Stochastic Free Vibration Behavior of Multi-Layered Helicoidal Laminated Composite Shells Under Thermal Conditions

Abstract

The present work aims to quantify the influence of uncertainties in the ply orientation of multi-layered bio-inspired helicoidal laminated composite conical, hemispherical, and toroidal shells under thermal conditions. Any change in the ply orientation affects the free vibration behavior of the laminates. The present investigation focuses on the different levels of uncertainties in the ply orientation on the free vibration behavior of the shell. Moreover, the study also focuses on the sensitivity of the uncertainties in ply orientations on the free vibration behavior of the shells, which is also quantified. To quantify the stochastic free vibration behavior of the shells, the Gaussian process regression (GPR) machine learning algorithm-based surrogate model is developed to predict the frequencies of the shells. The surrogate is created in the framework of higher-order shear deformation theory. The uncertainties in the ply orientations are introduced using bootstrapping. The present results are compared with the stochastic frequencies obtained using Monte Carlo simulations (MCS) to determine the model’s accuracy. The study highlights the influence of the temperature, type of shell, and end conditions on the stochastic free vibration behavior of bio-inspired laminated shells.

1. Introduction

The availability of extensive research on analyzing the behavior of laminated composite structures highlights their importance in constructing various engineering structures such as aircraft, automobiles, defense structures, thin-walled structures, etc. [1,2]. Laminated composites are layered structures in which all the layers are bonded together using resin. Each layer is made up of orthotropic material [3]. Thus, the effective stiffness or strength of the laminated composite structure can be altered by changing the orientation or thickness of the layers within the laminate. This property of laminates is termed tailor-ability [4,5].

The orientation of the layer within the laminate widely affects the behavior of the structure [6]. With the help of the first-order perturbation technique, Tripathi et al. [7] predicted uncertainty in the free vibration behavior of laminated conical shells. With the help of the D-optimal design approach, Dey et al. [8] studied the influence of uncertainties in material properties and ply orientation on the free vibration behavior of laminated conical shells. The bottom-up surrogate approach for studying the influence of uncertainties on the free vibration behavior of laminates was proposed by Dey et al. [9]. Using a first-order shear deformation theory-based Kriging surrogate model was proposed by Mukhopadhyay et al. [10] They predicted the influence of noise on the free vibration behavior of shallow laminated composite shells. Dey et al. [11] predicted the stochastic free vibration response of laminated shallow doubly curved shells using the Kriging surrogate model. With the help of the discrete singular convolution method, Seçgin and Kara [12] predicted the influence of randomness in the properties of the laminates on their free vibration behavior. Using a reproducing kernel particle method in aid of meshless solutions, Wang et al. [13] studied the influence of randomness in material properties on the vibration behavior of carbon fiber-reinforced plates. In aid of Flügge’s theory for orthotropic shells, Huo et al. [14] proposed a discrete analytical method for quantifying the influence of randomness on the free vibration behavior of thin orthotropic cylindrical shells. Employing first-order shear deformation theory along with the pseudo-excitation method, Zuo et al. [15] predicted the free vibration behavior of conical–cylindrical shell structure under random thermal conditions. Considering stochasticity in the thermal gradient, Parviz and Fakoor [16] predicted its effect on the free vibration behavior of laminated structures. Chen et al. [17] studied the stochastic vibration behavior of laminates using the Kriging model. Hakula et al. [18] considered uncertainties in various properties of joined cylindrical shells subjected to free vibration conditions.

Singh et al. [19] predicted stochasticity in the free vibration behavior of laminates containing material uncertainty using the Monte Carlo simulations. The influence of uncertainties in material properties on the free vibration behavior of conical laminated shells was carried out by Lal et al. [20] using a higher-order shear deformation theory-based first-order perturbation technique. A similar methodology was adopted by Parhi and Singh [21] for predicting the influence of uncertainties in the free vibration behavior of spherical and cylindrical shells under thermal conditions. Pseudo-excitation method-based analytical solutions were proposed by Chen et al. [22] for studying the stochastic free vibration behavior of thin cylindrical shells. Singh et al. [23] studied the influence of randomness in material properties on the free vibration behavior of laminated cylindrical shells. Employing higher-order shear deformation theory along with Monte Carlo simulations, Garg and Li [24] predicted the influence of uncertainties in ply angle on the free vibration behavior of helicoidal laminated composite cylindrical shells. Huo et al. [25] considered uncertainties in the geometrical and material properties of laminated shells and predicted their influence on the deformation studies of laminates. Belarbi et al. [26] employed Monte Carlo to quantify the stochasticity in the free vibration behavior of functionally graded beams.

Amongst the studies available in the literature on the stochastic free vibration behavior of laminates, ply orientation is found to be one of the most important properties affecting their behavior. To exploit the strength of the laminates up to their peak, researchers and scientists tried to mock up the layer orientation as observed in several living organisms, such as the dactyl club of mantis shrimp, scales of fishes, nacre, etc. [27]. The layup configuration found in biological beings is similar to a helix in appearance and is commonly called a helicoid lamination scheme [28]. In a helicoidal or Bouligand lamination scheme, each new layer is rotated by a specific angle concerning the previous layer, thus giving rise to the overall circular staircase-like structure. The helicoidal laminates in living beings increase their toughness and energy absorption [29,30]. Recently, researchers carried out free vibration [31,32], bending [33,34], and buckling [35,36,37,38,39] analysis of bio-inspired helicoidal laminated composite structures with various helicoidal layup schemes. By changing the helicoidal layup scheme, the behavior and mode shape of the laminate are found to be affected.

To estimate the distribution of sample statistics and generate random sampling or resampling of the data, bootstrapping is amongst the most preferred statistical methods [40]. Compared to parametric methods, such as ANOVA, no assumption is required by bootstrapping and it does not involve complex mathematical formulations [41]. Monte Carlo simulation generates samples based on the probability model and thus depends on the accuracy of the model developed [42]. From the review work on the stochasticity of the free vibration behavior of laminates, Kriging and first-order perturbation techniques are amongst the most widely used techniques for uncertainty quantification and statistical interference [43]. However, Kriging required special assumptions during its application, i.e., assumptions involved for the specific distribution of the data [44]. Also, the Kriging model necessitates fitting a variogram, which, in turn, is a complex and time-consuming exercise [45]. Considering the first-order perturbation technique, the bootstrap can handle non-linear data effectively and efficiently and can be applied to any statistical model [46,47]. The straightforwardness, effectiveness with small sample size, and lack of complex statistics are some of the other advantages of bootstrapping compared to the Kriging and first-order perturbation techniques [48,49,50]. The present work proposes the use of bootstrapping along with the Gaussian Process Regression (GPR) machine learning algorithm [51] for studying the influence of the uncertainties in the orientation of angle of layers for 20- and 28-layered bio-inspired helicoidal clamped laminated composite cylindrical, hemispherical, and toroid shells. The layup schemes adopted during the present study are presented in

Table 1. Orientation of layers in bio-inspired helicoidal laminated composite panels.

Nomenclature	20—Layered
Helicoidal Exponential: ${\phi }_1/{\phi }_2/{\phi }_3/\dots /{\phi }_n={\xi }^n$
HE1	[2/4/8/16/32/2/4/8/16/32/32/16/8/4/2/32/16/8/4/2]
HE2	[2.5/6.3/15.6/39/97.7/2.5/6.3/15.6/39/97.7/97.7/39/15.6/3.6/2.5/97.7/39/15.6/6.3/2.5]
HE3	[3/9/27/81/243/3/9/27/81/243/243/81/27/9/3/243/81/27/9/3]
Helicoidal Recursive: ${\phi }_1/{\phi }_2/{\phi }_3/\dots /{\phi }_n={\phi }_{n-1}+\xi \left(n-1\right)$
HR1	[0/1/3/6/10/15/21/28/36/45/45/36/28/21/15/10/6/3/1/0]
HR2	[0/2/6/12/20/30/42/56/72/90/90/72/56/42/30/20/12/6/2/0]
HR3	[0/3/9/18/30/45/63/84/108/135/135/108/84/63/45/30/18/9/3/0]
Semi-circular: ${\phi }_1/{\phi }_2/{\phi }_3/\dots /{\phi }_n=\sqrt{{\xi }_{max}^2-{\left(\gamma \left(n-1\right)-{\xi }_{max}^2\right)}^2}$
HS1	[0/20.6/28.3/33.5/37.4/40.3/42.4/43.9/44.7/45/45/44.7/43.9/42.4/40.3/37.4/33.5/28.3/20.6/0], γ = 5
HS2	[0/41.2/56.5/67.1/74.8/80.6/84.9/87.7/89.4/90/90/89.4/87.7/84.9/80.6/74.8/67.1/56.5/41.2/0], γ = 10
HS3	[0/82.5/113.1/134/149.7/161.2/169.7/175.5/178.9/180/180/178.9/175.5/169.7/161.2/149.7/134/113.1/82.5/0], γ = 20
Linear Helicoidal (LH)
LH1	[0/4/8/12/16/20/24/28/32/36/36/32/28/24/20/16/12/8/4/0]
LH2	[0/5/10/15/20/25/30/35/40/45/45/40/35/30/25/20/15/10/5/0]
LH3	[0/6/12/18/24/30/36/42/48/54/54/48/42/36/30/24/18/12/6/0]
Fibonacci Helicoidal (FH)
FH	[0/10/10/20/30/50/80/130/210/340/340/210/130/80/50/30/20/10/10/0]
28—Layered
Helicoidal Exponential: ${\phi }_1/{\phi }_2/{\phi }_3/\dots /{\phi }_n={\xi }^n$
HE1	[2/4/8/16/32/64/128/2/4/8/16/32/64/128/128/64/32/16/8/4/2/128/64/32/16/8/4/2]
HE2	[2.5/6.3/15.6/39/97.7/244.1/610.4/2.5/6.3/15.6/39/97.7/244.1/610.4/610.4/244.1/97.7/39/15.6/3.6/2.5/610.4/244.1/97.7/39/15.6/6.3/2.5]
HE3	[3/9/27/81/243/729/2187/3/9/27/81/243/729/2187/2187/729/243/81/27/March 9, 2187/729/243/81/27/9/3]
Helicoidal Recursive: ${\phi }_1/{\phi }_2/{\phi }_3/\dots /{\phi }_n={\phi }_{n-1}+\xi \left(n-1\right)$
HR1	[0/1/3/6/10/15/21/28/36/45/55/66/78/91/91/78/66/55/45/36/28/21/15/10/6/3/1/0]
HR2	[0/2/6/12/20/30/42/56/72/90/110/132/156/182/182/156/132/110/90/72/56/42/30/20/12/6/2/0]
HR3	[0/3/9/18/30/45/63/84/108/135/165/198/234/273/273/234/198/165/135/108/84/63/45/30/18/9/3/0]
	Semi-circular: ${\phi }_1/{\phi }_2/{\phi }_3/\dots /{\phi }_n=\sqrt{{\xi }_{max}^2-{\left(\gamma \left(n-1\right)-{\xi }_{max}^2\right)}^2}$
HS1	[0/17.4/24.1/28.9/32.6/35.6/38.1/40.1/41.7/42.9/43.9/44.5/44.8/45/45/44.8/44.5/43.9/42.9/41.7/40.1/38.1/35.6/32.6/28.9/24.1/17.4/0] γ = 3.5
HS2	[0/34.8/48.2/57.8/65.2/71.2/76.1/80.1/83.3/85.9/87.7/89/89.8/90/90/89.8/89/87.7/85.9/83.3/80.1/76.1/71.2/65.2/57.8/48.2/34.8/0], γ = 7
HS3	[0/69.6/96.4/115.6/130.5/142.5/152.3/160.2/166.7/171.7/175.5/178.1/179.6/180/180/179.6/178.1/175.5/171.7/166.7/160.2/152.3/142.5/130.5/115.6/96.4/69.6/0], γ = 14
Linear Helicoidal (LH)
LH1	[0/4/8/12/16/20/24/28/32/36/40/44/48/52/52/48/44/40/36/32/28/24/20/16/12/8/4/0]
LH2	[0/5/10/15/20/25/30/35/40/45/50/55/60/65/65/60/55/50/45/40/35/30/25/20/15/10/5/0]
LH3	[0/6/12/18/24/30/36/42/48/54/60/66/72/78/78/72/66/60/54/48/42/36/30/24/18/12/6/0]
Fibonacci Helicoidal (FH)
FH	[0/10/10/20/30/50/80/130/210/340/190/170/360/170/170/360/170/190/340/210/130/80/50/30/20/10/10/0]

. For generating random samples, bootstrapping is used. The dataset for training the GPR model is obtained using higher-order shear deformation theory. For randomly generated uncertain samples, a trained GPR surrogate is used to predict the frequencies of the shells. The methodology adopted during the present study is presented in Figure 1. The present results are compared with the results obtained using the Monte Carlo simulations. The proposed methodology can effectively predict the stochastic behavior of the laminated shells without a large dataset, thus incorporating the benefits of both bootstrapping and GPR.

2. Mathematical Formulation

2.1. Quasi-3D Theory

The generalized displacement field for laminated composite shell structures using quasi-3D shear deformation theory can be stated as [52,53]

$$\begin{array}{c}U\left(x,y,z\right)=u_0-z{\varphi }_x+f\left(z\right){\phi }_x\\ V\left(x,y,z\right)=v_0-z{\varphi }_y+f\left(z\right){\phi }_y\\ W\left(x,y,z\right)=w_0+g\left(z\right){\phi }_z\end{array}$$

(1)

where the mid-plane displacements are denoted by $u_0$, $v_0$, and $w_0$ along X-, Y-, and-axes, respectively. The shear deformations of the mid-plane are denoted by ${\phi }_x$, ${\phi }_y$, and ${\phi }_z$. The shape function $f\left(z\right)$ governs the distribution of strains across the thickness of the shell. In the present study, parabolic function is employed, which is expressed as [54]

$$f\left(z\right)=z\left(1-{\displaystyle \frac{3z^2}{2h^2}}+{\displaystyle \frac{2z^4}{5h^4}}\right)$$

(2)

The Green–Lagrange non-linear strain-displacement relationships can be stated as

$$\begin{array}{c}{\epsilon }_{xx}=\left({\displaystyle \frac{\mathit{\partial }U}{\mathit{\partial }x}}+{\displaystyle \frac{W}{R_x}}\right)+{\displaystyle \frac{1}{2}}\left\{{\left({\displaystyle \frac{\mathit{\partial }U}{\mathit{\partial }x}}+{\displaystyle \frac{W}{R_x}}\right)}^2+{\left({\displaystyle \frac{\mathit{\partial }V}{\mathit{\partial }x}}+{\displaystyle \frac{W}{R_{xy}}}\right)}^2+{\left({\displaystyle \frac{\mathit{\partial }W}{\mathit{\partial }x}}-{\displaystyle \frac{U}{R_x}}\right)}^2\right\}\\ {\epsilon }_{yy}=\left({\displaystyle \frac{\mathit{\partial }V}{\mathit{\partial }y}}+{\displaystyle \frac{W}{R_y}}\right)+{\displaystyle \frac{1}{2}}\left\{{\left({\displaystyle \frac{\mathit{\partial }U}{\mathit{\partial }y}}+{\displaystyle \frac{W}{R_{xy}}}\right)}^2+{\left({\displaystyle \frac{\mathit{\partial }V}{\mathit{\partial }y}}+{\displaystyle \frac{W}{R_y}}\right)}^2+{\left({\displaystyle \frac{\mathit{\partial }W}{\mathit{\partial }y}}-{\displaystyle \frac{V}{R_y}}\right)}^2\right\}\\ {\epsilon }_{zz}=\left({\displaystyle \frac{\mathit{\partial }W}{\mathit{\partial }z}}\right)+{\displaystyle \frac{1}{2}}\left\{{\left({\displaystyle \frac{\mathit{\partial }U}{\mathit{\partial }z}}\right)}^2+{\left({\displaystyle \frac{\mathit{\partial }V}{\mathit{\partial }z}}\right)}^2+{\left({\displaystyle \frac{\mathit{\partial }W}{\mathit{\partial }z}}\right)}^2\right\}\\ {\gamma }_{xz}=\left({\displaystyle \frac{\mathit{\partial }U}{\mathit{\partial }z}}+{\displaystyle \frac{\mathit{\partial }W}{\mathit{\partial }x}}-{\displaystyle \frac{U}{R_x}}-{\displaystyle \frac{V}{R_{xy}}}\right)+[\left\{{\displaystyle \frac{\mathit{\partial }U}{\mathit{\partial }z}}\right\}\left\{{\displaystyle \frac{\mathit{\partial }U}{\mathit{\partial }x}}+{\displaystyle \frac{W}{R_x}}\right\}+\left\{{\displaystyle \frac{\mathit{\partial }V}{\mathit{\partial }z}}\right\}\left({\displaystyle \frac{\mathit{\partial }V}{\mathit{\partial }x}}+{\displaystyle \frac{W}{R_{xy}}}\right)+\\ \left\{{\displaystyle \frac{\mathit{\partial }W}{\mathit{\partial }z}}\right\}\left({\displaystyle \frac{\mathit{\partial }W}{\mathit{\partial }x}}-{\displaystyle \frac{U}{R_x}}\right)]\\ {\gamma }_{yz}=\left({\displaystyle \frac{\mathit{\partial }V}{\mathit{\partial }z}}+{\displaystyle \frac{\mathit{\partial }W}{\mathit{\partial }y}}-{\displaystyle \frac{V}{R_y}}-{\displaystyle \frac{U}{R_{xy}}}\right)+\left[\left\{{\displaystyle \frac{\mathit{\partial }U}{\mathit{\partial }z}}\right\}\left({\displaystyle \frac{\mathit{\partial }U}{\mathit{\partial }y}}+{\displaystyle \frac{W}{R_{xy}}}\right)+\left\{{\displaystyle \frac{\mathit{\partial }V}{\mathit{\partial }z}}\right\}\left({\displaystyle \frac{\mathit{\partial }Y}{\mathit{\partial }y}}+{\displaystyle \frac{W}{R_y}}\right)+\left\{{\displaystyle \frac{\mathit{\partial }W}{\mathit{\partial }z}}\right\}\left({\displaystyle \frac{\mathit{\partial }W}{\mathit{\partial }y}}-{\displaystyle \frac{V}{R_y}}\right)\right]\\ {\gamma }_{xy}=\left({\displaystyle \frac{\mathit{\partial }V}{\mathit{\partial }x}}+{\displaystyle \frac{\mathit{\partial }U}{\mathit{\partial }y}}+{\displaystyle \frac{2W}{R_{xy}}}\right)+{\displaystyle \frac{z}{2}}\left({\displaystyle \frac{1}{R_x}}-{\displaystyle \frac{1}{R_y}}\right)\left({\displaystyle \frac{\mathit{\partial }{u}_0}{\mathit{\partial }y}}-{\displaystyle \frac{\mathit{\partial }{v}_0}{\mathit{\partial }x}}\right)+[\left\{{\displaystyle \frac{\mathit{\partial }U}{\mathit{\partial }y}}+{\displaystyle \frac{W}{R_{xy}}}\right\}\left\{{\displaystyle \frac{\mathit{\partial }U}{\mathit{\partial }x}}+{\displaystyle \frac{W}{R_x}}\right\}+\\ \left\{{\displaystyle \frac{\mathit{\partial }V}{\mathit{\partial }x}}+{\displaystyle \frac{W}{R_{xy}}}\right\}\left\{{\displaystyle \frac{\mathit{\partial }V}{\mathit{\partial }y}}+{\displaystyle \frac{W}{R_y}}\right\}+\left\{{\displaystyle \frac{\mathit{\partial }W}{\mathit{\partial }y}}-{\displaystyle \frac{V}{R_y}}\right\}\left\{{\displaystyle \frac{\mathit{\partial }W}{\mathit{\partial }x}}-{\displaystyle \frac{U}{R_x}}\right\}]\end{array}$$

(3)

The strain matrix is the sum of linear and nonlinear strain matrices as

$$\left\{\epsilon \right\}=\left\{{\epsilon }^L\right\}-\left\{\alpha \right\}\Delta T+\left\{{\epsilon }^{NL}\right\}$$

(4)

Using generalized strain-displacement relationship, the linear strain vector can be expressed as

$$\left\{{\epsilon }^L\right\}=\left[V^L\right]\left\{{\overline{\epsilon }}^L\right\}$$

(5)

where, $\left\{{\overline{\epsilon }}^L\right\}={\left\{{\epsilon }_{xx}^{L_a} {\epsilon }_{yy}^{L_a} {\epsilon }_{yz}^{L_a} {\epsilon }_{xz}^{L_a} {\epsilon }_{xy}^{L_a} k_{xx}^{L_b} k_{yy}^{L_b} k_{yz}^{L_b} k_{xz}^{L_b} k_{xy}^{L_b} k_{xx}^{L_c} k_{yy}^{L_c} k_{yz}^{L_c} k_{xz}^{L_c} k_{xy}^{L_c}{k}_{xx}^{L_d} k_{yy}^{L_d} k_{yz}^{L_d} k_{xz}^{L_d} k_{xy}^{L_d} k_{zz}^{L_e}\right\}}^T$ and $\left[V^L\right]$ is the matrix containing terms related to thickness coordinates of the shell. In terms of nodal degrees of freedom, the $\left\{{\epsilon }^L\right\}$ can be re-stated as

$$\left\{{\epsilon }^L\right\}=\left[V^L\right]\left[B\right]\left\{q\right\}$$

(6)

In the similar manner, the nonlinear strain vector can be expressed as

$$\left\{{\epsilon }^{NL}\right\}={\displaystyle \frac{1}{2}}\left[A\right]\left\{\vartheta \right\}$$

(7)

where, $\left\{\vartheta \right\}={\left\{{\displaystyle \frac{\mathit{\partial }{u}_0}{\mathit{\partial }x}} {\displaystyle \frac{\mathit{\partial }{u}_0}{\mathit{\partial }y}} {\displaystyle \frac{\mathit{\partial }{v}_0}{\mathit{\partial }x}} {\displaystyle \frac{\mathit{\partial }{v}_0}{\mathit{\partial }y}} {\displaystyle \frac{\mathit{\partial }{w}_0}{\mathit{\partial }x}} {\displaystyle \frac{\mathit{\partial }{w}_0}{\mathit{\partial }y}} {\displaystyle \frac{\mathit{\partial }{\varphi }_x}{\mathit{\partial }x}} {\displaystyle \frac{\mathit{\partial }{\varphi }_x}{\mathit{\partial }y}} {\displaystyle \frac{\mathit{\partial }{\varphi }_y}{\mathit{\partial }x}} {\displaystyle \frac{\mathit{\partial }{\varphi }_y}{\mathit{\partial }y}} {\displaystyle \frac{\mathit{\partial }{\phi }_x}{\mathit{\partial }x}} {\displaystyle \frac{\mathit{\partial }{\phi }_x}{\mathit{\partial }y}} {\displaystyle \frac{\mathit{\partial }{\phi }_y}{\mathit{\partial }x}} {\displaystyle \frac{\mathit{\partial }{\phi }_y}{\mathit{\partial }y}} {\displaystyle \frac{\mathit{\partial }{\phi }_z}{\mathit{\partial }x}} {\displaystyle \frac{\mathit{\partial }{\phi }_z}{\mathit{\partial }y}} u_0 v_0 w_0 {\varphi }_x {\varphi }_y {\phi }_x {\phi }_y {\phi }_x\right\}}^T$, which can be re-stated as $\left\{\vartheta \right\}=\left[G\right]\left\{d\right\}$, where $\left\{d\right\}$ is the vector containing degrees of freedom $\left\{u_0 v_0 w_0 {\varphi }_x {\varphi }_y {\phi }_x {\phi }_y {\phi }_x\right\}$.

The stress–strain relationship can be expressed as

$$\left\{\sigma \right\}=\left[\overline{Q}\right]\left\{\epsilon \right\}$$

(8)

where, $\left\{\sigma \right\}$ and $\left\{\epsilon \right\}$ are the stress and strain vectors, respectively, and $\left[\overline{Q}\right]$ is the transformed rigidity matrix.

Nine-noded isoparametric finite element with eight degrees of freedom per node $\left(u_0 v_0 w_0 {\varphi }_x {\varphi }_y {\phi }_x {\phi }_y {\phi }_z\right)$ is considered. Considering the penalty parameter $\left(\xi \right)$, the Lagrange equation of motion can be expressed as

$${\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{\mathit{\partial }T}{\mathit{\partial }d}}\right)+{\displaystyle \frac{\mathit{\partial }U}{\mathit{\partial }d}}+{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }d}}\left({\displaystyle \frac{\xi }{2}}{\int }_V{\left\{{\phi }_x-{\displaystyle \frac{\mathit{\partial }{w}_0}{\mathit{\partial }x}}\right\}}^2+{\left\{{\phi }_y-{\displaystyle \frac{\mathit{\partial }{w}_0}{\mathit{\partial }y}}\right\}}^{2}dV\right)=0$$

(9)

where $U$ and $T$ are the strain and kinetic energies of the shell, respectively, which can be computed as

$$U={\displaystyle \frac{1}{2}}{\int }_V{\left\{\epsilon \right\}}^T\left\{\sigma \right\}dV, T={\displaystyle \frac{1}{2}}{\int }_V\rho \left\{{\left(\dot{U}\right)}^2+{\left(\dot{V}\right)}^2+{\left(\dot{W}\right)}^2\right\}dV$$

(10)

Substituting the Equation (9) in Equation (8), we can get the governing equation as

$$\left[\left[K^L\right]+{\displaystyle \frac{\xi }{2}}\left[K^p\right]+{\displaystyle \frac{1}{2}}\left[K_1^{NL}\right]+\left[K_2^{NL}\right]+{\displaystyle \frac{1}{2}}\left[K_3^{NL}\right]\right]\left\{d\right\}={\omega }^2\left[M\right]\left\{d\right\}$$

(11)

where $\left[K^L\right]=ʃʃʃ{\left[B\right]}^T{\left[V^L\right]}^T\left[\overline{Q}\right]\left[V^L\right]\left[B\right]dxdydz$ is the global linear stiffness matrix, $\left[K^p\right]$ represents the artificial constraint stiffness matrix, $\left[K_1^{NL}\right]=ʃʃʃ{\left[B\right]}^T{\left[V^L\right]}^T\left[\overline{Q}\right]\left[V^{NL}\right]\left[A\right]\left[G\right]dxdydz$, $\left[K_2^{NL}\right]=ʃʃʃ{\left[G\right]}^T{\left[A\right]}^T{\left[V^{NL}\right]}^T\left[\overline{Q}\right]\left[V^L\right]\left[B\right]dxdydz$, and $\left[K_3^{NL}\right]=ʃʃʃ{\left[G\right]}^T{\left[A\right]}^T{\left[V^{NL}\right]}^T\left[\overline{Q}\right]\left[V^{NL}\right]\left[A\right]\left[G\right]dxdydz$. The temperature-dependent material properties were adopted while evaluating the stiffness matrices.

2.2. Gaussian Process Regression

For a collection of a finite set of random variables $X=\left[x_1, x_2, x_3,\dots ,x_n\right]$ having $n$ input–output pairs and possessing Gaussian distribution as $f=\left[f\left(x_1\right), f\left(x_2\right), f\left(x_3\right),\dots ,f\left(x_n\right)\right]$, the general form of the model built by GPR can be stated as

$$P\left(f\left|X\right.\right)~\mathcal{G}\mathcal{P}\left(f\left|\mu ,K\right.\right)$$

(12)

where $K=k\left(x,x^{\prime }\right)$ represents the covariance function and $k$ denotes the kernel function which governs the smoothness of the GPR function. The mean function for the GP model is expressed as $\mu =\left[m\left(x_1\right),m\left(x_2\right),m\left(x_3\right),\dots ,m\left(x_n\right)\right]$. For a normalized dataset $\left\{f\left(x\right),xϵ{ℝ}^d\right\}$, the mean function equals to zero, then the Gaussian and covariance can be stated as

$$E\left(f\left(x\right)\right)=m\left(x\right)$$

(13)

and

$$Cov\left[f\left(x\right),f\left(x\right)\right]=E\left[\left\{f\left(x\right)-m\left(x\right)\right\}\left\{f\left(x^{\prime }\right)-m\left(x^{\prime }\right)\right\}\right]=k\left(x,x^{\prime }\right)$$

(14)

For predicting the output for any unseen dataset $\left(X_{*}\right)$, the mean function can be expressed as $f\left(X_{*}\right)$. The joint distribution of $f$ and $f_{*}$ can be expressed as

$$\left[\begin{array}{c}f\\ f_{*}\end{array}\right]~ \mathcal{G}\mathcal{P}\left(\left[\begin{array}{c}m\left(X\right)\\ m\left(X_{*}\right)\end{array}\right],\left[\begin{array}{cc}K& K_{*}\\ K_{*}^T& K_{**}\end{array}\right]\right)$$

(15)

For regression distribution $P\left(f_{*}\left|f,X,X_{*}\right.\right)$ over $f_{*}$, the output can be expressed as

$$f,f_{*}\left|X,X_{*}\right.~\mathcal{G}\mathcal{P}\left(K_{*}^T K f,K_{**}-K_{*}^T{K}^{-1}{K}_{*}\right)$$

(16)

GPR can handle complex relationships in data without assuming a specific mathematical formula for how the data is related. It can work with different types of data distributions and is especially helpful when dealing with data that is not linear and has a lot of noise. The output data has the noise as $y=f\left(x\right)+c$ and is identically distributed and independent with covariance that ${\sigma }_n^2$ can be expressed as $Cov\left(y\right)=K+{\sigma }_n^{2}I$. Thus, the predicted value can be stated as

$$\left(\begin{array}{c}y\\ f_{*}\end{array}\right)~\mathcal{G}\mathcal{P}\left(0,\left[\begin{array}{cc}K+{\sigma }_n^{2}I& K_{*}\\ K_{*}^T& K_{**}\end{array}\right]\right)$$

(17)

Thus, the final predictive equation can be expressed as

$$\overline{f_{*}}\left|X,y,X_{*}\right.~\mathcal{G}\mathcal{P}\left(\overline{f_{*}},cov\left(f_{*}\right)\right)$$

(18)

2.3. Bootstrapping

A statistical method of resampling that allows estimation of sampling distribution of statistics by replacing the same from original data. Let the original sample be

$$X=\left[x_1, x_2, x_3,\dots ,x_n\right]$$

(19)

For generating large number of bootstrap samples $B$, each bootstrap sample $X_b^{*}$ can be stated as

$$X_b^{*}=\left[x_{1b}^{*},x_{2b}^{*},x_{3b}^{*},\dots ,x_{nb}^{*}\right]$$

(20)

This data is then fed to a trained GPR surrogate to predict the frequency of the bio-inspired laminated composite shells.

3. Results and Discussion

This section presents the results for the stochastic free vibration behavior of clamped bio-inspired laminated composite conical, hemispherical, and toroid shells with uncertainties in ply orientation for 20- and 28-layered shells. The material properties are taken as temperature-dependent are presented in

Table 2. Temperature-dependent material properties adopted during the present study (G₁₃ = G₁₂, G₂₃ = 0.5 × G₁₂, ν₁₂ = 0.3, ρ = 1627 kg/m³, α₁ = −0.3 × 10⁻⁶/K, and α₂ = 28.1 × 10⁻⁶/K) [55].

Elastic Moduli (GPa)	Temperature (T) (in K)
	300	325	350	375	400	425
E1	130	130	130	130	130	130
E2	9.5	8.5	8.0	7.5	7.0	6.75
G12	6.0	6.0	5.5	5.0	4.75	4.5

. The wide dependence and variation of ply angles can be observed from the same.

3.1. Validation Study

No study is available in the literature regarding the free vibration analysis of bio-inspired helicoidal laminated composite shells. The present results are first validated with the higher-order zigzag theory-based results for natural frequencies of helicoidal laminated composite plates under thermal conditions published by Paruthi et al. [31]. For the same, the cross [0°/90°/6°/84°/12°/78°/…/78°/12°/84°/6°/90°/0°] and double [0°/6°/12°/18°/…/84°/90°]_s helicoidal laminated composite plate is considered with clamped at all edges (a/h = 5). The present results (

Table 3. Validation study.

Temperature (in K)	Source	Double Helicoidal	Cross Helicoidal
300	Present	0.4563	0.5139
	Paruthi et al. [31]	0.4517	0.5052
325	Present	0.4511	0.5096
	Paruthi et al. [31]	0.4489	0.5047
350	Present	0.4384	0.4905
	Paruthi et al. [31]	0.4335	0.4882
375	Present	0.4200	0.4700
	Paruthi et al. [31]	0.4172	0.4705
400	Present	0.4092	0.4607
	Paruthi et al. [31]	0.4079	0.4610
425	Present	0.4002	0.4518
	Paruthi et al. [31]	0.3990	0.4512

) are found to be in good agreement with the results published by Paruthi et al. [31].

After validating the present model, the stochasticity in the natural frequencies of the bio-inspired helicoidal laminated composite shells due to uncertainties in the ply orientation is predicted. For generating the dataset, different geometric parameters, such as the height and radius of the conical shells, the radius of the hemispherical shell, and the inner and outer radius of toroidal shells, along with their different thickness, are adopted for having generalized stochastic behavior of the shells.

3.2. Influence of Uncertainties in Ply Orientation on the Free Vibration Behavior of Bio-Inspired Helicoidal Laminated Composite Conical Shells

In this section, the influence of uncertainties in ply orientation for 20- and 28-layered clamped bio-inspired helicoidal laminated composite conical shells is carried out using the proposed methodology. The present results are compared with the stochastic results obtained using Monte Carlo simulations (MCS) with 10,000 samples. The results for the statistics of stochastic non-dimensional natural frequency are reported in

Table 4. Statistical data for uncertainties in ply angles for 20-layered bio-inspired clamped helicoidal laminated composite conical shells were obtained using the present methodology (700 samples) and MCS simulations (10,000 samples).

Noise Level	$\mathit{T}$	Present					MCS
		Mean	Standard Deviation	Kurtosis	Skewness	Time (s)	Mean	Standard Deviation	Kurtosis	Skewness	Time (s)
0	300	4.0318	0.0566	2.9489	0.0917	10.59	4.0316	0.0562	2.9461	0.0901	420.72
	325	4.0002	0.0591	2.9436	0.1157	10.87	3.9998	0.0585	2.9426	0.1142	419.93
	350	3.9191	0.0618	2.9421	0.1024	10.99	3.9192	0.0614	2.9418	0.1020	421.25
	375	3.8342	0.0645	2.9459	0.0838	11.53	3.8344	0.0645	2.9434	0.0840	423.69
	400	3.7801	0.0670	2.9667	0.0819	10.72	3.7793	0.0671	2.9657	0.0819	427.72
	425	3.7321	0.0684	2.9889	0.0719	11.21	3.7328	0.0688	2.9872	0.0712	425.01
0.01	300	4.0310	0.0562	2.9669	0.1091	11.56	4.0307	0.0559	2.9668	0.1094	439.57
	325	3.9985	0.0581	2.9573	0.1311	11.67	3.9989	0.0582	2.9571	0.1312	440.31
	350	3.9185	0.0611	2.9572	0.1198	12.04	3.9184	0.0611	2.9569	0.1197	431.09
	375	3.8339	0.0640	2.9594	0.1020	11.95	3.8336	0.0642	2.9584	0.1017	428.87
	400	3.7810	0.0675	2.9678	0.0722	11.58	3.7805	0.0672	2.9675	0.0719	429.65
	425	3.7300	0.0690	3.0018	0.0843	11.86	3.7303	0.0689	3.0016	0.0840	430.24
0.05	300	4.0209	0.0573	2.9861	0.1440	12.86	4.0209	0.0574	2.9858	0.1442	452.28
	325	3.9895	0.0597	2.9781	0.1515	12.74	3.9891	0.0597	2.9774	0.1515	453.31
	350	3.9087	0.0621	2.9795	0.1447	12.54	3.9086	0.0624	2.9793	0.1448	451.29
	375	3.8239	0.0655	2.9822	0.1301	12.02	3.8238	0.0653	2.9820	0.1309	456.60
	400	3.7857	0.0725	2.9551	0.0245	12.33	3.7859	0.0725	2.9549	0.0241	455.57
	425	3.7165	0.0748	2.9126	0.1024	12.51	3.7166	0.0748	2.9129	0.1023	456.68
0.1	300	4.0264	0.0630	2.9461	0.0967	12.05	4.0265	0.0629	2.9460	0.0964	483.36
	325	3.9945	0.0649	2.9461	0.1142	12.62	3.9946	0.0645	2.9462	0.1139	485.62
	350	3.9147	0.0674	2.9504	0.1001	12.41	3.9141	0.0675	2.9502	0.1004	482.33
	375	3.8297	0.0705	2.9553	0.0821	12.33	3.8293	0.0706	2.9552	0.0824	489.92
	400	3.8058	0.0690	2.9705	0.0410	12.42	3.8055	0.0690	2.9702	0.0407	487.52
	425	3.7048	0.0707	2.9661	0.1256	11.98	3.7050	0.0709	2.9662	0.1252	488.80
0.5	300	4.0415	0.1360	3.0821	0.1874	11.69	4.0417	0.1367	3.0826	0.1876	492.26
	325	4.0098	0.1358	3.0878	0.2062	11.57	4.0098	0.1362	3.0881	0.2064	491.05
	350	3.9291	0.1371	3.0834	0.1919	12.05	3.9293	0.1379	3.0838	0.1922	493.33
	375	3.8447	0.1388	3.0799	0.1721	11.89	3.8445	0.1393	3.0792	0.1725	489.92
	400	3.6524	0.1048	2.9711	−0.0255	11.67	3.6527	0.1049	2.9715	−0.0251	492.20
	425	3.6832	0.1937	2.0975	−0.1215	12.50	3.6830	0.1942	2.9072	−0.1209	493.36
1	300	3.6351	0.2751	3.2695	−0.4881	11.85	3.6352	0.2757	3.2700	−0.4885	505.57
	325	3.6031	0.2761	3.2614	−0.4774	12.54	3.6033	0.2765	3.2618	−0.4778	506.81
	350	3.5228	0.2747	3.2657	−0.4828	12.65	3.5228	0.2747	3.2657	−0.4827	505.59
	375	3.4382	0.2730	3.2685	−0.4864	12.47	3.4380	0.2731	3.2688	−0.4866	507.81
	400	3.9755	0.2724	2.9013	−0.1400	12.55	3.9753	0.2721	2.9011	−0.1402	509.97
	425	3.6808	0.2933	3.0725	−0.3338	11.27	3.6807	0.2931	3.0723	−0.3339	510.07
2	300	4.7475	0.6601	2.9375	0.0391	11.57	4.7471	0.6598	2.9372	0.0393	531.25
	325	4.7155	0.6631	2.9358	0.0381	11.84	4.7153	0.6633	2.9363	0.0386	530.27
	350	5.4641	0.6647	2.9359	0.0410	12.01	4.6348	0.6645	2.9363	0.0406	534.62
	375	4.5501	0.6651	2.9351	0.0435	12.56	4.5500	0.6659	2.9363	0.0426	538.84
	400	4.3482	0.4717	2.9514	−0.0738	11.89	4.3488	0.4720	2.9516	−0.0741	534.64
	425	2.6632	0.3517	2.9341	−0.1231	12.00	2.6633	0.3517	2.9343	−0.1235	537.71

and

Table 5. Statistical data for uncertainties in ply angles for 28-layered bio-inspired clamped helicoidal laminated composite conical shells were obtained using the present methodology (700 samples) and MCS simulations (10,000 samples).

Noise Level	$\mathit{T}$	Present					MCS
		Mean	Standard Deviation	Kurtosis	Skewness	Time (s)	Mean	Standard Deviation	Kurtosis	Skewness	Time (s)
0	300	4.1304	0.0531	2.9878	−0.0921	13.67	4.1306	0.0530	2.9884	−0.0928	561.08
	325	4.0956	0.0556	2.9991	−0.0442	13.57	4.0957	0.0556	2.9994	−0.0443	564.28
	350	4.0168	0.0590	3.0184	−0.0672	12.98	4.0168	0.0592	3.0182	−0.0673	566.41
	375	3.9340	0.0625	3.0041	−0.0674	13.05	3.9339	0.0628	3.0045	−0.0676	558.94
	400	3.8791	0.0651	2.9994	−0.0642	13.92	3.8797	0.0655	2.9998	−0.0646	564.21
	425	3.8291	0.0665	3.0099	−0.0752	14.05	3.8292	0.0665	3.0096	−0.0753	558.87
0.01	300	4.1355	0.0532	3.0414	−0.1042	13.98	4.1351	0.0531	3.0412	−0.1044	572.89
	325	4.0942	0.0557	2.9983	−0.0442	12.65	4.0941	0.0557	2.9983	−0.0443	573.61
	350	4.0169	0.0598	2.9815	−0.0752	14.30	4.0165	0.0590	2.9807	−0.0754	571.60
	375	3.9312	0.0635	3.0432	−0.0992	13.54	3.9310	0.0631	3.0434	−0.0990	578.92
	400	3.8771	0.0651	3.0368	−0.0956	12.98	3.8774	0.0658	3.0373	−0.0953	581.07
	425	3.8265	0.0668	3.0549	−0.1064	14.05	3.8269	0.0667	3.0548	−0.1059	582.64
0.05	300	4.1171	0.0566	3.0071	−0.1805	12.69	4.1172	0.0566	3.0075	−0.1809	599.64
	325	4.0972	0.0581	3.0236	−0.0139	13.55	4.0977	0.0576	3.0232	−0.0136	608.84
	350	4.0185	0.0544	3.0221	0.0091	12.98	4.0183	0.0548	3.0223	0.0095	611.25
	375	3.9274	0.0628	2.9712	−0.1272	12.04	3.9275	0.0628	2.9711	−0.1270	614.32
	400	3.8736	0.0651	2.9687	−0.1235	12.69	3.8738	0.0655	2.9692	−0.1229	611.25
	425	3.8236	0.0664	2.9734	−0.1337	13.58	3.8234	0.0663	2.9735	−0.1333	612.71
0.1	300	4.1124	0.0493	3.0495	−0.2126	14.68	4.1122	0.0492	3.0491	−0.2125	625.61
	325	4.1400	0.0596	2.9301	−0.0769	12.65	4.1405	0.0598	2.9307	−0.0772	623.84
	350	3.9985	0.0736	2.9391	−0.0328	13.62	3.9986	0.0735	2.9390	−0.0330	6.26.83
	375	3.9176	0.0671	2.9731	−0.1091	12.07	3.9172	0.0663	2.9720	−0.1083	620.58
	400	3.8631	0.0682	2.9688	−0.1057	13.71	3.8636	0.0689	2.9694	−0.1060	625.39
	425	3.8129	0.0691	2.9835	−0.1293	12.67	3.8132	0.0694	2.9831	−0.1289	629.57
0.5	300	4.2669	0.1095	2.9859	−0.0526	12.96	4.2666	0.1091	2.9833	−0.0515	653.98
	325	4.0450	0.1082	2.9426	0.0245	13.25	4.0450	0.1083	2.9424	0.0241	654.71
	350	4.1826	0.1481	3.0868	−0.2876	11.99	4.1828	0.1485	3.0866	−0.2877	659.25
	375	4.1257	0.1248	3.0786	0.2586	12.24	4.1253	0.1254	3.0776	0.2582	657.23
	400	4.0719	0.1269	3.0836	0.2678	12.67	4.0717	0.1263	3.0834	0.2670	660.20
	425	4.0215	0.1289	3.0784	0.2580	12.99	4.0212	0.1286	3.0783	0.2571	658.88
1	300	4.3822	0.2460	2.9209	0.0795	13.51	4.3822	0.2461	2.9207	0.0797	672.54
	325	4.1599	0.2251	3.0181	0.2841	13.02	4.1589	0.2253	3.0185	0.2845	675.20
	350	3.7091	0.1643	2.9983	0.0261	12.69	3.7093	0.1640	2.9980	0.0268	673.45
	375	3.8104	0.2735	2.9553	0.0561	12.40	3.8108	0.2732	2.9551	0.0564	678.98
	400	3.7578	0.2736	2.9552	0.0579	12.36	3.7572	0.2731	2.9548	0.0575	675.24
	425	3.7069	0.2752	2.9548	0.0528	11.89	3.7067	0.2749	2.9544	0.0532	673.34
2	300	4.6512	0.4395	2.9255	−0.0399	13.33	4.6518	0.4397	2.9257	−0.0402	692.52
	325	4.1745	0.5751	2.9327	0.0473	12.69	4.1745	0.5750	2.9329	0.0477	695.87
	350	3.8154	0.4900	2.9413	0.0222	12.06	3.8155	0.4904	2.9412	0.0226	696.66
	375	3.2091	0.4823	2.9853	−0.0742	12.35	3.2092	0.4823	2.9851	−0.0740	692.05
	400	3.1558	0.4835	2.9853	−0.0731	11.80	3.1556	0.4830	2.9851	−0.0738	694.41
	425	3.1051	0.4824	2.9854	−0.0710	12.90	3.1052	0.4827	2.9858	−0.0714	691.06

for 20- and 28-layered shells, respectively, for different levels of noise and temperature (T). The results obtained using the present methodology with only 700 samples are the results obtained using MCS. Also, the time taken by the present methodology is much less when compared to the time taken by the MCS. Increasing the noise level, the mean of the simulated outputs decreases for the 20-layered shell and increases for the 28-layered shell. This indicates the non-linear relationship between ply orientation and frequency. At lower noise levels and near room temperature (T = 300 K), the value of skewness is close to zero, which indicates the even distribution of the frequencies around the mean value. This is also supplemented by the probability distribution curves as given in Figure 2 and Figure 3 for 20- and 28-layered shells, respectively. However, negative skewness is observed for the shell made up of 28 layers, which indicates that the tail on the left side is longer and fatter on the right side (Figure 3). Positive skewness may indicate potential high risks. Therefore, the 20-layered shells are more sensitive to the uncertainties in ply angles compared to the 28-layered shells.

From the probability density functions (PDF) of the non-dimensional natural frequency for clamped 20-layered bio-inspired helicoidal laminated composite conical shells under different thermal conditions and varying noise levels in ply orientation as reported in Figure 2, it has been observed that across all temperatures, the trend is consistent. The PDF is narrowest and highest at noise level 0.00, indicating low uncertainty. As noise levels increase, the distribution broadens and flattens, indicating increased uncertainty in the non-dimensional natural frequency. At higher noise levels, there is a noticeable skewness in the distribution, indicating that the noise is not symmetrically distributed. Figure 3 depicts the PDF for 28-layered shells. Both sets show increased uncertainty with higher noise, but the 28-layered shells seem to maintain a slightly narrower distribution, suggesting better stability and lower variability. However, the effect of temperature on the spread and mean shift is consistent across both sets of data.

The importance scores/sensitivities for different plies vary significantly with temperature changes and are presented in Figure 4 and Figure 5 for 20- and 28-layered shells, respectively. For instance, at lower temperatures (300 K and 325 K), certain plies such as ply10 in the 20-layered and ply 17 in the 28-layered shell show higher importance scores (Figure 4). As the temperature increases to 400 K and 425 K, the importance scores for specific plies (e.g., ply 10, ply 17) become even more pronounced in the 20-layered shell, whereas the pattern is more evenly distributed among various plies in the 28-layered shell. Higher noise levels (1.00 and 2.00) consistently show higher importance scores compared to lower noise levels (0.01, 0.05, 0.10). For both shell types, the zero-noise level (Noise Level 0.00) generally shows the lowest importance scores across all plies, indicating that noise introduces variability and sensitivity in the ply orientations. In 20-layered shells, specific plies (e.g., ply 10, ply 12, ply 17) exhibit high sensitivity across all temperatures and noise levels, suggesting these plies play a critical role in the vibration behavior. In 28-layered shells, the importance scores are more distributed, but certain plies (e.g., ply 10, ply 17, ply 26) still show significant sensitivity, indicating critical plies but with a more distributed influence compared to the 20-layered shells.

3.3. Influence of Uncertainties in Ply Orientation on the Free Vibration Behavior of Bio-Inspired Helicoidal Laminated Composite Hemispherical Shells

This section presents the influence of the uncertainties in the ply angle for clamped bio-inspired helicoidal laminated composite hemispherical shells under thermal conditions with different levels of noise. The results for the same are presented in

Table 6. Statistical data obtained for uncertainties in ply angles for 20-layered bio-inspired clamped helicoidal laminated composite hemispherical shells were obtained using the present methodology (700 samples) and MCS simulations (10,000 samples).

Noise Level	$\mathit{T}$	Present					MCS
		Mean	Standard Deviation	Kurtosis	Skewness	Time (s)	Mean	Standard Deviation	Kurtosis	Skewness	Time (s)
0	300	2.9754	0.0876	2.9781	0.1976	11.05	2.9757	0.0873	2.9789	0.1977	431.56
	325	2.9483	0.0895	2.9770	0.2021	10.87	2.9470	0.0901	2.9771	0.2023	432.98
	350	2.8905	0.0934	2.9765	0.2007	11.17	2.8904	0.0930	2.9765	0.2006	435.61
	375	2.8316	0.0963	2.9765	0.1973	10.98	2.8314	0.0961	2.9760	0.1971	431.28
	400	2.7930	0.0985	2.9751	0.1969	11.25	2.7927	0.0987	2.9750	0.1967	440.27
	425	2.7611	0.1010	2.9751	0.1671	11.64	2.7612	0.1004	2.9747	0.1945	439.89
0.01	300	2.9755	0.0865	2.9584	0.2002	11.32	2.9750	0.0869	2.9590	0.2001	441.25
	325	2.9465	0.0899	2.9561	0.2014	10.58	2.9462	0.0897	2.9550	0.2056	436.57
	350	2.8891	0.0925	2.9562	0.2036	10.63	2.8897	0.0926	2.9556	0.2031	435.95
	375	2.8304	0.0951	2.9575	0.2001	11.78	2.8307	0.0957	2.9562	0.1996	433.65
	400	2.7910	0.0985	3.6858	0.1994	11.64	2.7919	0.0982	2.9553	0.1992	435.63
	425	2.7624	0.1014	2.9551	0.1991	11.99	2.7605	0.0999	2.9556	0.1971	457.32
0.05	300	2.9653	0.0881	2.9591	0.2165	12.04	2.9650	0.0877	2.9585	0.2160	425.95
	325	2.9368	0.0901	2.9531	0.2169	11.60	2.9363	0.0906	2.9536	0.2165	443.14
	350	2.8802	0.0936	2.9557	0.2144	11.87	2.8797	0.0934	2.9548	0.2149	423.56
	375	2.8214	0.0972	2.9568	0.2118	12.30	2.8208	0.0965	2.9562	0.2123	436.56
	400	2.7826	0.0999	2.9550	0.2119	12.51	2.7820	0.0990	2.9553	0.2110	441.69
	425	2.7501	0.1001	2.9561	0.2084	12.59	2.7505	0.1007	2.9560	0.2091	443.82
0.1	300	2.9710	0.0948	2.9381	0.1895	12.69	2.9708	0.0945	2.9378	0.1892	448.92
	325	2.9421	0.0981	2.9364	0.1966	12.57	2.9420	0.0970	2.9367	0.1965	446.05
	350	2.8858	0.1010	2.9371	0.1938	12.36	2.8855	0.0999	2.9372	0.1934	447.25
	375	2.8269	0.1028	2.9389	0.1899	13.41	2.8265	0.1030	2.9380	0.1894	445.02
	400	2.7881	0.1058	2.9384	0.1896	13.08	2.7877	0.1054	2.9379	0.1893	449.87
	425	2.7568	0.1077	2.9386	0.1860	13.55	2.7563	0.1072	2.9383	0.1869	447.67
0.5	300	2.9864	0.1376	3.0415	0.1810	13.98	2.9860	0.1369	3.0399	0.1820	443.51
	325	2.9578	0.1379	3.0364	0.1952	12.47	2.9573	0.1371	3.0360	0.1959	439.81
	350	2.9001	0.1395	3.0342	0.1784	13.60	2.9007	0.1384	3.0337	0.1794	440.04
	375	2.8412	0.1401	3.0324	0.1610	13.84	2.8418	0.1400	3.0318	0.1616	435.65
	400	2.8035	0.1410	3.0299	0.1568	13.66	2.8030	0.1410	3.0294	0.1563	438.57
	425	2.7715	0.1421	3.0294	0.1468	12.99	2.7715	0.1419	3.0287	0.1466	450.09
1	300	2.5796	0.2587	3.2210	−0.4541	12.58	2.5793	0.2598	3.2221	−0.4536	446.58
	325	2.5509	0.2587	3.2147	−0.4440	14.10	2.5506	0.2600	3.2140	−0.4443	449.87
	350	2.4948	0.2574	3.2135	−0.4440	14.36	2.4941	0.2587	3.2131	−0.4440	448.56
	375	2.4356	0.2571	3.2118	−0.4436	15.00	2.4351	0.2575	3.2119	−0.4432	442.36
	400	2.3975	0.2568	3.2095	−0.4341	14.39	2.3963	0.2569	3.2088	−0.4398	440.09
	425	2.3645	0.2562	3.2081	−0.4395	13.85	2.3648	0.2562	3.2079	−0.4390	447.80
2	300	3.6920	0.6874	2.9386	0.0468	13.67	3.6913	0.6885	2.9385	0.0465	443.39
	325	3.6629	0.6915	2.9379	0.0457	13.79	3.6626	0.6913	2.9379	0.0469	438.71
	350	3.6075	0.6929	2.9385	0.0467	13.58	3.6061	0.6928	2.9380	0.0488	450.01
	375	3.5475	0.6938	2.9388	0.0509	13.64	3.5471	0.6944	2.9381	0.0508	449.90
	400	3.5083	0.6957	2.9384	0.0526	15.60	3.5083	0.6961	2.9379	0.0520	440.08
	425	3.4762	0.6974	2.9380	0.0529	14.87	3.4768	0.6971	2.9380	0.0530	439.99

and

Table 7. Statistical data obtained for uncertainties in ply angles for 28-layered bio-inspired clamped helicoidal laminated composite hemispherical shells were obtained using the present methodology (700 samples) and MCS simulations (10,000 samples).

Noise Level	$\mathit{T}$	Present					MCS
		Mean	Standard Deviation	Kurtosis	Skewness	Time (s)	Mean	Standard Deviation	Kurtosis	Skewness	Time (s)
0	300	3.1035	0.0839	2.9641	0.0631	13.81	3.1037	0.0837	2.9643	0.0630	568.81
	325	3.0755	0.0862	2.9594	0.0851	13.51	3.0756	0.0861	2.9595	0.0854	569.45
	350	3.0287	0.0896	2.9573	0.0625	12.87	3.0287	0.0895	2.9570	0.0621	568.99
	375	2.9705	0.0942	2.9587	0.0621	12.65	2.9701	0.0930	2.9592	0.0611	567.25
	400	2.9345	0.0984	2.9586	0.0594	11.98	2.9344	0.0956	2.9578	0.0602	569.98
	425	2.9348	0.0961	2.9585	0.0609	12.40	2.9050	0.0977	2.9580	0.0561	570.34
0.01	300	3.1014	0.0845	2.9541	0.0486	13.65	3.1014	0.0843	2.9542	0.0481	567.50
	325	3.0738	0.0872	2.9438	0.0701	14.00	3.0733	0.0867	2.9443	0.0706	559.84
	350	3.0268	0.0908	2.9481	0.0477	13.87	3.0264	0.0901	2.9476	0.0474	562.52
	375	2.9671	0.0931	2.9475	0.0469	12.83	2.9678	0.0936	2.9503	0.0467	568.78
	400	2.9617	0.0968	2.9487	0.0451	12.31	2.9321	0.0963	2.9492	0.0463	587.56
	425	2.9021	0.0989	2.9484	0.0417	13.69	2.9027	0.0983	2.9506	0.0419	562.52
0.05	300	3.0986	0.0831	2.9481	0.0271	13.51	3.0980	0.0837	2.9482	0.0276	573.52
	325	3.0684	0.0859	2.9448	0.0428	14.21	3.0698	0.0864	2.9452	0.0432	560.47
	350	3.0235	0.0901	2.9449	0.0256	13.72	3.0229	0.0897	2.9445	0.0249	562.34
	375	2.9648	0.0938	2.9461	0.0279	13.60	2.9644	0.0931	2.9458	0.0277	565.95
	400	2.9292	0.0950	2.9458	0.0260	12.88	2.9287	0.0957	2.9450	0.0275	563.34
	425	2.8990	0.0971	2.9451	0.0241	13.07	2.8992	0.0977	2.9454	0.0249	569.84
0.1	300	3.0884	0.0886	2.9389	0.0312	13.99	3.0879	0.0870	2.9382	0.0308	562.35
	325	3.0604	0.0894	2.9347	0.0525	14.21	3.0598	0.0891	2.9343	0.0518	547.58
	350	3.0135	0.0936	2.9284	0.0326	14.37	3.0129	0.0930	2.9276	0.0321	559.54
	375	2.9549	0.0969	2.9351	0.0305	13.26	2.9544	0.0962	2.9340	0.0296	528.62
	400	2.9181	0.0992	2.9326	0.0294	12.99	2.9187	0.0988	2.9327	0.0293	555.54
	425	2.8887	0.1001	2.9342	0.0258	14.04	2.8892	0.1008	2.9338	0.0251	559.84
0.5	300	3.2951	0.1358	3.0980	0.2810	14.69	3.2959	0.1363	3.0988	0.2806	563.34
	325	3.2679	0.1364	3.1061	0.2975	15.21	3.2678	0.1371	3.1065	0.2978	564.78
	350	3.2215	0.1380	3.1054	0.2900	15.17	3.2209	0.1388	3.1057	0.2904	565.95
	375	3.1621	0.1419	3.1054	0.2895	14.85	3.1624	0.1413	3.1049	0.2902	563.38
	400	3.1264	0.1435	3.1068	0.2929	14.60	3.1267	0.1428	3.1061	0.2923	569.94
	425	3.0977	0.1445	3.1056	0.2921	14.08	3.0972	0.1441	3.1053	0.2901	571.21
1	300	2.9815	0.2777	2.9502	0.0825	14.90	2.9812	0.2777	2.9497	0.0823	564.43
	325	2.9531	0.2789	2.9506	0.0858	14.08	2.9531	0.2781	2.9498	0.0852	566.36
	350	2.9068	0.2780	2.9508	0.0841	14.66	2.9062	0.2779	2.9505	0.0837	561.25
	375	2.8475	0.2789	2.9509	0.0889	13.59	2.8477	0.2783	2.9509	0.0884	566.37
	400	2.8120	0.2795	2.9518	0.0908	13.98	2.8120	0.2785	2.9513	0.0902	559.94
	425	2.7856	0.2799	2.9522	0.0919	13.57	2.7825	0.2786	2.9515	0.0917	562.22
2	300	2.3799	0.4788	2.9868	−0.0751	13.87	2.3798	0.4782	2.9863	−0.0752	572.25
	325	2.3525	0.04791	2.9872	−0.0764	13.90	2.3517	0.4791	2.9868	−0.0769	553.31
	350	2.3052	0.04821	2.9878	−0.0748	13.87	2.3047	0.4814	2.9862	−0.0755	555.57
	375	2.2461	0.4819	2.9881	−0.0749	13.98	2.2462	0.4813	2.9869	−0.0753	561.25
	400	2.2114	0.4836	2.9883	−0.0759	13.75	2.2105	0.4822	2.9871	−0.0754	553.31
	425	2.1819	0.4847	2.9889	−0.0761	14.57	2.1811	0.4827	2.9873	−0.0750	563.97

for shells made of 20- and 28-layers, respectively. With an increase in noise level, both the mean and standard deviation generally increase for both 20- and 28-layered configurations. The computational time required for the MCS method is significantly higher compared to the present methodology. Figure 6 and Figure 7 illustrate the PDFs of the non-dimensional natural frequency for clamped bio-inspired helicoidal laminated composite hemispherical shells under different thermal conditions and noise levels in ply orientation with 20- and 28-layers, respectively. In both the 20-layered and 28-layered configurations, the PDFs show that the natural frequency distribution broadens and flattens with increasing noise levels, indicating increased variability. Both configurations show that as noise levels increase, the distributions broaden and flatten, indicating greater variability in natural frequencies. The 28-layered shells consistently exhibit slightly higher and narrower peaks at lower noise levels compared to the 20-layered shells, suggesting less variability in their natural frequencies under low-noise conditions. Across all temperatures, the trend of increasing noise leading to broader distributions is observed, with higher noise levels causing more significant shifts and spreading in the peak frequency. This pattern highlights the substantial impact of noise on the vibrational characteristics of these composite shells, increasing uncertainty in their natural frequencies.

Figure 8 and Figure 9 show the variable importance or sensitivity of the orientation of each ply on the stochastic free vibration behavior of 20- and 28-layered clamped bio-inspired helicoidal laminated composite hemispherical shells at various temperatures (300 K, 325 K, 350 K, 375 K, 400 K, 425 K) and different noise levels (0.00, 0.01, 0.05, 0.10, 0.50, 1.00, 2.00). As the temperature increases from 300 K to 425 K, there is a noticeable increase in the importance scores for certain plies, particularly under higher noise levels. The sensitivity of certain plies (e.g., ply 3, ply 8, ply 12, ply 20 in the 20-layered configuration) tends to increase significantly at higher temperatures and higher noise levels. For the 28-layered configuration, plies such as ply 7, ply 12, ply 18, and ply 25 show significant sensitivity changes at higher temperatures and noise levels. Higher noise levels (1.00 and 2.00) consistently result in higher importance scores across many plies, indicating that noise has a strong impact on the sensitivity of ply orientation. At lower noise levels (0.00, 0.01, 0.05), the importance scores are generally lower, suggesting a more stable system with less sensitivity to ply orientation changes. Noise levels of 0.50 and above significantly amplify the importance scores, highlighting the system’s increased sensitivity to perturbations in ply orientation. The 20-layered configuration shows distinct peaks in importance scores for specific plies, especially under high-noise and high-temperature conditions. The 28-layered configuration exhibits a more distributed sensitivity pattern, with several plies showing notable importance scores under similar conditions. Certain plies in the 28-layered configuration exhibit high sensitivity at moderate noise levels (e.g., ply 7, ply 12, ply 18), which might indicate critical structural roles or positions within the composite structure.

3.4. Influence of Uncertainties in Ply Orientation on Free Vibration Behavior of Bio-Inspired Helicoidal Laminated Composite Toroid Shells

Under this section, a study of the influence of uncertainties in ply orientation on free vibration behavior of bio-inspired helicoidal laminated composite toroid shells is carried out. The statistics of non-dimensional natural frequencies are presented in

Table 8. Statistical data obtained for uncertainties in ply angles for 20-layered bio-inspired clamped helicoidal laminated composite toroidal shells were obtained using the present methodology (700 samples) and MCS simulations (10,000 samples).

Noise Level	$\mathit{T}$	Present					MCS
		Mean	Standard Deviation	Kurtosis	Skewness	Time (s)	Mean	Standard Deviation	Kurtosis	Skewness	Time (s)
0	300	2.2072	0.0378	2.9392	−0.0041	13.89	2.2070	0.0362	2.9357	0.0055	440.52
	325	2.1905	0.0394	2.9212	−0.0026	13.54	2.1898	0.0384	2.9199	−0.0021	443.61
	350	2.1559	0.0397	2.9218	−0.0047	12.87	2.1553	0.0388	2.9209	−0.0050	445.69
	375	2.1179	0.0399	2.9226	−0.0079	12.65	2.1178	0.0391	2.9220	−0.0082	439.57
	400	2.0936	0.0408	2.9220	−0.0111	12.59	2.0931	0.0399	2.9211	−0.0107	441.20
	425	2.0726	0.0412	2.9223	−0.0125	13.60	2.0720	0.0400	2.9216	−0.0124	443.06
0.01	300	2.2130	0.0369	2.9236	−0.0118	13.51	2.2127	0.0367	2.9231	−0.0107	441.05
	325	2.1895	0.0391	2.9106	0.0229	13.15	2.1888	0.0388	2.9093	0.0225	445.95
	350	2.1535	0.0399	2.9126	0.0121	14.05	2.1542	0.0392	2.9103	0.0197	446.67
	375	2.1160	0.0405	2.9106	0.0129	12.96	2.1168	0.0396	2.9097	0.0125	448.28
	400	2.0925	0.0409	2.9109	0.0139	13.56	2.0920	0.0403	2.9099	0.0139	449.65
	425	2.0725	0.0412	2.9112	0.0126	13.99	2.0710	0.0404	2.9104	0.0122	443.32
0.05	300	2.2156	0.0310	2.9578	0.0182	14.51	2.2152	0.0315	2.9560	0.0172	442.85
	325	2.1793	0.0400	2.9551	0.0059	12.78	2.1790	0.0403	2.9543	0.0045	447.77
	350	2.1449	0.0415	2.9568	0.0021	13.58	2.1445	0.0407	2.9554	0.0030	443.92
	375	2.1078	0.0421	2.9572	0.0005	13.00	2.1070	0.0411	2.9567	0.0011	444.52
	400	2.0820	0.0432	2.9548	−0.0021	13.69	2.0823	0.0419	2.9549	−0.0005	448.95
	425	2.0613	0.0436	2.9562	−0.0029	12.59	2.0613	0.0420	2.9556	−0.0015	440.57
0.1	300	2.2038	0.0391	2.9702	−0.0696	13.78	2.2038	0.0378	2.9692	−0.0689	463.15
	325	2.1849	0.0420	2.9312	−0.6569	12.98	2.1846	0.0416	2.9304	−0.0556	451.03
	350	2.1503	0.0423	2.9315	−0.0569	12.56	2.1500	0.0420	2.9309	−0.0582	449.68
	375	2.1129	0.0429	2.9326	−0.0625	12.30	2.1126	0.0422	2.9315	−0.0612	456.21
	400	2.0881	0.0431	2.9321	−0.0631	12.41	2.0878	0.0430	2.9312	−0.0619	451.02
	425	2.0672	0.0439	2.9311	−0.0639	13.20	2.0668	0.0431	2.9314	−0.0636	452.38
0.5	300	2.0968	0.1678	2.9284	0.0392	13.15	2.0963	0.1676	2.9296	0.0380	453.04
	325	2.1999	0.1309	3.1321	0.3035	13.67	2.1996	0.1303	3.1301	0.3026	440.09
	350	2.1656	0.1312	3.1316	0.3031	13.55	2.1650	0.1301	3.1305	0.3025	443.95
	375	2.1275	0.1300	3.1319	0.3025	14.05	2.1276	0.1299	3.1309	0.3019	463.96
	400	2.1020	0.1298	3.1326	0.3049	14.30	2.1028	0.1298	3.1319	0.3041	452.25
	425	2.0815	0.1291	3.1335	0.3045	13.99	2.0818	0.1297	3.1320	0.3035	447.63
1	300	1.9540	0.2801	2.9678	0.0952	13.62	1.9545	0.2808	2.9661	0.0948	446.32
	325	1.7936	0.3015	3.1889	−0.3821	13.04	1.7935	0.3011	3.1882	−0.3817	443.02
	350	1.7592	0.3016	3.1896	−0.3816	13.07	1.7590	0.3012	3.1881	−0.3811	441.01
	375	1.7216	0.3008	3.1899	−0.3812	12.56	1.7215	0.3012	3.1881	−0.3808	447.78
	400	1.6969	0.3005	3.1867	−0.3802	12.88	1.6967	0.3011	3.1878	−0.3799	449.90
	425	1.6748	0.3002	3.1872	−0.3800	14.09	1.6757	0.3010	3.1879	−0.3798	443.03
2	300	2.3499	0.4252	2.9699	0.0126	14.52	2.3492	0.4250	2.9699	0.0126	452.00
	325	2.9059	0.6551	2.9381	−0.0069	14.22	2.9054	0.6553	2.9388	−0.0068	440.05
	350	2.8712	0.6559	2.9378	−0.0061	13.65	2.8708	0.6554	2.9387	−0.0065	453.04
	375	2.8339	0.6561	2.9381	−0.0065	13.99	2.8334	0.6555	2.9386	−0.0062	447.75
	400	2.8092	0.6562	2.9378	−0.0060	14.27	2.8086	0.6559	2.9384	−0.0058	446.31
	425	2.7882	0.6569	2.9375	−0.0054	13.62	2.7876	0.6560	2.9384	−0.0056	441.07

and

Table 9. Statistical data obtained for uncertainties in ply angles for 28-layered bio-inspired clamped helicoidal laminated composite toroidal shells were obtained using the present methodology (700 samples) and MCS simulations (10,000 samples).

Noise Level	$\mathit{T}$	Present					MCS
		Mean	Standard Deviation	Kurtosis	Skewness	Time (s)	Mean	Standard Deviation	Kurtosis	Skewness	Time (s)
0	300	2.2069	0.0361	2.9296	−0.1126	15.92	2.2260	0.0324	2.9821	−0.1159	569.58
	325	2.2145	0.0334	2.9811	−0.1205	15.63	2.2135	0.0339	2.9802	−0.1201	562.35
	350	2.1805	0.0346	2.9836	−0.1259	15.48	2.1795	0.0342	2.9832	−0.1245	561.22
	375	2.1434	0.0349	2.9878	−0.1305	14.58	2.1425	0.0345	2.9863	−0.1293	572.25
	400	2.1205	0.0359	2.9881	−0.1348	14.66	2.1192	0.0351	2.9871	−0.1334	573.36
	425	2.1005	0.0361	2.9959	−0.1315	15.30	2.0984	0.0352	2.9945	−0.1308	571.00
0.01	300	2.2248	0.0362	3.0482	−0.1445	15.06	2.2242	0.0329	3.0479	−0.1444	569.55
	325	2.2111	0.0351	3.0455	−0.1475	15.70	2.2108	0.0343	3.0451	−0.1471	564.52
	350	2.1769	0.0349	3.0491	−0.1518	15.63	2.1768	0.0347	3.0483	−0.1513	568.80
	375	2.1410	0.0351	3.0525	−0.1562	15.33	2.1399	0.0350	3.0517	−0.1559	569.63
	400	2.1170	0.0356	3.0529	−0.1599	14.87	2.1165	0.0355	3.0523	−0.1594	561.57
	425	2.0951	0.0362	3.0351	−0.1415	13.98	2.0944	0.0362	3.0335	−0.1408	555.95
0.05	300	2.2216	0.0401	3.0512	−0.1999	14.08	2.2210	0.0394	3.0506	−0.1901	561.29
	325	2.2077	0.0412	3.0479	−0.1958	14.51	2.2075	0.0408	3.0471	−0.1931	563.34
	350	2.1768	0.0419	3.0502	−0.1963	14.16	2.1736	0.0412	3.0497	−0.1968	561.07
	375	2.1369	0.0413	3.0536	−0.2021	14.30	2.1366	0.0415	3.0526	−0.2010	563.95
	400	2.1136	0.0422	3.0538	−0.2056	13.78	2.1133	0.0421	3.0528	−0.2042	564.48
	425	2.0948	0.0392	2.9912	−0.1468	13.67	2.0944	0.0384	2.9906	−0.1449	564.25
0.1	300	2.2112	0.0405	2.9662	−0.0405	14.56	2.2105	0.0399	2.9651	−0.0407	563.59
	325	2.1968	0.0425	2.9654	−0.0482	14.36	2.1971	0.0411	2.9662	−0.0495	564.58
	350	2.1641	0.0429	2.9658	−0.0569	14.44	2.1631	0.0414	2.9672	−0.0575	563.09
	375	2.1269	0.0432	2.9689	−0.0669	14.60	2.1262	0.0417	2.9682	−0.0654	567.05
	400	2.1035	0.0435	2.9699	−0.0735	14.95	2.1028	0.0423	2.9691	−0.0721	568.82
	425	2.1005	0.0433	2.9478	0.0525	16.05	2.0991	0.0434	2.9471	0.0519	563.38
0.5	300	2.4192	0.1062	2.9858	0.0769	15.86	2.4183	0.1054	2.9847	0.0766	564.89
	325	2.4059	0.1062	2.9789	0.0859	15.78	2.4049	0.1053	2.9780	0.0847	561.37
	350	2.3715	0.0514	2.9858	0.0899	15.05	2.3709	1.0544	2.9881	0.0870	561.28
	375	2.3352	0.1056	2.9889	0.0902	15.69	2.3340	0.1055	2.9893	0.0891	563.22
	400	2.3112	0.1069	2.9899	0.0936	15.99	2.3106	0.1055	2.9906	0.0924	565.59
	425	2.2259	0.1425	2.9921	0.0025	15.48	2.2255	0.1359	3.0077	−0.0820	565.20
1	300	2.1045	0.2896	2.9419	0.0362	15.09	2.1041	0.2895	2.9416	0.0352	563.34
	325	2.0915	0.2869	2.9421	0.0369	16.25	2.0907	0.2862	2.9416	0.0363	567.72
	350	2.0578	0.2899	2.9419	0.0378	14.69	2.0567	0.2894	2.9417	0.0363	569.52
	375	2.0199	0.2899	2.9425	0.0375	14.26	2.0198	0.2892	2.9418	0.0362	568.25
	400	1.9978	0.2896	2.9427	0.0379	14.50	1.9965	0.2892	2.9418	0.0364	563.25
	425	1.8015	0.2519	2.9839	−0.1532	14.63	1.7993	0.2512	2.9836	−0.1541	564.28
2	300	1.5029	24682	2.9772	−0.0739	14.87	1.5026	0.4675	2.9785	−0.0736	566.75
	325	1.4902	0.4692	2.9779	−0.0751	14.52	1.4892	0.4680	2.9786	−0.0744	563.48
	350	1.4569	0.4695	2.9781	−0.0755	13.86	1.4552	0.4684	2.9786	−0.0744	564.87
	375	1.4196	0.4689	2.9789	−0.0756	15.37	1.4183	0.4689	2.9786	−0.0744	565.36
	400	1.4005	0.4702	2.9788	−0.0759	14.58	1.3949	0.4693	2.9786	−0.0747	567.00
	425	1.2069	0.5109	2.9651	−0.0762	15.04	2.2044	0.5102	2.9642	0.1115	568.24

for 20- and 28-layered shells, respectively. The mean values are generally consistent between the present methodology and MCS, with standard deviations slightly higher in MCS. The time taken for MCS is significantly higher than for the present methodology. The mean values for 28-layered shells are generally higher than those for 20-layered shells, with differences also observed in standard deviations, kurtosis, and skewness. Overall, both methodologies provide similar mean values, but MCS is more computationally intensive. Figure 10 and Figure 11 present the PDFs for 20- and 28-layered clamped bio-inspired helicoidal laminated composite toroid shells, respectively, for different levels of noise and temperature conditions. In both figures, the probability density functions exhibit a peak around a non-dimensional natural frequency of approximately 2, with the peak height varying depending on the noise level. As the noise level increases, the peak tends to broaden and decrease in height, indicating a wider distribution of natural frequencies. However, the 28-layered composite shell (Figure 11) generally shows higher peak probability densities compared to the 20-layered composite shell (Figure 10), suggesting a more concentrated distribution of natural frequencies. Additionally, the 28-layered shell exhibits a higher sensitivity to noise, with more pronounced changes in the probability density functions as the noise level increases. This comparison highlights the influence of the number of layers on the dynamic behavior of the composite shells under thermal and noise conditions. Figure 12 shows the sensitivity of the uncertainty in ply angles on the free vibration behavior of toroid shells. For 20-layered shells, it has been observed that the inner layers are found to affect the free vibration behavior of the shell compared to the outer layers. In case of 28-layered shells, the outer layers are found to affect the free vibration behavior of shells containing uncertainties in the ply orientations (Figure 13).

4. Conclusions

The present work proposed the use of a bootstrap-based Gaussian Process Regression machine learning algorithm for predicting the influence of uncertainties in the ply angle on the free vibration behavior of bio-inspired clamped helicoidal laminated composite conical, hemispherical, and toroid shells. For training the surrogate model, quasi-3D shear deformation theory in the framework of the finite element method is used. The following important points are noted from the present study:

• The proposed methodology is found to be computationally efficient compared to the conventional Monte Carlo simulation model, as the present methodology requires very little random data.
• Uncertainty in ply angles widely governs the free vibration behavior of conical, hemispherical, and toroid bio-inspired clamped laminated composite shells.
• Higher temperatures and noise levels tend to increase the sensitivity of the ply orientations, indicating that the structural behavior of these composite shells is more affected by perturbations under such conditions.
• The distribution of sensitivity across plies varies between the 20- and 28-layered configurations, reflecting the complexity and interactions within the composite structure.
• Both 20-layered and 28-layered clamped bio-inspired helicoidal laminated composite conical shells show increased uncertainty with higher noise levels. The 28-layered shells maintain slightly better stability and lower variability in their natural frequency measurements.
• Specific plies within all three configurations exhibit higher sensitivity, suggesting critical layers that might require more precise control or monitoring in practical applications.