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Linear buckling of plates with general boundary conditions

1Geometric stiffness

1.1Geometric stiffness for beams

Using the full nonlinear Green-Lagrange strain relation, the axial strain of a beam can be written as:

εxx=∂u∂x+12[(∂u∂x)2+(∂v∂x)2+(∂w∂x)2]\varepsilon_{xx} = \frac{\partial u}{\partial x} + \frac{1}{2} \left[ \left(\frac{\partial u}{\partial x}\right)^2 + \left(\frac{\partial v}{\partial x}\right)^2 + \left(\frac{\partial w}{\partial x}\right)^2 \right]

The first variation becomes, using ∂(⋅)/∂x=(⋅),x\partial(\cdot)/\partial x = (\cdot)_{,x}

δεxx=δu,x+u,xδu,x+v,xδv,x+w,xδw,x\delta \varepsilon_{xx} = \delta u_{,x} + u_{,x}\delta u_{,x} + v_{,x}\delta v_{,x} + w_{,x}\delta w_{,x}

In terms of nodal displacements (or Ritz coefficients) uˉ\boldsymbol{\bar{u}}:

δεxx=Bˉδuˉ=(BL+BNL)δuˉ\delta \varepsilon_{xx} = \boldsymbol{\bar{B}} \delta \boldsymbol{\bar{u}} = (\boldsymbol{B}_L + \boldsymbol{B}_{NL}) \delta \boldsymbol{\bar{u}}

with:

BL=∂Su∂xBNL=∂u∂x∂Su∂x+∂v∂x∂Sv∂x+∂w∂x∂Sw∂x\boldsymbol{B}_L = \frac{\partial \boldsymbol{S}^u}{\partial x} \boldsymbol{B}_{NL} = \frac{\partial u}{\partial x}\frac{\partial \boldsymbol{S}^u}{\partial x} + \frac{\partial v}{\partial x}\frac{\partial \boldsymbol{S}^v}{\partial x} + \frac{\partial w}{\partial x}\frac{\partial \boldsymbol{S}^w}{\partial x}

The second variation becomes:

δ2εxx=δ2u,x+δu,xδu,x+u,xδ2u,x+δv,xδv,x+v,xδ2v,x+δw,xδw,x+w,xδ2w,x\delta^2 \varepsilon_{xx} = \delta^2 u_{,x} + \delta u_{,x}\delta u_{,x} + u_{,x}\delta^2 u_{,x} + \delta v_{,x}\delta v_{,x} + v_{,x}\delta^2 v_{,x} + \delta w_{,x}\delta w_{,x} + w_{,x}\delta^2 w_{,x}

Note that δ2(⋅),x≪δ(⋅),x\delta^2(\cdot)_{,x} \ll \delta(\cdot)_{,x}, leading to:

δ2εxx=δu,xδu,x+δv,xδv,x+δw,xδw,x\delta^2 \varepsilon_{xx} = \delta u_{,x}\delta u_{,x} + \delta v_{,x}\delta v_{,x} + \delta w_{,x}\delta w_{,x}

with δ2εxx\delta^2 \varepsilon_{xx} defined as:

δ2εxx=δu,xδu,x+δv,xδv,x+δw,xδw,x\delta^2 \varepsilon_{xx} = \delta u_{,x}\delta u_{,x} + \delta v_{,x}\delta v_{,x} + \delta w_{,x}\delta w_{,x}

using the finite element (or Ritz method) shape functions:

δ2εxx=δuˉ⊤[(∂Su∂x)⊤(∂Su∂x)+(∂Sv∂x)⊤(∂Sv∂x)+(∂Sw∂x)⊤(∂Sw∂x)]δuˉ\delta^2 \varepsilon_{xx} = \delta \boldsymbol{\bar{u}}^\top \left[ \left(\frac{\partial \boldsymbol{S}^u}{\partial x}\right)^\top \left(\frac{\partial \boldsymbol{S}^u}{\partial x}\right) + \left(\frac{\partial \boldsymbol{S}^v}{\partial x}\right)^\top \left(\frac{\partial \boldsymbol{S}^v}{\partial x}\right) + \left(\frac{\partial \boldsymbol{S}^w}{\partial x}\right)^\top \left(\frac{\partial \boldsymbol{S}^w}{\partial x}\right) \right] \delta \boldsymbol{\bar{u}} \nonumber

Then:

δuˉ⊤KGδuˉ=∫Ωσ^xxδ2εxx dΩ=δuˉ⊤∫Ωσ^xx(∂Su∂x)⊤(∂Su∂x)+σ^xx(∂Sv∂x)⊤(∂Sv∂x)+σ^xx(∂Sw∂x)⊤(∂Sw∂x)dΩ δuˉ\begin{split} \delta \boldsymbol{\bar{u}}^\top \boldsymbol{K}_G \delta \boldsymbol{\bar{u}} = \int_{\Omega} \hat{\sigma}_{xx}\delta^2 \varepsilon_{xx} \, d\Omega \\ = \delta \boldsymbol{\bar{u}}^\top \int_{\Omega} \hat{\sigma}_{xx} \left(\frac{\partial \boldsymbol{S}^u}{\partial x}\right)^\top \left(\frac{\partial \boldsymbol{S}^u}{\partial x}\right) + \hat{\sigma}_{xx} \left(\frac{\partial \boldsymbol{S}^v}{\partial x}\right)^\top \left(\frac{\partial \boldsymbol{S}^v}{\partial x}\right) + \hat{\sigma}_{xx} \left(\frac{\partial \boldsymbol{S}^w}{\partial x}\right)^\top \left(\frac{\partial \boldsymbol{S}^w}{\partial x}\right) d\Omega \, \delta \boldsymbol{\bar{u}} \end{split}

σ^xx\hat{\sigma}_{xx} can be given or calculated from an initial nodal displacement state u^\boldsymbol{\hat{u}} (pre-buckling of fundamental state):

σ^xx=E(BL+12BNL)u^\hat{\sigma}_{xx} = E \left( \boldsymbol{B}_L + \frac{1}{2}\boldsymbol{B}_{NL} \right) \boldsymbol{\hat{u}} \nonumber

1.2Geometric stiffness for plates

For plates, using the full nonlinear Green-Lagrange strain relation:

εxx=∂u∂x+12[(∂u∂x)2+(∂v∂x)2+(∂w∂x)2]\varepsilon_{xx} = \frac{\partial u}{\partial x} + \frac{1}{2} \left[ \left(\frac{\partial u}{\partial x}\right)^2 + \left(\frac{\partial v}{\partial x}\right)^2 + \left(\frac{\partial w}{\partial x}\right)^2 \right] \nonumber
εyy=∂v∂y+12[(∂u∂y)2+(∂v∂y)2+(∂w∂y)2]\varepsilon_{yy} = \frac{\partial v}{\partial y} + \frac{1}{2} \left[ \left(\frac{\partial u}{\partial y}\right)^2 + \left(\frac{\partial v}{\partial y}\right)^2 + \left(\frac{\partial w}{\partial y}\right)^2 \right] \nonumber
γxy=∂u∂y+∂v∂x+(∂u∂x∂u∂y+∂v∂x∂v∂y+∂w∂x∂w∂y)\gamma_{xy} = \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} + \left( \frac{\partial u}{\partial x}\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x}\frac{\partial v}{\partial y} + \frac{\partial w}{\partial x}\frac{\partial w}{\partial y} \right) \nonumber

The first variation becomes, using ∂(⋅)/∂x=(⋅),x\partial(\cdot)/\partial x = (\cdot)_{,x}:

δεxx=δu,x+u,xδu,x+v,xδv,x+w,xδw,xδεyy=δv,y+u,yδu,y+v,yδv,y+w,yδw,yδγxy=δu,y+δv,x+δu,xu,y+u,xδu,y+δv,xv,y+v,xδv,y+δw,xw,y+w,xδw,y\begin{aligned} \delta \varepsilon_{xx} &= \delta u_{,x} + u_{,x}\delta u_{,x} + v_{,x}\delta v_{,x} + w_{,x}\delta w_{,x} \\ \delta \varepsilon_{yy} &= \delta v_{,y} + u_{,y}\delta u_{,y} + v_{,y}\delta v_{,y} + w_{,y}\delta w_{,y} \\ \delta \gamma_{xy} &= \delta u_{,y} + \delta v_{,x} + \delta u_{,x}u_{,y} + u_{,x}\delta u_{,y} + \delta v_{,x}v_{,y} + v_{,x}\delta v_{,y} + \delta w_{,x}w_{,y} + w_{,x}\delta w_{,y} \end{aligned}

The second variation becomes:

δ2εxx=δ2u,x+δu,xδu,x+u,xδ2u,x+δv,xδv,x+v,xδ2v,x+δw,xδw,x+w,xδ2w,xδ2εyy=δ2v,y+δu,yδu,y+u,yδ2u,y+δv,yδv,y+v,yδ2v,y+δw,yδw,y+w,yδ2w,yδ2γxy=δ2u,y+δ2v,x+δ2u,xu,y+2δu,xδu,y+u,xδ2u,y+δ2v,xv,y+2δv,xδv,y+v,xδ2v,y+δ2w,xw,y+2δw,xδw,y+w,xδ2w,y\begin{aligned} \delta^2 \varepsilon_{xx} &= \delta^2 u_{,x} + \delta u_{,x}\delta u_{,x} + u_{,x}\delta^2 u_{,x} + \delta v_{,x}\delta v_{,x} + v_{,x}\delta^2 v_{,x} + \delta w_{,x}\delta w_{,x} + w_{,x}\delta^2 w_{,x} \\ \delta^2 \varepsilon_{yy} &= \delta^2 v_{,y} + \delta u_{,y}\delta u_{,y} + u_{,y}\delta^2 u_{,y} + \delta v_{,y}\delta v_{,y} + v_{,y}\delta^2 v_{,y} + \delta w_{,y}\delta w_{,y} + w_{,y}\delta^2 w_{,y} \\ \delta^2 \gamma_{xy} &= \delta^2 u_{,y} + \delta^2 v_{,x} + \delta^2 u_{,x}u_{,y} + 2\delta u_{,x}\delta u_{,y} + u_{,x}\delta^2 u_{,y} \\ &+ \delta^2 v_{,x}v_{,y} + 2\delta v_{,x}\delta v_{,y} + v_{,x}\delta^2 v_{,y} + \delta^2 w_{,x}w_{,y} \\ &+ 2\delta w_{,x}\delta w_{,y} + w_{,x}\delta^2 w_{,y} \end{aligned}

Note that δ2(⋅),x≪δ(⋅),x\delta^2(\cdot)_{,x} \ll \delta(\cdot)_{,x} and δ2(⋅),y≪δ(⋅),y\delta^2(\cdot)_{,y} \ll \delta(\cdot)_{,y}, leading to:

δ2εxx=δu,xδu,x+δv,xδv,x+δw,xδw,x\delta^2 \varepsilon_{xx} = \delta u_{,x}\delta u_{,x} + \delta v_{,x}\delta v_{,x} + \delta w_{,x}\delta w_{,x}
δ2εyy=δu,yδu,y+δv,yδv,y+δw,yδw,y\delta^2 \varepsilon_{yy} = \delta u_{,y}\delta u_{,y} + \delta v_{,y}\delta v_{,y} + \delta w_{,y}\delta w_{,y}
δ2γxy=2δu,xδu,y+2δv,xδv,y+2δw,xδw,y\delta^2 \gamma_{xy} = 2\delta u_{,x}\delta u_{,y} + 2\delta v_{,x}\delta v_{,y} + 2\delta w_{,x}\delta w_{,y}

with δ2ε\delta^2 \boldsymbol{\varepsilon} defined as:

δ2εxx=δu,xδu,x+δv,xδv,x+δw,xδw,x\delta^2 \varepsilon_{xx} = \delta u_{,x}\delta u_{,x} + \delta v_{,x}\delta v_{,x} + \delta w_{,x}\delta w_{,x}

δ2εyy=δu,yδu,y+δv,yδv,y+δw,yδw,y\delta^2 \varepsilon_{yy} = \delta u_{,y}\delta u_{,y} + \delta v_{,y}\delta v_{,y} + \delta w_{,y}\delta w_{,y}

δ2γxy=2δu,xδu,y+2δv,xδv,y+2δw,xδw,y\delta^2 \gamma_{xy} = 2\delta u_{,x}\delta u_{,y} + 2\delta v_{,x}\delta v_{,y} + 2\delta w_{,x}\delta w_{,y}

using finite element or Ritz shape functions:

δ2εxx=δuˉ⊤[(∂Su∂x)⊤(∂Su∂x)+(∂Sv∂x)⊤(∂Sv∂x)+(∂Sw∂x)⊤(∂Sw∂x)]δuˉδ2εyy=δuˉ⊤[(∂Su∂y)⊤(∂Su∂y)+(∂Sv∂y)⊤(∂Sv∂y)+(∂Sw∂y)⊤(∂Sw∂y)]δuˉδ2γxy=δuˉ⊤[(∂Su∂x)⊤(∂Su∂y)+(∂Su∂y)⊤(∂Su∂x)+(∂Sv∂x)⊤(∂Sv∂y)+(∂Sv∂y)⊤(∂Sv∂x)+(∂Sw∂x)⊤(∂Sw∂y)+(∂Sw∂y)⊤(∂Sw∂x)]δuˉ\begin{gather} \delta^2 \varepsilon_{xx} = \delta \boldsymbol{\bar{u}}^\top \left[ \left(\frac{\partial \boldsymbol{S}^u}{\partial x}\right)^\top \left(\frac{\partial \boldsymbol{S}^u}{\partial x}\right) + \left(\frac{\partial \boldsymbol{S}^v}{\partial x}\right)^\top \left(\frac{\partial \boldsymbol{S}^v}{\partial x}\right) + \left(\frac{\partial \boldsymbol{S}^w}{\partial x}\right)^\top \left(\frac{\partial \boldsymbol{S}^w}{\partial x}\right) \right] \delta \boldsymbol{\bar{u}} \\ \delta^2 \varepsilon_{yy} = \delta \boldsymbol{\bar{u}}^\top \left[ \left(\frac{\partial \boldsymbol{S}^u}{\partial y}\right)^\top \left(\frac{\partial \boldsymbol{S}^u}{\partial y}\right) + \left(\frac{\partial \boldsymbol{S}^v}{\partial y}\right)^\top \left(\frac{\partial \boldsymbol{S}^v}{\partial y}\right) + \left(\frac{\partial \boldsymbol{S}^w}{\partial y}\right)^\top \left(\frac{\partial \boldsymbol{S}^w}{\partial y}\right) \right] \delta \boldsymbol{\bar{u}} \\ \delta^2 \gamma_{xy} = \delta \boldsymbol{\bar{u}}^\top \left[ \left(\frac{\partial \boldsymbol{S}^u}{\partial x}\right)^\top \left(\frac{\partial \boldsymbol{S}^u}{\partial y}\right) + \left(\frac{\partial \boldsymbol{S}^u}{\partial y}\right)^\top \left(\frac{\partial \boldsymbol{S}^u}{\partial x}\right) \right. \\ + \left(\frac{\partial \boldsymbol{S}^v}{\partial x}\right)^\top \left(\frac{\partial \boldsymbol{S}^v}{\partial y}\right) + \left(\frac{\partial \boldsymbol{S}^v}{\partial y}\right)^\top \left(\frac{\partial \boldsymbol{S}^v}{\partial x}\right) \\ \left. + \left(\frac{\partial \boldsymbol{S}^w}{\partial x}\right)^\top \left(\frac{\partial \boldsymbol{S}^w}{\partial y}\right) + \left(\frac{\partial \boldsymbol{S}^w}{\partial y}\right)^\top \left(\frac{\partial \boldsymbol{S}^w}{\partial x}\right) \right] \delta \boldsymbol{\bar{u}} \end{gather}

Then:

δuˉ⊤KGδuˉ=δuˉ⊤∫Ωσ^xx(∂Su∂x)⊤(∂Su∂x)+σ^xx(∂Sv∂x)⊤(∂Sv∂x)+σ^xx(∂Sw∂x)⊤(∂Sw∂x)\delta \boldsymbol{\bar{u}}^\top \boldsymbol{K}_G \delta \boldsymbol{\bar{u}} = \delta \boldsymbol{\bar{u}}^\top \int_{\Omega} \hat{\sigma}_{xx} \left(\frac{\partial \boldsymbol{S}^u}{\partial x}\right)^\top \left(\frac{\partial \boldsymbol{S}^u}{\partial x}\right) + \hat{\sigma}_{xx} \left(\frac{\partial \boldsymbol{S}^v}{\partial x}\right)^\top \left(\frac{\partial \boldsymbol{S}^v}{\partial x}\right) + \hat{\sigma}_{xx} \left(\frac{\partial \boldsymbol{S}^w}{\partial x}\right)^\top \left(\frac{\partial \boldsymbol{S}^w}{\partial x}\right) \nonumber
+σ^yy(∂Su∂y)⊤(∂Su∂y)+σ^yy(∂Sv∂y)⊤(∂Sv∂y)+σ^yy(∂Sw∂y)⊤(∂Sw∂y)+ \hat{\sigma}_{yy} \left(\frac{\partial \boldsymbol{S}^u}{\partial y}\right)^\top \left(\frac{\partial \boldsymbol{S}^u}{\partial y}\right) + \hat{\sigma}_{yy} \left(\frac{\partial \boldsymbol{S}^v}{\partial y}\right)^\top \left(\frac{\partial \boldsymbol{S}^v}{\partial y}\right) + \hat{\sigma}_{yy} \left(\frac{\partial \boldsymbol{S}^w}{\partial y}\right)^\top \left(\frac{\partial \boldsymbol{S}^w}{\partial y}\right) \nonumber
+τ^xy(∂Su∂x)⊤(∂Su∂y)+τ^xy(∂Su∂y)⊤(∂Su∂x)+ \hat{\tau}_{xy} \left(\frac{\partial \boldsymbol{S}^u}{\partial x}\right)^\top \left(\frac{\partial \boldsymbol{S}^u}{\partial y}\right) + \hat{\tau}_{xy} \left(\frac{\partial \boldsymbol{S}^u}{\partial y}\right)^\top \left(\frac{\partial \boldsymbol{S}^u}{\partial x}\right) \nonumber
+τ^xy(∂Sv∂x)⊤(∂Sv∂y)+τ^xy(∂Sv∂y)⊤(∂Sv∂x)+ \hat{\tau}_{xy} \left(\frac{\partial \boldsymbol{S}^v}{\partial x}\right)^\top \left(\frac{\partial \boldsymbol{S}^v}{\partial y}\right) + \hat{\tau}_{xy} \left(\frac{\partial \boldsymbol{S}^v}{\partial y}\right)^\top \left(\frac{\partial \boldsymbol{S}^v}{\partial x}\right) \nonumber
+τ^xy(∂Sw∂x)⊤(∂Sw∂y)+τ^xy(∂Sw∂y)⊤(∂Sw∂x)dΩ δuˉ+ \hat{\tau}_{xy} \left(\frac{\partial \boldsymbol{S}^w}{\partial x}\right)^\top \left(\frac{\partial \boldsymbol{S}^w}{\partial y}\right) + \hat{\tau}_{xy} \left(\frac{\partial \boldsymbol{S}^w}{\partial y}\right)^\top \left(\frac{\partial \boldsymbol{S}^w}{\partial x}\right) d\Omega \, \delta \boldsymbol{\bar{u}} \nonumber

σ^xx,σ^yy,τ^xy\hat{\sigma}_{xx}, \hat{\sigma}_{yy}, \hat{\tau}_{xy} can be calculated for plate elements from an initial nodal displacement state u^\boldsymbol{\hat{u}} (fundamental or pre-buckling state) as:

{σ^xxσ^yyτ^xy}=1h[ABBD](BL+12BNL)u^\begin{Bmatrix} \hat{\sigma}_{xx} \\ \hat{\sigma}_{yy} \\ \hat{\tau}_{xy} \end{Bmatrix} = \frac{1}{h} \begin{bmatrix} \boldsymbol{A} & \boldsymbol{B} \\ \boldsymbol{B} & \boldsymbol{D} \end{bmatrix} \left( \boldsymbol{B}_L + \frac{1}{2}\boldsymbol{B}_{NL} \right) \boldsymbol{\hat{u}} \nonumber

applying the van K'arm'an simplifications, δ2ε\delta^2 \boldsymbol{\varepsilon} is defined as:

δ2εxx=δw,xδw,xδ2εyy=δw,yδw,yδ2γxy=2δw,xδw,y\begin{aligned} \delta^2 \varepsilon_{xx} &= \delta w_{,x}\delta w_{,x} \delta^2 \varepsilon_{yy} &= \delta w_{,y}\delta w_{,y} \delta^2 \gamma_{xy} &= 2\delta w_{,x}\delta w_{,y} \end{aligned}

using finite element or Ritz shape functions:

δ2εxx=δuˉ⊤[(∂Sw∂x)⊤(∂Sw∂x)]δuˉ\delta^2 \varepsilon_{xx} = \delta \boldsymbol{\bar{u}}^\top \left[ \left(\frac{\partial \boldsymbol{S}^w}{\partial x}\right)^\top \left(\frac{\partial \boldsymbol{S}^w}{\partial x}\right) \right] \delta \boldsymbol{\bar{u}}
δ2εyy=δuˉ⊤[(∂Sw∂y)⊤(∂Sw∂y)]δuˉ\delta^2 \varepsilon_{yy} = \delta \boldsymbol{\bar{u}}^\top \left[ \left(\frac{\partial \boldsymbol{S}^w}{\partial y}\right)^\top \left(\frac{\partial \boldsymbol{S}^w}{\partial y}\right) \right] \delta \boldsymbol{\bar{u}}
δ2γxy=δuˉ⊤[(∂Sw∂x)⊤(∂Sw∂y)+(∂Sw∂y)⊤(∂Sw∂x)]δuˉ\delta^2 \gamma_{xy} = \delta \boldsymbol{\bar{u}}^\top \left[ \left(\frac{\partial \boldsymbol{S}^w}{\partial x}\right)^\top \left(\frac{\partial \boldsymbol{S}^w}{\partial y}\right) + \left(\frac{\partial \boldsymbol{S}^w}{\partial y}\right)^\top \left(\frac{\partial \boldsymbol{S}^w}{\partial x}\right) \right] \delta \boldsymbol{\bar{u}}

Then:

δuˉ⊤KGδuˉ=δuˉ⊤∫Ωσ^xx(∂Sw∂x)⊤(∂Sw∂x)\delta \boldsymbol{\bar{u}}^\top \boldsymbol{K}_G \delta \boldsymbol{\bar{u}} = \delta \boldsymbol{\bar{u}}^\top \int_{\Omega} \hat{\sigma}_{xx} \left(\frac{\partial \boldsymbol{S}^w}{\partial x}\right)^\top \left(\frac{\partial \boldsymbol{S}^w}{\partial x}\right) \nonumber
+σ^yy(∂Sw∂y)⊤(∂Sw∂y)+ \hat{\sigma}_{yy} \left(\frac{\partial \boldsymbol{S}^w}{\partial y}\right)^\top \left(\frac{\partial \boldsymbol{S}^w}{\partial y}\right) \nonumber
+τ^xy(∂Sw∂x)⊤(∂Sw∂y)+τ^xy(∂Sw∂y)⊤(∂Sw∂x)dΩ δuˉ+ \hat{\tau}_{xy} \left(\frac{\partial \boldsymbol{S}^w}{\partial x}\right)^\top \left(\frac{\partial \boldsymbol{S}^w}{\partial y}\right) + \hat{\tau}_{xy} \left(\frac{\partial \boldsymbol{S}^w}{\partial y}\right)^\top \left(\frac{\partial \boldsymbol{S}^w}{\partial x}\right) d\Omega \, \delta \boldsymbol{\bar{u}} \nonumber

σ^xx,σ^yy,τ^xy\hat{\sigma}_{xx}, \hat{\sigma}_{yy}, \hat{\tau}_{xy} can be calculated for plate elements from an initial nodal displacement state u^\boldsymbol{\hat{u}} (fundamental or pre-buckling state) as:

{σ^xxσ^yyτ^xy}=1h[ABBD](BL+12BNL)u^\begin{Bmatrix} \hat{\sigma}_{xx} \\ \hat{\sigma}_{yy} \\ \hat{\tau}_{xy} \end{Bmatrix} = \frac{1}{h} \begin{bmatrix} \boldsymbol{A} & \boldsymbol{B} \\ \boldsymbol{B} & \boldsymbol{D} \end{bmatrix} \left( \boldsymbol{B}_L + \frac{1}{2}\boldsymbol{B}_{NL} \right) \boldsymbol{\hat{u}} \nonumber

2Buckling of a plate using full 3D elasticity

For the 3D elasticity case, the following expression can be used to calculate the geometric stiffness matrix for plates, using van Kármán kinematics:

KG=∭x,y,z[σ^xx(∂Sw∂x)⊤(∂Sw∂x)+σ^yy(∂Sw∂y)⊤(∂Sw∂y)+σ^xy(∂Sw∂x)⊤(∂Sw∂y)+σ^xy(∂Sw∂y)⊤(∂Sw∂x)]dxdydz\begin{split} \mathbf{K}_G = \iiint_{x,y,z} \biggl[ \hat{\sigma}_{xx} \left( \frac{\partial \boldsymbol{S}^w}{\partial x} \right)^\top \left( \frac{\partial \boldsymbol{S}^w}{\partial x} \right) \\ + \hat{\sigma}_{yy} \left( \frac{\partial \boldsymbol{S}^w}{\partial y} \right)^\top \left( \frac{\partial \boldsymbol{S}^w}{\partial y} \right) \\ + \hat{\sigma}_{xy} \left( \frac{\partial \boldsymbol{S}^w}{\partial x} \right)^\top \left( \frac{\partial \boldsymbol{S}^w}{\partial y} \right) \\ + \hat{\sigma}_{xy} \left( \frac{\partial \boldsymbol{S}^w}{\partial y} \right)^\top \left( \frac{\partial \boldsymbol{S}^w}{\partial x} \right) \biggr] dxdydz \end{split}

An example on how to implement buckling of aplate using full 3D elasticity and the Ritz Method can be found in this notebook.

This is also available in this web version of the documentation, see: this page.

3Buckling of a plate using the CLPT, FSDT or TSDT

For all 3 equivalent single-layer (ESL) theories previously discussed, the following expression can be used to calculate the geometric stiffness matrix for plates, using van Kármán kinematics:

KG=∬x,y□N^xx(∂Sw∂x)T(∂Sw∂x)+N^yy(∂Sw∂y)T(∂Sw∂y)+N^xy(∂Sw∂x)T(∂Sw∂y)+N^xy(∂Sw∂y)T(∂Sw∂x)dxdy\begin{split} \boldsymbol{K}_G = \iint\limits_{x,y}^{\square} \widehat{N}_{xx} \left(\frac{\partial \boldsymbol{S}^w}{\partial x}\right)^{\mathsf{T}} \left(\frac{\partial \boldsymbol{S}^w}{\partial x}\right) \\ + \widehat{N}_{yy} \left(\frac{\partial \boldsymbol{S}^w}{\partial y}\right)^{\mathsf{T}} \left(\frac{\partial \boldsymbol{S}^w}{\partial y}\right) \\ + \widehat{N}_{xy} \left(\frac{\partial \boldsymbol{S}^w}{\partial x}\right)^{\mathsf{T}} \left(\frac{\partial \boldsymbol{S}^w}{\partial y}\right) \\ + \widehat{N}_{xy} \left(\frac{\partial \boldsymbol{S}^w}{\partial y}\right)^{\mathsf{T}} \left(\frac{\partial \boldsymbol{S}^w}{\partial x}\right) dxdy \end{split}

An example on how to implement buckling of aplate using the FSDT and the Ritz Method can be found in this notebook.

This is also available in this web version of the documentation, see: this page.

Buckling handbook
Practice, deflection of a plate using the TSDT
Buckling handbook
Practice, 3D semi-analytical plate buckling analysis via the Ritz Method