An Evaluation and Comparison of Models for Maximum Deflection of Stiffened Plates Using Finite...

An Evaluation and Comparison of Models for Maximum Deflectionof Stiffened Plates Using Finite Element Analysis

Lior Banai1 and Omri Pedatzur2

Stiffened plates form the backbone of most of a ship’s structure. Today, finite element (FE) models areused to analyze the behavior of such structural elements for different types of loads. In the past, whenusage of computers and FE models were not used very much, analytical analysis methods were required.Two well-known methods have been developed for analyses of stiffened plates under lateral loading(uniform pressure), based on two different models, namely, the orthotropic plate model and the grillagemodel. Both models can give estimations for the maximum plate deflection under uniform lateral pressure.The objective of this paper is to present the two methods, evaluate and compare the methods using thefinite element method, and finally implement the methods as a computer program for quick estimations ofthe maximum deflection of stiffened plates. The degree of accuracy of the two methods when comparedto FE is discussed in some detail.

1. IntroductionTHE MAIN TYPE of framing system found in ships nowadays

is a longitudinal one that has stiffeners running in two or-thogonal directions. Structures of this kind are usually calledcross-stiffened plates/panels, or grillages. Deck and bottomstructures of this type are reinforced mainly in the longitu-dinal direction, with widely spaced, heavier, transverse stiff-eners, as shown in Fig. 1. The stiffeners used in the longitu-dinal direction are T-beams, angles, bulbs, and flat bars,while the transverse stiffeners are typically chosen as T-section beams.

During the first half of the 20th century, with the rapidgrowth in shipbuilding and deployment of ships, the needarose for a way to evaluate the response of the ships’ struc-ture to different types of sea loads. At that time, the use ofcomputers was limited. Numerical methods, such as finiteelement analysis (FEA) and other variational techniquesmade their first steps into the world of engineering, and com-putation time was unacceptably long.

Having these limitations in mind, many engineers and sci-entists sought a way to bypass the numerical difficulty bydeveloping analytical methods for the analysis of shell struc-tures (mostly, analytical methods were not exact solutions,but approximated solutions of the partial differential equations,or PDEs, that described the given engineering problem).

The first analytical method, developed by H.A Schade dur-ing the 1940s, enables the evaluation of the maximum staticdeflection of a stiffened plate subjected to a uniform lateralpressure on its surface. The related work described in thispaper is based on a number of papers (Schade 1940, 1941,1951). The second method is based on the grillage model asdeveloped by Clarkson et al. (1959).

2. Part 1: Summaries and evaluations of models2.1. The orthotropic plate model—Summary oftheory and Schade’s method theory

In many structures, the elastic properties are not isotropic;that is, they are not the same in all directions. Even steel

plate, which is commonly regarded as isotropic, can exhibitdifferent elastic behavior in different directions. An ortho-tropic plate is one that has different elastic properties inthree (or two) orthogonal directions, and those properties areuniform within each direction. The fundamental equation ofthe static response of an orthotropic plate can be written asfollows:

Dx

�4w

�x4 + 2H�4w

�x2�y2 + Dy

�4w

�y4 = p�x, y� (1)

where Dx and Dy are bending stiffnesses in the x and y di-rections, respectively. H is function of the bending stiffnessesand the torsional rigidity (Dxy). p(x,y) is the load per unitwidth acting on the plate.

The determination of the bending stiffnesses is relativitystraightforward, but determination of the torsional rigidity isnot as easy. The value of H can be determined theoretically orby experiment. While some authors have given expressionsfor the coefficient H that accounts for shear deflection,Schade’s work itself does not include that deflection compo-nent, which for instance may be relevant in a stiffened paneldue to the effect of shear distortion of the webs. The effect ofshear deflection is normally a reduction in the torsional ri-gidity Dxy.

The earliest solution of equation (1) seems to have beenthat of Huber, who showed that the solution involved twonondimensional parameters that are functions of the plateaspect ratio and the rigidity parameters Dx and Dy. Thoseparameters are designated by the letters � and �:

� =H

2�DxDy

� =ab�4 Dy

Dx. (2)

Schade has developed an approximate e xpression for thecoefficient H, and using that he also obtained the followingexpression for �:

� =�Ipa

Ina�

Ipb

Inb(3)

where Ipa (Ipb) is the moment of inertia of plating only, work-ing with long (short) stiffeners and Ina (Inb) is the moment ofinertia of repeating long stiffeners.

1 The School of Mechanical Engineering, Tel Aviv University, TelAviv, Israel.

2 The Naval Architecture and Marine Engineering Department,Israeli Navy, Israel.

Manuscript received at SNAME headquarters July 2006.

© Marine Technology, Vol. 44, No. 4, October 2007, pp. 212–225

212 OCTOBER 2007 MARINE TECHNOLOGY0025-3316/07/4404-021200.00/0

2.2. Calculation of maximum deflection

The Schade method is composed of charts and equations,which, when followed in sequential steps, give the maximumdeflection at the center of a stiffened plate. The most com-monly used geometric configuration is a cross-stiffened plate,as shown in Figs. 1 and 2.

In this paper, the adapted notations for describing themodels are the same as the notations used in the originalpapers (Schade 1941, 1951, 1951 & 1953); the long and shortaxes are denoted by a and b, respectively, and the distances

between the long and short stiffeners are denoted by Sa andSb, respectively. As required by the model, the stiffeners ineach direction are evenly spaced. Moreover, the axes are as-sumed orthogonal and thus entirely independent.

The boundary conditions for the plate are selected fromfour prescribed boundary conditions, as follows:

Case 1—All four edges are simply supported (rotations al-lowed at the edges).

Case 2—Both short edges are fixed, and both long edges aresimply supported.

Fig. 1 Cross-stiffened plate

NomenclatureOrthotropic plate model

a � length of long edge of plateb � length of short edge of plateE � Young’s Modulus of elasticityIa � moment of inertia of central

long stiffener includingeffective breadth of plating.

ia � moment of inertia of thestiffeners per unit width oflong axis

Ib � moment of inertia of centralshort stiffener includingeffective breadth ofplating.

ib � moment of inertia of thestiffeners per unit width ofshort axis

Ina � moment of inertia ofrepeating long stiffenersincluding effective breadthof plating.

Inb � moment of inertia ofrepeating short stiffenersincluding effective breadthof plating.

Ipa � moment of inertia of effectivebreadth of plating onlyworking with long stiffeners.

Ipb � moment of inertia of effectivebreadth of plating only,working with shortstiffeners.

K � dimensionless coefficientused to calculate themaximum static deflection

L � general notation of plate’sedges

Na � number of longitudinalstiffeners

Nb � number of transversestiffeners

P � uniform pressure acting onplate

S � general notation of stiffeners’spacing

Sa � spacing of longitudinalstiffeners

Sb � spacing of transversestiffeners

t � plate thicknessWmax � maximum deflection of plate

� � torsion coefficient� � effective breadth of plate� � Poisson’s ratio� � virtual side ratio

Grillage model

a � length of R beamsb � length of S beamsc � spacing of S beamsd � spacing of R beamsIr � moment of inertia of R beam

including effective breadthof plating equal to halfbeam spacing

Is � moment of inertia of S beamincluding effective breadthof plating equal to halfbeam spacing

Kr � edge clamping coefficient forR beams

Ks � edge clamping coefficient forS beams

P � uniformr � number of R beamss � number of S beams

w+ � central deflection

OCTOBER 2007 MARINE TECHNOLOGY 213

Case 3—Both long edges are fixed, and both short edges aresimply supported.

Case 4—All four edges are clamped (all six degrees of free-dom are fixed at the edges).

The calculation process begins with the evaluation of theeffective breadth of plating.

2.2.1. Step 1—Evaluating the effective breadth of plat-ing. The effective breadth of plating (�) is defined as thebreadth of plating that, when used in calculating the momentof inertia of the cross section, will give the correct maximumstress at the junction of the web and flange, using simplebeam theory. Table 1 gives the breadths for various aspectratios.

After calculating the effective breadths for both axes, thenext step is calculation of moment of inertia of the crosssection of the plate and stiffeners and calculating the unitstiffness for both axes.

It should be noted that for all the tested cases the resultswere insensitive to the value of the effective breadth. For adetailed explanation regarding the effective breadth concept,please refer back to the author’s original papers (Schade 1951& 1953).

2.2.2. Step 2—Calculating the unit stiffness in both di-rections. The following notations are used for the momentof inertia of plate and stiffeners:

Ina � moment of inertia of repeating long stiffeners, includ-ing effective breadth of plating.

Inb � moment of inertia of repeating short stiffeners, includ-ing effective breadth of plating.

Ia � moment of inertia of central long stiffener, includingeffective breadth of plating.

Ib � moment of inertia of central short stiffener, includingeffective breadth of plating.

Ipa � moment of inertia of effective breadth of plating only,working with long stiffeners.

Ipb � moment of inertia of effective breadth of plating only,working with short stiffeners.

Before calculating the unit stiffness for each axis, there isa need to obtain the moment of inertia of the cross sections.The unit stiffnesses are the moment of inertia of the stiffen-ers per unit width, and these parameters are given by thefollowing equations:

ia =Ina

Sa+ 2 � �Ia−Ina

b � ib =Inb

Sb+ 2 � �Ib−Inb

a � . (4)

If the central stiffener is similar to the repeating stiffeners(e.g., Ia � Ina), then these equations are reduced to the fol-lowing equations:

ia =Ina

Saib =

Inb

Sb. (5)

For an unstiffened plate the following equations are to beused:

ia = ib =t3

12 � �1 − �2�. (6)

Table 1 Effective breadth of stiffened panel (Schade 1951 & 1953)

L/S 0.5 1.0 2.0 4.0 6.0 8.0 10.0�/S for

uniformload 0.196 0.369 0.737 0.989 1.045 1.069 1.080

L � span of simply supported plate or distance between points ofzero bending moment for fixed ends (0.58 × the span for uniformlydistributed load). S � spacing of stiffeners

Fig. 2 Cross-stiffened plate and notations. L is a or b, and S is Sa or Sb.

214 OCTOBER 2007 MARINE TECHNOLOGY

For a plate with single stiffening (stiffeners in short directiononly), the following equations are valid:

ia = 0 ib =Inb

Sb. (7)

2.2.3. Step 3—Calculating the virtual side ratio andthe torsion coefficient. As presented previously, the vir-tual aspect ratio (�) is the actual plate ratio, a/b, modified bythe ratio of the unit stiffness in the two directions. The vir-tual side ratio is always equal to or greater than 1 and isgiven in the following definition:

� =ab�4 ib

ia. (8)

The torsion stiffness coefficient (�) accounts for horizontalshear stress in the plating and is defined roughly as the ratioof the inertia of the material subject to horizontal shearstress to the inertia of the material subject to bending.Schade provided the following expression:

� =�Ipa

Ina�

Ipb

Inb. (9)

In an unstiffened plate, all the material is subjected to bothhorizontal shear and bending and � � 1. In stiffened platestructures, mostly only the plating is subjected to horizontalshear, but both plating and stiffeners are subjected to bend-ing, so 0 < � < 1.

2.2.4. Step 4—Evaluating the dimensionless coeffi-cient K and the maximum static deflection at center ofplate. The last step is selection of dimensionless coefficientK (which depends on �, �, and the boundary condition), whichis used to calculate the maximum static deflection using thefollowing equation:

Wmax�static� = KP � b4

E � ib, (10)

where P is the uniform pressure acting on the plate and E ismodulus elasticity of the material of the plate and stiffeners.The value for K is selected from Fig. 3. The applicable geom-etries and the relevant equations are summarized in theTable 2. For this study only the first geometry was tested,namely, the cross-stiffening geometry.

2.3. Grillage model

The implementation of the grillage model is based on thepaper of Clarkson et al. (1959). A summary of their work ispresented in this section. Here, i, j refer to the ith beam of setR and jth beam of set S, respectively, and n, m refer to thenth beam of set R and mth beam of set S, respectively.

In the grillage model, the uniform pressure is considered tobe an equivalent concentrated load at each intersection of thebeams.

The solution method used in Clarkson et al. (1959) is thatof equating intersection point deflections in order to obtain aset of simultaneous equations in the statically indeterminatereactions at the intersection of the beams. The number ofsimultaneous equations is thus equal to the number of inter-sections. By applying the Euler-Bernoulli beam theory to theR beams, the deflection of intersection ij may be written asfollows:

wij = Br �m=1

1�2�s+1�

�P − Rim��mj, (11)

where:

�mj = ��3�4j�s + 1�2�s + 1 − �s + 1 − j�Kr�

− j3�s + 1��12�s + 1�4;m = 1�2�s + 1�

�6mj�s + 1 − m��s + 1 − �s + 1 − j�Kr�− 2j3�s + 1��12�s + 1�4;

m � 1�2�s + 1�,m j

�6mj�s + 1 − j��s + 1 − �s + 1 − m�Kr�− 2m3�s + 1��12�s + 1�4;

m � 1�2�s + 1�,m j

(12)

and

Br =a3

EIr, Kr = �0 simply supported

1 fixed supported.

In the same way, the deflection of the same intersection pointon the S beam is:

wij = Bs �n=1

1�2�r+1�

�P − Rnj��in , (13)

where Bs � a3/EIs and �in is given in equation (13) with r, i,n, and Ks in place of s, j, m, and Kr respectively.

By equating the two set of equations, the reactions Rij canbe obtained and the deflections in each intersection of thebeams can be calculated. For a more detailed presentation,please refer back to authors’ original paper (Clarkson et al.1959).

2.4. Calculation of maximum deflection

The grillage model requires fewer steps than the ortho-tropic plate model. Like the orthotropic plate model, it beginswith calculating the spacing of the stiffeners. Figure 2 showsa representative stiffened plate for the grillage model by re-placing a, b, Sa and Sb with b, a, c, and d, respectively. Thesteps for computing the maximum deflection of a stiffenedplate using the grillage model are as follows:

2.4.1. Calculating the spacing of the beams.

The spacing of the longitudinal stiffeners is c � a/(s + 1).The spacing of the transverse stiffeners is d � b/(r + 1).

s denotes the longitudinal stiffeners and r denotes the trans-verse stiffeners.

2.4.2. Calculating the moment of inertia of the beams.

Ir � Moment of inertia of r beams, including effective breadthof plating equal half beam spacing.

Is � Moment of inertia of s beam, including effective breadthof plating equal half beam spacing.

2.4.3. Calculating the independent stiffness ratio vari-able �.

� =�r + 1�b3Ir

�s + 1�a3Is

. (14)

2.4.4. Calculating the static m aximum deflection ofthe plate. Using the stiffness ratio variable calculated pre-viously and extracting the value corresponding to � from Fig.4, the maximum deflection is then calculated as follows:

W = �value from chart� ��s + 1�pcda3

EIr, (15)

where E is the modulus of elasticity of the material of theplate and stiffeners.


2.5. Error estimation of software predictions

In order to evaluate the accuracy of the two models, a totalof 400 different finite elements analyses have been tested;those tests were performed only for the similar geometricconfigurations of both models (since the orthotropic platemodel is more diverse) in order to use those results as refer-ence data for the comparison process. The stiffened plateswere modeled and solved with ADINA® Finite Element soft-ware using nine node shell elements. The following remarksshould be noted:

• Only cross-stiffened plates were checked, meaning atleast one stiffener was used per axis.

• Out of the 400 cases, only 320 were used for evalua-

tion. The other 80 were dropped due to significanterrors in their results (in most cases, the stiffeners weretoo stiff compared to the plate and the maximum de-flections were not at the center of plate but betweenthe stiffeners) or for not being applicable for both mod-els.

• Cases where the maximum deflections were not at thecenter but also did not cause high relative errors wereincluded in the acceptable cases for evaluations.

• For 290 cases, more than one stiffener was used in eachaxis. For 30 cases, one longitudinal stiffener was used,and these cases are not included in this study.

• Only odd numbers of stiffeners were checked.• For simplicity, all the stiffeners used were blade shaped

(rectangular cross section).

Fig. 3 Deflection at the center of a stiffened plate (Schade 1941)


Before detailing the results by boundary condition, all thecases were plotted as a bar graph. This graph is shown inFig. 5.

Generally speaking, both models give good results that canbe used for preliminary design and/or for verification of re-sults obtained by other techniques. For a cross-stiffenedplate, the grillage model is shown to be the better model, butfor geometric configurations (such as those depicted in Table2) that cannot be represented by the grillage mode, the or-thotropic plate can be used for some cases with acceptableaccuracy (as was checked by FEA out of the scope of thispaper).

2.6. Case studies of stiffened plates

Inspecting the number of independent parameters of themodels shows that there are quite a lot of combinations. Be-ginning with the plate itself, there are three independentparameters (breadth, length, thickness). While only oddnumbers of stiffeners are valid, any number of stiffeners canbe used for each axis. Each axis can also have different typesof stiffeners. For each stiffened plate, there are four bound-ary conditions.

Because there are, in fact, countless stiffened plates avail-able for testing it was decided to assemble all the case studiesfrom three plate parameters. The plate parameters were var-ied, and for each plate a different count of stiffeners weretested (3, 5, 7, and 9 stiffeners with different combinationsfor each axis). The distances between the stiffeners were inthe range of 0.5 to 1.5 m, but for most of the analyses thedistance was kept at 1 m. The tested plates were in the rangeof 2 to 8 m. Both symmetrical and asymmetric plates weretested, but neither one showed better results.

The plate thickness was varied in order to examine thesensitivity of the model to the plate thickness. This wastested on small plates (3 × 3 m, 4 × 4 m) and large plates (8× 8 m, 8 × 6 m). The thickness started at 2 up to 10 mm withincrements of 1 mm per analysis. In those cases when onlythe thickness varied there was a value, for both models, whenthe relative error fell to its lowest value and got higher as thethickness grew. For the tested cases, this value was in the

range of 4 to 6 mm, but this is not always so for every case asthis depends on the stiffener count and properties.

Since the only distinguishing parameter that was shown toaffect the relative errors in a clear manner, in this study, isthe boundary condition; breaking down the results perboundary condition is perhaps the appropriate way to displaythe results. This is discussed in section 7.

2.7. Boundary conditions comparison

The following figures plot the results for each boundarycondition. As previously presented, the four applicableboundary conditions are as follows:

Case 1—All four edges are simply supported (allows rota-tions at the edges).

Case 2—Both short edges fixed and both long edges simplysupported.

Case 3—Both long edges fixed and both short edges simplysupported.

Case 4—All four edges are clamped (all six degrees of free-dom are fixed at the edges).

In order not to clutter the figure, each boundary condition(BC) is plotted in a separate figure.

The first BC is plotted in Fig. 6. For a simple supportedplate, both models give good results (up to 12% of relativeerror). No model seems to be the favorable one, and eachmodel can be used with high degree of confidence in theirresults.

The second and third boundary conditions are plotted inFigs. 7 and 8. For those boundary conditions, the grillagemodel gives slightly better results. Although the grillagemodel is better for those boundary conditions, the orthotropicplate model (OPM) is not too bad either; the results for theOPM are centered in the 12 to 16% range (60% of the resultsare inside this interval) with Gaussian-like distributions forthe rest.

It is clear that the grillage model should be preferred whenall the plate edges are fixed/clamped (Fig. 9).

Table 2 Types of stiffening with applicable formulas

Cross StiffeningTransverse Stiffeners With

Central Longitudinal Stiffeners Transverse Stiffeners Only Unstiffened Plate

ia =Ina

Sa+ 2 � �Ia − Ina

b � ia = 2Ia

bia = 0; ib =

Inb

Sb

ib =Inb

Sb+ 2 � �Ib − Inb

a � ib =Inb

Sb+ 2 � �Ib − Inb

a � � = �ia = ib =

t3

12�1 − �3�

� =ab

��4 ib

ia� =

ab

��ib

ia

� � indeterminate, butthere is no need tocalculate � because allthe values of K aresimilar at � � �.

� =ab

� =�Ipa

Ina

Ipb

Inb� = 0.124 ��Ipb�2

IaInb

bSa

� = 1.0


While the grillage model shows better results, its geomet-ric configurations are limited. The plate requires at least onestiffener in each axis, and when more than one stiffener isused the central stiffener must be the same as the noncentralstiffeners (while the OPM allows different central stiffenerand some other geometric configuration as specified inTable 2).

3. Part 2: Implementation of models and briefsoftware presentation

3.1. Implementation

This part presents the computerization and implementa-tion of the models as a computer program for quick estima-tion of the maximum deflection of stiffened plates/panels. Inorder to computerize the models, the design curves of eachmodel had to be converted in a meaningful way, that is, onewith which computers can work. While the obvious way to doso was to perform curve fitting to points on the curves, dif-ferent interpolation methods were needed for each model.

3.2. Computerization of the orthotropic plate model

Since most of the model consists of simple equations, theentire computerization process boils down to computerization

of only two parameters: effective breadths (Table 1) andcurves for K values (Fig. 3).

The discrete values of Table 1 can be connected usingfourth-order polynomial, as presented in equation (16) andplotted in Fig. 10.

�

S= −0.0004 � �L

S�4

+ 0.0112 � �LS�3

− 0.1243 � �LS�2

+ 0.6149 � �LS�1

− 0.0991. (16)

As can be seen, the polynomial fits quite well to the discretevalues, and it is very adequate for our purpose.

Since the curves of Fig. 3 do not represent physical prop-erties or natural behavior, the only need from the interpola-tions is to be able to follow the original curves as closely aspossible. The interpolation data were acquired by handpick-ing points on the curves. The number of points taken were insufficient quantities so the new curves would, in theory, tracethe original curves with a high degree of accuracy; by super-imposing the interpolated curves on top the original curves ofFig. 3, it can be shown that this is indeed the case with onlyslight deviation in a few places.

Fig. 4 Deflection at the center of a stiffened plate (Clarkson et al. 1959)


Fig. 6 Results of first boundary condition

Fig. 5 Results of entire case studies


Fig. 7 Results of second boundary condition

Fig. 8 Results of third boundary condition


After developing the new curves, the computation processis completed. The polynomial of Fig. 10 is used for calculationof the moment of inertia of plate, and the polynomials of Fig.3 are used for extracting the value of the coefficient K for themaximum deflection equation [equation (10)].

3.3. Computerization of the grillage model

While implementation of the orthotropic plate model, usingup to sixth-order interpolation polynomials, was quite goodenough with respect to the original curves, that was not thecase with the grillage model. Since the curves of the grillagemode are presented in a log scale, a curve fit of polynomial

type for the entire data is not adequate, so a different type ofinterpolation was needed.

As with the orthotropic plate model, the interpolated datawas chosen manually from the design curves themselves, andthose points were interpolated using cubic spline interpola-tions for each interval between two adjacent points.

3.4. Program presentation

The software was written using Visual Basic .NET pro-gramming language. This section gives a brief overview ofthe software.

Fig. 9 Results of last boundary condition

Fig. 10 Effective breadth of stiffened plates


In general, the computation process is composed of fourmajor steps:

Step 1—Defining the plate’s parameters (length, breadth andthickness)

Step 2—Selecting the geometric configuration (for the ortho-tropic plate model only)

Step 3—Defining the stiffeners’ properties (this step takesmost of the evaluation time)

Step 4—Defining the uniform pressure acting on the plate,the boundary conditions, and the material properties of theplate and stiffeners (Young’s modulus and Poisson’s ratio).

Once the data are entered entirely, the user gets the evalu-ated maximum deflection as well as the error estimation forthe selected problem (geometric configuration and boundarycondition). As in most engineering tools, the metric unit sys-tem (SI system) is used here.

The use of the computer code is demonstrated using thefollowing example:

Plate dimensions:

• Length: 8 m• Breadth: 4 m• Thickness: 6 mm

Stiffeners:

• Number of longitudinal stiffeners: Na � 5• Number of transverse stiffeners: Nb � 7

Both axes have the same stiffeners, and the central stiffeneris similar to the noncentral stiffeners. The stiffeners areblade type with the following dimension: height: 80 mm,thickness: 8 m.

The uniform pressure is taken as 1,000 Pa, the modulus ofelasticity of the material is 207 GPa, Poisson’s ratio is 0.3,and the plate is simply supported at all four edges.

The first steps are straightforward; The user inputs the plateparameters and selects the geometric configuration if needed.The most time-consuming step is defining the stiffeners’properties and types. The user needs to define the number ofstiffeners in each axis, to select the desired stiffeners, and todefine their properties using three available options:

Select from a predefined database that contains all the com-monly used beams and cross sections

Enter the exact numeric values of the stiffeners’ properties(e.g., moment of inertia of cross section, total area of crosssection, and the centroid of cross section)

Define the geometric parameters as shown in Fig. 11.

The first item to do at step 3 in this example is to set thenumber of stiffeners for each axis and to check all the stiff-eners’ checkboxes (since all the stiffeners are the same), thento choose the third option from the three methods availableand to enter the stiffeners parameters as can be seen inFig. 11.

Once the desired values are inputted, the “X” mark neareach type of stiffener is replaced with a check mark.

The last step is shown in Fig. 12. Selecting the “finish”button completes the process. The user than gets a summa-rization window that shows all the input data; this is notshown here.

Once the data are confirmed, selecting the “Maximum De-flection Result” icon from the list on the left gives the user theresult of the problem he or she wanted to check. Additionally,the user gets a text message, which specifies the upper errorbound that was estimated using comparison of many models’

Fig. 11 Step three


results by finite element analysis and a figure that shows theerror distributions for the specific boundary condition prob-lem (Fig. 13). Other features included in this screen are theability to save all the calculations of the problem and, more

important, the ability to create the same geometric modeland settings (pressure, boundary conditions, physical prop-erties, etc.) for ADINA finite element software as a Sessionfile to run from the ADINA GUI.

Fig. 12 Step four

Fig. 13 Result of the desired problem and error distribution


The result of our problem, according to the developed soft-ware using the orthotropic plate model, is 12.6 mm, while thegrillage model gives a value of 13.18 mm.

The FEA result of the same problem is presented in Fig. 14.The problem was solved using ADINA finite element soft-ware. Both the plate and the stiffeners were represented bynine node shell elements.

As presented previously, the maximum displacement ob-tained with the FEA is 13.19 mm, which is higher by 6.6%than the result of the orthotropic plate model and smaller by2.2% than the grillage model.

While in real physical structure, the displacement (and/orfailure of the structure) can be caused by both lateral dis-placement and buckling of the structure’s elements (stiffen-ers only, stiffened plate/panel only, stiffeners and plate, etc),both models neglected the additional displacement due tobuckling and assumed the structure kept its original geom-etry, beside being displaced in the lateral direction. The FEanalyses made the same omissions and ignored the bucklingof the structure.

4. Concluding remarks

This paper presents the implementation and evaluation ofthe most common models used for analysis of stiffened plates/panels. While additional work could and should be done (forexample, testing the model for stress evaluation), the evalu-ation of the maximum lateral deflection due to uniform pres-sure can be considered to be complete.

The research performed in this paper assumes that themaximum deflection occurs at the center of the plate (or atleast near it). If the stiffeners are too stiff compared with theplate, the maximum deflection would, most likely, be be-tween the stiffeners, and the models could be considerablywrong (in fact, some of those cases were added, nevertheless,to the results in Figs. 6 to 9). A study of stiffeners-plate ratiofor which the deflection occurs at the center has not beenperformed yet. The user should have a general feeling for thestiffened plate he or she is about to test using those models.

Each model has its own limitations and benefits. The or-thotropic plate model is highly diverse with respect to theapplicable geometric configurations it can work with. Whilethis paper presented only results of geometric configurationsthat both models can work with (cross-stiffened plates), it

should be noted that all other configurations were also testedbut with a lower number of FEA.

Some rough remarks can be made for those configurations:

• For small plates (when the plate edges were 2 m andbelow, for example 2 × 1 m plate), the relative errors forthe two first boundary conditions were up to 20%, andthe software predictions were smaller than the FEA re-sults. For the last two boundary conditions, the errorswere significantly higher in the area of 35% and there-fore should not be used for deflection estimations at all.Additionally, a small number of stiffeners were used (1,3, 5, 7).

• When the plate length was greater (e.g., 6 × 3 m), theerrors for the two first boundary conditions were up to10% relative error and for the last two boundary condi-tion were around 15%. For those plates, nine stiffenertests were added.

• Because a small number of analyses were performed forthose configurations, additional tests should be made inorder to validate these results.

For cross-section plates, the following remarks can bemade with a high degree of confidence:

• For the boundary condition of case 4 (all edges are fixed/clamped), the orthotropic plate model should not be usedat all. The relative errors of 30 to 35% are just too highto be accepted. Since more than 90% of the results areconcentrated in that range, a correction factor might beconsidered for incorporating in future versions of thesoftware.

• For the boundary condition of case 3, the orthotropicplate model has Gaussian-like distribution centered at12 to 16%. For many results of the higher percentage,the maximum deflection did not occur at the center, sowhen the stiffeners are not too stiff compared to theplate the result would most likely be less than 20%.

• For the first boundary condition, the model can be usedfor all types of cross-stiffened plates when the plate isreasonably configured (meaning, a reasonable combina-tion of stiffeners-plate is selected).

• The grillage is limited to only cross-stiffened plates inwhich the central stiffener is equal to the other stiffen-ers in each axis (the central stiffener, for example, can-

Fig. 14 Finite element analysis displacement solution of desired problem


not be heavier to reduce the deflection at the center,while the OPM allows it). For this limitation, the modelcompensates by being more accurate: for all boundaryconditions the model gave better results. For the firsttwo BCs the results were up to 16% relative error, whichfor a preliminary design tool is a great value. For the lastboundary conditions (cases 3 and 4), the relative errorsare less than great. However, since they are distributedat the entire range and many FEA agreed quite wellwith the software predictions, the model can still be usedwith acceptable accuracy, especially if one is to drop thefew analyses at the 25 to 30% mark, in which the deflec-tions were not at the center of the plate (which bring usback to relative errors of up to 20%).

Conclusively, for reasonably configured plates the grillagemodel, when applicable, should be preferred over the ortho-tropic plate model. When the stiffeners are not too stiff withrespect to the plate itself, the relative errors would mostlikely be in an acceptable range of up to 20% at most. Whena different type of stiffened plates is needed (e.g., when thecentral stiffener is heavier/bigger) the orthotropic platemodel is to be used, also with acceptable results when theplate is not fixed or clamped.

On a final note, an important remark regarding the soft-ware should be made. The software can save a great deal of

time for the design process of stiffened plates. It can be usedto generate ADINA® models (with applied loads and bound-ary conditions) for many purposes other than deflectionevaluations. In the example above, the results of the modelswere obtained in less than a minute time (including the gen-eration of the ADINA model). The FE method, on the otherhand, required significant time and knowledge; That timemust be spent on building the geometry model, making edu-cated decisions regarding the meshing and the suitable ele-ments, applying boundary conditions and desired materials,and some other FE-related issues. Finally, additional com-puter time is required for solving the problem.

ReferencesBANAI, L. AND PEDATZUR, O. 2006 Computer implementation of the ortho-

tropic plate model for quick estimation of the maximum deflection ofstiffened plates, Ships and Offshore Structures, 1, 4, 323–333

CLARKSON, J., WILSON, L. B., AND MCKEEMAN, J. L. 1959 Data sheets forthe elastic design of flat grillages under uniform pressure, EuropeanShipbuilding, 8, 174–198.

SCHADE, H. A. 1940 The orthogonally stiffened plate under uniform lat-eral load, Journal of Applied Mechanics, 7, 4, 139–158.

SCHADE, H. A. 1941 Design curves for cross-stiffened plating, Trans.SNAME, 49, 154–182.

SCHADE, H. A. 1951 The effective breadth of stiffened plating under bend-ing loads, Trans. SNAME, 59, 127–141.

SCHADE, H. A. 1951 & 1953 The effective breadth concept in ship struc-ture design, Trans. SNAME.


An Evaluation and Comparison of Models for Maximum Deflection of Stiffened Plates Using Finite...

Documents

Transcript of An Evaluation and Comparison of Models for Maximum Deflection of Stiffened Plates Using Finite...