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TMR 4305.Week 37.2007

Shell theory (repetition)

y 8 beam element approx. p

x E, A, I

Curved structures Characteristic feature:

Load carrying by membrane, not bending

-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

1

Shell element - Circular arch (repetition)

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

S = kv + S 0

k = B H HBdV = BT H T HBdV = ds BT H T HBdA

T V V s A T

S 0 = N qds

s

T

= H Nv = HBv

= E

2

Shell theory -Shell elements

· Circular arches -Elements - straight beam element, B31

w = cubic for lateral u = linear axial displacement

- curved elements, C3q

with exact circular geometry w = cubic polynomial u = polynomial of degree q - curved elements, C3qS1, C3qS2 selective/reduced integration

3

Example study

Circular arch with point load

- 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.

4

Table 8.1 Finite element approximations of circular arch with point load

Element type B31 Number of Displ.at A elements (10-2) 0.7586 4 0.7751 8 0.7798 16 0.496 C31 4 0.1402 8 0.3464 16 0.7795 D3151 4 0.7810 8 0.7814 16 0.7795 C3152 4 0.7810 8 0.7814 16 0.5177 C32 4 0.7478 8 0.7791 16 0.3719 C33 2 0.7659 4 0.7808 8 EXACT 0.7814

1)

Moment at A1) Conventional

5.6 82.6 292.2 300.7 302.6 292.2 300.7 302.6 225.0 290.8 301.7 119.2 251.2 300.8 303.0

Generalized 287.4 299.1 302.1 60.4 99.4 178.3 310.8 305.0 303.5 287.2 299.1 302.0 304.4 302.7 303.0 283.5 302.3 303.0

Axial force at A1) Conven-tional Generalized 0.918 0.918 0.918 7.3 0.954 10.0 0.926 0.914 0.910 56.4 0.916 20.9 0.918 4.6 0.934 1.2 0.922 1.2 0.919 1.2 0.897 42.8 0.917 22.7 0.918 6.0 0.960 47.9 0.919 35.3 0.918 23.6 0.918

5

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

Shell elements

Plane 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

v k = u k v k , w k , xk , yk

[

] = [v

T

T m

vT p

]

k

Figure 8.7 Shell element made up of a triangular plate element with 9 d.o.f. (T9) and constant strain triangle (CST). 6

Curved shells

based on assumed displacements - approximate geometry

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

u=-zw,x z,w dx P z o h/2 x,u w

o

u=-z0x

Midsurface

w

w ,x

P

w

0x

P

Midsurface z

o

z w ,x

h/2

x,u w ,x

w

x,u w ,x

Thin plate theory (Kirchhoff theory)

Thick plate theory (Mindlin-Reissner theory)

a) Differential element of a thin plate before loading

b) After loading: deformations

7

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 = x and w,y = y (i.e. xz = yz = 0) Thick plate theory (Mindlin-Reissner theory) - assume interpolation polynomials for the lateral displacement w, and the rotations, x, y of the normals to the mean surface Degenerate solid element

8

Other shell element formulations

· Shell theory - thin, thick shell ­ analogous to plate formulations - 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

9

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 are S4R5, 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

10

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

For example, -S4R is a 4-node, quadrilateral, stress/displacement shell element with reduced integration and a large-strain formulation; and - SC8R is an 8-node, quadrilateral, first-order interpolation, stress/displacement continuum shell element with reduced integration.

11

Numerical Study: finite element modelling of shell structures.

- Effect of mesh viz characteristic length, Rh of a cylindrical shell with radius, R and plate thickness, h.

q h = 20 mm

Local bending R = 1000 mm occurs adjacent to the radial load L = 1000

-

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).

12

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

13

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

Finite element values Coarse Mesh 1 Max. nodal displacement z (mm) Min. element bending moment Mxx (N*mm) Min. element shear force Nyz (N) Max. element shear force Nyz (N) 0.434 Mesh 2 Fine Mesh 3 Theoretical value 0.433

0.434

0.434

-4380

-3640

-3540

-3546

-88.5

-96.7

-99.1

-100

23.9

21.2

20.9

20.77

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 14

.45 .4 .35 .3 .25 .2 .15 .1 .05 0

DISPLACEMENT, z

.5

R STRESS

0 .2 -.5 -1 -1.5 -2 -2.5 -3 .4 .6

DISTANCE

Rh

Rh

.2 .4 .6

-3.5 -4

-.05

DISTANCE

-4.5

a)Displacement, w (mm)

b)Moment, Mx(NM)

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

15

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 ?

16

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