Read Microsoft PowerPoint  TMR4305.Lecture.Week 37.2007 text version
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
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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
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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, C3q
with exact circular geometry w = cubic polynomial u = polynomial of degree q  curved elements, C3qS1, C3qS2 selective/reduced integration
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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 wavelength approximately equal to the radius, R.
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Table 8.1 Finite element approximations of circular arch with point load
Element type B31 Number of Displ.at A elements (102) 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) Conventional 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
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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 inplane 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 (MindlinReissner theory)
a) Differential element of a thin plate before loading
b) After loading: deformations
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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 = x and w,y = y (i.e. xz = yz = 0) Thick plate theory (MindlinReissner theory)  assume interpolation polynomials for the lateral displacement w, and the rotations, x, y of the normals to the mean surface Degenerate solid element
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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
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ABAQUS/Standard shell elements for structural analysis
generalpurpose 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
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Threedimensional shell elements Threedimensional shell elements in ABAQUS are named as follows:
For example, S4R is a 4node, quadrilateral, stress/displacement shell element with reduced integration and a largestrain formulation; and  SC8R is an 8node, quadrilateral, firstorder interpolation, stress/displacement continuum shell element with reduced integration.
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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 WoinoskiKrieger (1959).
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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
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
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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 ?
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