Sanjay Kantrao Kulkarni
© 2025 The author. This article is published by IIETA and is licensed under the CC BY 4.0 license (http://creativecommons.org/licenses/by/4.0/).
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Orthotropic plates are used in advanced structures where directional stiffness is critical. Under nonlinear thermal loading their bending behaviour becomes complex and demands advanced modelling techniques. The primary goal of this study is to formulate and validate a numerical approach for evaluating the thermally induced cylindrical bending of orthotropic plate under nonlinear temperature distribution. The study investigates the influence of nonlinear thermal loading on the cylindrical bending behaviour of the orthotropic plate. The applied thermal profile comprises a base temperature field, a linear gradient across plate’s thickness and an additional nonlinear or complex variation along the same dimension. Such a composite temperature distribution is essential for capturing the realistic thermal response of orthotropic laminated structures subjected to cylindrical bending. To analyse the resulting thermal deformations a higher order trigonometric plate theory is employed under both linear and nonlinear thermal conditions. For comparative evaluation, a parabolic plate theory is also utilized, specifically to assess the accuracy and performance of the trigonometric model when exposed to nonlinear or complex thermal gradients. The mathematical model is derived using energy principles and solved analytically via Navier’s type series expansion. A simulation tool is developed in FORTRAN to analyse the thermal stresses and deflections under nonlinear thermal loadings. Simulation results demonstrate that a nonlinear thermal load intensify the results of axial and transverse displacements. The developed analytical framework effectively captures the nonlinear thermal effects in orthotropic plate undergoing cylindrical bending.
thermal analysis, thermal deformations, nonlinear thermal load, higher order theory, cylindrical bending
The cylindrical bending idealization treats a laminated plate as infinitely long in one direction and simply supported along its edges, enabling a clean separation of variables for studying flexure under mechanical or thermal fields. Modern plate and elasticity theories have significantly sharpened our understanding of how composite laminates respond in such one-dimensional bending states, especially when the temperature field is nonuniform and the coupling is nonlinear. Building on earlier nth-order and higher-order formulations applied to orthotropic laminates and sandwich constructions [1-3], as well as exact three-dimensional and discrete-layer solutions that exposed the limits of classical plate theory (CPT) [4, 5], subsequent work has spanned dynamic behavior, piezoelectric coupling, and refined shear/normal-deformation kinematics in cylindrical bending [6-18].
In the last few years, several trends stand out. First, a notable advancement in the field is the refinement of semi-analytical and isogeometric scaled boundary finite element methods. These techniques have proven highly effective for analyzing cylindrical bending in thick, layered composite structures, offering excellent precision while maintaining computational efficiency number of through a reduced number of degrees of freedom [19-25]. Second, realistic service conditions—moisture diffusion, temperature gradients, and elastic foundations—are now embedded directly in cylindrical-panel models; for example, quasi-static hygrothermal bending of laminated cylindrical panels has been analyzed with fully coupled field equations [20]. Third, thermal boundary complexities are being resolved explicitly: laminated plates with arbitrary edge restraints subjected to non-uniform temperature fields admit efficient analytical–numerical solutions that are immediately useful for validation and benchmarking [21]. Fourth, data-assisted and energy-variational solvers are being used to capture geometric nonlinearity in laminated plates with high fidelity, pointing to robust surrogates for demanding bending cases [22]. Fifth, coupled thermo-mechanical analyses for advanced constituents—such as graphene-reinforced functionally graded plates—now quantify post-buckling and large-deflection responses under prescribed temperature fields, informing design under severe environments [23-28]. The study of material orthotropy and length to thickness ratio on transverse displacement of sandwich beam was studied and presented by Kulkarni [29] using parabolic shear deformation theory (PSDT).
Against this backdrop, the analysis of an orthotropic plate strip under a nonlinear through-thickness temperature profile in cylindrical bending remains comparatively under-reported. The present study addresses that gap by quantifying how a nonlinear thermal field modifies the stress resultants and interlaminar stress distributions of an orthotropic plate subjected to cylindrical bending.
The following Figure 1 illustrates an orthotropic plate strip with its coordinate system. The strip extends significantly in the y direction, while its length in the x direction is limited and denoted by “a”. The z axis points downward through the thickness of the plate strip.
Figure 1. A coordinate system of an orthotropic plate strip
In structural mechanics, higher order theories play a vital role in the analysis of plates that experience a significant shear effect under nonlinear thermal load. In present case higher order trigonometric and Parabolic plate theories are used to understand the effect of nonlinear thermal load on orthotropic plate in cylindrical bending. The displacement field of trigonometric plate theory is represented by the following Eqs. (1) and (2). Trigonometric and parabolic plate theories describe how plates deform due to shear forces. These are known as shear deformation theories. From this point forward, these theories are referred to as Trigonometric Shear Deformation Theory (TSDT) and PSD.
$u(x,z)=-z\frac{\partial w(x)}{\partial x}+\frac{h}{\pi }\sin \frac{\pi z}{h}\varphi (x)$ (1)
$w(x)=w(x)$ (2)
In the referenced expression (1) and (2), the variable u denotes displacement along the axial (x axis), whereas w represents displacement in transverse direction (z axis). The function $\varnothing $ in Eq. (1) is unknown rotation to be determined. The strains are evaluated with reference to elasticity theory as given by following Eq. (3).
${{\varepsilon }_{x}}=\frac{\partial u}{\partial x},{{\gamma }_{zx}}=\frac{\partial u}{\partial z}+\frac{\partial w}{\partial x}$ (3)
The relationship between stress and strain for an orthotropic plate strip is represented by below Eq. (4).
$\left\{\begin{array}{c}\sigma_x \\ \tau_{z x}\end{array}\right\}=\left[\begin{array}{cc}\bar{Q}_{11} & 0 \\ 0 & \bar{Q}_{55}\end{array}\right]\left\{\begin{array}{c}\varepsilon_x-\alpha_x T \\ \gamma_{z x}\end{array}\right\}$ (4)
The reduced stiffness coefficients are represented by ${{\bar{Q}}_{ij}}$ in Eq. (4) are defined in Eq. (5).
${{\bar{Q}}_{11}}={{Q}_{11}}=\frac{{{E}_{1}}}{1-{{\mu }_{12}}{{\mu }_{21}}},{{\bar{Q}}_{55}}={{Q}_{55}}={{G}_{13}}$ (5)
In the above Eq. (5), the parameters are defined as follows: E represents the modulus of elasticity, $\mu $ denotes Poisson’s ratio, $G$ is the shear modulus, ${{\alpha }_{x}}$ indicates the coefficient of thermal expansion along the x axis and T corresponds to the change in temperature. Furthermore, Eq. (6) describes the temperature distribution within the system, where the temperature at any specific location is a function of both the horizontal coordinate (x) and the vertical coordinate (z), expressed as T (x, z).
$T(x,z)={{T}_{1}}(x)+\frac{z}{h}{{T}_{2}}(x)+\frac{\psi (z)=\frac{h}{\pi }\sin \frac{\pi z}{h}}{h}{{T}_{3}}(x)$ (6)
The above Eq. (6) characterizes how temperature varies along the depth of the beam structure. This thermal profile is influenced by three disctinct thermal loads, denoted as ${{T}_{1}},{{T}_{2}}~\text{and}~{{T}_{3}}$. Among these the component ${{T}_{2}}$ contrubutes to a linearly varying temperature field, reflecting a uniform thermal gradient across the beam’s thickness. In contrast, the influence of ${{T}_{3}}$ introduces a nonlinear variation mathematically described by the function $\left( \psi \left( z \right)=\frac{h}{\pi }\sin \frac{\pi z}{h} \right),$ where, h is the total depth of the beam and z is vertical coordinate. This sinusoidal term captures the oscillatory nature of the temperature distribution induced by ${{T}_{3}}$, offering a more complex thermal behaviour compared to the linear case. In this analysis, the thermal load applied follows a sinusoidal pattern.
2.1 Equations of motion
The equations of motion are derived using the principle of virtual work, which asserts that for a system subjected to infinitesimal virtual displacements, the external work performed must be exactly balanced by the internal work generated by the system’s internal forces. This principle forms the foundation of the variational approach, a powerful technique for formulating the governing differential equations of structural behaviour. Once these equations are established, Navier’s method employed to compute the resulting displacements and stress fields under thermal loadings. By applying this principle to an orthotropic plate strip, one can derive the corresponding governing equations or equation of motion that describe its mechanical behaviour. The following Eq. (7) represents an application of the virtual work principle to an orthotropic plate in cylindrical bending.
$\int_{-\frac{h}{2}}^{\frac{h}{2}}{\int_{0}^{a}{\left( {{\sigma }_{x}}\delta {{\varepsilon }_{x}}+{{\tau }_{zx}}\delta {{\gamma }_{zx}} \right)}}dxdz=0$ (7)
To solve the above Eq. (7), the method of integration by parts is employed, which facilitates the transformation of the variational expression into a more tractable form. This mathematical technique allows the redistribution of derivatives between functions ultimately leading to the derivation of the systems’ governing equations. As a result of this process, the fundamental equations of motion are obtained and these are represented by the following Eqs. (8) and (9).
$\begin{aligned} & \delta w: Q_{11} \frac{h^3}{12} \frac{\partial^4 w}{\partial x^4}-Q_{11} \frac{2 h^3}{\pi^3} \frac{\partial^3 \varphi}{\partial x^3} +\alpha_x Q_{11} \frac{h^2}{12} \frac{\partial^2 T_1}{\partial x^2}+\alpha_x Q_{11} \frac{2 h^2}{\pi^3} \frac{\partial^2 T_3}{\partial x^2}=0\end{aligned}$ (8)
$\begin{aligned} & \delta \varphi: Q_{11} \frac{2 h^3}{\pi^3} \frac{\partial^3 w}{\partial x^3}-Q_{11} \frac{h^3}{2 \pi^2} \frac{\partial^2 \varphi}{\partial x^2}+Q_{55} \frac{h}{2} \varphi +\alpha_x Q_{11} \frac{2 h^2}{\pi^3} \frac{\partial T_2}{\partial x}+\alpha_x Q_{11} \frac{h^2}{2 \pi^2} \frac{\partial T_3}{\partial x}=0\end{aligned}$ (9)
2.2 Material properties
The material properties used in the analysis are given in Eq. (10) below.
$\frac{{{E}_{1}}}{{{E}_{2}}}=25,{{G}_{12}}={{G}_{13}}=0.5{{E}_{2}},{{\mu }_{12}}=0.25,\frac{{{\alpha }_{2}}}{{{\alpha }_{1}}}=3$ (10)
The moderate modular ratio and low ratio of thermal expansion coefficients is considered for analysis. E1 and E2 represent Young’s modulus in fibre (1) and transverse (2) directions are shown in Figure 2. A ratio of 25 indicates the material is highly stiff along the fibre axis compared to the transverse direction. This is typically a fibre reinforced composite where fibres dominate stiffeness in one direction. Such anisotropy is very crucial in applications demanding directional strength, like aerospace panels.
Figure 2. Orientation of fibre and transverse directions
The shear modulus is denoted by G12and G13 in the planes involving the fibre direction. Setting these shear moduli to half of E2indicates moderate resistance to shear deformation especially in the directions involving the weaker transverse axis. This balance helps to ensure the material can tolerate torsional load without being over rigid. This helps to prevent cracking under complex stresses. The value of Poisson’s ratio indicates that when the material is stretched along fibre direction, it contracts 25% as much in the transverse direction. The symbols $\alpha_1$ and $\alpha_2$ indicate how much a material streaches with heat along and across the fibre directions, respectively. A ratio of coefficient of thermal expansion of 3 implies that the material expands 3 times more in transverse direction (2) than along the fibre direction (1) or axis when heated. The low coefficient of thermal expansion minimizes warping in the structural member.
2.3 Navier’s solution
The Navier’s solution technique is an analytical approach used to solve problems in elasticity and plate theory. It is especially useful in thermal stress analysis where temperature stresses are induced under simply supported boundary conditions. This method is very effective in scenarios which involves temperature variations, where uneven heating leads to internal stresses and structural distortion. By expressing displacement and stress components as series expansions, Navier’s technique simplifies the complex equations governing plate deformation under thermal influence. The solution fulfills the conditions as given in the below Eq. (11).
$w=0, M_x=0, N_x=0, M_x^s=0$ (11)
In the above equation, the term w is the transverse displacement which is zero; this implies the simply supported boundary conditions. The term $M_x$ is bending moment in x direction has a value of zero, indicating that no resistance to bending since simply supported edge. The term $N_x$ is in-plane normal force in x direction and has the value zero. This indicates no axial force along x direction at the boundary. This is important in thermal stress analysis where expansion or contraction may occur.
The unknowns given below are represented in trigonometric form which satisfy the exact boundary conditions. Thermal load expansion is expressed using a single sine term from the Fourier series, as illustrated below in the Eqs. (12) to (16).
$w(x)=\sum\limits_{m=1}^{\infty }{{{w}_{m}}}\sin \frac{m\pi x}{a}$ (12)
$\varphi (x)=\sum\limits_{m=1}^{\infty }{{{\varphi }_{m}}}\cos \frac{m\pi x}{a}$ (13)
${{T}_{1}}(x)=\sum\limits_{m=1}^{\infty }{{{T}_{1m}}}\sin \frac{m\pi x}{a}$ (14)
${{T}_{2}}(x)=\sum\limits_{m=1}^{\infty }{{{T}_{2m}}}\sin \frac{m\pi x}{a}$ (15)
${{T}_{3}}(x)=\sum\limits_{m=1}^{\infty }{{{T}_{3m}}}\sin \frac{m\pi x}{a}$ (16)
The first harmonic of a Fourier sine series or first mode of sinusoidal thermal distribution is represented by $T_{1 m}=T_{2 m}= T_{3 m}=T_1=T_2=T_3, m=1$. This implies that only the first term in the series is active representing a single sinusoidal wave across the thickness in one direction.
Substitution of the solution into the equations of motion results into a set of algebraic equations. This can be represented by following Eq. (17).
$[K]\{\delta \}=\{f\}$ (17)
The symmetric stiffness is represented by [K]. The coefficients of the stiffness matrix [K] are as given in the following Eq. (18).
$\left[\begin{array}{ll}K_{11} & K_{12} \\ K_{21} & K_{22}\end{array}\right]=\left[\begin{array}{cc}D_1 \frac{m^4 \pi^4}{a^4} & -D_4 \frac{m^3 \pi^3}{a^3} \\ -D_4 \frac{m^3 \pi^3}{a^3} & D_9 \frac{m^2 \pi^2}{a^2}+D_{11}\end{array}\right]$ (18)
The $\{\delta\}$ in Eq. (17) is given by the following Eq. (19).
$\{\delta\}=\left\{w_m, \varphi_m\right\}$ (19)
The force vector is denoted by $\{f\}$ in the above Eq. (17). The elements of the thermal load vector $\{f\}$ are given below Eq. (20).
$\left\{\begin{array}{l}f_1 \\ f_2\end{array}\right\}=\left\{\begin{array}{c}R_1 \frac{m^2 \pi^2}{a^2} T_{2 m}+R_2 \frac{m^2 \pi^2}{a^3} T_{3 m} \\ -R_5 \frac{m \pi}{a} T_{2 m}-R_6 \frac{m \pi}{a} T_{3 m}\end{array}\right\}$ (20)
By solving this system of equation, it becomes possible to determine the values represented by $\{\delta\}$, which in turn allows for the calculation of stresses and resulting displacements within the structure.
Thermal deformation behaviour has been analysed using four distinct plate theories: higher order trigonometric (TSDT), PSDT first order shear deformation (FSDT) and CPT. These models are applied to orthotropic plate undergoing cylindrical bending under linear and nonlinear thermal loading conditions. The calculations are performed for aspect ratios of 4 and 10. The aspect ratio (a/h) is denoted by S in the result tables. To ensure consistency and clarity all stress and displacement outcomes are presented in the normalized form in Eqs. (21)-(24).
$\bar{u}\left( 0,-\frac{h}{2} \right)=\frac{u}{{{\alpha }_{1}}{{T}_{0}}{{a}^{2}}}$ (21)
$\bar{w}\left( \frac{a}{2},0 \right)=\frac{10\times h\times w}{{{\alpha }_{1}}{{T}_{0}}{{a}^{2}}}$ (22)
${{\bar{\sigma }}_{x}}\left( \frac{a}{2},-\frac{h}{2} \right)=\frac{{{\sigma }_{x}}}{{{E}_{2}}{{\alpha }_{1}}{{T}_{0}}{{a}^{2}}}$ (23)
$\bar{\tau }_{zx}^{EE}(0,0)=\frac{{{\tau }_{zx}}}{{{E}_{2}}{{\alpha }_{1}}{{T}_{0}}{{a}^{2}}}$ (24)
The transverse shear stresses are critical under nonlinear thermal loading. To capture the exact thermal behaviour of these stresses, equation of equilibrium is used and given as below Eq. (25).
${{\tau }_{zx}}=-\cdot \int_{z=-\frac{h}{2}}^{+\frac{h}{2}}{\frac{\partial {{\sigma }_{x}}}{\partial x}}\cdot dz+C$ (25)
In the above equation the constant of integration “C” is calculated by using boundary conditions and upon substitution in the Eq. (25), shear stress can be calculated. In the above Eq. (25) the term $\left(\tau_{z x}\right)$ represents the shear force per unit area acting along $x$ axis on a surface normal to the $z$ axis. The term ($\sigma_x$) is normal stress in the $x$ direction and varies due to temperature gradients. The term $\left(\frac{\partial \sigma_x}{\partial x}\right)$ in the above Eq. (25) represents spatial derivative of normal stress which captures how the normal stress changes along the length. Since these stresses vary through the thickness under thermal loading, integrating across the thickness of the plate from bottom ($h / 2$) to top $(-h / 2)$ captures the cumulative effect of these variations across the total depth $(h)$. Figure 3 illustrates the upper and lower surfaces of the plate undergoing cylindrical bending.
Figure 3. The cross section of plate in cylindrical bending
The integration constant accounts for boundary conditions and ensure the solution satisfies equilibrium. The above Eq. (25) says that to maintain equilibrium under thermal loading, the transverse shear stress must balance the gradient of normal stress through the thickness.
The results of thermal deformations in dimensionless form under linear (L) and nonlinear (NL) thermal loadings are presented in Tables 1 and 2 below. For linear $(\mathrm{L})$ thermal load $T_2=1$ and $T_1=T_3=0$. In case of nonlinear (NL) thermal load the thermal loads are $T_1=0$ and $T_2=T_3=1$. The importance of Tables 1 and 2 is that both tables show how temperature variations affect displacements and stresses under linear and nonlinear thermal loads.
Table 1. Normalized displacements values for an orthotropic plate under cylindrical bending influenced by linear (L) and nonlinear thermal (NL) loads with aspect ratios 4 and 10
|
Theory |
|
$\overline{\boldsymbol{u}}$ |
|
$\overline{\boldsymbol{w}}$ |
|
||
|
|
S |
L |
NL |
% Difference |
L |
NL |
% Difference |
|
TSDT |
4 |
0.1592 |
0.2814 |
55.46 |
1.0132 |
1.7635 |
54.04 |
|
PSDT |
0.1595 |
0.2828 |
55.76 |
1.0238 |
1.8126 |
55.61 |
|
|
FSDT |
0.1592 |
- |
- |
1.0132 |
- |
- |
|
|
CPT |
0.1592 |
- |
- |
1.0132 |
- |
- |
|
|
|
|
|
|
|
|
|
|
|
TSDT |
10 |
0.1592 |
0.2822 |
55.73 |
1.0132 |
1.7918 |
55.55 |
|
PSDT |
0.1594 |
0.2829 |
55.84 |
1.0200 |
1.8071 |
55.68 |
|
|
FSDT |
0.1592 |
- |
- |
1.0132 |
- |
- |
|
|
CPT |
0.1592 |
- |
- |
1.0132 |
- |
- |
|
Table 2. Normalized stress values for an orthotropic plate under cylindrical bending influenced by linear (L) and nonlinear thermal (NL) loads with aspect ratios 4 and 10
|
Theory |
|
$\overline{\boldsymbol{\sigma}}_{\boldsymbol{x}}$ |
$\overline{\boldsymbol{\tau}}_{\boldsymbol{zx}}^{\boldsymbol{E} \boldsymbol{E}}$ |
|
||
|
|
S |
L |
NL |
L |
NL |
Remark |
|
TSDT |
4 |
0.0003 |
-1.6426 |
0.0000 |
-0.0859 |
Values are less than zero |
|
PSDT |
-0.0267 |
-1.7601 |
-0.0004 |
-0.0904 |
Values are less than zero |
|
|
FSDT |
0.0000 |
- |
0.0000 |
- |
- |
|
|
CPT |
0.0000 |
- |
0.0000 |
- |
- |
|
|
|
|
|
|
|
|
|
|
TSDT |
10 |
0.0001 |
-1.7053 |
0.0000 |
-0.0357 |
Values are less than zero |
|
PSDT |
-0.0171 |
-1.7463 |
-0.0001 |
-0.0361 |
Values are less than zero |
|
|
FSDT |
0.0000 |
- |
0.0000 |
- |
- |
|
|
CPT |
0.0000 |
- |
0.0000 |
- |
- |
|
In linear thermal load case (L), all theories give identical results of axial displacements for an aspect ratio 4 and 10. This implies that axial displacement does not get affected under linear thermal load. In case of nonlinear load (NL), higher theories TSDT and PSDT predicts higher values of axial displacements. Nonlinearity in thermal load increases axial displacement. First order and classical do not provide results under nonlinear thermal load. There is no significant variation in the results of transverse displacement evaluated by all theories for an aspect ratio 4 and 10 under linear thermal load (L). However, nonlinear thermal loading (NL) greatly amplifies transverse displacements for thick and thin orthotropic plate under cylindrical bending. Aspect ratio has negligible effect on axial displacements. The percentage difference between the values of axial and transverse displacements under linear and nonlinear thermal load is 55% as shown in Table 1. The nonlinear thermal loading leads to large temperature gradients through the thickness. These gradients include higher thermal strains which ultimately amplify the bending and stretching of the plate under cylindrical bending. As temperature increases nonlinearity, the material expands more aggressively this causes significant deformation which linear model fails to capture. The 55% rise in displacements is a direct result of this intensified thermal expansion. In linear model, temperature is assumed to vary mildly. But in reality, nonlinear profiles cause uneven stress distribution leading to larger deflections. Higher order theories like TSDT and PSDT include higher order terms in the displacement field. These terms allow the model to account for shear deformation and thickness wise variation in strain. Thus, the higher order theories are essential to capture realistic stress and displacement analysis in advanced materials. Whereas FSDT and CPT do not show nonlinear results since they lack necessary terms to capture these effects.
The results of normal stresses are very small under linear thermal load (L) and nonlinear thermal load (NL). Nonlinearity in thermal load does not affect normal stress significantly. The transverse shear stresses are affected strongly than normal stresses by the influence of aspect ratio. Slender plate in cylindrical bending experiences reduced shear stresses under nonlinear thermal load (NL). The results of transverse shear stresses appear to be highly sensitive to the choice of theory. It is noted that nonlinear thermal effects reduce shear stresses as the plate becomes thinner. Aspect ratio plays an important role under nonlinear thermal load, where slender plate relieves thermal shear stress more effectively. Higher order theories (TSDT and PSDT) provide consistent and realistic prediction of normal stresses and shear stresses. The results evaluated by higher theories confirm that higher order plate theories are essential for analysis under thermal loadings.
The variation of these displacements and stresses are shown in Figures 4-6. The variations of axial displacement, normal stress and shear stress across the thickness (h) evaluated by trigonometric shear deformation theory (TSDT) under linear and nonlinear thermal loadings is shown in Figures 4-6 respectively. The higher values of axial displacement and nonlinear variation of normal stress across thickness (h) is seen under nonlinear thermal load and shown in Figures 4 and 5. The transverse shear stresses evaluated by 3D equations of equilibrium are zero at upper and lower surfaces of plate under cylindrical bending. The nonlinear variation and realistic curve of shear stresses is seen across the thickness (h) of plate as shown in Figure 6. Table 2 shows negative stresses under nonlinear thermal loading which indicates that nonlinear thermal effects induce compressive type stresses in the plate undergoing cylindrical bending. On the other hand, CPT and FSDT yield zero stress prediction under same conditions. This shows their limitations in capturing nonlinear response.
Figure 4. Variation of axial displacement $(\bar{u})$ through the thickness $(h)$ under linear and nonlinear thermal load for an aspect ratio 4
Figure 5. Variation of normal stress $\left(\bar{\sigma}_x\right)$ across the thickness $(h)$ under linear and nonlinear thermal load for an aspect ratio 4
Figure 6. Variation of transverse shear stress $\left(\bar{\tau}_{z x}\right)$ across the thickness $(h)$ under linear and nonlinear thermal load for an aspect ratio 4
The study explores the thermal behaviour of an orthotropic laminated plate subjected to cylindrical bending under both linear and nonlinear thermal loading conditions, employing trigonometric shear deformation theory. The finding revels a pronounced increase in thermal displacements and stress levels when the thermal load transitions from a linear to nonlinear profile. This shift underscores the sensitivity of plate’s structural response to the nature of thermal input.
Furthermore, under nonlinear thermal loading the results obtained using trigonometric shear deformation theory show only a slight deviation from those predicted by PSTD. This implies that both higher order theories are capable of capturing the intricate thermal effects with reasonable accuracy.
In real world applications, thermal loads are seldom uniform or linear. Nonlinear thermal loading reflects a more realistic and complex distribution of heat across the structure which significantly influences the mechanical response. By incorporating these nonlinear effects, the analysis yields a more precise prediction of plate’s behaviour.
Higher order theories, such as trigonometric and parabolic models are better equipped to account for these complexities. Their ability to represent the through thickness variation of temperature and shear strains make them especially valuable in evaluating the performance of laminated composite structures under thermal stress. The comparative results reinforce the importance of using advanced modelling approaches when dealing with nonlinear thermal environment as they provide deeper insights into the structural integrity and reliability of such systems. A key limitation of this study is the assumption of simply supported boundary conditions which may not fully represent the complexity of real-world structural constraints. One of key area for future studies is the integration of coupled thermo-mechanical effects were both temperature variation and mechanical forces acts on the laminates undergoing cylindrical bending. Another important area for future studies is the extension of this work and study of optimization. This will improve overall structural efficiency.
The author expresses sincere gratitude to Symbiosis Institute of Technology, Pune, for their unwavering support and motivation throughout the course of this work. Their consistent encouragement played a vital role in the successful completion of this study.
|
$\varepsilon_x$ |
normal strain |
|
$\gamma_{z x}$ |
shear strain |
|
E |
modulus of elasticity |
|
G |
shear modulus |
|
$\bar{u}$ |
normalized axial displacement |
|
$\bar{w}$ |
normalized transverse |
|
$\bar{\sigma}_x$ |
normalized normal stress |
|
$\bar{\tau}_{Z x}^{E E}$ |
normalized transverse shear stress obtained by using equilibrium equation |
|
$[K]$ |
stiffness matrix |
|
$\left(\bar{Q}_{i j}\right)$ |
reduced stiffness coefficient |
|
$\{f\}$ |
force vector |
|
$\left[\frac{E_1}{E_2}\right]$ |
modular ratio |
|
$E_1$ |
modulus of elasticity in fibre direction |
|
$E_2$ |
modulus of elasticity in transverse direction |
|
L |
linear thermal load |
|
NL |
nonlinear thermal load |
|
S |
aspect ratio $\left(S=\frac{a}{h}\right)$ |
|
h |
thickness of plate |
|
a |
length of plate along x axis |
|
b |
length of plate along y axis |
|
Eq. (18) |
$D_1=Q_{11} \frac{h^3}{12}, D_4=Q_{11}\left(0.0645 h^3\right)$ |
|
Eq. (19) |
$D_9=Q_{11}\left(0.0506 h^3\right), D_{11}=Q_{55}(0.5 h)$ |
|
Eq. (20) |
$R_1=\frac{h^2}{12}\left(\alpha_x Q_{11}\right), R_2=\frac{2 h^2}{\pi^3}\left(\alpha_x Q_{11}\right)$ |
|
Eq. (20) |
$R_5=\frac{2 h^2}{\pi^3}\left(\alpha_x Q_{11}\right), R_6=\frac{h^2}{2 \pi^2}\left(\alpha_x Q_{11}\right)$ |
|
Greek symbols |
|
|
$\left[\frac{\alpha_2}{\alpha_1}\right]$ |
ratio of coefficient of thermal expansion |
|
$\alpha_2$ |
coefficient of thermal expansion in transverse direction |
|
$\alpha_1$ |
coefficient of thermal expansion in fiber direction |
|
$\theta$ |
angle between fiber axis and reference axis x. In present case $\theta$ = 0. Hence, $\alpha_1$ = $\alpha_x$ and $\alpha_2=\alpha_y$ |
|
$\delta$ |
variational operator |
|
$\mu_{i j}$ |
Poisson’s ratio |
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