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.


図 1. Node Shell Local Reference Frame
The vector normal to the plane of the element is defined as:(1)
n = g 1 × g 2 g 1 × g 2
The vector defining the local x direction is defined as edge 1-2:(2)
t 1 = 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 × t 1

Time Step

The characteristic length for computing the critical time step is defined by:(4)
L = 2 a r e a max ( 12 ¯ , 23 ¯ , 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 = I = 1 3 ϕ I v x I
(9)
v y = I = 1 3 ϕ I v y I
(10)
v z = I = 1 3 ϕ I v z I
(11)
ω x = I = 1 3 ϕ I ω x I
(12)
ω y = I = 1 3 ϕ I ω y I
(13)
v x x = I = 1 3 ϕ I x v x I
(14)
v x y = I = 1 3 ϕ I 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)
{ e ˙ } m = { e ˙ x , e ˙ y , 2 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)
{ e ˙ } m = [ B ] m { v } m
Where the [ B ] m matrix of shape function gradients is defined as:(18)
[ B ] m = [ ϕ 1 x 0 ϕ 2 x 0 0 0 0 ϕ 1 y 0 ϕ 2 y 0 ϕ 3 y ϕ 1 y ϕ 1 x ϕ 2 y ϕ 2 x ϕ 3 y ϕ 3 x ]

Where ϕ 3 x = 0 for a shell element.

The overall bending strain or curvature rate is computed by:(19)
{ e ˙ } b = { k ˙ x , k ˙ y , 2 k ˙ x y , 2 e ˙ z x , 2 e ˙ y z }
(20)
{ v } b = { ω y 1 , ω x 1 , ω y 2 , ω x ˙ 2 ω y 3 , ω x 3 , v z 1 , v z 2 , v z 3 }
(21)
{ e ˙ } b = [ B ] b { v } b m
Where,(22)
[ B ] = [ ϕ 1 x 0 ϕ 2 x 0 0 0 0 0 0 0 ϕ 1 y 0 ϕ 2 y 0 ϕ 3 y 0 0 0 ϕ 1 y ϕ 1 x ϕ 2 y ϕ 2 x ϕ 3 y ϕ 3 x 0 0 0 ϕ 1 0 ϕ 2 0 ϕ 3 0 ϕ 1 x ϕ 2 x ϕ 3 x 0 ϕ 1 0 ϕ 2 0 ϕ 3 ϕ 1 y ϕ 2 y ϕ 3 y ]

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 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaaaa@36BC@
Shell thickness
A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaaaa@36BC@
Reference plane surface area
The mass moment of inertia about all axes is the same:(24)
I x x = m ( 2 A 6 + 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.


図 2. Mass Distribution
The mass and inertia at node i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGPbaaaa@39C5@ are given by:(27)
m i = α i π m ; I i = α i π I