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

Marine Technology, Vol. 44, No. 4, October 2007, pp. 212 225 An Evaluation and Comparison of Models for Maximum Deflection of Stiffened Plates Using Finite Element Analysis Lior Banai 1 and Omri Pedatzur 2 Stiffened plates form the backbone of most of a ship s structure. Today, finite element (FE) models are used to analyze the behavior of such structural elements for different types of loads. In the past, when usage 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 grillage model. 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 the finite element method, and finally implement the methods as a computer program for quick estimations of the maximum deflection of stiffened plates. The degree of accuracy of the two methods when compared to FE is discussed in some detail. 1. Introduction THE MAIN TYPE of framing system found in ships nowadays is a longitudinal one that has stiffeners running in two orthogonal directions. Structures of this kind are usually called cross-stiffened plates/panels, or grillages. Deck and bottom structures of this type are reinforced mainly in the longitudinal direction, with widely spaced, heavier, transverse stiffeners, as shown in Fig. 1. The stiffeners used in the longitudinal 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 rapid growth in shipbuilding and deployment of ships, the need arose for a way to evaluate the response of the ships structure to different types of sea loads. At that time, the use of computers was limited. Numerical methods, such as finite element analysis (FEA) and other variational techniques made their first steps into the world of engineering, and computation time was unacceptably long. Having these limitations in mind, many engineers and scientists sought a way to bypass the numerical difficulty by developing analytical methods for the analysis of shell structures (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 during the 1940s, enables the evaluation of the maximum static deflection of a stiffened plate subjected to a uniform lateral pressure on its surface. The related work described in this paper is based on a number of papers (Schade 1940, 1941, 1951). The second method is based on the grillage model as developed by Clarkson et al. (1959). 2. Part 1: Summaries and evaluations of models 2.1. The orthotropic plate model Summary of theory 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 1 The School of Mechanical Engineering, Tel Aviv University, Tel Aviv, Israel. 2 The Naval Architecture and Marine Engineering Department, Israeli Navy, Israel. Manuscript received at SNAME headquarters July 2006. plate, which is commonly regarded as isotropic, can exhibit different elastic behavior in different directions. An orthotropic plate is one that has different elastic properties in three (or two) orthogonal directions, and those properties are uniform within each direction. The fundamental equation of the static response of an orthotropic plate can be written as follows: 4 w D x x + 2H 4 w 4 x 2 y + D 4 w 2 y = p x, y (1) 4 y where D x and D y are bending stiffnesses in the x and y directions, respectively. H is function of the bending stiffnesses and the torsional rigidity (D xy ). p(x,y) is the load per unit width acting on the plate. The determination of the bending stiffnesses is relativity straightforward, but determination of the torsional rigidity is not as easy. The value of H can be determined theoretically or by experiment. While some authors have given expressions for the coefficient H that accounts for shear deflection, Schade s work itself does not include that deflection component, which for instance may be relevant in a stiffened panel due to the effect of shear distortion of the webs. The effect of shear deflection is normally a reduction in the torsional rigidity D xy. The earliest solution of equation (1) seems to have been that of Huber, who showed that the solution involved two nondimensional parameters that are functions of the plate aspect ratio and the rigidity parameters D x and D y. Those parameters are designated by the letters and : H = = a D y. (2) 2 D x D b 4 y D x Schade has developed an approximate e xpression for the coefficient H, and using that he also obtained the following expression for : = I pa I na I pb I nb (3) where I pa (I pb ) is the moment of inertia of plating only, working with long (short) stiffeners and I na (I nb ) is the moment of inertia of repeating long stiffeners. 212 OCTOBER 2007 0025-3316/07/4404-021200.00/0 MARINE TECHNOLOGY

Fig. 1 Cross-stiffened plate 2.2. Calculation of maximum deflection The Schade method is composed of charts and equations, which, when followed in sequential steps, give the maximum deflection at the center of a stiffened plate. The most commonly used geometric configuration is a cross-stiffened plate, as shown in Figs. 1 and 2. In this paper, the adapted notations for describing the models are the same as the notations used in the original papers (Schade 1941, 1951, 1951 & 1953); the long and short axes are denoted by a and b, respectively, and the distances between the long and short stiffeners are denoted by S a and S b, respectively. As required by the model, the stiffeners in each direction are evenly spaced. Moreover, the axes are assumed orthogonal and thus entirely independent. The boundary conditions for the plate are selected from four prescribed boundary conditions, as follows: Case 1 All four edges are simply supported (rotations allowed at the edges). Case 2 Both short edges are fixed, and both long edges are simply supported. Orthotropic plate model a length of long edge of plate b length of short edge of plate E Young s Modulus of elasticity I a moment of inertia of central long stiffener including effective breadth of plating. i a moment of inertia of the stiffeners per unit width of long axis I b moment of inertia of central short stiffener including effective breadth of plating. i b moment of inertia of the stiffeners per unit width of short axis I na moment of inertia of repeating long stiffeners including effective breadth of plating. I nb moment of inertia of repeating short stiffeners including effective breadth of plating. Nomenclature I pa moment of inertia of effective breadth of plating only working with long stiffeners. I pb moment of inertia of effective breadth of plating only, working with short stiffeners. K dimensionless coefficient used to calculate the maximum static deflection L general notation of plate s edges N a number of longitudinal stiffeners N b number of transverse stiffeners P uniform pressure acting on plate S general notation of stiffeners spacing S a spacing of longitudinal stiffeners S b spacing of transverse stiffeners t plate thickness W max maximum deflection of plate torsion coefficient effective breadth of plate Poisson s ratio virtual side ratio Grillage model a length of R beams b length of S beams c spacing of S beams d spacing of R beams I r moment of inertia of R beam including effective breadth of plating equal to half beam spacing I s moment of inertia of S beam including effective breadth of plating equal to half beam spacing K r edge clamping coefficient for R beams K s edge clamping coefficient for S beams P uniform r number of R beams s number of S beams w+ central deflection OCTOBER 2007 MARINE TECHNOLOGY 213

Fig. 2 Cross-stiffened plate and notations. L is a or b, and S is S a or S b. Case 3 Both long edges are fixed, and both short edges are simply supported. Case 4 All four edges are clamped (all six degrees of freedom are fixed at the edges). The calculation process begins with the evaluation of the effective breadth of plating. 2.2.1. Step 1 Evaluating the effective breadth of plating. The effective breadth of plating ( ) is defined as the breadth of plating that, when used in calculating the moment of inertia of the cross section, will give the correct maximum stress at the junction of the web and flange, using simple beam theory. Table 1 gives the breadths for various aspect ratios. After calculating the effective breadths for both axes, the next step is calculation of moment of inertia of the cross section of the plate and stiffeners and calculating the unit stiffness for both axes. It should be noted that for all the tested cases the results were insensitive to the value of the effective breadth. For a detailed 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 directions. The following notations are used for the moment of inertia of plate and stiffeners: I na moment of inertia of repeating long stiffeners, including effective breadth of plating. I nb moment of inertia of repeating short stiffeners, including effective breadth of plating. I a moment of inertia of central long stiffener, including effective breadth of plating. I b moment of inertia of central short stiffener, including effective breadth of plating. 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 uniform load 0.196 0.369 0.737 0.989 1.045 1.069 1.080 L span of simply supported plate or distance between points of zero bending moment for fixed ends (0.58 the span for uniformly distributed load). S spacing of stiffeners I pa moment of inertia of effective breadth of plating only, working with long stiffeners. I pb moment of inertia of effective breadth of plating only, working with short stiffeners. Before calculating the unit stiffness for each axis, there is a need to obtain the moment of inertia of the cross sections. The unit stiffnesses are the moment of inertia of the stiffeners per unit width, and these parameters are given by the following equations: i a = I na S a + 2 I a I na b i b = I nb S b + 2 I b I nb a. (4) If the central stiffener is similar to the repeating stiffeners (e.g., I a I na ), then these equations are reduced to the following equations: i a = I na S a i b = I nb S b. (5) For an unstiffened plate the following equations are to be used: t 3 i a = i b = 12 1 2. (6) 214 OCTOBER 2007 MARINE TECHNOLOGY

For a plate with single stiffening (stiffeners in short direction only), the following equations are valid: i a = 0 i b = I nb S b. (7) 2.2.3. Step 3 Calculating the virtual side ratio and the torsion coefficient. As presented previously, the virtual aspect ratio ( ) is the actual plate ratio, a/b, modified by the ratio of the unit stiffness in the two directions. The virtual side ratio is always equal to or greater than 1 and is given in the following definition: = a i b. (8) b 4 i a The torsion stiffness coefficient ( ) accounts for horizontal shear stress in the plating and is defined roughly as the ratio of the inertia of the material subject to horizontal shear stress to the inertia of the material subject to bending. Schade provided the following expression: = I pa I pb. (9) I na I nb In an unstiffened plate, all the material is subjected to both horizontal shear and bending and 1. In stiffened plate structures, mostly only the plating is subjected to horizontal shear, but both plating and stiffeners are subjected to bending, so 0 < <1. 2.2.4. Step 4 Evaluating the dimensionless coefficient K and the maximum static deflection at center of plate. The last step is selection of dimensionless coefficient K (which depends on,, and the boundary condition), which is used to calculate the maximum static deflection using the following equation: P b4 W max static = K, (10) E i b where P is the uniform pressure acting on the plate and E is modulus elasticity of the material of the plate and stiffeners. The value for K is selected from Fig. 3. The applicable geometries and the relevant equations are summarized in the Table 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 the paper of Clarkson et al. (1959). A summary of their work is presented in this section. Here, i, j refer to the ith beam of set R and jth beam of set S, respectively, and n, m refer to the nth beam of set R and mth beam of set S, respectively. In the grillage model, the uniform pressure is considered to be an equivalent concentrated load at each intersection of the beams. The solution method used in Clarkson et al. (1959) is that of equating intersection point deflections in order to obtain a set of simultaneous equations in the statically indeterminate reactions at the intersection of the beams. The number of simultaneous equations is thus equal to the number of intersections. By applying the Euler-Bernoulli beam theory to the R beams, the deflection of intersection ij may be written as follows: 1 2 s+1 w ij = B r m=1 P R im mj, (11) where: + 1 mj = 3 4j s 2 s + 1 s + 1 j K r j 3 s + 1 12 s + 1 4 ; 6mj s + 1 m s + 1 s + 1 j K r 2j 3 s + 1 12 s + 1 4 ; and 6mj s + 1 j s + 1 s + 1 m K r 2m 3 s + 1 12 s + 1 4 ; B r = a3 EI r, K r = 0 simply supported 1 fixed supported. m = 1 2 s + 1 m 1 2 s + 1, m j m 1 2 s + 1, m j (12) In the same way, the deflection of the same intersection point on the S beam is: 1 2 r+1 w ij = B s n=1 P R nj in, (13) where B s a 3 /EI s and in is given in equation (13) with r, i, n, and K s in place of s, j, m, and K r respectively. By equating the two set of equations, the reactions R ij can be obtained and the deflections in each intersection of the beams 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 orthotropic plate model. Like the orthotropic plate model, it begins with calculating the spacing of the stiffeners. Figure 2 shows a representative stiffened plate for the grillage model by replacing a, b, S a and S b with b, a, c, and d, respectively. The steps for computing the maximum deflection of a stiffened plate 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 transverse stiffeners. 2.4.2. Calculating the moment of inertia of the beams. I r Moment of inertia of r beams, including effective breadth of plating equal half beam spacing. I s Moment of inertia of s beam, including effective breadth of plating equal half beam spacing. 2.4.3. Calculating the independent stiffness ratio variable. = r + 1 b3 I r s + 1 a 3 I s. (14) 2.4.4. Calculating the static m aximum deflection of the plate. Using the stiffness ratio variable calculated previously and extracting the value corresponding to from Fig. 4, the maximum deflection is then calculated as follows: s + 1 pcda3 W = value from chart, (15) EI r where E is the modulus of elasticity of the material of the plate and stiffeners. OCTOBER 2007 MARINE TECHNOLOGY 215

Fig. 3 Deflection at the center of a stiffened plate (Schade 1941) 2.5. Error estimation of software predictions In order to evaluate the accuracy of the two models, a total of 400 different finite elements analyses have been tested; those tests were performed only for the similar geometric configurations of both models (since the orthotropic plate model is more diverse) in order to use those results as reference data for the comparison process. The stiffened plates were modeled and solved with ADINA Finite Element software using nine node shell elements. The following remarks should be noted: Only cross-stiffened plates were checked, meaning at least one stiffener was used per axis. Out of the 400 cases, only 320 were used for evaluation. The other 80 were dropped due to significant errors in their results (in most cases, the stiffeners were too stiff compared to the plate and the maximum deflections were not at the center of plate but between the stiffeners) or for not being applicable for both models. Cases where the maximum deflections were not at the center but also did not cause high relative errors were included in the acceptable cases for evaluations. For 290 cases, more than one stiffener was used in each axis. 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). 216 OCTOBER 2007 MARINE TECHNOLOGY

Table 2 Types of stiffening with applicable formulas Cross Stiffening Transverse Stiffeners With Central Longitudinal Stiffeners Transverse Stiffeners Only Unstiffened Plate i a = I na + 2 S a I a I na b i a = 2 I a b i a = 0; i b = I nb S b + 2 I b I nb a i b = I nb S b + 2 I b I nb a = = a b 4 i b i a = a b i b i a = I pa I na I pb I nb = 0.124 I pb 2 I a I nb i b = I nb S b indeterminate, but there is no need to = a b calculate because all the values of K are b = 1.0 similar at. S a t 3 i a = i b = 12 1 3 Before detailing the results by boundary condition, all the cases were plotted as a bar graph. This graph is shown in Fig. 5. Generally speaking, both models give good results that can be used for preliminary design and/or for verification of results obtained by other techniques. For a cross-stiffened plate, the grillage model is shown to be the better model, but for geometric configurations (such as those depicted in Table 2) that cannot be represented by the grillage mode, the orthotropic plate can be used for some cases with acceptable accuracy (as was checked by FEA out of the scope of this paper). 2.6. Case studies of stiffened plates Inspecting the number of independent parameters of the models shows that there are quite a lot of combinations. Beginning with the plate itself, there are three independent parameters (breadth, length, thickness). While only odd numbers of stiffeners are valid, any number of stiffeners can be used for each axis. Each axis can also have different types of stiffeners. For each stiffened plate, there are four boundary conditions. Because there are, in fact, countless stiffened plates available for testing it was decided to assemble all the case studies from three plate parameters. The plate parameters were varied, and for each plate a different count of stiffeners were tested (3, 5, 7, and 9 stiffeners with different combinations for each axis). The distances between the stiffeners were in the range of 0.5 to 1.5 m, but for most of the analyses the distance was kept at 1 m. The tested plates were in the range of 2 to 8 m. Both symmetrical and asymmetric plates were tested, but neither one showed better results. The plate thickness was varied in order to examine the sensitivity of the model to the plate thickness. This was tested on small plates (3 3m,4 4 m) and large plates (8 8m,8 6 m). The thickness started at 2 up to 10 mm with increments of 1 mm per analysis. In those cases when only the thickness varied there was a value, for both models, when the relative error fell to its lowest value and got higher as the thickness 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 as this depends on the stiffener count and properties. Since the only distinguishing parameter that was shown to affect the relative errors in a clear manner, in this study, is the boundary condition; breaking down the results per boundary condition is perhaps the appropriate way to display the results. This is discussed in section 7. 2.7. Boundary conditions comparison The following figures plot the results for each boundary condition. As previously presented, the four applicable boundary conditions are as follows: Case 1 All four edges are simply supported (allows rotations at the edges). Case 2 Both short edges fixed and both long edges simply supported. Case 3 Both long edges fixed and both short edges simply supported. Case 4 All four edges are clamped (all six degrees of freedom 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 supported plate, both models give good results (up to 12% of relative error). No model seems to be the favorable one, and each model can be used with high degree of confidence in their results. The second and third boundary conditions are plotted in Figs. 7 and 8. For those boundary conditions, the grillage model gives slightly better results. Although the grillage model is better for those boundary conditions, the orthotropic plate model (OPM) is not too bad either; the results for the OPM are centered in the 12 to 16% range (60% of the results are inside this interval) with Gaussian-like distributions for the rest. It is clear that the grillage model should be preferred when all the plate edges are fixed/clamped (Fig. 9). OCTOBER 2007 MARINE TECHNOLOGY 217

Fig. 4 Deflection at the center of a stiffened plate (Clarkson et al. 1959) While the grillage model shows better results, its geometric configurations are limited. The plate requires at least one stiffener in each axis, and when more than one stiffener is used the central stiffener must be the same as the noncentral stiffeners (while the OPM allows different central stiffener and some other geometric configuration as specified in Table 2). 3. Part 2: Implementation of models and brief software presentation 3.1. Implementation This part presents the computerization and implementation of the models as a computer program for quick estimation of the maximum deflection of stiffened plates/panels. In order to computerize the models, the design curves of each model had to be converted in a meaningful way, that is, one with which computers can work. While the obvious way to do so was to perform curve fitting to points on the curves, different interpolation methods were needed for each model. 3.2. Computerization of the orthotropic plate model Since most of the model consists of simple equations, the entire computerization process boils down to computerization of only two parameters: effective breadths (Table 1) and curves for K values (Fig. 3). The discrete values of Table 1 can be connected using fourth-order polynomial, as presented in equation (16) and plotted in Fig. 10. S S = 0.0004 L 4 + 0.0112 L S 3 0.1243 L S 2 + 0.6149 L S 1 0.0991. (16) As can be seen, the polynomial fits quite well to the discrete values, and it is very adequate for our purpose. Since the curves of Fig. 3 do not represent physical properties or natural behavior, the only need from the interpolations is to be able to follow the original curves as closely as possible. The interpolation data were acquired by handpicking points on the curves. The number of points taken were in sufficient quantities so the new curves would, in theory, trace the original curves with a high degree of accuracy; by superimposing the interpolated curves on top the original curves of Fig. 3, it can be shown that this is indeed the case with only slight deviation in a few places. 218 OCTOBER 2007 MARINE TECHNOLOGY

Fig. 5 Results of entire case studies Fig. 6 Results of first boundary condition OCTOBER 2007 MARINE TECHNOLOGY 219

Fig. 7 Results of second boundary condition Fig. 8 Results of third boundary condition 220 OCTOBER 2007 MARINE TECHNOLOGY

Fig. 9 Results of last boundary condition Fig. 10 Effective breadth of stiffened plates After developing the new curves, the computation process is completed. The polynomial of Fig. 10 is used for calculation of the moment of inertia of plate, and the polynomials of Fig. 3 are used for extracting the value of the coefficient K for the maximum deflection equation [equation (10)]. 3.3. Computerization of the grillage model While implementation of the orthotropic plate model, using up to sixth-order interpolation polynomials, was quite good enough with respect to the original curves, that was not the case with the grillage model. Since the curves of the grillage mode are presented in a log scale, a curve fit of polynomial type for the entire data is not adequate, so a different type of interpolation was needed. As with the orthotropic plate model, the interpolated data was chosen manually from the design curves themselves, and those points were interpolated using cubic spline interpolations for each interval between two adjacent points. 3.4. Program presentation The software was written using Visual Basic.NET programming language. This section gives a brief overview of the software. OCTOBER 2007 MARINE TECHNOLOGY 221

In general, the computation process is composed of four major steps: Step 1 Defining the plate s parameters (length, breadth and thickness) Step 2 Selecting the geometric configuration (for the orthotropic plate model only) Step 3 Defining the stiffeners properties (this step takes most of the evaluation time) Step 4 Defining the uniform pressure acting on the plate, the boundary conditions, and the material properties of the plate and stiffeners (Young s modulus and Poisson s ratio). Once the data are entered entirely, the user gets the evaluated maximum deflection as well as the error estimation for the selected problem (geometric configuration and boundary condition). As in most engineering tools, the metric unit system (SI system) is used here. The use of the computer code is demonstrated using the following example: Plate dimensions: Length: 8 m Breadth: 4 m Thickness: 6 mm Stiffeners: Number of longitudinal stiffeners: N a 5 Number of transverse stiffeners: N b 7 Both axes have the same stiffeners, and the central stiffener is similar to the noncentral stiffeners. The stiffeners are blade type with the following dimension: height: 80 mm, thickness: 8 m. The uniform pressure is taken as 1,000 Pa, the modulus of elasticity 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 plate parameters 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 of stiffeners in each axis, to select the desired stiffeners, and to define their properties using three available options: Select from a predefined database that contains all the commonly 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 cross section, 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 the number of stiffeners for each axis and to check all the stiffeners checkboxes (since all the stiffeners are the same), then to choose the third option from the three methods available and to enter the stiffeners parameters as can be seen in Fig. 11. Once the desired values are inputted, the X mark near each 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 summarization window that shows all the input data; this is not shown here. Once the data are confirmed, selecting the Maximum Deflection Result icon from the list on the left gives the user the result of the problem he or she wanted to check. Additionally, the user gets a text message, which specifies the upper error bound that was estimated using comparison of many models Fig. 11 Step three 222 OCTOBER 2007 MARINE TECHNOLOGY

Fig. 12 Step four results by finite element analysis and a figure that shows the error distributions for the specific boundary condition problem (Fig. 13). Other features included in this screen are the ability to save all the calculations of the problem and, more important, the ability to create the same geometric model and settings (pressure, boundary conditions, physical properties, etc.) for ADINA finite element software as a Session file to run from the ADINA GUI. Fig. 13 Result of the desired problem and error distribution OCTOBER 2007 MARINE TECHNOLOGY 223

The result of our problem, according to the developed software using the orthotropic plate model, is 12.6 mm, while the grillage 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 software. Both the plate and the stiffeners were represented by nine node shell elements. As presented previously, the maximum displacement obtained with the FEA is 13.19 mm, which is higher by 6.6% than the result of the orthotropic plate model and smaller by 2.2% than the grillage model. While in real physical structure, the displacement (and/or failure of the structure) can be caused by both lateral displacement and buckling of the structure s elements (stiffeners only, stiffened plate/panel only, stiffeners and plate, etc), both models neglected the additional displacement due to buckling and assumed the structure kept its original geometry, beside being displaced in the lateral direction. The FE analyses made the same omissions and ignored the buckling of the structure. 4. Concluding remarks This paper presents the implementation and evaluation of the most common models used for analysis of stiffened plates/ panels. While additional work could and should be done (for example, testing the model for stress evaluation), the evaluation of the maximum lateral deflection due to uniform pressure can be considered to be complete. The research performed in this paper assumes that the maximum deflection occurs at the center of the plate (or at least near it). If the stiffeners are too stiff compared with the plate, the maximum deflection would, most likely, be between the stiffeners, and the models could be considerably wrong (in fact, some of those cases were added, nevertheless, to the results in Figs. 6 to 9). A study of stiffeners-plate ratio for which the deflection occurs at the center has not been performed yet. The user should have a general feeling for the stiffened plate he or she is about to test using those models. Each model has its own limitations and benefits. The orthotropic plate model is highly diverse with respect to the applicable geometric configurations it can work with. While this paper presented only results of geometric configurations that both models can work with (cross-stiffened plates), it should be noted that all other configurations were also tested but 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 and below, for example 2 1 m plate), the relative errors for the two first boundary conditions were up to 20%, and the software predictions were smaller than the FEA results. For the last two boundary conditions, the errors were significantly higher in the area of 35% and therefore 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), the errors for the two first boundary conditions were up to 10% relative error and for the last two boundary condition were around 15%. For those plates, nine stiffener tests were added. Because a small number of analyses were performed for those configurations, additional tests should be made in order to validate these results. For cross-section plates, the following remarks can be made 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 used at all. The relative errors of 30 to 35% are just too high to be accepted. Since more than 90% of the results are concentrated in that range, a correction factor might be considered for incorporating in future versions of the software. For the boundary condition of case 3, the orthotropic plate model has Gaussian-like distribution centered at 12 to 16%. For many results of the higher percentage, the maximum deflection did not occur at the center, so when the stiffeners are not too stiff compared to the plate the result would most likely be less than 20%. For the first boundary condition, the model can be used for all types of cross-stiffened plates when the plate is reasonably configured (meaning, a reasonable combination of stiffeners-plate is selected). The grillage is limited to only cross-stiffened plates in which the central stiffener is equal to the other stiffeners in each axis (the central stiffener, for example, can- Fig. 14 Finite element analysis displacement solution of desired problem 224 OCTOBER 2007 MARINE TECHNOLOGY

not be heavier to reduce the deflection at the center, while the OPM allows it). For this limitation, the model compensates by being more accurate: for all boundary conditions the model gave better results. For the first two BCs the results were up to 16% relative error, which for a preliminary design tool is a great value. For the last boundary conditions (cases 3 and 4), the relative errors are less than great. However, since they are distributed at the entire range and many FEA agreed quite well with the software predictions, the model can still be used with acceptable accuracy, especially if one is to drop the few analyses at the 25 to 30% mark, in which the deflections were not at the center of the plate (which bring us back to relative errors of up to 20%). Conclusively, for reasonably configured plates the grillage model, when applicable, should be preferred over the orthotropic plate model. When the stiffeners are not too stiff with respect to the plate itself, the relative errors would most likely be in an acceptable range of up to 20% at most. When a different type of stiffened plates is needed (e.g., when the central stiffener is heavier/bigger) the orthotropic plate model is to be used, also with acceptable results when the plate is not fixed or clamped. On a final note, an important remark regarding the software should be made. The software can save a great deal of time for the design process of stiffened plates. It can be used to generate ADINA models (with applied loads and boundary conditions) for many purposes other than deflection evaluations. In the example above, the results of the models were obtained in less than a minute time (including the generation of the ADINA model). The FE method, on the other hand, required significant time and knowledge; That time must be spent on building the geometry model, making educated decisions regarding the meshing and the suitable elements, applying boundary conditions and desired materials, and some other FE-related issues. Finally, additional computer time is required for solving the problem. References BANAI, L. AND PEDATZUR, O. 2006 Computer implementation of the orthotropic plate model for quick estimation of the maximum deflection of stiffened plates, Ships and Offshore Structures, 1, 4, 323 333 CLARKSON, J., WILSON, L. B., AND MCKEEMAN, J. L. 1959 Data sheets for the elastic design of flat grillages under uniform pressure, European Shipbuilding, 8, 174 198. SCHADE, H. A. 1940 The orthogonally stiffened plate under uniform lateral 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 bending loads, Trans. SNAME, 59, 127 141. SCHADE, H. A. 1951 & 1953 The effective breadth concept in ship structure design, Trans. SNAME. OCTOBER 2007 MARINE TECHNOLOGY 225