Stress analysis of the five layer circular sandwich plate subjected to uniform distributed load by layerwise theory along with second order shear deformation theory

ABSTRACT

Layerwise theory (LT) along with the second order shear deformation theory (SSDT) is used to determine the stress distribution in a simply supported circular sandwich plate subjected to a uniformly distributed load. Two adhesive layers are used to adhere the core to their neighbouring functionally graded face sheets. According to the results, the finite element analysis (FEA) findings give almost the same estimations on planar stresses compared to first order shear deformation theory (FSDT) as well as SSDT. Additionally, the out-of-plane shear stresses obtained by FSDT, are slightly different from those of FEA. The differences are decreased by using SSDT.

KEYWORDS:

• Layerwise theory
• second order shear deformation theory
• first order shear deformation theory

Nomenclature

Bn	=	Coefficient for the pressure components $q_0$, Eq. (6b)(6) $q\left(r\right)=\sum _{n=1}^{\mathrm{\infty }}{B}_{n}sin\left(\frac{n\pi }{2R}r\right)\text{break}\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}{B}_n=\left\{\begin{array}{cc}\frac{4q_o}{n\pi }& \mathrm{n}\mathrm{i}\mathrm{s}\mathrm{o}\mathrm{d}\mathrm{d}.\\ 0& \mathrm{n}\mathrm{i}\mathrm{s}\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}.\end{array}\right.$(6)
c	=	Ceramic
E	=	Elastic Modulus
E(k)	=	Modulus of elasticity of the kth layer
G(k)	=	Shear modulus of the kth layer
h	=	Overall thickness of the sandwich plate
h1	=	Thickness of each face sheets
h2	=	Thickness of each adhesive layers
h3	=	Thickness of the core
m	=	Metal
n	=	Material number
$q_0$	=	Uniformly distributed transverse load
$\left[Q_{ij}^{\left(k\right)}\right]$	=	The stiffness matrix of the kth layer
R	=	Circular plate’s radius
u	=	Displacement component along r direction
$u_n$	=	Coefficient for displacement components u, Eq. (5a(5a) $u=\sum _{n=1}^{\mathrm{\infty }}{u}_{n}sin\left(\frac{n\pi }{2R}r\right)$(5a) )
u(k)	=	Displacement Component along r direction in the kth layer.
Vc	=	Volume fraction of ceramic composition in the metal
w	=	Displacement component along z direction
$w_n$	=	Coefficient for displacement components w, Eq. (5a(5a) $u=\sum _{n=1}^{\mathrm{\infty }}{u}_{n}sin\left(\frac{n\pi }{2R}r\right)$(5a) )
w(k)	=	Displacement Component along z direction in the kth layer
${\gamma }_{rz}^{(k)}$	=	The coefficients of the strain r-z for the kth layer
${\gamma }_{r\theta }^{(k)}$	=	The coefficients of the strain r-θ for the kth layer
${\epsilon }_r^{(k)}$	=	The coefficients of the strain r for the kth layer
${\epsilon }_z^{(k)}$	=	The coefficients of the strain z for the kth layer
${\epsilon }_{\theta }^{(k)}$	=	The coefficients of the strain θ for the kth layer
${\nu }^{(k)}$	=	Poisson’s ratio of the kth layer
$\pi$	=	Potential energy
${\sigma }_r$	=	Normal stress along r direction
${\sigma }_{\theta }$	=	Normal stress along θ direction
${\tau }_{rz}$	=	Shear stress along r-z plane
${\varphi }_r^{(k)}n$	=	Coefficient for the curvature component, Eq. (5c(5c) ${\varphi }_r^{\left(k\right)}={\displaystyle \sum _{n=1}^{\infty }\varphi }{}_r{}^{\left(k\right)}\mathrm{s}in\left(\frac{n\pi }{2R}r\right)$(5c) )
${\psi }_r^{(k)}n$	=	Coefficient for the curvature component, Eq. (5d(5d) ${\psi }_r^{\left(k\right)}={\displaystyle \sum _{n=1}^{\infty }{\psi }_r^{\left(k\right)}}n\sin \left(\frac{n\pi }{2R}r\right)$(5d) )

Acknowledgments

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Notes on contributors

Hamed Raissi

Hamed Raissi has completed Ph.D and M.Sc of mechanical engineering and B.Sc of aerospace engineering. Currently, he teaches mechanical engineering courses in Shahid Chamran University of Ahvaz. He graduated his B.Sc, M.Sc and Ph.D at Amirkabir University (2007), Shahid Beheshti (2009) and Shahid Chamran University of Ahvaz (2018), respectively.

Mohammad Shishehsaz

Mohammad Shishehsaz is a professor of mechanical engineering, Shahid Chamran University of Ahvaz.

Shapour Moradi

Shapour Moradi is a professor of mechanical engineering, Shahid Chamran University of Ahvaz.