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Reissner-Mindlin Flat shell theoryReissner-Mindlin Flat shell element

Results and VerificationsFuture Work

References

Department of Aerospace EngineeringDevelopment of Flat Shell Element

April 28, 2015M. Tech Thesis Presentation

Surendra Verma | 10AE30018 Department of Aerospace Engineering



References

Outline

1 Reissner-Mindlin Flat shell theory

2 Reissner-Mindlin Flat shell elementDiscretization of displacement fieldDiscretization of generalized strain fieldElement stiffness equationAssembly of stiffness equationsNumerical Integration

3 Results and VerificationsScordelis-Lo roof problem

4 Future Work

5 References




References

Stress-Strain Relation

Resultant stresses & Generalized constitutive matrix

σ̂′ =

σ̂′mσ̂′bσ̂′s

=

Nx′Ny′Nx′y′Mx′My′Mx′y′Qx′Qy′

=

∫ t2−t2

ST (D′ε′)dz′ = D̂ε̂′

D̂′ =

∫ t2−t2

STD′Sdz′ =

∫ t2−t2

D′p −z′D′p 0

−z′D′p −z′2D′p 0

0 0p D′s

dz′ =

D̂′m D̂′mb 0

D̂′mb D̂′b 0

0 0 D̂′s

D̂′m =

∫ t2−t2

D′pdz′; D̂′mb = −

∫ t2−t2

z′D′pdz′; D̂′b =

∫ t2−t2

z′2D′pdz

′;

D̂′s =

[k11D̄

′s11 k12D̄

′s12

k12D̄′s12 k22D̄

′s22

]; D̄′sij

=

∫ t2−t2

D′sijdz′




References

Principle of Virtual Work done

∫ ∫A

(δε̂′Tm σ̂′m + δε̂

′Tb σ̂′b + δε̂

′Ts σ̂′s )dA =

∫ ∫Aε̂′Tσ̂′dA

where V and A are shell volume and area of the shell surface respectively,

t′ =[fx′ , fy′ , fz′ ,mx′ ,my′

]Tis the distributed surface load vector in the local coordinate directions x′ ,y′,z′.

P′i =

[Px′

i, Py′

i, Pz′

i,Mx′

i,Mx′

i

]T

are concentrated loads and moments in local coordinate system.




References

Discretization of displacement fieldDiscretization of generalized strain fieldElement stiffness equationAssembly of stiffness equationsNumerical Integration

Local displacement vector

u′ =n∑

i=1

= [N1,N2, ...,Nn ]

a′(e)1

a′(e)2...

a′(e)n

= Na′(e)

where

Ni =

Ni 0 0 0 00 Ni 0 0 00 0 Ni 0 00 0 0 Ni 00 0 0 0 Ni

; a′(e)i

=

[u′oi, v′oi

,w′oi, θx′

i, θy′

i

]T;




References


Local strain vector

ε̂′ =

ε̂′m...ε̂′b...ε̂′s

=

∂u′o∂x′∂v′o∂y′

∂u′o∂y′ +

∂v′o∂x′

.......∂θ

x′∂x′∂θy′∂x′

∂θx′

∂y′ +∂θy′∂x′

.......∂w′ox′ − θx′∂w′oy′ − θy′

=n∑

i=1

∂Ni∂x′ u

′oi

∂Ni∂y′ v

′oi

∂Ni∂y′ u

′oi

+∂Ni∂x′ v

′oi

.......∂Ni∂x′ θx′i∂Ni∂x′ θy′i

∂Ni∂y′ θx′i

+∂Ni∂x′ θy′i

.......∂Nix′ w′oi

− Niθx′i

∂Niy′ w′oi

− Niθy′i

=n∑

i=1

B′i a′(e) =

[B′1, B

′2, ..., B

′n

]a1′(e)

a′(e)2...

a′(e)3

= B′a′(e)




References


Local strain vector

B′i =

B′miB′biB′si

; B′mi=

∂Ni∂x′ 0 0 0 0

0∂Ni∂y′ 0 0 0

∂Ni∂y′

∂Ni∂x′ 0 0 0

B′bi=

0 0 0

∂Ni∂x′ 0

0 0 0 0∂Ni∂y′

0 0 0∂Ni∂y′

∂Ni∂x′

B′si=

0 0∂Ni∂x′ −Ni 0

0 0∂Ni∂y′ 0 −Ni

where B′mi,B′bi

and B′siare membrane,bending and transverse shear strain matrices respectively.




References


Why Python?

Open source, free

Widely used and growing, active scientific community

Competitive array math package and plotting packages

Clean language design

Object oriented, dynamically typed, garbage collected,bytecode compiled

Efficient

Enforced indentation!




References


Why Python?

Open source, free





Efficient





References


Why Python?

Open source, free





Efficient





References

Scordelis-Lo roof problem

Model Defination

Geometric Data :

Length (L) : 6 m

Radius (R) : 3 m

Material Properties :

Modulus of Elasticity (E) : 3x1010 Pa

Poisson’s ratio (ν) : 0

Loading :

Uniformly distribute load of 6250 Pa/Area is applied to roof.

Constraints :

Straight edges are free.

Curved edges are supported on rigid diaphragms.




References


2×2 Meshing

Figure: Discretization of a Scordelis-Lo roof problem 2×2 mesh




References


4×4 Meshing





References


8×8 Meshing





References


12×12 Meshing





References


Results

Figure: Contour Plot of a Scordelis-Lo roof problem 12×12 mesh




References


Results

Figure: Contour Plot of a Scordelis-Lo roof problem 12×12 mesh




References


Result Convergence

Meshing Wc (in m) Wb (in m) % Error Wb % Error Wc

2X2 6.2883e-3 -4.2847e-2 18.7 16.2

4X4 5.0848e-3 -3.4182e-2 5.3 6.0

6X6 5.2174e-3 -3.4888e-2 3.3 3.5

8X8 5.3137e-3 -3.5431e-2 1.8 1.7

10X10 5.3807e-3 -3.5797e-2 0.8 0.5

12X12 5.4346e-3 -3.6077e-2 0.06 0.45

Ref. [HCM] 5.41e-3 -3.63211e-2 - -

Table: Scordelis-Lo Roof,Convergence of Wc and Wb




References

Future Work

There are many future work that can be derived :

Development of 4 and 8 Noded Degenerated Shell Element.




References

Reference

HCM D. HAMADI, R. CHEBILI and M. MELLAS, Numerical andExperimental Investigation of an Elliptical Paraboloid Shell Model.


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