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Handout 3 TMR 4305/4505 Advanced Structural Analysis

Lectures following the text on pages 8.1-8.7 in the Lecture Notes by T.M

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Shell theory (repetition)

x

y

p

8 beam element approx.

E, A, I

Load carrying by membrane, not bending

Characteristic feature:-Load carrying by membrane and bending interaction

-Equilibrium, by stresses, foreces-Kinematic Compatibility, strains (curvature) expressed by displacement-Hookes law, relation between stresses and strains

Curved structures

F. TMR 4305.13.09.2007

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Shell element - Circular arch (repetition)

Stiffness relation (between nodal forces and displacements) may be written as

0SkvS +=TT T T T T

V V s A

k B H HBdV B H HBdV ds B H HBdA= = =∫ ∫ ∫ ∫T0

s

ds= ∫S N q

εσ E=

= =ε HΔN HBv v

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Shell theory -Shell elements• Circular arches -Elements

- straight beam element, B31 w = cubic for lateral u = linear axial displacement

- curved elements, C3qwith exact circular geometryw = cubic polynomial u = polynomial of degree q

- curved elements, C3qS1, C3qS2selective/reduced integration

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Example study

- The maximum bending stress is approximately 80 times the axial stress. - The exact axial strain is nearly constant while

the curvature varies slightly with a wave-length approximately equal to the radius, R.

Circular arch with point load

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Table 8.1 Finite element approximations of circular arch with point load

1) Conventional = differentiating the displacement fields.Generalized = obtained from nodal forces by the stiffness relation

Moment at A1) Axial force at A1) Element

type Number of elements

Displ.at A (10-2) Conventional Generalized Conven-tional Generalized

B31 4 8 16

0.7586 0.7751 0.7798

287.4 299.1 302.1

0.918 0.918 0.918

C31 4 8 16

0.496 0.1402 0.3464

5.6 82.6

60.4 99.4 178.3

7.3 10.0

0.954 0.926 0.914

D3151 4 8 16

0.7795 0.7810 0.7814

292.2 300.7 302.6

310.8 305.0 303.5

56.4 20.9 4.6

0.910 0.916 0.918

C3152 4 8 16

0.7795 0.7810 0.7814

292.2 300.7 302.6

287.2 299.1 302.0

1.2 1.2 1.2

0.934 0.922 0.919

C32 4 8 16

0.5177 0.7478 0.7791

225.0 290.8 301.7

304.4 302.7 303.0

42.8 22.7 6.0

0.897 0.917 0.918

C33 2 4 8

0.3719 0.7659 0.7808

119.2 251.2 300.8

283.5 302.3 303.0

47.9 35.3 23.6

0.960 0.919 0.918

EXACT 0.7814 303.0 0.918

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Shell elementsPlane shell elements-a simple flat shell formulation can be obtained by using

the Morley plate element together with the constant strain triangle

Other plane plate bending and membrane elements can be combined to form shell elements

For plane element there is no coupling between in-plane and bending behaviour The stiffness relation for a plane shell element therefore can be established by superimposing the plate and membrane stiffness relations.For the element in Figure 8.7 the d.o.f. for each node, k are

Figure 8.7 Shell element made up of a triangular plate element with 9 d.o.f. (T9) and constant strain triangle (CST).

[ ] [ ]k

Tp

Tm

Tykxkkkkk wvu vvv == θθ ,,,

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Curved shells- based on assumed displacements- approximate geometry

-Shell (element) = membrane + plate (element)- plate bending is the main challenge (Ch.7)

h/2P

h/2

o

zx,u

z,w dx

w o w,x

,xw

x,u

,xww

P

Midsurface

z

,xu=-zw

w

w x0

zP

o

,xw

x,u

Midsurface

u=-z0x

Thin plate theory Thick plate theory (Kirchhoff theory) (Mindlin-Reissner theory)

a) Differential element of b) After loading: deformations a thin plate before loading

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Curved shells (continued)

- based on assumed displacements- approximate geometry

-Shell (element) = membrane + plate (element)- plate bending is the main challenge (repetition of Ch.7)

Thin plate theory (Kirchhoff theory) - analytical formulation- discrete Kirchhoff in selected points

Based on thick shell formulation and the Kirchhoff constraints imposed as follows:(i) At corner nodes: w,x = θx and w,y = θy (i.e. γxz = γyz = 0)

Thick plate theory (Mindlin-Reissner theory)- assume interpolation polynomials for

the lateral displacement w, and the rotations, θx, θy of the normals to the mean surface

Degenerate solid element 10

Other shell element formulations

• Shell theory- thin, thick shell – analogous to plateformulations

- strain (for a thin shell: mean strain, curvature;for a thick shell : starins incl. shear deformation)

- small strain or finite strain• Approximation

- assumed displacements & interpolation polynomial- assumed strains- number of nodes and degrees of freedom- numerical integration over the surface and thickness

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ABAQUS/Standard shell elements for structural analysis� general-purpose elements, as well as elements specifically suitable for the analysis

of “thick” or “thin” shells;- Element types S3/S3R (finite strain), S3RS, S4, S4R, S4RS, S4RSW, - allow transverse shear deformation- transition from thick shell theory to discrete Kirchhoff thin shell elements

as the thickness decreases;

� general thick shell elements- element types S8R and S8RT

� general thin shell elements- thin shell element that solves thin shell theory is STRI3 and is a flat, faceted element

- The elements that impose the Kirchhoff constraint numerically areS4R5, STRI65, S8R5, S9R5 (all are five d.o.f elements),

� elements that use five degrees of freedom per node where possible

� continuum shell elements.- SC6R, and SC8R

READ Guidelines, Sect.23.6.1 Shell elements; Overview ; Sect. 23.6.2 Choosing a shell element of Theory Manual 12

Three-dimensional shell elementsThree-dimensional shell elements in ABAQUS are named as follows:

For example, -S4R is a 4-node, quadrilateral, stress/displacement shell element with reducedintegration and a large-strain formulation; and

- SC8R is an 8-node, quadrilateral, first-order interpolation, stress/displacementcontinuum shell element with reduced integration.

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Numerical Study:finite element modelling of shell structures.- Effect of mesh viz characteristic length, of a cylindrical shell with radius, R and plate thickness, h.

- Cylindrical shell with radius, R and plate thickness, h.-A cylindrical shell with symmetric radial (and axial) loading -- Exact solutions are given in Timoshenko and Woinoski-Krieger (1959).

Rh

L = 1000

R = 1000 mm

q h = 20 mmLocal bending occurs adjacent to the radial load

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Medium Finite element mesh for Case a) covering a sector of 100 of the cylinder. Each element spans a sector of 2.5.

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Table 8.2 Comparison of finite element results with the exact solution for Case a

20.7720.921.223.9Max. element shear force Nyz(N)

-100-99.1-96.7-88.5Min. elementshear force Nyz(N)

-3546-3540-3640-4380Min. element bending moment Mxx (N*mm)

0.4330.4340.4340.434Max. nodaldisplacement z (mm)

Theoretical value

Fine Mesh 3Mesh 2Coarse Mesh 1

Finite element values

Note: The element size in the longitudinal direction is 25,50 and 100 mm for the fine, medium and coarse mesh, respectively. The characteristic length: Rh = 141.4 mm

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Figure 8.12 Displacement and moment distribution in the longitudinal direction for Case a) with a medium mesh

DISTANCE

a)Displacement, w (mm)

DISPLACEMENT, z.45

.4

.35

.3

.25

.2

.15

.1

.05

0

-.05.2 .4 .6

b)Moment, Mx(N⋅M)

R STRESS .5

-4

-.5

-1

-1.5

-2

-2.5

-3

-3.5

0

-4.5

.2 .4 .6 DISTANCE

Rh

Rh

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Use of shell elements in the analysis of flat panels

Shell (element) = membrane + plate (element)

--what is the stress vriation (gradient) ?--how good is the membrane part ?- & the bending part ?