19/07/2007 1

Lecture 5 – Plate and Shell Elements and 3D Problems

Yan ZhugeCIVE 3011 Structural Analysis and

Computer Applications

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Plate and Shell Elements

Plate bending element is used to model plate type structures where the thickness is very small compared to the other dimensions and when the plate structure is subjected to loads which are normal to its surface. As a consequence of this, flexural effects dominate.

Note the similarity of 1D straight beam model to 2D flat plate model.

Examples: floor slabs, bridge decks, sides of rectangular water tanks and other fluid retaining structures.

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Plate and Shell ElementsForces and Moments acting on the plate:

Relations between Forces and Stresses:Bending moments (per unit length)

∫−=2/

2/

t

t xx zdzM σ(N m/m)

(1)

∫−=2/

2/

t

t yy zdzM σ

(N m/m)

(2)

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Plate and Shell ElementsTwisting moment (per unit length)

Shear Forces (per unit length)

(N /m)

∫−=2/

2/

t

t xyxy zdzM τ

∫−=2/

2/

t

t xzx dzQ τ

∫−=2/

2/

t

t yzy dzQ τ(4)

(3)

(5)(N /m)

Maximum bending stresses

2max6

)(tM x

x ±=σ

2max

6)(

tM y

y ±=σ (6)

• Maximum stress is always at z = ±t/2.• No bending stresses at midsurface (similar to the

beam model)

(N m/m)

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Finite Element for PlateA plate is a thin solid and might be modelled by 3D solid elements. But a solid element is wasteful of d.o.f. as it computes transverse normal stress and transverse shear stresses, all of which are considered negligible in a thin plate. In sketches, thickness t may appear to be zero, but the physically correct value is used in formulating element stiffness matrices.

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Finite Element for Plate

The plate element normally has 3 d.o.f. per node which are the lateral deflection and rotation about the x and y axes, resulting in 12 d.o.f. per element. The displacement function is therefore of the form:

w x y x xy y xx y xy y xy xy= + + + + + +

+ + + + +

β β β β β β β

β β β β β1 2 3 4

25 6

27

3

82

92

103

113

123

(7)

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Shells and Shell Elements

Shells – Thin structures which span over curved surfaces.

Examples:• Sea shell, egg shell; • Containers, pipes, tanks;• Car bodies;• Roofs, buildings (dome), etc.

Forces in shells:

Membrane forces + Bending Moments

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Shells and Shell ElementsA thin shell can be very strong if membrane action dominates. However, no shell is completely free of bending stresses.

Example: A cylindrical Container

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Shell Elements

The most direct way to obtain a shell element is to combine a membrane element and a bending element.

DOF at each node:

Q4 or Q8 shell element

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Curved Shell Elements

• Based on shell theories;• Most general shell elements (flat shell and plate

elements are subsets);• Complicated in formulation.

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Solid ElementsThe 3D brick elements are used to model structures in which the thickness is substantial when compared to the other dimensions. Examples: valve and gear housings, bearing races, thick welded joints, and thick walled cylinders.

For 3D solid element, all six possible stresses (three normal and three shear) must be taken into account. By using a number of brick elements through the thickness of the part, the stress distribution can be accurately obtained.

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Solid Elements

A 3D brick element normally has 8 nodes, each with 3 d.o.f., x, y and z translation. However, some programs support variable-node brick elements which can support as many as 21 nodes where a high degree of accuracy is required.

x

Y

Z

Problems of beam bending, plane stress, plates, and so on, can all be regarded as special cases of a 3D solid. However, 3D models are the hardest to prepare, the most tedious to check for errors, and the most demanding of computer resources.

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Typical 3D Elements

Penta

Avoid using this element in 3D stress analysis (Inaccurate! But it is OK for dynamic analysis).

Tetrahedron

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Stress-strain Relations of 3D Model

(1)

or (2)

Displacements u within an element are interpolated from nodal d.o.f. d in the usual way; that is, u = Nd, where N is the shape function matrix. If nodes have only translational d.o.f. and n is the number of nodes per element, N has 3n columns for a 3D element.

σσστττ

ν ν

ν ν νν ν νν ν ν

ν

ν

ν

εεεγγγ

x

y

z

xy

yz

zx

x

y

z

xy

yz

zx

E

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪

⎫

⎬

⎪⎪⎪

⎭

⎪⎪⎪

=+ −

−−

−−

−

−

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪

⎫

⎬

⎪⎪⎪

⎭

⎪⎪⎪

( )( )1 1 2

1 0 0 01 0 0 0

1 0 0 0

0 0 01 2

20 0

0 0 0 01 2

20

0 0 0 0 01 2

2

σ = Eε

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Displacement Field

The Strain vector

(3)

(4)

u = Nd

(5)

uvw

N NN N

N N

uvwuvw

x1 x N

Nx1

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪

=

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎧

⎨

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎫

⎬

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

( ) ( )

( )

3

1 2

1 2

1 2 3 3

1

1

1

2

2

2

3

0 0 0 00 0 0 00 0 0 0

L

L

L

M

or

ε = Bd(6x1) (6x3N) x (3Nx1)

Stiffness Matrix k B EBdvT

v

= ∫(3xN) (3Nx6)x(6x6)x(6x3N)

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3D Solid Elements

Please note that 3D elements usually do not use rotational d.o.f.

Loads: Distributed loads ⇒ Nodal forces

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Selection of the element typesBefore defining the nodes, it is necessary to determine what type of element is appropriate for the problem. The selection of elements will influence the definition of the nodes.

• Structures consisting of beams/trusses should be modelled with beam or truss elements.

• If the program does not provide truss element, trusses can be modelled as beam elements without moment capability. The most common method is to simply specify the moments of inertia as zero.

• In general, any object whose length is much greater than its depth or width can be modelled with beam elements.

• Modelling beam elements is straightforward as long as the sectional properties are properly defined and the beam is oriented properly.

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Examples of truss/beam elements

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Selection of the element types

The choice between plate and brick elements is not always as straightforward, particularly in problems involving bending. Plate elements assume no variation in stress through the thickness of the plate. This assumption is valid provided the thickness is much less than the other dimensions. In general, plate elements are appropriate when the thickness is no more than 10% of the other dimensions.

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Selection of the element typesIf your problem does not fit the criteria for beam or plate elements, the first step is to evaluate possible simplifying assumptions:

Are plane strain or axisymmetry assumptions applicable? If so, you can still create a 2D model using either plate or axisymmetric elements. If you intend to use plane strain or axisymmetry, you should redraw your preliminary sketch to reflect this. If these assumptions do not apply, your only recourse is a 3D element model.

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Examples of 2D elements

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Selection of the element types

Finally, you may require spring or stiffness elements to model support points of the structure. If the stiffness of any part of the structure is known, this part can be modelled with spring or stiffness elements. If other special elements are required, such as lumped mass, these should be noted on the preliminary sketch.