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

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4 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) 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 < < 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 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 s ; 6mj s + 1 m s + 1 s + 1 j K r 2j 3 s s ; and 6mj s + 1 j s + 1 s + 1 m K r 2m 3 s s ; 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) 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: 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 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 Calculating the independent stiffness ratio variable. = r + 1 b3 I r s + 1 a 3 I s. (14) 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

5 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

6 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 = 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 = 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) 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 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

7 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 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 = L L S L S L S (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

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

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

10 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)] 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 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

11 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

12 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

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