FEM analysis of shell structures CMCE Lecture 5/1, Civil Engineering, II cycle, specialty BEC

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FEM analysis of shell structures

CMCE Lecture 5/1, Civil Engineering, II cycle, specialty BEC

Adam Wosatko

Institute for Computational Civil Engineering Civil Engineering Department, Cracow University of Technology

e-mail: JPamin@L5.pk.edu.pl

With thanks to:

Maria Radwańska, Anna Stankiewicz, Jerzy Pamin, Piotr Pluciński

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Layout

Membranes (plane stress) Deep cantilever beam

Plates – conforming FE Conforming FE Square plate Corner plate

Shells

Some theoretical aspects Shell FE

Cylinder container

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FE Q4 for membranes

Quadrilateral

u3

v₃ v4

u4

b

η = 2 y /b

ξ = 2 x /a X Y

a

u2

v2

v1

u1

4 3

1 2

NDOFN = 2 NNE = 4

NDOFE = NDOFN × NNE = 8

Displacement vectors for node and element:

qn= {un, vn}^T

q^e= {u1, v1|u2, v2|u3, v3|u4, v4}^T for n = 1, ..., NNE , e = 1, ..., NE

For approximation ofboth displacementsu(ξ, η) i v (ξ, η) we use shape functions N_i, i = 1, 2, 3, 4, which are bilinear (liniear in each direction for standard coordinates ξ, η ∈ [−1, +1]):

uⁿ(ξ, η)_(2×1)= {u(ξ, η), v (ξ, η)} = Nⁿ_(2×8)q^{e n}_(8×1) → u(ξ, η) = N₁u₁+ N₂u₂+ N₃u₃+ N₄u₄

v (ξ, η) = N₁v₁+ N₂v₂+ N₃v₃+ N₄v₄

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Bilinear shape functions

for quadrilateral Q4

N₁=¹₄(1 − ξ)(1 − η) N₂= ¹₄(1 + ξ)(1 − η)

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Stiffness matrix of FE Q4 for membranes

Stiffness matrix:

k^{e n}_(8×8)= Z Z

A^e

B^nT_(8×3)Dⁿ_(3×3)Bⁿ_(3×8)dA

Kinematic relation matrix:

Bⁿ_(3×8)=







∂N₁

∂x 0 ... ^∂N_∂x⁴ 0 0 ^∂N_∂y¹ ... 0 ^∂N_∂y⁴

∂N1

∂y

∂N1

∂x ... ^∂N_∂y⁴ ^∂N_∂x⁴







Constitutive matrix:

Dⁿ_(3×3)= Dⁿ





1 ν 0

ν 1 0

0 0 ^1−ν₂





Membrane stiffness: Dⁿ= _1−ν^{E h}2

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Deep cantilever beam (ROBOT)

Membrane example

Mesh 4 x 16 Deformation

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Deep cantilever beam

Contour plots

displacement UX [cm] displacement UY [cm]

stress σx [MPa] stress τxy [MPa]

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Rectangular FE for plates

Rectangular, conforming

Degrees of freedom of a node:

q_w

[4×1]

= {w , ϕ_x, ϕ_y, χ}_w= {w , ∂w /∂y , −∂w /∂x , ∂²w /∂x ∂y }_w Degrees of freedom of an element:

q^e

[16×1]

= {q₁, q₂, q₃, q₄} = {w1, ϕ_{x 1}, ϕ_{y 1}, χ₁|w2. . . | . . . | . . . χ₄}

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Approximation of deflection field

Dimensionless surface coordinates: ξ = 2 ^x_a − 1, η = 2 ^y_b − 1 Polynomial approximation:

w (ξ, η) = (α1+ α2ξ + α3ξ²+ α4ξ³)(β1+ β2η + β3η²+ β4η³) = C1+

C2ξ + C3η +

C4ξ²+ C5ξη + C6η²+

C7ξ³+ C8ξ²η + C9ξη²+ C10η³+ C11ξ³η + C12ξ²η²+ C13ξη³+ C₁₄ξ³η²+ C₁₅ξ²η³+

C₁₆ξ³η³ u^e

[1×1]= {w (ξ, η)} = N

[1×16](ξ, η) · q^e

[16×1]

=

= {N₁¹, N₁², N₁³, N₁⁴, |N₂¹. . . | . . . | . . . N₄⁴} · {w₁, ϕ_{x 1}, ϕ_{y 1}, χ₁|w₂. . . | . . . | . . . χ₄}^T

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Bicubic shape functions

for first node of rectangular plate FE

N₁¹ corresponds to w₁ N₁²corresponds to ϕ_{x 1} N₁³corresponds to ϕ_{y 1}

N₁⁴corresponds to χ₁

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Stiffness matrix of FE Q4 for plates

Stiffness matrix:

k^{e m}_(16×16) = Z Z

A^e

B^mT_(16×3)D^m_(3×3)B^m_(3×16)dA

Kinematic relation matrix:

B^m_(3×16)=







−^∂_∂x²^N2¹¹ −^∂_∂x²^N2¹² −^∂_∂x²^N2¹³ −^∂_∂x²^N2¹⁴ ... −^∂_∂x²^N2⁴⁴

−^∂_∂y²^N₂¹¹ −^∂_∂y²^N₂¹² −^∂_∂y²^N₂¹³ −^∂_∂y²^N₂¹⁴ ... −^∂_∂y²^N₂⁴⁴

−2_{∂x ∂y}^∂²^N¹¹ −2_{∂x ∂y}^∂²^N²¹ −2_{∂x ∂y}^∂²^N³¹ −2_{∂x ∂y}^∂²^N⁴¹ ... −2^∂_{∂x ∂y}²^N⁴⁴







Constitutive matrix:

D^m_(3×3)= D^m





1 ν 0

ν 1 0

0 0 ^1−ν₂





Bending stiffness: D^m= _12(1−ν^Eh³ 2)

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Square plate

5 7

4 K

Y

Z

X

Data:

I dimensions LX = LY = 3.0 m

I thickness h = 0.12 m

I load intensity

pz = q = −14.4 kN/m²

I Young modulus E = 32.0 · 10⁶ kPa

I Poisson coefficient ν = 0 Bending stiffness D^m=_12(1−ν^Eh³ ₂₎ = 4608 kNm.

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Sign convention – ROBOT

Bending moments and transverse forces

Axis 0Z is oriented upwards, so positive bending moments cause tensile stresses at the top surface.

z

x

y tx

mx my

ty

m_yx mxy

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Square plate – results

w7 mx 7 mx 5 my 5 my 4

[mm] _kNm

m

_kNm

m

_kNm

m

_kNm

m

Eng. comp. -2.303 -11.37 -6.58 -2.19 15.23 Package Mesh

ALGOR 8 × 8 -2.316 -11.27 -6.53 -2.15 13.20 GRAITEC 8 × 8 -2.344 -11.51 -6.64 -2.31 12.27 ROBOT 8 × 8 -2.355 -11.63 -6.73 -2.30 14.96 ALGOR 32 × 32 -2.344 -11.39 -6.59 -2.19 14.75 GRAITEC 32 × 32 -2.392 -11.44 -6.66 -2.18 14.47 ROBOT 32 × 32 -2.394 -11.50 -6.67 -2.21 15.19 ROBOT 128 × 128 -2.410 -11.47 -6.68 -2.16 15.29

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Square plate – results

Contour plots mx and my

mx

my

ALGOR GRAITEC ROBOT

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Corner plate (ROBOT) – data, applied meshes

I length of segment L = 4.0 m

I Young modulus E = 10.0 · 10⁶kPa

I Poisson coefficient ν = 0.17 Deformation

Regular mesh Mesh refined at corner

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Moments m

x

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Moments m

y

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Moments m

xy

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Principal stress directions – top surface

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How many times should mesh be refined?

Element Concave corner mesh mx ,max= my ,maxkNm

m

regular 38.12

refined once 47.22 refined many times 133.93

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Engineering shell structures

Curvilinear roofs

http://www.ketchum.org Drilling vessel supports

Branch shells

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Shell classification – slenderness

Shells:

I) thin → apply Kirchhoff-Love theory,

II) moderately thin → apply Mindlin-Reissner theory,

III) thick → compute 3D continuum model, e.g. problem of thick-walled pipe.

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Three-parameter shell theory

Kirchhoff-Love – thin shells

Thin shell

Thickness is much smaller than minimum curvature radius:

h R_min < 1

20÷ 1 30. Kinematic hypothesis →

independent kinematic fields in shells are three displacements:

u1(ξ1, ξ2), u2(ξ1, ξ2), w (ξ1, ξ2), in local directions (e1, e2, n), introduced at point of middle surface.

Static hypothesis →

stresses σ_n, τ_1n, τ_2n in thin shells are much smaller than others.

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Boundary conditions

Boundary conditions (static and/or kinematic) can be imposed:

I at points,

I along lines,

I on surfaces.

At edge points we can apply boundary loads which correspond to local boundary coordinates.

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Types of kinematic constraints

Type of kinematic constraints

Free edge → different displacement constraints → fully clamped edge

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Membrane state in shells

Membrane state – stress distribution along thickness is homogeneous.

This state is optimal in shell from the viewpoint of material use.

However, we must ensure suitable boundary conditions and loading.

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Disturbances of membrane state

induce bending state

I Bending moments and transverse forces increase quickly as their origin is approached

Membrane state and its disturbances

p

boundary conditions boundary loads not smooth changes of thickness

p p

p

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Shell FEs – classification

I FEs based on shell structures theory

I FEs 1D – geometrically one-dimensional

I Flat FEs 2D – three- or four-noded

I FEs 2D – curvilinear, based on Kirchhoff-Love theory for thin shells or Mindlin-Reissner theory for moderately thin shells

I so-called degenerate FEs, based on equations for 3D continuum, but with hypotheses as adopted for shells, in accordance with

five-parametric Mindlin-Reissner theory

I FEs for 3D solids – thick shells We consider:

I number of approximated fields in FE,

I number and type of degrees of freedom at node,

I formulas for internal energy and types of integration for stiffness matrix and external load vector.

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Shell FE 1D

for axisymmetric problems

Geometrically one-dimensional finite element models meridian of

axisymmetric shell, but all functions are developed in trigonometric series in circumferential direction.

u(ξ, Θ) =



 u1

u2

w



=



 u v w



=

=







J

P

j =1

u^j(ξ) cos(j Θ)

J

P

j =1

v^j(ξ) sin(j Θ)

J

Pw^j(ξ) cos(j Θ)







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Flat FE 2D for shells

three- and four-noded

Flat finite elements for shells with 3 or 4 nodes are formulated as a result of superposition of membrane and bending states.

+ =

We obtain the following degrees of freedom in node:

qⁿ_w= {u, v }_w q^m_w= {w , ϕ₁, ϕ₂}w q_w= {qⁿq^m}w= {u, v , w , ϕ₁, ϕ₂}w

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Flat FE 2D for shells

implmented in ROBOT package

We should introduce third rotational degree of freedom in shells, especially if we compute shells with sharp edges or branching: ϕ_nis this sixth degree of freedom, hence:

qw= {u, v , w , ϕ1, ϕ2| ϕn}w.

Angleϕnis rotation around the normal to middle surface at node and is useful in tranformation of complete DOF set into other coordinate

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Curvilinear shell FE 2D

based on classical three-parameter theory for thin shells

Approximation of displacement field (3 translation variables):

u(ξ₁, ξ₂) = {u₁(ξ₁, ξ₂), u₂(ξ₁, ξ₂), w (ξ₁, ξ₂)} =

= {u(ξ₁, ξ₂), v (ξ₁, ξ₂), w (ξ₁, ξ₂)} = N

[3×NDOFE ]

· q^e

[NDOFE ×1]

DOF vector at node:

qw= {u, v , w , ϕ1, ϕ2}w or qw= {u, v , w , ϕ1, ϕ2| ϕn}w

NDOFN = 6 NNE = 6 NDOFE = 36

NDOFN = 6 NNE = 8 NDOFE = 48

Elastic energy (internal):

U = Uⁿ+ U^m=¹₂RR

A

e^nTsⁿ dA + ¹₂RR

A

e^mTs^m dA

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Cylinder container

Fourth-order displacement differential equation for axisymmetric membrane-bending state and its solution

w^IV(x ) + 4β⁴w (x ) = 1

D^m[pn+ ν R(

Z

p1dx + C1)]

where: β = r 1

Rh ·p⁴

3(1 − ν²)

w (x ) = w^o(x )

| {z }

general solution

+ w (x )

| {z }

particular solution

=

= e^−βx[B₁cos(βx ) + B₂sin(βx )]

| {z }

quickly increase with x

+ e^+βx[B₃cos(βx ) + B₄sin(βx )]

| {z }

quickly decrease with x

+ w (x )

w (x ) – normal displacement for membrane axisymmetric state w^o(x ) – normal displacement for local bending state

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Cylinder container

Geometry, material, loading

I height L = H = 20.0 m

I radius R = 10.0 m

I Young modulus E = 15.0 · 10⁶kPa

I Poisson coefficient ν = ¹₆≈ 0.1667

I dead load

p₁= −γ_bh = −24 · 0.12 = −2.88 kPa

I hydrostastic pressure

pn= γc(L − x ) = 10(20 − x ) kPa

I bending stiffness

D^m=_12(1−ν^Eh³ 2) = 2221.7 kNm

I boundary conditions w (0) = 0, w⁰(0) = 0, m1(L) = 0, t1(L) = 0

Long shell?

β = 1.193 _m¹, λ = ^π_β ≈ 2.63 m λ - length of half-wave 3 · λ < L ?

≈ 7.9 m < 20 m ⇒ YES

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Cylinder container – membrane state

Analytical solution

Normal dispalcement w

0 5 10 15 20

0 0.2 0.4 0.6 0.8 1 1.2

x [m]

w(x) [cm]

Meridional force n1 Circumferential force n2

10 15 20

x [m] 10

15 20

x [m]

Normal displacement function:

w (x ) = w (x ) = R

Eh(γ_cR + νγ_bh)(L−x )

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Cylinder container – disturbance of membrane state

Analytical solution

Normal displacement w Transverse force t1 Meridional moment m1

0 5 10 15 20

0 0.2 0.4 0.6 0.8 1 1.2

x [m]

w(x) [cm]

0 5 10 15 20

−20 0 20 40 60 80 100 120 140 160

x [m]

t1(x) [kN/m]

0 5 10 15 20

−70−60−50−40−30−20−10 0 10 20

x [m]

m1(x) [kNm/m]

m1(0) = −67.6^kNm_m

0 5 10 15 20

−60−50 −40 −30 −20 −10 0

x [m]

n1(x) [kN/m]

0 5 10 15 20

0 500 1000 1500 2000

x [m]

n2(x) [kN/m]

n2(2) = 1841^kN_m

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Cylinder container – disturbance of membrane state

Numerical solution – ROBOT

Circumferential force n2 Transverse force t1 Meridional moment m1

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Cylinder container – disturbance of membrane state

Numerical solution – ANSYS

Meridional moment m1 Circumferential moment m2

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Cylinder container – disturbance of membrane state

Summary

Analytical solution

wmax n2,max n2,min m1,max m1,min

[cm] _kN

m

_kN

m

_kNm

m

_kNm

m

position of x [m] 2.3 2.32 0.0 1.3 0.0

value 1.028 1841.3 -9.60 14.68 -67.64 Numerical solution

ANKA (FE 1D) 1.024 1835.9 -14.25 13.96 -67.67 ROBOT (FE 2D) 1.024 1846.8 -9.97 14.32 -60.52 ANSYS (FE 2.5D) 1.020 1847.8 -7.10 13.72 -58.19

Element type NE Active NDOFS

ANKA SRSK – 1D 20 80

ROBOT flat – 2D 1280 7680

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References

M. Radwańska. Ustroje powierzchniowe. Podstawy teoretyczne oraz rozwiązania analityczne i numeryczne. Skrypt PK, Kraków, 2009 (in Polish).

R.D. Cook, D.S. Malkus, M.E. Plesha, R.J Witt. Concepts and Applications of Finite Element Analysis.

University of Wisconsin–Madison, John Wiley&Sons, 2002.

O.C. Zienkiewicz, R.L. Taylor, J.Z. Zhu. The Finite Element Method:

Its Basis and Fundamentals. VI edition, Elsevier Butterworth Heineman, 2005.

M. Radwańska, A. Stankiewicz, A. Wosatko, J. Pamin. Plate and Shell Structures. Selected Analytical and Finite Element Solutions. John Wiley &

Sons, 2017.