matrix [D]E.

7. The stiffened panel unit for analysis of ultimate strength

Fig. 9 shows an ISUM stiffened panel unit. The panel with stiffeners in eitherone or both directions, bounded by support members, is modeled as a unit by meansof the ISUM stiffened panel unit. When the stiffeners are relatively strong so thatthey would not fail prior to plating between stiffeners, the same panel may also bemodeled as an assembly of rectangular plate units and beam–column units, the latterbeing without attached plating.

For a panel with relatively weak stiffeners so that the overall panel buckling mode

Fig. 9. The ISUM stiffened panel unit (�: nodal points).


may be predominant, or in the general case where it can be accommodated, however,the ISUM stiffened panel unit may be more appropriate.

Like the rectangular plate unit, the stiffened panel unit has four nodal points, i.e.,each at every corner. The relationship between the nodal force increment vector{�R} and the nodal displacement increment vector {�U} is given by Eq. (2) where{�R} and {�U} are defined by Eqs. (3a) and (3b).

In formulating the stiffened panel unit, six types of failure modes need to beconsidered: overall collapse, biaxiall compressive collapse, beam-column type col-lapse, stiffener web buckling, tripping and gross yielding [4].

8. The stiffened panel unit for analysis of collision and grounding mechanics

The geometry and dimensions of the ISUM stiffened panel unit for analysis ofcollision and grounding mechanics are similar to that of Fig. 9. The unit has fournodal points, i.e., each at every corner point. Each node has the three translationaldegrees of freedom since the deflected panel is treated as an imaginary undeflected(flat) panel but with reduced in-plane stiffness. The force and displacement vectorsof the unit are again given by Eqs. (3a) and (3b). The idealizations of the ISUMstiffened panel unit for analysis of the collision and grounding mechanics can bemade in a manner similar to that of the rectangular plate unit as described in Section6 or Fig. 8.

9. The gap/contact unit

In collision or grounding accidents, the interface between the striking and thestruck bodies will change as the collision or grounding proceeds. In this case, thegap and contact conditions between the two bodies are modeled by means of thegap/contact unit.

The gap/contact unit is a special type of the beam–column unit described in Sec-tion 4. The gap unit has two nodal points and each node has three translationaldegrees of freedom. In the structural modeling using the gap/contact unit, one nodemay be positioned at the striking body and the other at the struck body. Thegap/contact unit may also be used for connecting two nodes which both are in thestruck body as well.

Two such nodes are then connected by a gap/contact unit with nonlinear character-istics. As the striking body is pushed into the struck body, the length of the unit inthe gap condition will be reduced without resistance, and eventually both nodes willcome into contact. Coordinates of both nodes are updated at every incremental load-ing step. It is considered that the unit is in the contact condition if the length of theunit becomes smaller than a prescribed tolerance.

The gap/contact units have only axial stiffness. In the gap condition, the unit doesnot carry any external forces, but under contact it behaves as a rigid body. Therefore,nearly zero stiffness is assumed for the gap condition, while a very large stiffness


is assumed for the contact condition. The stiffness equation of the gap/contact unitmay then be given, as follows [13]

��Rx1

�Rx2� �

EgcAL �1 �1

�1 1��u1

�u2� (4)

where A is the cross-sectional area. It may be assumed that Egc = E×10�7 for thegap condition or Egc = E×107 for the contact condition, where E is the elastic modu-lus.

In practice, some axial stresses may develop in the gap/contact unit because theassumed stiffness is neither zero nor infinite, but this effect is generally quite smalland negligible. Also, the node positioned at the striking ship and the other node atthe struck ship may ‘overlap’ if the increment of axial deformation is too large, butthis problem can be overcome by using small load increments.

10. Treatment of dynamic/impact load effects

Ship collision or grounding accidents are dynamic in nature, and this fact willaffect the crushing and rupture response of structures. Three aspects of a dynamicloading situation are possibly relevant, namely material strain rate sensitivity, inertiaeffects and dynamic frictional effects. With increase in the strain rate for steel, theyield strength of the material increases and the rupture strain may decrease. Due toinertia effects, deformation patterns may be varied. It is known that as the speed ofdynamic loading increases, the coefficient of friction becomes lower.

When dynamic loads are applied, the crushing response is mainly affected by thematerial strain rate sensitivity. During crushing response, the material strain rategenerally varies with displacements. For simplicity, however, the average value ofthe strain rate during the dynamic loading phase may be used. The average strainrate for rectangular plates or stiffened plates can be approximately estimated con-sidering that the loading speed is linearly reduced to zero.

To estimate the dynamic yield strength of the material, the Cowper–Symondsequation has been widely used. The crushing effects and yield strength increase asthe collision speed gets faster, while any rupture or tearing of steel occurs earlier.The dynamic rupture strain is obtained by inverting the Cowper–Symonds equationfor the dynamic yield stress.

The inertia effects may be ignored when the strain rates are less than about 50s�1. Friction effects may play an important role during the grounding process, whilethey can be ignored for crushing of structures since the relative velocity betweenstriking and struck bodies is then normally comparatively small.

In the illustrative ISUM calculations given below of crushing and rupture behaviortaking account of dynamic effects, theoretical developments derived for a quasi-staticcondition are employed, but with the use of the dynamic yield stress or the dynamicfracture strain in place of their static counterparts. Hence certain effects such asdynamic amplification are not included in the calculation of structural response. Forfurther details, the interested reader may refer to [4,16].


11. Application examples

In the following sections, some typical examples for nonlinear analysis of steelplated structures are illustrated. The ISUM method is used in the calculations. Resultsfrom ISUM are compared against conventional numerical FE analysis or experi-mental results for the same problem to illustrate the accuracy achieved.

For purposes of these comparative calculations, the ISUM theory we used hasbeen implemented within a computer program called ALPS/ISUM which stands fornonlinear Analysis of Large Plated Structures using Idealized Structural Unit Method.For convenience, ALPS/ISUM is divided into three sub-programs, namelyALPS/GENERAL, ALPS/HULL and ALPS/SCOL. ALPS/GENERAL is a programfor analysis of progressive collapse behavior of general types of steel plated struc-tures. ALPS/HULL is a special purpose program for analysis of progressive collapsebehavior of ship hulls. ALPS/SCOL is a program for analysis of internal mechanicsof steel plated structures in collision or grounding.

11.1. Progressive collapse analysis of a cantilever box-girder

Ultimate strength of a cantilever box-girder is analyzed in this case by ISUM andconventional nonlinear ANSYS FEM [23]. The box-girder is fixed at one end andfree at the other end, and it is subjected to concentrated load at the free end, asshown in Fig. 10. The cross section of the structure is square, and three transversediaphragms are arranged within. The dimensions, material properties and initialimperfections of individual plate elements are as follows:

� plate units: a × b×t = 1000 × 1000 × 15(mm);� Young’s modulus: E = 205.8 GPa;� yield stress: sY = 352.8 MPa;� Poisson’s ratio: n = 0.3;� initial deflection: wopl / t = 0.15;� welding residual stresses: srcx = srcy = 0.0.

Fig. 10. An internally stiffened cantilever box-girder.


Figs. 11 and 12 show typical examples of structural modeling for use in thiscase by ANSYS and ALPS/GENERAL, respectively. In ISUM modeling, each plateelement is modeled as one ISUM rectangular plate unit, while a number of finemeshes are used in conventional FE modeling where a half of the box-girder withrespect to the center line is taken as the extent of conventional nonlinear FE analysisconsidering the symmetric behavior. Fig. 13 shows the applied load versus deflectioncurves at the free end of the structure, as obtained by ANSYS andALPS/GENERAL computations.

For reference, the plastic collapse load of a cantilever beam can be predicted whenlocal buckling is not accounted for [4], as follows:

Pc �Mp

L�

14000

364

(10003�9703) � 9.8 � 1925.56 kN

Fig. 11. (a) Undeformed shape for the conventional nonlinear FE model for the cantilever box-girder.(b) Deformed shape for the conventional nonlinear FE model for the cantilever box-girder at the ultimatelimit state.


Fig. 12. ALPS/GENERAL model for the cantilever box-girder.

Fig. 13. The force versus deflection curves at the free end of the cantilever box-girder, as obtained byISUM and conventional nonlinear FEM.

Fig. 13 also represents the selected structural failure events. As the applied loadsincrease, the compressed flange near the fixed end collapses and the box-girderreaches the ultimate strength it the transverse floors near the fixed end fail. It is abenefit of structural designers to have detailed information about the progressive


collapse behavior until the ultimate strength is reached. It is seen from Fig. 13 thatthe ISUM results correlate well with the more refined nonlinear FE analysis. Ofinterest, the computing time used was 2 min for the ISUM analysis and 4.5 h forthe FE analysis using a Pentium III personal computer. The structural modeling effortfor ISUM analysis is of course much smaller than that for FE analysis.

11.2. Progressive collapse analysis of a box-column

Ultimate strength of a box column is now analyzed in this example by ISUM andconventional nonlinear ANSYS FEM [23]. The structure is simply supported at bothends, as shown in Fig. 14. It has a number of transverse diaphragms (floors). Thedimensions, material properties and initial imperfections of the structure are as fol-lows:

� plate elements: a × b × t = 500 × 500 × 7.5(mm);� Young’s modulus: E = 205.8 GPa;� yield stress:sY = 352.8 MPa;� Poisson’s ratio: n = 0.3;� initial deflection function of plate elements: Ao / t = 0.05;� column type initial deflection function for the whole structure: wo = dosin(px /

L) where do /L = 0.0015;� welding induced residual stresses: srcx = srcy = 0.05 for ISUM analysis and srcx

= srcy = 0.0 for ANSYS analysis.

Figs. 15 and 16 show typical examples of structural modeling for this case, byconventional ANSYS FEM [23] and ALPS/GENERAL, respectively. In ISUMmodeling, each plate element is modeled as one ISUM rectangular plate unit anddue to the symmetry, a half length of the box column is taken as the extent ofanalysis. For conventional nonlinear FE analysis, a quarter of the box column issubdivided into a number of fine meshes. For convenience related to computationaleffort, some coarse meshing was adopted for the conventional nonlinear FE analysis

Fig. 14. A box column with both ends simply supported.


Fig. 15. Conventional FE model for a quarter region of the box column.

Fig. 16. ALPS/GENERAL model for a half length of the box column.

such that local plate buckling effect (i.e., local geometric nonlinearity) is not takeninto consideration in the ANSYS analysis, although the influence of plasticity(material nonlinearity) and column type buckling (global geometric nonlinearity) areaccounted for. On the other hand, the ALPS/GENERAL takes into account bothlocal and global buckling effects as well as plasticity.

Fig. 17 shows the computed axial compressive load versus mid-span deflectioncurves for the box column. The Euler elastic column buckling load which intention-ally neglects the influence of local plate buckling is also plotted in the figure. Whenthe structure involves both local and global buckling, the ultimate buckling strengthof the box column is about 67% of the Euler column buckling load. It is evidentthat the effect of local geometric nonlinearity as well as global geometric nonlinearitycan be of crucial importance for nonlinear analysis of slender structures (e.g., boxcolumn). Of interest, the computing time used was 3 min for the ISUM analysis and6 h for the FE analysis using a Pentium III personal computer.


Fig. 17. The axial compressive load versus deflection curves of the box column, as obtained by ISUMand conventional nonlinear FEM.

11.3. Progressive collapse analysis of a ship hull

A physical test for investigating the progressive collapse characteristics under avertical sagging moment was undertaken on a welded steel frigate ship structuremodel with 1/3 scale to the original ship dimensions [24,25]. ALPS/HULL is nowused to analyze the progressive collapse behavior of the Dow test model and theresults are then compared with the experimental results. In the ALPS/HULL compu-tations, both sagging and hogging cases are considered.

Fig. 18 shows the ALPS/HULL model for the test structure. For simplicity, the

Fig. 18. (a) Mid-section of the Dow frigate test structure. (b) ALPS/HULL model for the progressivecollapse analysis of the Dow frigate test structure.


hull module between two transverse frames is taken as the extent of the presentanalysis, while it is not difficult to take the entire structure as the extent of the ISUManalysis if needed. Plating between stiffeners is modeled using the ISUM rectangularplate unit and stiffeners without attached plating are modeled using the ISUM beam–column unit. The webs of deep girders in bottom structures are also modeled usingthe ISUM rectangular plate units, while their flanges are modeled using the ISUMbeam–column units.

Fig. 19 shows the progressive collapse behavior of the Dow test structure undersagging or hogging moment, as obtained by ALPS/HULL. The Dow test result forsagging is also plotted. In the ALPS/HULL computations, the magnitude of initialimperfections is varied. It is seen from Fig. 19 that the effect of initial imperfectionson the progressive collapse behavior is of significance. Also, ALPS/HULL providesreasonably accurate results when compared with the experiment. Of interest, thecomputing time used was 2 min for the ISUM analysis using a Pentium III per-sonal computer.

11.4. Analysis of internal mechanics of a LNG carrier side structure

As a hypothetical collision scenario, it is now considered that the bow of anotherLNG carrier with the same size as the object LNG carrier in the full load conditionstrikes the mid-side structure of the object LNG carrier. At the initiation of thecollision event, the striking ship is assumed to be at 50% of the full design speed,while the struck LNG carrier is considered to be at standstill at a pier.

The penetrating depth of the striking ship from the initial contact to the boundaryof the LNG cargo tanks or to the bow position which may cause the fracture ofLNG cargo tanks varies in accordance with the fore-end shape of the striking ship

Fig. 19. Progressive collapse behavior of the Dow test structure under vertical moment.


Fig. 20. MSC/DYTRAN model for the LNG carrier–LNG carrier collision.

and the initial contact positions and related details for the two ships. The bulbousbow shape under the sea water level and/or the fore-end shape of the striking shipdetermines the initial contact position for the two ships and may also affect theabsorbed energy in the side structure of the struck ship.

Both the striking and struck LNG carriers are assumed to be in the full loadcondition. Two computer programs, namely MSC/DYTRAN [26] and ALPS/SCOL,are used to obtain the collision force–penetration curves as the collision proceeds.Figs. 20 and 21 represent the structural models used for MSC/DYTRAN andALPS/SCOL, respectively.

In the MSC/DYTRAN model, the entire ship structure is included using finemeshes around the impact location while coarser meshes are used at the other parts

Fig. 21. ALPS/SCOL model for the LNG carrier–LNG carrier collision.


away from the collision location. In the ALPS/SCOL model, a quarter of the mid-ship cargo hold of the struck ship including the LNG tank together with the strikingship bow is taken as the extent of the analysis, using ISUM plate and stiffened panelunits to model the struck structures and gap/contact units to represent the gap andcontact conditions between the struck ship side and the striking ship bow.

In the MSC/DYTRAN computations, the motion of both striking and struck shipswith the surrounding water is taken into consideration, while the ALPS/SCOL com-putations assume that the struck ship is fixed while the striking ship penetrates withthe constant speed. In both models, it is assumed that the striking ship bow is a rigidbody which does not dissipate any strain energy.

The fracture strain of the structural members in both ship hull and cargo tankstructures is supposed to be 10% in the ALPS/SCOL models which use large sizedelements, while it is assumed to be 20% in the MSC/DYTRAN models which usefine meshes. This is because the MSC/DYTRAN method using a fine mesh canautomatically handle the localized tearing behavior inside the plate members, whilethe ALPS/SCOL method using large sized elements must necessarily account forsuch behavior at a macroscopic level. In the failure state of finite elements, the strainis typically concentrated at the tip of crack or around fractured area. The representa-tive fracture strain should then become smaller in large sized elements to properlyaccount for the effect of localized fracture inside the plate unit. Hence in theALPS/SCOL model, a fracture strain that is likely to be more characteristic of weldmetal fracture is used. With the larger value of fracture strain, the energy absorptioncapability of the structure will of course increase.

The computations were continued until the struck LNG carrier reaches the acciden-tal limit states in the following two conditions, namely (a) the striking ship bowpenetrates the boundary of the struck ship cargo tank, and (b) the LNG cargo tanksstart to rupture. In reality, the energy dissipation capability of the struck LNG carriermust be evaluated at the earlier of these two limiting conditions.

Fig. 22 shows the deformed shape of the ALPS/SCOL model. Figs. 23 and 24show the collision force versus penetration curves and the absorbed energy–pen-etration curves, respectively, the two curves in each figure being obtained byMSC/DYTRAN and ALPS/SCOL. Oscillations in the force–penetration curvesobtained by MSC/DYTRAN are primarily due to the simulation being based on time-variant impact theory. Also, the behavior of localized failure including tearing andfolding is more accurately described by the MSC/DYTRAN method using very finemeshes, while the ALPS/SCOL method is more concerned with an ‘average’ natureof the failure response.

As evident, both MSC/DYTRAN and ALPS/ SCOL predictions correlate well withone another, and thus it may be taken with reasonable certainty that the ISUM com-puted results for the energy dissipation capability of the struck LNG carrier areaccurate enough for the purposes of design stage safety assessment. For other typesof accident scenarios varying the type of striking ship or loading condition, the inter-ested reader may refer to [17].

Of interest, the computing time used for the ALPS/SCOL analysis was about 4 husing a Pentium III personal computer, while it was more than 14.4 h for the


Fig. 22. Deformed shape of the ALPS/SCOL model immediately after the struck LNG cargo tank startsto fracture.

Fig. 23. The collision force–penetration curves (SS-F=collision of LNG carrier to LNG carrier in thefull load condition).


Fig. 24. The absorbed energy–penetration curves (SS-F=collision of LNG carrier to LNG carrier in thefull load condition).

MSC/DYTRAN analysis using the SGI Power Challenge XL Series computer. Asevident, the former will be of a benefit when a quick estimate is more important forcarrying out a series of collision analyses with a variety of accident scenarios.

12. Concluding remarks

While the ISUM has been recognized by researchers as an efficient and accuratemethodology for nonlinear analysis of large plated structures such as ships, offshoreplatforms, box girder bridges and other steel structures, the authors are of the opinionthat the number of structural designers who use ISUM are currently quite limited.This may be partly because an educational effort on details of the basic idea ofISUM is necessary, and the literature on the theory and application of ISUM is notsufficient for this purpose at the moment.

The twin aims of the present paper have thus been to summarize the ISUM theoryand to illustrate some examples of its application to nonlinear analysis of steel platedstructures. Important concepts for developments of the various ISUM units whichwould be typically needed by a designer to analyze the nonlinear behavior of steelplated structures are described. Some examples of ISUM application to analysis ofultimate strength and internal accident mechanics of ships and steel plated structuresare illustrated. The accuracy of ISUM predictions is studied by comparison with thecorresponding results from conventional nonlinear finite element analyses and fromexperimental results, as appropriate. The accuracy is illustrated to be good, and per-


haps more importantly, it is shown that a level of accuracy comparable to conven-tional methods can be achieved at a fraction of the cost and effort associated withthe conventional methods.

ISUM technology is now mature, and can clearly be useful to a structural designerin many contexts, including design assessment based on the ‘ true’ ultimate strengthand the general damage tolerant design of structures.

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Structural Unit Method for Nonlinear Analysis