International Journal of Trend in Scientific Research and Development (IJTSRD) Volume 5 Issue 6, September-October 2021 Available Online: www.ijtsrd.com e-ISSN: 2456 – 6470 @ IJTSRD | Unique Paper ID – IJTSRD47566 | Volume – 5 | Issue – 6 | Sep-Oct 2021 Page 1151 Limit States Solution to CSCS Orthotropic Thin Rectangular Plate Carrying Transverse Loads Bertram D. I., Okere C. E., Ibearugbulem O. M., Nwokorobia G. C. Department of Civil Engineering, Federal University of Technology, Owerri, Imo State, Nigeria ABSTRACT The analysis of thin rectangular orthotropic plate with two opposite edges clamped and the other two opposite edges simply supported (CSCS), carrying transverse loads was investigated in this study. The Ritz total potential energy functional was used. The minimization of the total potential energy functional produces the expression for the coefficient of deflection. The coefficient of deflection was used to obtain equation for the maximum lateral load of an orthotropic thin rectangular plate based on allowable deflection. Also, equation for the maximum lateral load of an orthotropic thin rectangular plate based on allowable stress was developed.Developed stiffness coefficients were substituted in the lateral load equations to obtain the maximum lateral load values for a CSCS plate. Numerical examples using permissible deflection of 10mm and yield strength of 250MPa, plate thickness (varying from 5mm to 12.5 mm with 0.5mm intervals) were done to determine the maximum lateral loads corresponding to an orthotropic thin rectangular CSCS plate carrying transverse loads (when n1 = Ey/Ex = 0.7 and n2 = G/Ex = 0.41) for aspect ratios (b/a) of 1.0, 1.25 and 1.50. KEYWORDS: Transverse Loads, Plates, Orthotropic, Direct Variation, Allowable deflection, Elastic yield stress How to cite this paper: Bertram D. I. | Okere C. E. | Ibearugbulem O. M. | Nwokorobia G. C. "Limit States Solution to CSCS Orthotropic Thin Rectangular Plate Carrying Transverse Loads" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-5 | Issue-6, October 2021, pp.1151-1157, URL: www.ijtsrd.com/papers/ijtsrd47566.pdf Copyright © 2021 by author (s) and International Journal of Trend in Scientific Research and Development Journal. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0) (http://creativecommons.org/licenses/by/4.0) Notations:a: Length of the plate, b: Width of the plate, w: Deflection equation of the plate, A:Coefficient of deflection of the plate,h :Shape function,
x: Normal stressin x - direction,
y: Normal stressin y - direction, xy: Shear stressin x-y direction,x : Direct strain along x – direction,: Direct strain along y – direction, : Shear Strain on x – y plane, µxy:Poisson ratio on x axis, µyx: Poisson ratio on y axis, Ex : Elastic modulus in the x direction, Ey: Elastic modulus in the y direction, Gxy: Shear modulus in the x-y plane, ∝: Aspect Ratio = b/a, t: Thickness of the plate, x:Primary axis of the plate, y:Secondary axis of the plate, z:Axis corresponding to the thickness of the plate,C: Clamped Support, S: Simply supported support,R: Non-dimensional parameter equal to x/a, Q:Non-dimensional parameter equal to y/b, q: Transverse load uniformly distributed, n1: Ratio of the young modulus in y direction to the young modulus in the x direction, n2: Ratio of the shear modulus in x-y plane to the young modulus in the x direction, n3: Ratio of the stress in the y direction to the stress in the x direction, n4: Ratio of the shear stress in x-y plane to the stress in the x direction. 1. Introduction Thin plates are structural elements that their thickness is smaller than its two dimensions. Among practical examples to describe the dimensions of these plates are roof, building windows, flat part of a table, manhole thin covering and panels. Plates are divided into two categories: thin plates with large deflections and thick plates (Boot, 1988 and Krysko, 2011). In thin plates, deflections and deformation of structural elements are usually considered and for ease of work, in structural plates, their deflections are studied under loading conditions. Geometrically, plates are bounded by either rectilinear or curved boundaries. Plates are widely used in the fields of aerospace, aeronautical, naval, marine, mechanical, architectural, structural, and highway engineering. The structural behavior of plates as a function of the type of loading acting on it can be classified as static flexure, dynamic flexure or buckling. The analysis of the plates for static flexure, IJTSRD47566 International Journal of Trend in Scientific Research and Development @ www.ijtsrd.com eISSN: 2456-6470 @ IJTSRD | Unique Paper ID – IJTSRD47566 | Volume – 5 | Issue – 6 | Sep-Oct 2021 Page 1152 dynamic flexure, and buckling has been extensively done in the works of Osadede et al. (2011), Ezeh et al. (2013), Aginam et al. (2012), Ibearugbulem et al. (2013), Ibearugbulem et al. (2011).Considerable research interest and activities have been generated on the analysis of plates, with varying methods being developed and used for specific cases. The methods for plate analysis can be grouped into two namely: analytical methods and numerical or approximate methods. Analytical methods target to obtain mathematical expressions valid for the entire plate region that identically solve the governing partial differential equations on the entire plate domain. This is usually subject to the geometric and boundary conditions at the plate edges. They are closed form mathematical solutions which exist for a limited number of plate problems, and do not exist for the vast majority of plate problems whether in static flexure, dynamic flexure or buckling (Szilard, 2004; Timoshenko and Woinowsky-Krieger, 1959). Numerical methods came up to address the need for approximate solutions for cases where closed form analytical solutions cannot be found. The double trigonometric or Fourier series method was one of the earliest methods of solving the plate problem. The method, applicable to plates with all edges simply supported, assumes that a double Fourier series can be developed to represent any distribution of the applied load p( x,y). By assuming that the deflection response can be represented by a double Fourier series of the same form as the loading, which is specifically constructed to satisfy the geometric and force boundary conditions at the simply supported edges, the governing fourth order biharmonic equation for flexural static analysis of thin plates is simplified to an algebraic problem, readily solved to find the unknown generalized displacement parameters. Thus, the internal forces are obtained from the internal force displacement relations. The energy method is another numerical method adopted in the solution to the vast plate problems. This was used extensively in the works of Ibearugbulem et al. (2014). 2. Theoretical Background The Ritz total potential energy functional for an orthotropic thin rectangular plate carrying both in-plane forces and transverse load was obtained by Bertram (2020) in his masters degree work. When only transverse load is considered, the result is as shown in Equation 1. 3. Methodology The procedure used in this work is outlined as follows: 3.1. Formulation of the Equation for the Maximum Transverse Load for an Orthotropic Thin Rectangular Plate Figure 1: Schematic Representation of CSCS plate carrying transverse uniform load (q). 3.1.1. Equation for the Maximum Transverse Load Parameter based on allowable deflection condition The Equation of Maximum Transverse Load Parameter based on allowable deflection is obtained by minimizing Equation 1 to obtain the deflection w, as a product of a coefficient of deflection A and a shape function h. Then, by direct variation of the total potential energy functional (that is, differentiating with respect to the coefficient b S a C C S x y q International Journal of Trend in Scientific Research and Development @ www.ijtsrd.com eISSN: 2456-6470 @ IJTSRD | Unique Paper ID – IJTSRD47566 | Volume – 5 | Issue – 6 | Sep-Oct 2021 Page 1153 of deflection and equating to zero). The Equation of Maximum Transverse Load Parameter based on allowable deflection is as shown in Equation 2 3.1.2. Equation for the Maximum Transverse Load Parameter as a function of the elastic yield stress In an elastic body, for general state of stress, the expression of internal work is given as shown in Equation 4 If the constitutive relation Equations of strains and stressesare substituted in Equation 4, it gives Equation 5 Solving Equation 5 bearing in mind that the internal work must be less than the maximum allowable internal work of the plate, an expression for the Maximum Lateral Load Parameter as a function of the elastic yield stress is obtained as shown in Equation 6 Formulation of the Ratios n3 and n4 n3 is the ratio of the stress in y-direction to the stress in x- direction, while n4is the ratio of the shear stress to the stress in x-direction. 3.2. Numerical example The material properties of CSCS plate include: 3.2.1. Numerical example based on allowable deflection criteria Bertram, 2020 obtained the stiffness coefficient Kmfor a CSCS plate as International Journal of Trend in Scientific Research and Development @ www.ijtsrd.com eISSN: 2456-6470 @ IJTSRD | Unique Paper ID – IJTSRD47566 | Volume – 5 | Issue – 6 | Sep-Oct 2021 Page 1154 Simplifying Equation 13 gives Bertram 2020, obtained hmax for CSCS plate as 0.0195312. If Equations 12 and the value of hmax are substituted into Equation 2, the lateral load parameter equation gives Numerical example based on maximum stress condition The shape function h for a CSCS plate is given in the work of Ibearugbulem et. al (2014) as Substituting Equations 19 and 20 into Equation 7 gives Substituting Equations 19 and 20 into Equation 9 gives Substituting Equations 19, 20 and 21 into Equation 11 gives International Journal of Trend in Scientific Research and Development @ www.ijtsrd.com eISSN: 2456-6470 @ IJTSRD | Unique Paper ID – IJTSRD47566 | Volume – 5 | Issue – 6 | Sep-Oct 2021 Page 1155 Substituting Equations 12, 23 and 26 into Equation 6 gives The maximum load can readily be calculated from Equation 28 The load that can cause failure need also be checked at the fixed edges where failure might occur first. Hence, from Equation 7, Also, from Equation 9, From Equation 11, Hence, the equation for the maximum load is obtained by substituting Equations 12, 29 and 31 into Equation 6 4. Results and Discussion Plate thickness varying from 5mm to 12.5mm (with 0.5mm intervals), allowable deflection of 10mm, yield strength of 250MPa, aspect ratio values varying from 1.0 to 2.25 (with 0.25 intervals) were used for the computation of the maximum Lateral load and the results presented on Table 1. The results presented on Table 1 were plotted for a square plate as shown on Figure 2. From Figure 2 and Table 1, it could be observed that, for plate thickness equal to and less than 10mm, the deflection condition governs the design as the values of the lateral load that will make the plate fail in deflection is the lowest. As the thickness reaches 10.5mm and above, the critical case becomes the stress condition at the clamped edge and this load value now governs the design. At each time, the smallest of the loads that will cause failure governs the design. Also, From Figure 2 and Table 1, it can be seen that in general, the value of the lateral load that will cause failure increases as the plate thickness increases. This is so because the plate tends to sustain more loads when the thickness is increased due to increase International Journal of Trend in Scientific Research and Development @ www.ijtsrd.com eISSN: 2456-6470 @ IJTSRD | Unique Paper ID – IJTSRD47566 | Volume – 5 | Issue – 6 | Sep-Oct 2021 Page 1156 in the stiffness of the plate. It can also be seen from Table 1, that for aspect ratios higher than 1.0, the design is governed always by the deflection case as the maximum lateral load that would cause failure is always smallest for the deflection condition. Table 1a: MaximumLateral load (q) for permissible deflection of 10mm, span of 1m and yield strength of 250 MPa of thin rectangular orthotropic CCCC plate in bending (when n1 = Ey/Ex = 0.5, n2 = G/Ex = 0.4, µxy = 0.3) for aspect ratios (b/a) of 1.0, 1.25 and 1.50 t (mm) q (KN/m 2 ) b/a = 1.0 b/a = 1.25 b/a = 1.50 Deflection Criteria Yield stress criteria Deflection criteria Yield stress criteria Deflection criteria Yield stress criteria Midspan Midspan Edge Midspan Midspan Edge Midspan Midspan Edge 5 6.20046505 17.9395998 12.936593 3.65755652 13.0870967 11.923581 2.64701936 10.6405748 12.426112 5.5 8.25281899 21.7069158 15.653278 4.86820773 15.835387 14.427533 3.52318277 12.8750955 15.035596 6 10.7144036 25.8330237 18.628694 6.32025767 18.8454193 17.169957 4.57404945 15.3224277 17.893602 6.5 13.6224217 30.3179237 21.862842 8.03565167 22.1171934 20.150852 5.81550154 17.9825714 21.00013 7 17.0140761 35.1616156 25.355723 10.0363351 25.6507095 23.370219 7.26342113 20.8555266 24.35518 7.5 20.9265696 40.3640995 29.107335 12.3442533 29.4459676 26.828058 8.93369034 23.9412933 27.958753 8 25.3971049 45.9253755 33.117678 14.9813515 33.5029676 30.524368 10.8421913 27.2398715 31.810848 8.5 30.4628848 51.8454434 37.386754 17.9695752 37.8217095 34.45915 13.0048061 30.7512612 35.911465 9 36.1611122 58.1243033 41.914562 21.3308696 42.4021933 38.632403 15.4374169 34.4754624 40.260604 9.5 42.5289898 64.7619553 46.701101 25.0871802 47.2444191 43.044129 18.1559058 38.412475 44.858266 10 49.6037204 71.7583992 51.746373 29.2604522 52.3483868 47.694325 21.1761549 42.5622992 49.70445 10.5 57.4225069 79.1136351 57.050376 33.8726309 57.7140965 52.582994 24.5140463 46.9249349 54.799156 11 66.0225519 86.827663 62.613111 38.9456618 63.3415481 57.710134 28.1854622 51.5003821 60.142384 11.5 75.4410583 94.9004829 68.434578 44.5014902 69.2307416 63.075745 32.2062846 56.2886407 65.734135 12 85.7152289 103.332095 74.514776 50.5620613 75.381677 68.679828 36.5923956 61.2897109 71.574408 12.5 96.8822665 112.122499 80.853707 57.1493206 81.7943544 74.522383 41.3596775 66.5035925 77.663203 Table 1b: MaximumLateral load (q) for permissible deflection of 10mm, span of 1m and yield strength of 250 MPa of thin rectangular orthotropic CSCS plate in bending (when n1 = Ey/Ex = 0.5, n2 = G/Ex = 0.4, µxy = 0.3) for aspect ratios (b/a) of 1.75, 2.0 and 2.25 t (mm) q (KN/m 2 ) b/a = 1.75 b/a = 2.0 b/a = 2.25 Deflection Criteria Yield stress criteria Deflection criteria Yield stress criteria Deflection criteria Yield stress criteria Midspan Midspan Edge Midspan Midspan Edge Midspan Midspan Edge 5 2.16385019 9.29233565 13.826076 1.90104362 8.49245715 15.865279 1.74396526 7.98710113 18.420377 5.5 2.88008461 11.2437261 16.729552 2.53028906 10.2758731 19.196988 2.32121776 9.66439237 22.288656 6 3.73913313 13.3809633 19.909549 3.28500338 12.2291383 22.846002 3.01357197 11.5014256 26.525343 6.5 4.75397888 15.7040472 23.366068 4.17659284 14.3522526 26.812322 3.83149168 13.4982009 31.130437 7 5.93760493 18.2129779 27.099108 5.2164637 16.645216 31.095947 4.78544068 15.6547182 36.103939 7.5 7.3029944 20.9077552 31.10867 6.41602223 19.1080286 35.696878 5.88588276 17.9709775 41.445848 8 8.86313039 23.7883793 35.394754 7.78667468 21.7406903 40.615115 7.14328171 20.4469789 47.156165 8.5 10.630996 26.85485 39.957359 9.33982732 24.5432012 45.850657 8.56810133 23.0827223 53.234889 9 12.6195743 30.1071675 44.796485 11.0868864 27.5155612 51.403504 10.1708054 25.8782077 59.682021 9.5 14.8418485 33.5453317 49.912133 13.0392582 30.6577703 57.273658 11.9618577 28.8334351 66.497561 10 17.3108016 37.1693426 55.304303 15.208349 33.9698286 63.461117 13.9517221 31.9484045 73.681508 10.5 20.0394166 40.9792002 60.972994 17.605565 37.451736 69.965881 16.1508623 35.223116 81.233862 11 23.0406769 44.9749045 66.918206 20.2423125 41.1034926 76.787951 18.5697421 38.6575695 89.154624 11.5 26.3275653 49.1564556 73.13994 23.1299978 44.9250983 83.927327 21.2188253 42.251765 97.443794 12 29.9130651 53.5238533 79.638196 26.280027 48.9165532 91.384008 24.1085758 46.0057025 106.10137 12.5 33.8101593 58.0770978 86.412973 29.7038066 53.0778572 99.157995 27.2494572 49.9193821 115.12736 International Journal of Trend in Scientific Research and Development @ www.ijtsrd.com eISSN: 2456-6470 @ IJTSRD | Unique Paper ID – IJTSRD47566 | Volume – 5 | Issue – 6 | Sep-Oct 2021 Page 1157 0 20 40 60 80 100 120 0 2 4 6 8 10 12 14 Maximum Lateral Load (kN)Thickness (mm) For deflection For stress at midspan For stress at Edge Figure 2: Graph of Maximum transverse load against thickness for a square plate Data Availability Statement All data that support the findings of this study are available from the corresponding author upon reasonable request. 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