ECIV 720 A Advanced Structural

Mechanics and Analysis

Lecture 20: Plates & Shells

Plates & Shells

Loaded in the transverse direction and may be assumed rigid (plates) or flexible (shells) in their plane.

Plate elements are typically used to model flat surface structural components

Shells elements are typically used to model curved surface structural components

Are typically thin in one dimension

Assumptions

Based on the proposition that plates and shells are typically thin in one dimension plate and shell bending deformations can be expressed in terms of the deformations of their midsurface

Assumptions

Stress through the thickness (perpendicular to midsurface) is zero.

As a consequence…

Material particles that are originally on a straight line perpendicular to the midsurface remain on a straight line after deformation

Plate Bending Theories

Kirchhfoff

Shear deformations are neglectedStraight line remains perpendicular to midsurface after deformations

Material particles that are originally on a straight line perpendicular to the midsurface remain on a straight line after deformation

Reissner/Mindlin

Shear deformations are includedStraight line does NOT remain perpendicular to midsurface after deformations

Kirchhoff Plate Theory

First Element developed for thin plates and shells

x

y

z

h

y

w

x

Transverse Shear deformations neglected

In plane deformations neglected

z

Strain Tensor

Strains

x

w

xzu

x

2

2

x

wz

x

ux

z

Strain Tensor

Strains

y

w

yzv

y

2

2

y

wz

y

vy

Strain Tensor

Shear Strains

yx

wz

x

v

y

uxy

2

0 zyzx

Strain Tensor

yx

wy

wx

w

z

xy

y

x

2

2

2

2

2

2

Moments

2/

2/

h

h

xx zdzM

2/

2/

h

h

yy zdzM

Moments

2/

2/

h

h

xyxy zdzM

Moments

2/

2/

h

hxy

y

x

xy

y

x

zdz

M

M

M

Stress-Strain Relationships

z

At each layer, z, plane stress conditions are assumed

h

xy

y

x

xy

y

xE

2

100

01

01

1 2

2/

2/

h

hxy

y

x

xy

y

x

zdz

M

M

M

yx

wy

wx

w

z

xy

y

x

2

2

2

2

2

2

Stress-Strain RelationshipsIntegrating over the thickness the generalized stress-strain matrix (moment-curvature) is obtained

2/

2/2

2

2

100

01

01

1

h

h

dzE

z

D

or

xy

y

x

xy

y

x

M

M

M

D

Generalized stress-strain matrix

2

100

01

01

112 2

3

Eh

D

Formulation of Rectangular Plate Bending Element

h

x

y

z

y

x

w

Node 1

Node 4

Node 2

Node 3

12 degrees of freedom

Pascal Triangle

1

x y

x2 xy y2

x3 x2y xy2 y3

x4 x3y x2y2 xy3 y4

…….

x5 x4y x3y2 x2y3 xy4 y5

Assumed displacement Field

312

311

310

29

28

37

265

24321

xyayxa

yaxyayxaxa

yaxyaxayaxaaw

Formulation of Rectangular Plate Bending Element

312

211

29

82

7542

3

232

yayxaya

xyaxayaxaax

wx

212

311

210

92

8653

33

22

xyaxaya

xyaxayaxaay

wy

For Admissible Displacement Field

iii yxww ,

y

xw

y

yxw iiix

, x

yxw iiiy

,

i=1,2,3,4 12 equations / 12 unknowns

Formulation of Rectangular Plate Bending Element

and, thus, generalized coordinates

a1-a12 can be evaluated…

Formulation of Rectangular Plate Bending Element

For plate bending the strain tensor is established in terms of the curvature

yx

wy

wx

w

xy

y

x

2

2

2

2

2

2

xy

y

x

xy

y

x

M

M

M

D

Formulation of Rectangular Plate Bending Element

xyayaxaax

w118742

2

6262

xyayaxaay

w1210962

2

6622

Formulation of Rectangular Plate Bending Element

yaxayayaayx

w12

211985

2

664422

Strain Energy

eV

Te dVU σDε

2

1

eA

Te dAU Dκκ

2

1

Substitute moments and curvature…

Element Stiffness Matrix

Shell Elements

x

y

z

h

y

w

x

u

v

Shell Element by superposition of plate element and plane stress

element

Five degrees of freedom per node

No stiffness for in-plane twisting

Stiffness Matrix

88

1212

2020~

0

0~

~

stressplane

plate

xshell k

kk

Kirchhoff Shell Elements

Use this element for the analysis of folded plate

structure

Kirchhoff Shell Elements

Use this element for the analysis of slightly curved shells

Kirchhoff Shell Elements

However in both cases transformation to Global CS is required

And a potential problem arises…

44

20202424

*

0

0~

~

0

kk

shell

xshell

2020

*

2424

~

TkTk shellT

xshell

Twisting DOF

Kirchhoff Shell Elements

… when adjacent elements are coplanar (or almost)

Singular Stiffness Matrix (or ill conditioned)

Zero Stiffness z

Kirchhoff Shell Elements

44

20202424

*

0

0~

~

I

kk

k

shell

xshell

Define small twisting stiffness k

Comments

Plate and Shell elements based on Kirchhoff

plate theory do not include transverse shear deformations

Such Elements are flat with straight edges and are used for the analysis of flat plates, folded plate structures and slightly curved shells. (Adjacent shell elements should not be co-planar)

Comments

Elements are defined by four nodes.

Elements are typically of constant thickness.

Bilinear variation of thickness may be considered by appropriate modifications to the system matrices. Nodal values of thickness need to be specified at nodes.

Plate Bending Theories

Kirchhfoff

Shear deformations are neglectedStraight line remains perpendicular to midsurface after deformations

Material particles that are originally on a straight line perpendicular to the midsurface remain on a straight line after deformation

Reissner/Mindlin

Shear deformations are includedStraight line does NOT remain perpendicular to midsurface after deformations

Reissner/Mindlin Plate Theory

x

y

z

h

y

w

x

Transverse Shear deformations ARE INCLUDED

In plane deformations neglected

Strain Tensor

zxzu

y

xz

x

u xx

xzx x

w

xz

x

w

Strain Tensor

zyzv

y

yz

y

v yy

yzy y

w

yz

y

w

Strain Tensor

Shear Strains

xy

zx

v

y

u yxxy

Transverse Shear assumed constant through thickness

xzx x

w

yzy y

w

xxz x

w

yyz y

w

Strain Tensor

xy

y

x

z

yx

y

x

xy

yy

xx

y

x

yz

xz

y

wx

w

Transverse Shear StrainPlane Strain

Stress-Strain Relationships

z

At each layer, z, plane stress conditions are assumed

h

Isotropic Material

Stress-Strain Relationships

xy

y

xE

z

yx

y

x

xy

y

x

2

100

01

01

1 2

Plane Stress

Stress-Strain Relationships

Transverse Shear Stress

y

x

yz

xz

y

wx

wE

)1(2

Strain Energy

Contributions from Plane Stress

dzdAE

U

xy

y

x

A

h

h xyyx

ps

2

100

01

01

12

12

2/

2/

Strain Energy

Contributions from Transverse Shear

dzdAEk

U

yz

xz

A

h

h yzxz

ts

122

2/

2/

k is the correction factor for nonuniform stress

(see beam element)

Stiffness Matrix

Contributions from Plane Stress

dzdAE

U

xy

y

x

A

h

h xyyxps

2

100

01

01

12

12

2/

2/

A

xy

y

x

xyyxps dAEh

2

100

01

01

1 2

3

k

Stiffness Matrix

Contributions from Plane Stress

A

y

x

yx

ts

dA

y

wx

wEhk

y

w

x

w

12

k

dzdAEk

Uyz

xz

A

h

h yzxzts

122

2/

2/

Stiffness Matrix

),,(),( yxtsyxps w kkk

Therefore, field variables to interpolate are

yxw ,,

Interpolation of Field Variables

For Isoparametric Formulation

Define the type and order of element

e.g.

4,8,9-node quadrilateral

3,6-node triangular

etc

Interpolation of Field Variables

q

i

iyiy N

1

q

i

ixix N

1

q

iiiwNw

1Where q is the number

of nodes in the

element

Ni are the appropriate

shape functions

Interpolation of Field Variables

In contrast to Kirchoff element, the same shape functions are used for the

interpolation of deflections and rotations

(Co continuity)

Comments

Elements can be used for the analysis of

general plates and shellsPlates and Shells with curved edges and faces are accommodated

The least order of recommended interpolation is cubic

i.e., 16-node quadrilateral

10-node triangular

Lower order elements show artificial stiffening

Due to spurious shear deformation modes

Shear Locking

Kirchhoff – Reissner/Mindlin Comparison

Kirchhoff:

Interpolated field variable is the deflection w

Reissner/Mindlin:

Interpolated field variables are

Deflection w

Section rotation x

Section rotation y

True Boundary Conditions are better represented

In addition to the more general nature of the

Reissner/Mindlin plate element note that

Shear Locking

Reduced integration of system matrices

To alleviate shear locking

Numerical integration is exact (Gauss)

Displacement formulation yields strain energy that is less than the exact and thus the stiffness of the system is overestimated

By underestimating numerical integration it is possible to obtain better results.

Shear Locking

The underestimation of the numerical integration compensates appropriately for the overestimation of the FEM stiffness matrices

FE with reduced integration

Before adopting the reduced integration element for practical use question its stability and convergence

Shear Locking & Reduced Integration

Kb correctly evaluated by quadrature (Pure bending or twist)

Ks correctly evaluated by 1 point quadrature only.

Shear Locking & Reduced Integration

Ks shows stiffer behavior =>Shear Locking

Shear Locking & Reduced Integration

Kb correctly evaluated by quadrature (Pure bending or twist)

Ks cannot be evaluated correctly

Shear Locking & Reduced Integration

Shear Locking – Other Remedies

Mixed Interpolation of Tensorial Components

MITCn family of elements

To alleviate shear locking

Reissner/Mindlin formulation

Interpolation of w, ,and

Good mathematical basis, are reliable and efficient

Interpolation of w, and is based on different order

Mixed Interpolation Elements

Mixed Interpolation Elements

Mixed Interpolation Elements

Mixed Interpolation Elements

Mixed Interpolation Elements

FETA V2.1.00

ELEMENT LIBRARY

Planning an Analysis

Understand the Problem

Survey of what is known and what is desired

Simplifying assumptions

Make sketches

Gather information

Study Physical Behavior

Time dependency/Dynamic

Temperature-dependent anisotropic materials

Nonlinearities (Geometric/Material)

Planning an Analysis

Devise Mathematical Model

Attempt to predict physical behaviorPlane stress/strain

2D or 3D

Axisymmetric

etc

Examine loads and Boundary Conditions

Concentrated/Distributed

Uncertain stiffness of supports or connections

etc

Data ReliabilityGeometry, loads BC, material properties etc

Planning an Analysis

Preliminary Analysis

Based on elementary theory, formulas from handbooks, analytical work, or experimental evidence

Know what to expect before FEA

Planning an Analysis

Start with Simple FE models and improve them

Planning an Analysis

Start with Simple FE models and improve them

Planning an Analysis

Check model and results

Checking the Model

• Check Model prior to computation

• Undetected mistakes lead to:– execution failure – bizarre results– Look right but are wrong

Common Mistakes

In general mistakes in modeling result from insufficient familiarity with:

a) The physical problem

b) Element Behavior

c) Analysis Limitations

d) Software

Common Mistakes

Null Element Stiffness Matrix

Check for common multiplier (e.g. thickness)

Poisson’s ratio = 0.5

Common Mistakes

Singular Stiffness Matrix• Material properties (e.g. E) are zero in all

elements that share a node• Orphan structure nodes• Parts of structure not connected to remainder• Insufficient Boundary Conditions• Mechanism exists because of inadequate

connections• Too many releases at a joint• Large stiffness differences

Common Mistakes

Singular Stiffness Matrix (cont’d)• Part of structure has buckled• In nonlinear analysis, supports or connections

have reached zero stiffness

Common Mistakes

Bizarre Results• Elements are of wrong type• Coarse mesh or limited element capability• Wrong Boundary Condition in location and type• Wrong loads in location type direction or

magnitude• Misplaced decimal points or mixed units• Element may have been defined twice• Poor element connections

Example

127

127 127

127 178 178

178 178

178

Unit: mm

74 o 74 o

11 11

11

17

17

12.7

(c) Instrumentation placement [7]

2440

Strain Gages

Survey Prism

DWT

25.4 mm = 1 inch

C L

A B C D

interior

exterior

Survey Prism

1133

0

center

17530

(c) Cross-bracing

1219 mm3962 mm

1219 mm

(e) Loading configuration

Mid-Spanx

z

X Z

Y

(a) Deck and girder(b) Stud pockets

(c) Cross-bracing

(a) Deformed shape

-8

-7

-6

-5

-4

-3

-2

-1

0

0 1 2 3 4 5 6 7 8 9

Distance from the End of Bridge (m)

Def

lec

tio

n (

mm

)

FEMTest 1Test 2

Mid-Span

Interior Girder

Center Girder Exterior

Girder

Center Girder Deflection

-8

-7

-6

-5

-4

-3

-2

-1

0

0 1 2 3 4 5 6 7 8 9

Distance from the End of Bridge (m)

Def

lect

ion

(m

m)

FEM

Test 1

Test 2

Interior Girder Deflection

-8

-7

-6

-5

-4

-3

-2

-1

0

0 1 2 3 4 5 6 7 8 9

Distance from the End of Bridge (m)

Def

lect

ion

(m

m)

FEM

Test 1

Test 2

Exterior Girder Deflection

-8

-7

-6

-5

-4

-3

-2

-1

0

0 1 2 3 4 5 6 7 8 9

Mid-Span

Interior Girder

Center Girder Exterior

Girder

Center Girder Deflection