3-Node Shell Elements

As for the four node shell element, a simple linear Mindlin Plate element formulation is used. Likewise, the use of one integration point and rigid body motion given by the time evolution of the local reference frame is applied. There is no hourglass mode in case of one integration point.

Local Reference Frame

The local reference frame for the three node shell element is shown in 図 1.

The vector normal to the plane of the element is defined as:(1)

n = \frac{g_{1} \times g_{2}}{‖ g_{1} \times g_{2} ‖}

The vector defining the local x direction is defined as edge 1-2:(2)

t_{1} = \frac{g_{1}}{‖ g_{1} ‖}

Hence, the vector defining the local y direction is found from the cross product of the two previous vectors:(3)

t_{2} = n \times t_{1}

Time Step

The characteristic length for computing the critical time step is defined by:(4)

L = \frac{2 a r e a}{\max (\bar{12}, \bar{23}, \bar{31})}

Three Node Shell Shape Functions

The three node shell has a linear shape functions defined as:(5)

ϕ_{1} = a_{1} + b_{1} x + c_{1} y

(6)

ϕ_{2} = a_{2} + b_{2} x + c_{2} y

(7)

ϕ_{3} = a_{3} + b_{3} x + c_{3} y

These shape functions are used to determine the velocity field in the element:(8)

v_{x} = \sum_{I = 1}^{3} ϕ_{I} v_{x I}

(9)

v_{y} = \sum_{I = 1}^{3} ϕ_{I} v_{y I}

(10)

v_{z} = \sum_{I = 1}^{3} ϕ_{I} v_{z I}

(11)

ω_{x} = \sum_{I = 1}^{3} ϕ_{I} ω_{x I}

(12)

ω_{y} = \sum_{I = 1}^{3} ϕ_{I} ω_{y I}

(13)

\frac{\partial v_{x}}{\partial x} = \sum_{I = 1}^{3} \frac{\partial ϕ_{I}}{\partial x} v_{x I}

(14)

\frac{\partial v_{x}}{\partial y} = \sum_{I = 1}^{3} \frac{\partial ϕ_{I}}{\partial y} v_{x I}

Membrane Behavior

The method used to calculate the membrane behavior and the membrane strain rates is exactly the same as that used for four node shell elements (Membrane Behavior).

Bending Behavior

The bending behavior and calculation of the bending strain rates (or curvature rates) is the exact same method used for four node shell elements (Bending Behavior).

Strain Rate Calculation

The strain rate calculation for the three node shell is the same as the method used for the four node shell. However, only three nodes are accounted for. This makes the vectors and matrices smaller. The overall membrane strain rate is calculated by:(15)

{\dot{e}}_{m} = {{\dot{e}}_{x}, {\dot{e}}_{y}, 2 {\dot{e}}_{x y}}

(16)

{v}_{m} = {v_{x}^{1}, v_{y}^{1}, v_{x}^{2}, v_{y}^{2}, v_{x}^{3}, v_{y}^{3}}

(17)

{\dot{e}}_{m} = {[B]}_{m} {v}_{m}

Where the

{[B]}_{m}

matrix of shape function gradients is defined as:(18)

{[B]}_{m} = [\begin{matrix} \frac{\partial ϕ_{1}}{\partial x} & 0 & \frac{\partial ϕ_{2}}{\partial x} & 0 & 0 & 0 \\ 0 & \frac{\partial ϕ_{1}}{\partial y} & 0 & \frac{\partial ϕ_{2}}{\partial y} & 0 & \frac{\partial ϕ_{3}}{\partial y} \\ \frac{\partial ϕ_{1}}{\partial y} & \frac{\partial ϕ_{1}}{\partial x} & \frac{\partial ϕ_{2}}{\partial y} & \frac{\partial ϕ_{2}}{\partial x} & \frac{\partial ϕ_{3}}{\partial y} & \frac{\partial ϕ_{3}}{\partial x} \end{matrix}]

Where $\frac{\partial ϕ_{3}}{\partial x} = 0$ for a shell element.

The overall bending strain or curvature rate is computed by:(19)

{\dot{e}}_{b} = {{\dot{k}}_{x}, {\dot{k}}_{y}, 2 {\dot{k}}_{x y}, 2 {\dot{e}}_{z x}, 2 {\dot{e}}_{y z}}

(20)

{v}_{b} = {ω_{y}^{1}, - ω_{x}^{1}, ω_{y}^{2}, - ω_{\dot{x}}^{2} ω_{y}^{3}, - ω_{x}^{3}, v_{z}^{1}, v_{z}^{2}, v_{z}^{3}}

(21)

{\dot{e}}_{b} = {[B]}_{b} {v}_{b m}

Where,(22)

{[B]}_{} = [\begin{matrix} \frac{\partial ϕ_{1}}{\partial x} & 0 & \frac{\partial ϕ_{2}}{\partial x} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \frac{\partial ϕ_{1}}{\partial y} & 0 & \frac{\partial ϕ_{2}}{\partial y} & 0 & \frac{\partial ϕ_{3}}{\partial y} & 0 & 0 & 0 \\ \frac{\partial ϕ_{1}}{\partial y} & \frac{\partial ϕ_{1}}{\partial x} & \frac{\partial ϕ_{2}}{\partial y} & \frac{\partial ϕ_{2}}{\partial x} & \frac{\partial ϕ_{3}}{\partial y} & \frac{\partial ϕ_{3}}{\partial x} & 0 & 0 & 0 \\ ϕ_{1} & 0 & ϕ_{2} & 0 & ϕ_{3} & 0 & \frac{\partial ϕ_{1}}{\partial x} & \frac{\partial ϕ_{2}}{\partial x} & \frac{\partial ϕ_{3}}{\partial x} \\ 0 & ϕ_{1} & 0 & ϕ_{2} & 0 & ϕ_{3} & \frac{\partial ϕ_{1}}{\partial y} & \frac{\partial ϕ_{2}}{\partial y} & \frac{\partial ϕ_{3}}{\partial y} \end{matrix}]

Mass and Inertia

The three node shell element is considered as an element with a lumped mass. Its mass is defined as:(23)

m = ρ A t

Where,

$ρ$: Material density
$t$: Shell thickness
$A$: Reference plane surface area

The mass moment of inertia about all axes is the same:(24)

I_{x x} = m (\frac{2 A}{6} + \frac{t^{2}}{12})

(25)

I_{z z} = I_{y y} = I_{x x}

(26)

I_{x y} = 0

When nodal masses need to be calculated, the distribution is determined by the shape of the element as shown in 図 2.

The mass and inertia at node

i

are given by:(27)

m_{i} = \frac{α_{i}}{π} m; I_{i} = \frac{α_{i}}{π} I