mindlin plate theory and abaqus uel implementationweb.mit.edu/npdhye/www/padhye-uel.pdfmindlin plate...

Nikhil Padhye, Subodh Kalia Implementation of Mindlin plate element

Mindlin Plate Theory and Abaqus UEL

Implementation

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Nikhil Padhye

[email protected]

Subodh Kalia

[email protected]

2


Contents

1 Mindlin Plate Theory 7

1.1 Stress and strain components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2 Shape functions and isoparametric element formulation . . . . . . . . . . . . . 13

1.3 Computation of Jacobian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.4 Calculation of stress and strain at integration points . . . . . . . . . . . . . . . . 18

1.5 Principle of virtual work and derivation of element stiffness matrix . . . . . . . 24

2 Sample Hand Calculations 29

2.1 Bending Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.2 Shear Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3 Some Consequences of Mindlin Plate Theory 37

4 ABAQUS UEL Implementation 40

4.1 UEL input variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.2 UEL output variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.3 Pseudo code for the UEL subroutine . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5 Benchmarking of UEL Implementation for Mindlin Plate Element 50

5.1 Single element test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.2 Testing multiple elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

6 Acknowledgments 59

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List of Figures

1 W , L and t represent the width, length and thickness of the plate, respectively. 7

2 Rotation of a material line (normal to the neutral plane in the undeformed

configuration), about Y-axis in anti-clockwise direction when viewed from the

positive Y-axis, and denoted by θy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Isoparametric formulation for 4-noded element using 2x2 quad integration. . . 14

4 Four-noded element with integration points in isoparametric formulation. . . . 30

5 UEL subroutine header from Abaqus 2016 documentation, Abaqus User Sub-

routines Reference Guide, Section 1.1.28. . . . . . . . . . . . . . . . . . . . . . . . 40

6 Input-Output block diagram for the UEL subroutine. . . . . . . . . . . . . . . . . 41

7 Boundary conditions for a single element test. . . . . . . . . . . . . . . . . . . . . 50

8 Reaction forces in Z-direction with Abaqus S4 element. . . . . . . . . . . . . . . 51

9 Reaction moments in X-direction with Abaqus S4 element. . . . . . . . . . . . . 52

10 Reaction moments in Y-direction with Abaqus S4 element. . . . . . . . . . . . . 52

11 Plots for reaction forces in Z-direction at the nodes with respect to nodal dis-

placement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

12 Plots for reaction moments in X-direction at the nodes with respect to nodal

displacements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

13 Plots of reaction moments in Y-direction at the nodes with respect to nodal

displacements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

14 Plate deflection from Matlab code. . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

15 Plate deflection from Abaqus S4 element analysis with a scale factor of 106. . . 57

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16 Plate deflection from UEL with a scale factor of 106. . . . . . . . . . . . . . . . . . 58

17 Comparison of deflections for a laterally loaded plate clamped at the edges. . . 58

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ABSTRACT

In this report, we present a model for plate bending based on the Mindlin plate theory for

small elastic deformations. The related field and constitutive equations, and finite element

discretization (FEM) at an element level are also presented. The implementation of the

Mindlin plate element is carried out for the UEL module of ABAQUS. The overall goal of

this technical report is to facilitate the understanding and implementation of finite element

discretization for the UEL module of Abaqus, so that users can define their own elements. The

overall procedure described in this technical report is expected to aid the implementation

of custom defined user elements for ABAQUS, a topic that is currently not discussed well

in the opinion of the authors. For example, other than official Abaqus documentation [1],

only one comprehensive documentation for UEL in Abaqus is available online [9]. The

current demonstration is carried out for small deformations, for the sake of simplicity, and

applications involving other plate/shell theories (along with large deformation or rotation

effects) will be done in the subsequent reports and publications 1. Although benchmarking

on simple test problems is performed in this study, detailed analyses on the performance of

current plate formulation such as patch test performance, shear locking issues, etc. are not

discussed. Reader is referred to [3], [7], [4], [2] and [6] for detailed discussions on these topics,

including more sophisticated finite element methods.

1The readers are encouraged to contact the authors in case they need any further clarification. Use of thisdocument or any content/code presented here for any commercial purposes is strictly prohibited. Readers arewelcome to cite this report if it is found useful to them in anyway.

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1 MINDLIN PLATE THEORY

L

W

t

Mid surface

Top surface

x

y

z

Figure 1: W , L and t represent the width, length and thickness of the plate, respectively.

A plate is commonly identified as a thin flat structure as shown in Figure 1. In order

to qualify as a plate, we require that the thickness (t) of the structure is small compared

to its length (L) and width (W ), i.e. t << L,W . This assumption simplifies the underlying

governing equilibrium equations and also allows for several simplifications on the kinematics

of deformation due to the “thin" nature of the structure at hand. The Mindlin plate theory

can account for homogeneous through thickness shear deformations (as opposed to another

elementary theory of Kirchhoff-Love plates).

The two key features of this theory are: (a) we assume by construction that σz = 0, and

(b) since this is a first order shear deformation theory; therefore, allowing σxz and σy z to be

non-zero. We estimate σxz , σy z from γxz , γy z which themselves are taken to be non-zero and

constant through the thickness of the plate. Membrane effects (i.e. in plane stretching of the

plate mid-surface) are not included in this theory. Some other consequences of such ad-hoc

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assumptions made in this theory are discussed later in Section 3.

1.1 Stress and strain components

The deformation in the Mindlin plate can be characterized by the displacement of the mid-

surface of the plate in the Z-direction (denoted by w), and rotation of the normal to the

mid-surface (denoted by θx and θy ) about X-axis and Y-axis, respectively. Thus we require

three degrees of freedom w,θx ,θy at any point on the mid-surface of the plate to describe the

complete deformation.

Let us consider a material line L1L2, across thickness of the plate, perpendicular to the

neutral plane in the undeformed configuration as shown in Figure 2. Upon deformation

we assume that points on the material line L1L2 still remain on a straight line l1l2 in the

deformed configuration. The material line L1L2 is rotated by an angle θy in the anti-clockwise

direction with respect to the Y-axis. For small deformations, the displacement of any material

point lying on the line L1L2, relative to the point ‘O’ on the mid-surface, in the X-direction,

can be approximated as 2:

ux = zθy , (1)

The extensional strain in the X-direction is given as:

εx = ∂ux

∂x= z

∂θy

∂x. (2)

2We have assumed no membrane action on the plate, i.e., no in-plane strains on the mid-surface of theplate, therefore the mid-surface displacement does not vary spatially, nor does it plays any role in strains acrossthickness as it shall merely enter as a constant sum to ux in equation 2.

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In the above equation, z indicates the coordinate of a material point along Z-axis in

the undeformed configuration, and for the sake of clarity we repeat that θy is measured in

anti-clockwise direction about Y-axis when observing from positive direction to the negative

direction.

It is worth highlighting that a rather tacit assumption has been made in this formulation

(while locating a material point along the thickness direction) - regarding no change in

thickness (implying εz = 0), which seems inconsistent with the prior assumption of σz = 0. In

opinion of the authors, it is quite important to understand the limitations of such classical

plate/shell theories (even with their more elegant extensions as made in the literature), since

they can lead to rather spurious predictions in scenarios where the underlying assumptions

are quite an overreach.

Following a similar procedure one can estimate uy =−zθx . Please note the minus sign in

the estimation of uy , which was not present in the case of ux . Thus, εy is given as

εy =∂uy

∂y=−z

∂θx

∂y(3)

and accordingly the shear strains in the XY-plane are given as

γx y = z

[−∂θx

∂x+ ∂θy

∂y

]. (4)

The shear strains across the thickness (γxz and γy z) can be calculated as

γxz = ∂ux

∂z+ ∂uz

∂x, (5)

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x

z

Center lineBottom surface

Top surfaceN

θy

z

ux = z θyy

L1

L2

N

L1

L2

‘O’

‘O’

Figure 2: Rotation of a material line (normal to the neutral plane in the undeformed configuration),about Y-axis in anti-clockwise direction when viewed from the positive Y-axis, and denoted by θy .

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substituting ux = z θy and uz = w(x, y) in equation 5, we get

γxz = θy (x, y)+ ∂w(x, y)

∂x. (6)

Similarly, we can obtain

γy z =−θx(x, y)+ ∂w(x, y)

∂y. (7)

Now we define a bending strain vector εb and shear strain vector εs as follows:

εb =

εx

εy

γx y

, (8)

εs =

γxz

γy z

. (9)

From linear elasticity we have

εz = 1

E

[σz −ν(σy +σx)

], (10)

where E is Young’s modulus and ν is Poisson’s ratio. Under the assumption of plane stress,

i.e. σz = 0, we get

εz = −νE

(σx +σy ). (11)

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Similarly, we get

εx = 1

E

[σx −νσy

], (12)

εy = 1

E

[σy −νσx

]. (13)

The shear strain γx y is given as

γx y =Gσx y , (14)

where G = E2(1+ν) is the shear modulus. Solving for σx and σy from equations 12 and 13,

we get

σx = E

1−ν2

[εx +νεy

], (15)

σy = E

1−ν2

[νεx +εy

]. (16)

Now we assemble all the (x and y) stress and strain components (and refer to them as

bending components):

σb =

σx

σy

σx y

= E

1−ν2

1 ν 0

ν 1 0

0 0 1−ν2

εx

εy

γx y

(17)

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The through thickness shear components of are given as

σxz =αG γxz (18)

σy z =αG γy z (19)

where α is some shear correction factor, and we choose α= 5/6 in this study. Shear stress

components in terms of shear strain components now can be assembled and written as

σs =

σxz

σy z

=G

1 0

0 1

γxz

γy z

. (20)

Thus far we have formulated stress and strain components for both the bending and

shear separately, and this was possible due to several kinematic assumptions and small

strain setting. Next, we present the finite element discretization at an element level such

that Mindlin theory can be simulated. We once again wish to highlight that the current finite

element plate formulation is rather naive, and more effective and consistent proposals have

been made in the literature. Our goal here is to focus on the simplicity so that implementation

aspects for UEL in Abaqus can be demonstrated easily.

1.2 Shape functions and isoparametric element formulation

We will use a four-noded bilinear isoparametric element where each node has three degrees

of freedom (w , θx and θy ). Four integration points (as per 2x2 quad integration rule) are used

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x

y

z

Node 1(x1, y1)

Node 2(x2, y2)

Node 3(x3, y3)

Node 4(x4, y4)

ζ

η

(-1, -1) (1, -1)

(1, 1)(-1, 1)

x x

x x

-1

3, -

1

3

1

3, -

1

3

-1

3,1

3

1

3,1

3

Figure 3: Isoparametric formulation for 4-noded element using 2x2 quad integration.

to capture bending response, and single integration point (reduced integration) is used to

account for shear response. The shape functions for this isoparametric element (with ξ and η

as isoparametric coordinates) are given as:

N1 = 1

4(1−ξ)(1−η) , (21)

N2 = 1

4(1+ξ)(1−η) , (22)

N3 = 1

4(1+ξ)(1+η) , (23)

N4 = 1

4(1−ξ)(1+η) . (24)

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We denote the shape function vector as N (which has a size of 4x1 and suppress ξ and η

for the sake of convenience in notation):

N =

N1

N2

N3

N4

. (25)

The derivatives of shape functions, denoted by d N , with respect to isoparametric coordi-

nates are written in a matrix form as:

d N =

∂N1∂ξ

∂N1∂η

∂N2∂ξ

∂N2∂η

∂N3∂ξ

∂N3∂η

∂N4∂ξ

∂N4∂η

= 1

4

−(1−η) −(1−ξ)

(1−η) −(1+ξ)

(1+η) (1+ξ)

−(1+η) (1−ξ)

. (26)

1.3 Computation of Jacobian

The derivatives of the shape functions (Ni ) with respect to the isoparametric coordinates can

be written as

∂Ni

∂ξ= ∂Ni

∂x

∂x

∂ξ+ ∂Ni

∂y

∂y

∂ξ, (27)

∂Ni

∂η= ∂Ni

∂x

∂x

∂η+ ∂Ni

∂y

∂y

∂η. (28)

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By combining equation 27 and 28 in a matrix form we obtain

∂Ni∂ξ

∂Ni∂η

=

∂x∂ξ

∂y∂ξ

∂x∂η

∂y∂η

∂Ni∂x

∂Ni∂y

. (29)

Now we define the Jacobian as

J =

∂x∂ξ

∂y∂ξ

∂x∂η

∂y∂η

(30)

where x and y are spatial coordinates of any point within the element. Using equation

30, the equation 29 can be re-written as

∂Ni∂x

∂Ni∂y

= J−1

∂Ni∂ξ

∂Ni∂η

(31)

Now we can assemble a matrix which contains the spatial derivatives of all the shape

functions in terms of the Jacobian and isoparametric shape function derivatives:

∂N1∂x

∂N2∂x

∂N3∂x

∂N4∂x

∂N1∂y

∂N2∂y

∂N3∂y

∂N4∂y

= J−1

∂N1∂ξ

∂N2∂ξ

∂N3∂ξ

∂N4∂ξ

∂N1∂η

∂N2∂η

∂N3∂η

∂N4∂η

= J−1d N>. (32)

We see from equation 31, that the derivatives of the shape functions with the x, y spatial

coordinates can be obtained from the derivates of the shape functions with respect to the

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isoparametric coordinates (ξ, η) if the Jacobian is known. Next we derive more explicit

expression for the Jacobian in terms of the nodal point coordinates.

Given the spatial nodal coordinates (xi , yi ), we can use the shape functions (Ni ) to map

any spatial point (x, y) in the element by using following equations

x =4∑

i=1Ni xi , (33)

y =4∑

i=1Ni yi . (34)

Using equation 33 we can write

∂x

∂ξ= ∂N1

∂ξx1 + ∂N2

∂ξx2 + ∂N3

∂ξx3 + ∂N4

∂ξx4 , (35)

∂x

∂η= ∂N1

∂ηx1 + ∂N2

∂ηx2 + ∂N3

∂ηx3 + ∂N4

∂ηx4 . (36)

Similarly, by using equation 34 we have

∂y

∂ξ= ∂N1

∂ξy1 + ∂N2

∂ξy2 + ∂N3

∂ξy3 + ∂N4

∂ξy4 , (37)

∂y

∂η= ∂N1

∂ηy1 + ∂N2

∂ηy2 + ∂N3

∂ηy3 + ∂N4

∂ηy4 . (38)

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Writing equations 35 - 38 in a matrix form we get

J =

∂x∂ξ

∂y∂ξ

∂x∂η

∂y∂η

=

∂N1∂ξ

∂N2∂ξ

∂N3∂ξ

∂N4∂ξ

∂N1∂η

∂N2∂η

∂N3∂η

∂N4∂η

.

x1 y1

x2 y2

x3 y3

x4 y4

. (39)

Using equation 39 we can calculate J from the known quantities on the right hand side.

The above equation can be written more compactly as

J === d N T X , (40)

where d N T and X are the first and second matrices on the right hand side of equation

39.

1.4 Calculation of stress and strain at integration points

If the nodal point displacements are given then the corresponding displacement field values

within the element can be obtained using the shape functions. The displacement values at

the integration points of the elements allow us to calculate strain and stress values at those

integration points and thereby leading to calculation of stiffness matrix through numerical

integration.

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The displacement fields are given as

w =n∑

i=1Ni wi , (41)

θx =n∑

i=1Niθxi , (42)

θy =n∑

i=1Niθyi . (43)

Using equations 41 to 43 in equation 8, the bending strain vector εb is computed as

εb = z

∂∂x

∑ni=1 Ni (ξ,η)θyi

− ∂∂y

∑ni=1 Ni (ξ,η)θxi

− ∂∂x

∑ni=1 Ni (ξ,η)θxi + ∂

∂y

∑ni=1 Ni (ξ,η)θyi

. (44)

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By expanding the terms in equation 44 we obtain

εb = z

∂N1θy1

∂x + ∂N2θy2

∂x + ∂N3θy3

∂x + ∂N4θy4

∂x

−∂N1θx1∂y − ∂N2θx2

∂y − ∂N3θx3∂y − ∂N4θx4

∂y

−∂N1θx1∂x − ∂N2θx2

∂x − ∂N3θx3∂x − ∂N4θx4

∂x + ∂N1θy1

∂y + ∂N2θy2

∂y + ∂N3θy3

∂y + ∂N4θy4

∂y

. (45)

We can split the matrix on the right hand side of equation 45 as product of a matrix Bb

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and nodal point displacement vector d as follows

εb = z

0 0 ∂N1∂x 0 0 ∂N2

∂x 0 0 ∂N3∂x 0 0 ∂N4

∂x

0 −∂N1∂y 0 0 −∂N2

∂y 0 0 −∂N3∂y 0 0 −∂N4

∂y 0

0 −∂N1∂x

∂N1∂y 0 −∂N2

∂x∂N2∂y 0 −∂N3

∂x∂N3∂y 0 −∂N4

∂x∂N4∂y

w1

θx1

θy1

w2

θx2

θy2

w3

θx3

θy3

w4

θx4

θy4

, (46)

i.e.,

εb = z Bb d . (47)

The spatial derivatives of the shape functions ∂Ni∂x and ∂Ni

∂y , in equation 46, can be obtained

from equation 31. A similar procedure can be adopted to compute the components of the

shear strain vector (εs).

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By substituting w,θx ,θy from equations 41 - 43 in equation 9 we obtain

εs =

γxz

γy z

=

∑n

i=1 Niθyi + ∂∂x

∑ni=1 Ni wi

−∑ni=1 Niθxi + ∂

∂y

∑ni=1 Ni wi

. (48)

Upon expanding the terms in equation 48

εs =

∂N1w1∂x +N1θy1 + ∂N2w2

∂x +N2θy2 + ∂N3w3∂x +N3θy3 + ∂N4w4

∂x +N4θy4

∂N1w1∂y −N1θx1 + ∂N2w2

∂y −N2θx2 + ∂N3w3∂y −N3θx3 + ∂N4w4

∂y −N4θx4

. (49)

The matrix on the right hand side in equation 49 can again be written as product of a

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matrix Bs and nodal displacement vectors d as

εs =

∂N1∂x 0 N1

∂N2∂x 0 N2

∂N3∂x 0 N3

∂N4∂x 0 N4

∂N1∂y −N1 0 ∂N2

∂y −N2 0 ∂N3∂y −N3 0 ∂N4

∂y −N4 0

w1

θx1

θy1

w2

θx2

θy2

w3

θx3

θy3

w4

θx4

θy4

, (50)

i.e.,

εs = Bs d . (51)

As before ∂Ni∂x and ∂Ni

∂y in equation 50 can be obtained from equation 31.

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1.5 Principle of virtual work and derivation of element stiffness matrix

Since we have considered small elastic deformations only, therefore both Kb and Ks (stiffness

matrices corresponding to bending and shearing response, respectively) are constant. Non-

linear material behavior or finite deformation may require updating of the element stiffness

matrix during iterative solution procedure of the global equilibrium equations.

At each node we have considered a total three degrees of freedom, one translational

(w in z direction) and two rotational (θx and θy about X and Y directions, respectively).

Corresponding to these degrees of freedom, at every node we have a work conjugate force

Fz and moments Mx and My . Thus, the nodal force vector of an element consists of 12

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components and we represent it as

F =

Fz1

Mx1

My1

Fz2

Mx2

My2

Fz3

Mx3

My3

Fz4

Mx4

My4

. (52)

To derive the expressions for bending stiffness and shear stiffness we decompose the

internal work done into bending and shear components. Let us consider an element which

has undergone a certain level of bending (εb) and shear deformation (εs ). In this state, the

associated stress vectors are denoted by σb and σs , the nodal point displacement vector is

represented by U and the external (equilibrating) nodal force vector by F . If we imagine a

virtual displacement of δU , then according to the principle of virtual work, external virtual

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work being equal to the internal virtual work, we can write

F>δU =∫

Vσ>

b δεbd v +∫

Vσ>

s δεs d v. (53)

Since εb = zBbU (using equation 47) and εs = zBsU (using equation 51), therefore

δεb = zBbδU (54)

and,

δεs = BsδU . (55)

Using σb(= Dbεb) = zDbBbU , σs (= Dsεs ) = Ds BsU and equations 54 and 55 on the right

hand side of equation 53 we obtain

F>δUb =∫

VU>B>

b D>b BbδU z2d v +

∫V

U>B>s D>

s BsδU d v. (56)

Since above equation holds good for all virtual displacements δU , thus we have

F> =∫

VU>B>

b D>b Bb z2d v +

∫V

U>B>s D>

s BsδU d v. (57)

By taking transpose of both sides in the above equation, and using F = KbU +KsU we can

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arrive at:

Kb =∫

VB>

b Db Bb z2d v (58)

and

Ks =∫

VB>

s Ds Bs d v. (59)

Now we shall analyze the computation of both Kb and Ks . In equation 58 the quantities

within the volume integral are not a function of thickness, except the ‘z term’ itself, and hence

it can be integrated across the thickness from −h/2 to h/2 to yield a result in terms of area

integral over the element as

Kb = h3

12

∫Ω

B>b Db Bb dΩ. (60)

In finite element analysis the integration of the stiffness matrix is done numerically, and

since we are using isoparametric elements, we can re-write the integral on the right hand

side of the equation 60 in terms of isoparametric coordinates as follows

Kb = h3

12

∫ 1

−1

∫ 1

−1Bb(ξ,η)> Db Bb(ξ,η) |J (ξ,η)| dξdη. (61)

Here, Bb and J depend on the integration point coordinates (ξ,η). We shall use 2x2

quad integration rule with four integration points with coordinates of (ξ1 = −1p3

,η1 = −1p3

),

(ξ2 = 1p3

,η2 = −1p3

), (ξ3 = 1p3

,η3 = 1p3

) and (ξ4 = −1p3

,η4 = 1p3

). The right hand side of the

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equation 61 can be numerically integrated as follows

Kb = h3

12

4∑i=1

Bb(ξi ,ηi )> Db Bb(ξi ,ηi ) |J (ξi ,ηi )|∆ξ∆η, (62)

where ∆ξ= 1 and ∆η= 1. For the sake of simplicity we express the above equation as

Kb =4∑

i=1I P i (63)

where,

I P i = h3

12Bb(ξi ,ηi )> Db Bb(ξi ,ηi ) |J (ξi ,ηi )|∆ξ∆η. (64)

Next, the integral on the right hand side of equation 59 for shear stiffness Ks can be

written as

Ks =∫Ω

∫ h/2

−h/2B>

s Ds Bs d z dΩ. (65)

As all the quantities inside the integral on the right hand side of the above equation are

independent of the thickness coordinate, therefore we can integrate along that direction to

arrive at

Ks = h∫Ω

B>s Ds Bs dΩ. (66)

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We can re-write right hand side of equation 66 in terms of isoparametric coordinates as

Ks = h∫ 1

−1

∫ 1

−1Bs(ξ,η)> Ds Bs(ξ,η) |J (ξ,η)| dξdη. (67)

In order avoid shear locking, we perform reduced numerical integration for the shear

stiffness by choosing a single integration point whose coordinates are given by ξ= 0, η= 0.

Thus, the expression for Ks can be written as

Ks = h Bs(ξ= 0,η= 0)> Ds Bs(ξ= 0,η= 0) |J (ξ= 0,η= 0)|∆ξ∆η (68)

where ∆ξ= 2 and ∆η= 2. Finally, overall stiffness K of the element in terms of bending

stiffness Kb and shear stiffness Ks can be written as

K = Kb +Ks . (69)

2 SAMPLE HAND CALCULATIONS

In this section we demonstrate through hand calculations how Kb and Ks are calculated for

an element. Since we are working under the assumption of small strain elasticity, the stiffness

matrix for each element will be the same and constant. E , ν and h are chosen as 2×108, 0.3

and 0.1, respectively throughout this study. The shear correction factor is taken to be 56 .

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2.1 Bending Stiffness

The expression for Kb was derived in equation 62 based on four integration points. The

isoparametric co-ordinates (ξi ,ηi ) for these four integration points are ( −1p3

, −1p3

), ( 1p3

, −1p3

),

( 1p3

, 1p3

) and ( −1p3

, 1p3

). We will show the calculation of I P 1 (i = 1 in equation 64) for the first

integration point with ξ= −1p3

and η= −1p3

. Substituting these values in equations 25 and 26 we

obtain

x

y

z

Node 1(x1, y1)

Node 2(x2, y2)

Node 3(x3, y3)

Node 4(x4, y4)

ζ

η

(-1, -1) (1, -1)

(1, 1)(-1, 1)

x x

x x

-1

3, -

1

3

1

3, -

1

3

-1

3,1

3

1

3,1

3

Figure 4: Four-noded element with integration points in isoparametric formulation.

N =

0.622008

0.166666

0.044658

0.166666

, (70)

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and

d N =

−0.39433 −0.39433

0.39433 −0.10566

0.10566 0.10566

−0.10566 0.39433

. (71)

If we choose the spatial nodal coordinates of the element nodes in X-Y coordinates as

(0,0), (1,0), (1,1) and (0,1) then we can construct X as

X =

0 0

1 0

1 1

0 1

. (72)

The Jacobian J according to equation 39 is computed as

J =

−0.39433 0.39433 0.10566 −0.10566

−0.39433 −0.10566 0.10566 0.39433

0 0

1 0

1 1

0 1

(73)

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i.e.,

J =

0.5 0

0 0.5

. (74)

J−1 is computed by inverting J and turns out to be

J−1 =

2 0

0 2

. (75)

Spatial derivatives of the all shape functions are denoted in a matrix X Y ′′′ and with aid of

equation 31 as

X Y ′′′ =

∂N1∂x

∂N1∂y

∂N2∂x

∂N2∂y

∂N3∂x

∂N3∂y

∂N4∂x

∂N4∂y

= d N J−>. (76)

The matrix of derivatives with respect to spatial coordinates (X Y ′′′) can be calculated

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based on equation 76 as

X Y ′′′ =

−0.78867 −0.78867

0.78867 −0.211325

0.211325 0.211325

−0.211325 0.78867

. (77)

Now we can assemble the elements of Bb matrix (as defined by equations 46 and 47) using

the elements of X Y ′′′ matrix obtained in equation 77,

Bb =

0 0 −.78867 0 0 .78867 0 0 .211325 0 0 −.211325

0 .78867 0 0 .211325 0 0 −.21135 0 0 −.78867 0

0 .78867 −.78867 0 −.78867 −.211325 0 −.211325 .21135 0 .211325 .78867

.

(78)

Now I P 1 (as defined by equation 64) can be calculated with all the known quantities.

Similar procedure can be carried to compute I P 2, I P 3 and I P 4, and finally Kb is calculated

according to equation 62. It is worth mentioning that Bb and J will vary as a function of (ξ,η)

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and will be different for each integration point. We get Kb as

Kb =

0 0 0 0 0 0 0 0 0 0 0 0

0 0.824 −0.298 0 0.091 −0.022 0 −0.412 0.298 0 −0.504 0.022

0 −0.298 0.824 0 0.022 −0.504 0 0.298 −0.412 0 −0.022 0.091

0 0 0 0 0 0 0 0 0 0 0 0

0 0.091 0.022 0 0.824 0.298 0 −0.504 −0.022 0 −0.412 −0.298

0 −0.022 −0.504 0 0.298 0.824 0 0.022 0.091 0 −0.298 −0.412

0 0 0 0 0 0 0 0 0 0 0 0

0 −0.412 0.298 0 −0.504 0.022 0 0.824 −0.298 0 0.091 −0.022

0 0.298 −0.412 0 −0.022 0.091 0 −0.298 0.824 0 0.022 −0.504

0 0 0 0 0 0 0 0 0 0 0 0

0 −0.504 −0.022 0 −0.412 −0.298 0 0.091 0.022 0 0.824 0.298

0 0.022 0.091 0 −0.298 −0.412 0 −0.022 −0.504 0 0.298 0.824

×104 (79)

2.2 Shear Stiffness

For calculation of stiffness associated with shear deformation, we use the reduced integration

approach, i.e., only one integration point with ξ = 0 and η = 0 is chosen. According to

equations 25 and 26 N and d N are computed as:

N =

0.25

0.25

0.25

0.25

, (80)

d N =

−0.25 −0.25

0.25 −0.25

0.25 0.25

−0.25 0.25

. (81)

Matrix of nodal coordinates X is chosen same as in equation 72, and the Jacobian J is

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calculated using the equation 39 as

J =

−0.25 0.25 0.25 −0.25

−0.25 −0.25 0.25 0.25

0 0

1 0

1 1

0 1

, (82)

J =

0.5 0

0 0.5

. (83)

J−1 turns out to be

J−1 =

2 0

0 2

. (84)

The matrix of X and Y derivatives of shape functions (X Y ′′′) is again calculated according

to equation 76 as

X Y ′′′ =

−0.5 −0.5

0.5 −0.5

0.5 0.5

−0.5 0.5

. (85)

Using the components of X Y ′′′ from equation 85, we can now construct the Bs matrix

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(based on equations 50 and 51) as

Bs =

−0.5 0 0.25 0.5 0 0.25 0.5 0 0.25 −0.5 0 0.25

−0.5 −0.25 0 −0.5 −0.25 0 0.5 −0.25 0 0.5 −0.25 0

.

Finally, the shear stiffness matrix Ks can be calculated from equation 68 and we get

Ks =

0.385 0.096 −0.096 0 0.096 −0.096 −0.385 0.096 −0.096 0 0.096 −0.096

0.096 0.048 0 0.096 0.048 0 −0.096 0.048 0 −0.096 0.048 0

−0.096 0 0.048 0.096 0 0.048 0.096 0 0.048 −0.096 0 0.048

0 0.096 0.096 0.385 0.096 0.096 0 0.096 0.096 −0.385 0.096 0.096

0.096 0.048 0 0.096 0.048 0 −0.096 0.048 0 −0.096 0.048 0

−0.096 0 0.048 0.096 0 0.048 0.096 0 0.048 −0.096 0 0.048

−0.385 −0.096 0.096 0 −0.096 0.096 0.385 −0.096 0.096 0 −0.096 0.096

0.096 0.048 0 0.096 0.048 0 −0.096 0.048 0 −0.096 0.048 0

−0.096 0 0.048 0.096 0 0.048 0.096 0 0.048 −0.096 0 0.048

0 −0.096 −0.096 −0.385 −0.096 −0.096 0 −0.096 −0.096 0.385 −0.096 −0.096

0.096 0.048 0 0.096 0.048 0 −0.096 0.048 0 −0.096 0.048 0

−0.096 0 0.048 0.096 0 0.048 0.096 0 0.048 −0.096 0 0.048

×107. (86)

The total stiffness matrix for the element is then obtained by summing the bending and

shear stiffness response as

K =

0.385 0.096 −0.096 0 0.096 −0.096 −0.385 0.096 −0.096 0 0.096 −0.096

0.096 0.048 −0.298 0.096 0.048 −0.022 −0.096 0.047 0.298 −0.096 0.047 0.022

−0.096 −0.298 0.048 0.096 0.022 0.047 0.096 0.298 0.047 −0.096 −0.022 0.048

0 0.096 0.096 0.385 0.096 0.096 0 0.096 0.096 −0.385 0.096 0.096

0.096 0.048 0.022 0.096 0.048 0.298 −0.096 0.047 −0.022 −0.096 0.047 −0.298

−0.096 −0.022 0.047 0.096 0.298 0.048 0.096 0.022 0.048 −0.096 −0.298 0.047

−0.385 −0.096 0.096 0 −0.096 0.096 0.385 −0.096 0.096 0 −0.096 0.096

0.096 0.047 0.298 0.096 0.047 0.022 −0.096 0.048 −0.298 −0.096 0.048 −0.022

−0.096 0.298 0.047 0.096 −0.022 0.048 0.096 −0.298 0.048 −0.096 0.022 0.047

0 −0.096 −0.096 −0.385 −0.096 −0.096 0 −0.096 −0.096 0.385 −0.096 −0.096

0.096 0.047 −0.022 0.096 0.047 −0.298 −0.096 0.048 0.022 −0.096 0.048 0.298

−0.096 0.022 0.048 0.096 −0.298 0.047 0.096 −0.022 0.047 −0.096 0.298 0.048

×107. (87)

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3 SOME CONSEQUENCES OF MINDLIN PLATE THEORY

Here we discuss consequences of the assumptions made in the Mindlin theory and associated

inconsistencies that arise in the formulation.

First we assumed that transverse normal stress σz = 0, by construction, through the plate

thickness. On the other hand we allowed for non-zero transverse shear stresses through

thickness. To investigate this let us study the equilibrium equation in the Z direction:

∂σzx

∂x+ ∂σz y

∂y+ ∂σz

∂z= 0, (88)

substituting σz = 0 in the above equation we obtain

∂σzx

∂x+ ∂σz y

∂y= 0. (89)

Substitute σzx =Gγzx and σz y =Gγz y in equation 89, where G is the shear modulus, to

arrive at

∂γzx

∂x+ ∂γz y

∂y= 0. (90)

From the previous kinematic assumptions made we have

γzx = ∂w

∂x+θy ,

and

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γz y = ∂w

∂y−θx .

Substituting γzx and γz y in equation 90 we obtain

∂

∂x

[∂w

∂x+θy

]+ ∂

∂y

[∂w

∂y−θx

]= 0 (91)

∂2w

∂x2+ ∂θy

∂x+ ∂2w

∂y2− ∂θx

∂y= 0. (92)

For small deformation, we can approximate ∂2w∂x2 = Kx (curvature in X-direction) and

∂2w∂y2 = Ky (curvature in Y-direction), and therefore the equation 92 becomes

Kx +Ky = ∂θx

∂y− ∂θy

∂x. (93)

In the Mindlin theory we have (from equations 2 and 3)

∂θx

∂y=−εy

z,

∂θy

∂x= εx

z. (94)

Thus using above expressions in equation 93 we obtain

z(Kx +Ky ) =−(εx +εy ). (95)

The above condition may not be exactly satisfied in general throughout the thickness and

is a consequence of ad-hoc assumptions made on the kinematics of deformation and stress

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state in the plate.

As mentioned in the beginning, the Mindlin theory does not take into account any mem-

brane strains, i.e. if we assume a case of homogeneous biaxial straining, with εx = εy = δ> 0,

then at z = 0 equation 95 is not satisfied. Lastly, we had assumed σz = 0, but this in turn

implies εz 6= 0 which again has not been accounted exactly.

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4 ABAQUS UEL IMPLEMENTATION

UEL is a programming interface provided with Abaqus Standard using which one can define

customized finite elements. In this section, we discuss several implementation details in

regards to the UEL. In order to work with UEL, one must start by using the default UEL

header from the Abaqus User Subroutines Reference Guide, Section 1.1.28 in Abaqus 2016

documentation, as shown in Figure 5. The user specified program is contained under this

header and the end of the UEL is marked by the ‘END’ syntax.

Figure 5: UEL subroutine header from Abaqus 2016 documentation, Abaqus User Subroutines Refer-ence Guide, Section 1.1.28.

Conceptually it is important to understand the overall operation of the ‘UEL’ block. Figure

6 shows a schematic describing the ‘input’ and ‘output’ flow from the UEL subroutine, that is

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UELUnew

dUold

State variables Updated state variables

RHS

AMATRX

COORDS

Figure 6: Input-Output block diagram for the UEL subroutine.

called for every element. For a current time increment, Abaqus provides the incremental and

final nodal point displacements. State variables at the start of that time increment are also

provided. UEL is then required to return the updated element stiffness matrix, internal force

and state variables all at the end of the current time step. Next we discuss in greater detail,

the incoming quantifies provided by Abaqus into the UEL and associated outgoing quantities

provided back to Abaqus.

4.1 UEL input variables

U : is a vector of size number-of-element-nodes×degrees-of-freedom-per-node, and contains

the degree of freedom values at each node at the end of the current time increment. This is

shown as Unew in Figure 6.

DU : is a vector of size number-of-nodes×degrees-of-freedom-per-node and contains the

incremental degree of freedom values for all the nodes in the previous increment. This is

dUol d in the Figure 6.

COORDS: is a matrix of size-of-spatial-dimensions×number-of-nodes and contains the

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coordinates of the nodes of an element, expressed in (x, y, z) format. A related parameter,

COORD I N AT ES needs to be defined in the UEL section of the Abaqus input file which sets

the value of ‘size-of-spatial-dimensions’.

4.2 UEL output variables

SV ARS: is a vector which stores the state variables of an element and needs to be passed

back to Abaqus after being updated. Typically stress and strains at all integration points

within an element are stored.

AM AT R X : is the stiffness matrix for an element. For our case, small deformation and linear

elasticity, the stiffness matrix is constant, however, if there are non-linear effects then the

stiffness matrix needs to be updated before returning to Abaqus.

R HS: is the internal force vector for the element which needs to be returned at the end of

the current increment. Since, we are solving a linear elasticity problem, for simplicity, the

internal force vector can be calculated as

R HS =−AM AT R X ×U (96)

4.3 Pseudo code for the UEL subroutine

The overall implementation of the UEL is summarized in form of a pseudo code as shown in

Algorithm 1. The pseudo code is based on the theory presented earlier and is self-explanatory.

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We also provide the actual UEL written in Fortran corresponding to this pseudo code subse-

quently.

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Algorithm 1 Pseudo code for UEL.

1: procedure UEL2: U ← Abaqus .D.O.F. values at current increment3: DU ← Abaqus . Previous incremental D.O.F. values4: SV ARS ← Abaqus . Previous state variables for the element5: COORDS ← Abaqus . Current nodal coordinates of the element6: X ←COORDS . Converted 2D nodal coordinates of the element7: Isoparametric formulation:8: Initialize I P _COORDS . Initialize integration point coordinates9: Db ← E ,ν

10: i = 1 . First integration point11: Kb = 0 . Initialize bending stiffness matrix to zero12: loop : . Loop over 4 integration point13: (ξi ,ηi ) ← I P _COORDS . Get (ξ,η) for the current integration point14: N ← (ξi ,ηi ) . Calculate shape functions15: d N ← (ξi ,ηi ) . Calculate derivatives of shape functions16: J ← d N>X . Calculate Jacobian17: J−1 . Calculate inverse of Jacobian18: X Y ′′′ ← d N J−> . Calculate XY derivatives of shape function19: Bb ← X Y ′′′ . Assemble B-matrix for bending

20: I P i ← h3

12 ×B>b Db Bb ×|J | . Stiffness contribution for one integration point

21: if i == 4 then22: break . Exit the loop once all four integration points are completed23: else24: Kb ← Kb + I P i . Form bending stiffness matrix25: i = i +126: goto loop . Iterate for next integration point

27: close;28: Ds ← E ,ν29: Ks = 0 . Initialize shear stiffness matrix to zero30: (ξ,η) ← (0,0) . Reduced integration for shear part31: N ← (ξ,η) . Shape function values for reduced integration at origin32: d N ← (ξ,η) .Derivatives of shape function33: J ← d N>X . Calculate Jacobian34: J−1 . Calculate inverse of Jacobian35: X Y ′′′ ← d N J−> . Calculate XY derivates of shape function36: Bs ← X Y ′′′ . Assemble B-matrix for shear37: Ks ← h ×B>

s Ds Bs ×|J |×4 . Calculate shear stiffness matrix38: AM AT R X ← Kb +Ks . Total stiffness39: R HS ←−AM AT R X U . Calculate internal force vector40:

41: SV ARS → To Abaqus .Updated state variables for the element42: AM AT R X → To Abaqus . Stiffness matrix for the element43: R HS → To Abaqus . Internal force vector

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1 CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC2 CCCCCCC THIS ROUTINE IS WRITTEN BY SUBODH KALIA, NIKHIL PADHYE CCCCCCCCC3 CCCCCCC FOR ABAQUS 2016. THE DISTRIBUTION OF THIS CODE WITHOUT PRIOR CCC4 CCCCCCC PERMISSION OR ANY KIND OF COMMERICAL USE IS STRICTLY CCCCCCCCCCC5 CCCCCCC PROHIBITED. PLEASE CITE THE WORK IF YOU WANT TO USE IT FOR CCCCC6 CCCCCCC YOUR RESEARCH. CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC7 CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC8 SUBROUTINE UEL(RHS,AMATRX,SVARS,ENERGY,NDOFEL,NRHS,NSVARS,9 1 PROPS,NPROPS,COORDS,MCRD,NNODE,U,DU,V,A,JTYPE,TIME,DTIME,

10 2 KSTEP,KINC,JELEM,PARAMS,NDLOAD,JDLTYP,ADLMAG,PREDEF,NPREDF,11 3 LFLAGS,MLVARX,DDLMAG,MDLOAD,PNEWDT,JPROPS,NJPROP,PERIOD)12 C13 INCLUDE 'ABA_PARAM.INC'14 C15 DIMENSION RHS(MLVARX,*),AMATRX(NDOFEL,NDOFEL),PROPS(*),16 1 SVARS(*),ENERGY(8),COORDS(MCRD,NNODE),U(NDOFEL),17 2 DU(MLVARX,*),V(NDOFEL),A(NDOFEL),TIME(2),PARAMS(*),18 3 JDLTYP(MDLOAD,*),ADLMAG(MDLOAD,*),DDLMAG(MDLOAD,*),19 4 PREDEF(2,NPREDF,NNODE),LFLAGS(*),JPROPS(*)20 21 Double precision, Dimension(12, 1) :: FORCE, V_TEMP22 Double precision, Dimension(12, 1) :: U_CONV, DU_CONV23 Double precision, Dimension(4, 1) :: N24 Double precision, Dimension(4, 2) :: DN, IP_COORDS, COORDS_T25 Double precision, Dimension(4, 2) :: XY_DERIVATIVES26 Double precision, Dimension(2, 4) :: XY_DERIVATIVES_T27 Double precision, Dimension(2, 4) :: DN_TRANSPOSE28 Double precision, Dimension(2, 2) :: J, INV_J29 Double precision, Dimension(3, 12) :: M_B_MAT_BENDING, M_TEMP30 Double precision, Dimension(12, 12) :: M_STIFFNESS, M_TEMP_STIFF31 Double precision, Dimension(12, 12) :: M_STIFF32 Double precision, Dimension(12, 3) :: M_B_MAT_BENDING_TRANSPOSE33 Double precision, Dimension(3, 1) :: V_STRAIN_INCREMENT_BENDING34 Double precision, Dimension(3, 1) :: V_STRESS_INCREMENT_BENDING35 Double precision, Dimension(3, 3) :: M_D_BENDING36 Double precision, Dimension(2, 12) :: M_B_MAT_SHEAR37 Double precision, Dimension(2, 1) :: V_STRAIN_INCREMENT_SHEAR38 Double precision, Dimension(2, 1) :: V_STRESS_INCREMENT_SHEAR39 Double precision, Dimension(12, 2) :: M_B_MAT_SHEAR_TRANSPOSE40 Double precision, Dimension(2, 2) :: M_D_SHEAR41 42 Double precision :: CHI, ETA, S_E, S_NU, S_INERTIA, S_THICKNESS43 Double precision :: S_J_DETERMINANT, S_KAPPA44 45 INTEGER :: NDOFEL, NSVARS, NPROPS, MCRD, NNODE, K1, K2, K3, NUM_IP46 INTEGER :: NDOF, NUM_SVAR_PER_IP, POSITION_POINTER47 48 PARAMETER(NUM_IP=4, NUM_SVAR_PER_IP=10)49 50 COORDS_T(1,1) = COORDS(1,1)51 COORDS_T(1,2) = COORDS(2,1)52 53 COORDS_T(2,1) = COORDS(1,2)54 COORDS_T(2,2) = COORDS(2,2)55 56 COORDS_T(3,1) = COORDS(1,3)57 COORDS_T(3,2) = COORDS(2,3)58 59 COORDS_T(4,1) = COORDS(1,4)60 COORDS_T(4,2) = COORDS(2,4)61 62 DO K1 = 1, 1263 U_CONV(K1, 1) = U(K1)64 DU_CONV(K1, 1) = DU(K1,1)65 END DO66 67 IF (LFLAGS(3).EQ.4) THEN68 DO K1=1, 1269 DO K2=1, 12


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70 AMATRX(K1, K2) = 0.D071 END DO72 AMATRX(K1, K1) = 1.D073 END DO74 GOTO 99975 END IF76 77 CCCCCCCCCCCCCCCCCCCCCCCCCC INITIALIZE PARAMETERS CCCCCCCCCCCCCCCCCCCCCCC78 DO K1 = 1, 1279 DO K2 = 1, 1280 M_STIFFNESS(K1, K2) = 0.D081 M_STIFF(K1, K2) = 0.D082 END DO83 END DO84 85 DO K1 = 1, 1286 FORCE(K1, 1) = 0.D087 V_TEMP(K1, 1) = 0.D088 END DO89 90 S_E = 20000000091 S_NU = 0.3D092 S_G = S_E/(2.D0*(1.D0 + S_NU))93 S_KAPPA = 6.D0/6.D094 S_THICKNESS = 0.1D095 S_INERTIA = (S_THICKNESS**3.D0)/12.D096 97 M_D_BENDING(1, 1) = 1.D098 M_D_BENDING(1, 2) = S_NU99 M_D_BENDING(1, 3) = 0.D0

100 M_D_BENDING(2, 1) = S_NU101 M_D_BENDING(2, 2) = 1.D0102 M_D_BENDING(2, 3) = 0.D0103 M_D_BENDING(3, 1) = 0.D0104 M_D_BENDING(3, 2) = 0.D0105 M_D_BENDING(3, 3) = (1.D0 - S_NU)/2.D0106 107 M_D_BENDING = (S_E/(1.D0 - (S_NU**2.D0)))*M_D_BENDING108 109 M_D_SHEAR(1, 1) = S_G110 M_D_SHEAR(1, 2) = 0.D0111 M_D_SHEAR(2, 1) = 0.D0112 M_D_SHEAR(2, 2) = S_G113 114 CALL TWO_BY_TWO_INTEGRATION_POINT_COORDINATES(IP_COORDS, NUM_IP)115 116 DO K1 = 1, NUM_IP117 CHI = IP_COORDS(K1, 1)118 ETA = IP_COORDS(K1, 2)119 CALL FOUR_NODED_BILINEAR_SHAPE_FUNCTIONS(CHI, ETA, N, DN)120 121 CALL GENERIC_TRANSPOSE_OF_MATRIX(DN, 4, 2, DN_TRANSPOSE)122 123 CALL GENERIC_MATRIX_MULTIPLY(DN_TRANSPOSE,2,4,COORDS_T,4,2,J)124 125 CALL MAT_INVERSE_TWO_BY_TWO(J, INV_J, S_J_DETERMINANT)126 127 CALL GENERIC_MATRIX_MULTIPLY(INV_J,2,2,DN_TRANSPOSE,128 1 2,4,XY_DERIVATIVES_T)129 130 CALL GENERIC_TRANSPOSE_OF_MATRIX(XY_DERIVATIVES_T,2,4,131 1 XY_DERIVATIVES)132 133 CCCCCCCCCCCCCCCCCCCCCC ASSEMBLE B_BENDING (3X12) CCCCCCCCCCCCCCCCCCCCCCC134 DO K2 = 1, 3135 DO K3 = 1, 12136 M_B_MAT_BENDING(K2, K3) = 0.D0137 END DO138 END DO


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139 M_B_MAT_BENDING(1, 3) = XY_DERIVATIVES(1, 1)140 M_B_MAT_BENDING(1, 6) = XY_DERIVATIVES(2, 1)141 M_B_MAT_BENDING(1, 9) = XY_DERIVATIVES(3, 1)142 M_B_MAT_BENDING(1, 12) = XY_DERIVATIVES(4, 1)143 144 M_B_MAT_BENDING(2, 2) = -XY_DERIVATIVES(1, 2)145 M_B_MAT_BENDING(2, 5) = -XY_DERIVATIVES(2, 2)146 M_B_MAT_BENDING(2, 8) = -XY_DERIVATIVES(3, 2)147 M_B_MAT_BENDING(2, 11) = -XY_DERIVATIVES(4, 2)148 149 M_B_MAT_BENDING(3, 2) = -XY_DERIVATIVES(1, 1)150 M_B_MAT_BENDING(3, 5) = -XY_DERIVATIVES(2, 1)151 M_B_MAT_BENDING(3, 8) = -XY_DERIVATIVES(3, 1)152 M_B_MAT_BENDING(3, 11) = -XY_DERIVATIVES(4, 1)153 154 M_B_MAT_BENDING(3, 3) = XY_DERIVATIVES(1, 2)155 M_B_MAT_BENDING(3, 6) = XY_DERIVATIVES(2, 2)156 M_B_MAT_BENDING(3, 9) = XY_DERIVATIVES(3, 2)157 M_B_MAT_BENDING(3, 12) = XY_DERIVATIVES(4, 2)158 CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC159 CALL GENERIC_MATRIX_MULTIPLY(M_B_MAT_BENDING, 3, 12,160 1 DU_CONV, 12, 1,161 2 V_STRAIN_INCREMENT_BENDING)162 163 CALL GENERIC_MATRIX_MULTIPLY(M_D_BENDING, 3, 3,164 1 V_STRAIN_INCREMENT_BENDING, 3, 1,165 2 V_STRESS_INCREMENT_BENDING)166 167 CALL GENERIC_MATRIX_MULTIPLY(S_INERTIA*S_J_DETERMINANT*168 1 M_D_BENDING, 3, 3, M_B_MAT_BENDING,169 2 3, 12, M_TEMP)170 171 CALL GENERIC_TRANSPOSE_OF_MATRIX(M_B_MAT_BENDING, 3, 12,172 1 M_B_MAT_BENDING_TRANSPOSE)173 174 CALL GENERIC_MATRIX_MULTIPLY(M_B_MAT_BENDING_TRANSPOSE, 12, 3,175 1 M_TEMP, 3, 12, M_TEMP_STIFF)176 177 CALL GENERIC_ADD_MATRIX(M_STIFFNESS, M_TEMP_STIFF, 12, 12,178 1 M_STIFF)179 180 M_STIFFNESS = M_STIFF181 182 END DO183 CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC184 CHI = 0.D0185 ETA = 0.D0186 CALL FOUR_NODED_BILINEAR_SHAPE_FUNCTIONS(CHI, ETA, N, DN)187 CALL GENERIC_TRANSPOSE_OF_MATRIX(DN, 4, 2, DN_TRANSPOSE)188 CALL GENERIC_MATRIX_MULTIPLY(DN_TRANSPOSE,2,4,COORDS_T,4,2,J)189 CALL MAT_INVERSE_TWO_BY_TWO(J, INV_J, S_J_DETERMINANT)190 CALL GENERIC_MATRIX_MULTIPLY(INV_J,2,2,DN_TRANSPOSE,191 1 2,4,XY_DERIVATIVES_T)192 193 CALL GENERIC_TRANSPOSE_OF_MATRIX(XY_DERIVATIVES_T,2,4,194 1 XY_DERIVATIVES)195 196 CCCCCCCCCCCCCCCCCCCCCC ASSEMBLE B_SHEAR (2X12) CCCCCCCCCCCCCCCCCCCCCCC197 DO K2 = 1, 2198 DO K3 = 1, 12199 M_B_MAT_SHEAR(K2, K3) = 0.D0200 END DO201 END DO202 M_B_MAT_SHEAR(1, 1) = XY_DERIVATIVES(1, 1)203 M_B_MAT_SHEAR(1, 4) = XY_DERIVATIVES(2, 1)204 M_B_MAT_SHEAR(1, 7) = XY_DERIVATIVES(3, 1)205 M_B_MAT_SHEAR(1, 10) = XY_DERIVATIVES(4, 1)206 207 M_B_MAT_SHEAR(1, 3) = N(1, 1)


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208 M_B_MAT_SHEAR(1, 6) = N(2, 1)209 M_B_MAT_SHEAR(1, 9) = N(3, 1)210 M_B_MAT_SHEAR(1, 12) = N(4, 1)211 212 M_B_MAT_SHEAR(2, 1) = XY_DERIVATIVES(1, 2)213 M_B_MAT_SHEAR(2, 4) = XY_DERIVATIVES(2, 2)214 M_B_MAT_SHEAR(2, 7) = XY_DERIVATIVES(3, 2)215 M_B_MAT_SHEAR(2, 10) = XY_DERIVATIVES(4, 2)216 217 M_B_MAT_SHEAR(2, 2) = -N(1, 1)218 M_B_MAT_SHEAR(2, 5) = -N(2, 1)219 M_B_MAT_SHEAR(2, 8) = -N(3, 1)220 M_B_MAT_SHEAR(2, 11) = -N(4, 1)221 CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC222 CALL GENERIC_MATRIX_MULTIPLY(M_B_MAT_SHEAR, 2, 12,223 1 DU_CONV, 12, 1,224 2 V_STRAIN_INCREMENT_SHEAR)225 226 CALL GENERIC_MATRIX_MULTIPLY(M_D_SHEAR, 2, 2,227 1 V_STRAIN_INCREMENT_SHEAR, 2, 1,228 2 V_STRESS_INCREMENT_SHEAR)229 230 CALL GENERIC_TRANSPOSE_OF_MATRIX(M_B_MAT_SHEAR, 2, 12,231 1 M_B_MAT_SHEAR_TRANSPOSE)232 233 234 CALL GENERIC_MATRIX_MULTIPLY(S_KAPPA*S_J_DETERMINANT*235 1 S_THICKNESS*4.D0*M_D_SHEAR, 2, 2, M_B_MAT_SHEAR,236 2 2, 12, M_TEMP)237 238 CALL GENERIC_MATRIX_MULTIPLY(M_B_MAT_SHEAR_TRANSPOSE, 12, 2,239 1 M_TEMP, 2, 12, M_TEMP_STIFF)240 241 CALL GENERIC_ADD_MATRIX(M_STIFFNESS, M_TEMP_STIFF, 12, 12,242 1 M_STIFF)243 244 M_STIFFNESS = M_STIFF245 246 CCCCCCCCCCCCCCCCCCCCCC UPDATE STRESS, STRAIN STATE VARIABLES CCCCCCCCC247 DO K1 = 1, NUM_IP248 249 POSITION_POINTER = (K1 - 1)*NUM_SVAR_PER_IP + 1250 C SIGMA_X251 SVARS(POSITION_POINTER) = SVARS(POSITION_POINTER)+252 1 V_STRESS_INCREMENT_BENDING(1, 1)253 C SIGMA_Y254 SVARS(POSITION_POINTER+1) = SVARS(POSITION_POINTER+1)+255 1 V_STRESS_INCREMENT_BENDING(2, 1)256 C SIGMA_XY257 SVARS(POSITION_POINTER+2) = SVARS(POSITION_POINTER+2)+258 1 V_STRESS_INCREMENT_BENDING(3, 1)259 C SIGMA_XZ260 SVARS(POSITION_POINTER+3) = SVARS(POSITION_POINTER+3)+261 1 V_STRESS_INCREMENT_SHEAR(1, 1)262 C SIGMA_YZ263 SVARS(POSITION_POINTER+4) = SVARS(POSITION_POINTER+4)+264 1 V_STRESS_INCREMENT_SHEAR(2, 1)265 C EPSILON_X266 SVARS(POSITION_POINTER+5) = SVARS(POSITION_POINTER+5)+267 1 V_STRAIN_INCREMENT_BENDING(1, 1)268 C EPSILON_Y269 SVARS(POSITION_POINTER+6) = SVARS(POSITION_POINTER+6)+270 1 V_STRAIN_INCREMENT_BENDING(2, 1)271 C EPSILON_XY272 SVARS(POSITION_POINTER+7) = SVARS(POSITION_POINTER+7)+273 1 V_STRAIN_INCREMENT_BENDING(3, 1)274 C EPSILON_XZ275 SVARS(POSITION_POINTER+8) = SVARS(POSITION_POINTER+8)+276 1 V_STRAIN_INCREMENT_SHEAR(1, 1)


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277 C EPSILON_YZ278 SVARS(POSITION_POINTER+9) = SVARS(POSITION_POINTER+9)+279 1 V_STRAIN_INCREMENT_SHEAR(2, 1)280 281 END DO282 283 CCCCCCCCCCCCCCCCCCCCCCCC CALCULATE FORCE CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC284 CALL GENERIC_MATRIX_MULTIPLY(M_STIFFNESS, 12, 12,285 1 U_CONV, 12, 1, FORCE)286 287 CCCCCCCCCCCCCCCCCCCCCCCCCCC RETURN TO ABAQUS CCCCCCCCCCCCCCCCCCCCCCCCCCC288 DO K1 = 1, 12289 RHS(K1, 1) = -FORCE(K1, 1)290 END DO291 292 DO K1 = 1, 12293 DO K2 = 1, 12294 AMATRX(K1, K2) = M_STIFFNESS(K1, K2)295 END DO296 END DO297 298 999 CONTINUE299 300 301 RETURN302 END


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5 BENCHMARKING OF UEL IMPLEMENTATION FOR MINDLIN PLATE

ELEMENT

In this section, we compare the results of plate bending from the Abaqus with the default S4

element, Matlab implementation of Mindlin plate from [5], and our developed 4-noded UEL

subroutine.

5.1 Single element test

Figure 7: Boundary conditions for a single element test.

A single element of length and width of 2 units each and a thickness of 0.1 units is

considered. Boundary conditions for this element are shown in Figure 7, where we have

restricted w,θx ,θy at nodes 1, 2, and 3. At node 4, we restrict θx and θy and apply a maximum

displacement of 0.1 in the positive Z-direction. Thus, this is a well defined displacement

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driven boundary value problem for which we seek to find the solution. It is important to note

that Abaqus S4 element has a total of 6 degrees of freedom at each node, therefore we simply

constrain the degrees of freedom other than w , θx and θy at all nodes.

Figure 8, 9 and 10 shows the reaction forces (RF3) in Z-direction, and reaction moments

RM1 and RM2 in X- and Y-directions, respectively from Abaqus S4 element analysis.

Figure 8: Reaction forces in Z-direction with Abaqus S4 element.

Results from the referenced Matlab code, ABAQUS UEL implementation of Mindlin plate,

and Abaqus S4 element analysis were gathered and compared for the nodal point reaction

forces and moments and shown in Figures 11, 12 and 13. Our UEL implementation yields

results exactly similar to those predicted by the referenced Matlab code. In general there

is a good agreement with Abaqus S4 element analysis, however, there is deviation in the

reaction forces at nodes 2 and 3. A detailed investigation reveals that reaction moments for

the UEL implementation and Matlab code were same in magnitude at four nodes, whereas

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Figure 9: Reaction moments in X-direction with Abaqus S4 element.

Figure 10: Reaction moments in Y-direction with Abaqus S4 element.

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in case of Abaqus S4 element the reaction moments in both directions varied within the

element. A simple explanation for this observation stems from the fact that in the Mindlin

plate element (UEL or Matlab implementation) the transverse shear strains are assumed to

be constant through the thickness, and interpolated based on the nodal point rotations and

displacements. The bending strains are interpolated based on nodal point rotations only.

In the current case, we have set the rotations and displacements at all the nodes equal to

zero except for the imposed displacement in the vertical direction at node 4. This causes the

vanishing of the bending strains all together and linearly varying (constant through thickness)

transverse shear strains. Thus the reaction moments at the nodes in the Mindlin element

appear to simply maintain the global moment-equilibrium and have the same magnitude in

both X- and Y-directions, at all nodes, due to symmetric shape of the element. On the other

hand the S4 element of Abaqus is more sophisticated and picks up the variation of bending

and shear more accurately.

Exact matching of the results between our UEL implementation and referenced Matlab

code, and overall consistency with respect to the Abaqus S4 element (foregoing certain

deviations) indicates that our procedure is working well.

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-350,000

-300,000

-250,000

-200,000

-150,000

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-50,000

0

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

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acti

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rce

in z

-dir

ect

ion

at

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1

Displacement

Abaqus S4 element

UEL 4-noded element

MATLAB 4-noded element

(a)

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Displacement

Abaqus S4 element

UEL 4-noded element


(b)

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Displacement

Abaqus S4 element

UEL 4-noded element


(c)

0

50,000

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400,000

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

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rce

in z

-dir

ect

ion

at

no

de

4

Displacement

Abaqus S4 element

UEL 4-noded element


(d)

Figure 11: Plots for reaction forces in Z-direction at the nodes with respect to nodal displacement.

-180,000

-160,000

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Displacement

Abaqus S4 element

UEL 4-noded element


(a)

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Displacement

Abaqus S4 element

UEL 4-noded element


(b)

-180,000

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3

Displacement

Abaqus S4 element

UEL 4-noded element


(c)

-200,000

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0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

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in x

-dir

ect

ion

at

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4

Displacement

Abaqus S4 element

UEL 4-noded element


(d)

Figure 12: Plots for reaction moments in X-direction at the nodes with respect to nodal displacements.

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0

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Displacement

Abaqus S4 element

UEL 4-noded element


(a)

0

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Abaqus S4 element

UEL 4-noded element


(b)

0

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Abaqus S4 element

UEL 4-noded element


(c)

0

20,000

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Abaqus S4 element

UEL 4-noded element


(d)

Figure 13: Plots of reaction moments in Y-direction at the nodes with respect to nodal displacements.

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5.2 Testing multiple elements

Now we shall compare the results from Abaqus, our UEL implementation and the Matlab

code on a laterally loaded plate, which is clamped at the edges, and made up of multiple

elements. 400 elements in a (20×20) grid along X and Y directions are employed. The overall

dimensions of the plate in X-, Y- and Z-directions are again taken as 2 units, 2 units and 0.1

units, respectively. Maximum lateral pressure of 1 unit per unit area is applied3.

Matlab code yields a maximum displacement of −1.15765×10−6 in Z-direction at the

central node. This is shown in Figure 14. With our implemented UEL subroutine,a similar

maximum displacement of −1.15765× 10−6 in Z-direction is noted, shown in Figure 16.

Using S4 element analysis in Abaqus, we get a maximum displacement of −1.15751×10−6

in Z-direction, Figure 15. For Abaqus and UEL results, we use a scale factor of 106 to give a

consistent visual comparison with Matlab generated displacements. These results reveal an

excellent agreement.

Figure 17 shows the variation of maximum displacement in Z-direction as we gradually

vary the lateral pressure to 1 unit. The obtained results confirm that our implementation is

accurate and working as desired.

3This is achieved by applying an equivalent level of concentrated force of 0.01 units over all the elementnodes.

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21.5

Y-axis

10.5

02

1.5

X-axis

1

0.5

#10-6

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0

Z-a

xis

Figure 14: Plate deflection from Matlab code.

Figure 15: Plate deflection from Abaqus S4 element analysis with a scale factor of 106.

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Figure 16: Plate deflection from UEL with a scale factor of 106.

-1.3E-06

-1.1E-06

-9E-07

-7E-07

-5E-07

-3E-07

-1E-07

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Max

imu

m d

isp

lace

me

nt

Lateral pressure

Abaqus S4 element

UEL 4-noded element


Figure 17: Comparison of deflections for a laterally loaded plate clamped at the edges.

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6 ACKNOWLEDGMENTS

Nikhil Padhye appreciates past technical discussions with his advisor Professor David M.

Parks for motivating the usage of Abaqus in finite element studies during the course of his

doctoral thesis [8].

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REFERENCES

[1] Ver ABAQUS. 6.14 documentation. Dassault Systemes Simulia Corporation, 2014.

[2] F Auricchio and RL Taylor. A shear deformable plate element with an exact thin limit.

Computer Methods in Applied Mechanics and Engineering, 118(3-4):393–412, 1994.

[3] Klaus-Jürgen Bathe and Eduardo N Dvorkin. A four-node plate bending element based

on mindlin/reissner plate theory and a mixed interpolation. International Journal for

Numerical Methods in Engineering, 21(2):367–383, 1985.

[4] Jean-Louis Batoz and Mabrouk Ben Tahar. Evaluation of a new quadrilateral thin

plate bending element. International Journal for Numerical Methods in Engineering,

18(11):1655–1677, 1982.

[5] Ferreira. MATLAB Codes for Finite Element Analysis - Solids and Structures. Solid mechan-

ics & it applications 157. Springer, 2009.

[6] Bo Häggblad and Klaus-Jürgen Bathe. Specifications of boundary conditions for reiss-

ner/mindlin plate bending finite elements. International Journal for Numerical Methods

in Engineering, 30(5):981–1011, 1990.

[7] Thomas JR Hughes and TEi Tezduyar. Finite elements based upon mindlin plate theory

with particular reference to the four-node bilinear isoparametric element. Journal of

applied mechanics, 48(3):587–596, 1981.

[8] Nikhil Padhye. Sub-Tg, solid-state, plasticity-induced bonding of polymeric films and

continuous forming. PhD thesis, Massachusetts Institute of Technology, 2015.

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[9] Erik Saether and Alexander Tessler. User-defined subroutine for implementation of

higher-order shell element in abaqus. Technical report, ARMY RESEARCH LAB ADELPHI

MD, 1993.

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1 *Heading2 ** Job name: abaqus_4_elements_lift_up Model name: Model-13 ** Generated by: Abaqus/CAE 2016.HF24 *Preprint, echo=NO, model=NO, history=NO, contact=NO5 **6 ** PARTS7 **8 *Part, name=Plate9 *Node

10 1, 0., 0., 0.11 2, 2., 0., 0.12 3, 0., 2., 0.13 4, 2., 2., 0.14 ******************************************************15 ******************************************************16 ******************************************************17 *USER ELEMENT, TYPE=U1, NODES=4, COORDINATES=3, VAR=40, 18 INTEGRATION=419 3,4,520 *ELEMENT,TYPE=U1,ELSET=SOLID21 1, 1,2,4,322 *UEL PROPERTY, ELSET=SOLID23 **************************************24 *End Part25 ** 26 **27 ** ASSEMBLY28 **29 *Assembly, name=Assembly30 ** 31 *Instance, name=Plate-1, part=Plate32 *End Instance33 ** 34 *Nset, nset=all_elements, instance=Plate-1, generate35 1, 4, 136 *Elset, elset=all_elements, instance=Plate-137 1,38 *Nset, nset=_PickedSet13, internal, instance=Plate-139 4,40 *Nset, nset=_PickedSet14, internal, instance=Plate-1, generate41 1, 3, 142 *End Assembly43 ** 44 ** 45 ** BOUNDARY CONDITIONS46 ** 47 ** Name: fix_nodes Type: Symmetry/Antisymmetry/Encastre48 *Boundary49 _PickedSet14, ENCASTRE50 ** ----------------------------------------------------------------51 ** 52 ** STEP: loading53 ** 54 *Step, name=loading, nlgeom=NO, inc=10000055 *Static, direct56 0.1, 1., 57 ** 58 ** BOUNDARY CONDITIONS59 ** 60 ** Name: lift_up Type: Displacement/Rotation61 *Boundary62 _PickedSet13, 3, 3, 0.163 _PickedSet13, 4, 464 _PickedSet13, 5, 565 ** 66 ** OUTPUT REQUESTS67 ** 68 *Restart, write, frequency=069 **


Abaqus job input file for UEL with single element

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70 ** FIELD OUTPUT: F-Output-271 ** 72 *Output, field73 *Node Output74 CF, RF, RM, RT, TF, VF75 *Element Output, directions=YES76 BF, CENTMAG, CENTRIFMAG, CORIOMAG, ESF1, GRAV, HP, NFORC, NFORCSO, P, ROTAMAG, SF,

TRNOR, TRSHR77 ** 78 ** FIELD OUTPUT: F-Output-179 ** 80 *Output, field, variable=PRESELECT81 ** 82 ** HISTORY OUTPUT: H-Output-283 ** 84 *Output, history85 *Element Output86 IRF1, IRF2, IRF3, IRM1, IRM2, IRM387 ** 88 ** HISTORY OUTPUT: H-Output-189 ** 90 *Output, history, variable=PRESELECT91 *End Step92


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1 %................................................................2 3 % MATLAB codes for Finite Element Analysis4 % problem19.m5 % Mindlin plate in bending6 % antonio ferreira 20087 8 % clear memory9 clc

10 clear *;11 clear all;12 colordef white;13 format long e;14 close all;15 16 % materials17 E = 200000000; poisson = 0.30; kapa=5/6;18 thickness=0.1;19 I=thickness^3/12;20 21 % matrix C22 % bending part23 C_bending=I*E/(1-poisson^2)*...24 [1 poisson 0;poisson 1 0;0 0 (1-poisson)/2];25 % shear part26 C_shear=kapa*thickness*E/2/(1+poisson)*eye(2);27 28 % load29 P = -1;30 31 %Mesh generation32 L = 2;33 numberElementsX=1;34 numberElementsY=1;35 numberElements=numberElementsX*numberElementsY;36 %37 [nodeCoordinates, elementNodes] = ...38 rectangularMesh(L,L,numberElementsX,numberElementsY);39 xx=nodeCoordinates(:,1);40 yy=nodeCoordinates(:,2);41 42 %drawingMesh(nodeCoordinates,elementNodes,'Q4','k-');43 %axis off44 numberNodes=size(xx,1);45 46 % GDof: global number of degrees of freedom47 GDof=3*numberNodes;48 49 % computation of the system stiffness matrix and force vector50 [stiffness]=...51 formStiffnessMatrixMindlinQ4(GDof,numberElements,...52 elementNodes,numberNodes,nodeCoordinates,C_shear,...53 C_bending,thickness,I);54 55 timestep = 1;56 57 U=[58 059 060 061 0.1*timestep62 063 064 065 066 067 068 069 0];


Modified Matlab main routine for single element

of 97

70 71 72 stiffness*U


of 97

1 *Heading2 ** Job name: push_center_Down Model name: Model-13 ** Generated by: Abaqus/CAE 2016.HF24 *Preprint, echo=NO, model=NO, history=NO, contact=NO5 **6 ** PARTS7 **8 *Part, name=Plate9 *Node

10 1, 0., 0., 0.11 2, 0.100000001, 0., 0.12 3, 0.200000003, 0., 0.13 4, 0.300000012, 0., 0.14 5, 0.400000006, 0., 0.15 6, 0.5, 0., 0.16 7, 0.600000024, 0., 0.17 8, 0.699999988, 0., 0.18 9, 0.800000012, 0., 0.19 10, 0.899999976, 0., 0.20 11, 1., 0., 0.21 12, 1.10000002, 0., 0.22 13, 1.20000005, 0., 0.23 14, 1.29999995, 0., 0.24 15, 1.39999998, 0., 0.25 16, 1.5, 0., 0.26 17, 1.60000002, 0., 0.27 18, 1.70000005, 0., 0.28 19, 1.79999995, 0., 0.29 20, 1.89999998, 0., 0.30 21, 2., 0., 0.31 22, 0., 0.100000001, 0.32 23, 0.100000001, 0.100000001, 0.33 24, 0.200000003, 0.100000001, 0.34 25, 0.300000012, 0.100000001, 0.35 26, 0.400000006, 0.100000001, 0.36 27, 0.5, 0.100000001, 0.37 28, 0.600000024, 0.100000001, 0.38 29, 0.699999988, 0.100000001, 0.39 30, 0.800000012, 0.100000001, 0.40 31, 0.899999976, 0.100000001, 0.41 32, 1., 0.100000001, 0.42 33, 1.10000002, 0.100000001, 0.43 34, 1.20000005, 0.100000001, 0.44 35, 1.29999995, 0.100000001, 0.45 36, 1.39999998, 0.100000001, 0.46 37, 1.5, 0.100000001, 0.47 38, 1.60000002, 0.100000001, 0.48 39, 1.70000005, 0.100000001, 0.49 40, 1.79999995, 0.100000001, 0.50 41, 1.89999998, 0.100000001, 0.51 42, 2., 0.100000001, 0.52 43, 0., 0.200000003, 0.53 44, 0.100000001, 0.200000003, 0.54 45, 0.200000003, 0.200000003, 0.55 46, 0.300000012, 0.200000003, 0.56 47, 0.400000006, 0.200000003, 0.57 48, 0.5, 0.200000003, 0.58 49, 0.600000024, 0.200000003, 0.59 50, 0.699999988, 0.200000003, 0.60 51, 0.800000012, 0.200000003, 0.61 52, 0.899999976, 0.200000003, 0.62 53, 1., 0.200000003, 0.63 54, 1.10000002, 0.200000003, 0.64 55, 1.20000005, 0.200000003, 0.65 56, 1.29999995, 0.200000003, 0.66 57, 1.39999998, 0.200000003, 0.67 58, 1.5, 0.200000003, 0.68 59, 1.60000002, 0.200000003, 0.69 60, 1.70000005, 0.200000003, 0.


Abaqus job input file for UEL with multiple elements

of 97

70 61, 1.79999995, 0.200000003, 0.71 62, 1.89999998, 0.200000003, 0.72 63, 2., 0.200000003, 0.73 64, 0., 0.300000012, 0.74 65, 0.100000001, 0.300000012, 0.75 66, 0.200000003, 0.300000012, 0.76 67, 0.300000012, 0.300000012, 0.77 68, 0.400000006, 0.300000012, 0.78 69, 0.5, 0.300000012, 0.79 70, 0.600000024, 0.300000012, 0.80 71, 0.699999988, 0.300000012, 0.81 72, 0.800000012, 0.300000012, 0.82 73, 0.899999976, 0.300000012, 0.83 74, 1., 0.300000012, 0.84 75, 1.10000002, 0.300000012, 0.85 76, 1.20000005, 0.300000012, 0.86 77, 1.29999995, 0.300000012, 0.87 78, 1.39999998, 0.300000012, 0.88 79, 1.5, 0.300000012, 0.89 80, 1.60000002, 0.300000012, 0.90 81, 1.70000005, 0.300000012, 0.91 82, 1.79999995, 0.300000012, 0.92 83, 1.89999998, 0.300000012, 0.93 84, 2., 0.300000012, 0.94 85, 0., 0.400000006, 0.95 86, 0.100000001, 0.400000006, 0.96 87, 0.200000003, 0.400000006, 0.97 88, 0.300000012, 0.400000006, 0.98 89, 0.400000006, 0.400000006, 0.99 90, 0.5, 0.400000006, 0.

100 91, 0.600000024, 0.400000006, 0.101 92, 0.699999988, 0.400000006, 0.102 93, 0.800000012, 0.400000006, 0.103 94, 0.899999976, 0.400000006, 0.104 95, 1., 0.400000006, 0.105 96, 1.10000002, 0.400000006, 0.106 97, 1.20000005, 0.400000006, 0.107 98, 1.29999995, 0.400000006, 0.108 99, 1.39999998, 0.400000006, 0.109 100, 1.5, 0.400000006, 0.110 101, 1.60000002, 0.400000006, 0.111 102, 1.70000005, 0.400000006, 0.112 103, 1.79999995, 0.400000006, 0.113 104, 1.89999998, 0.400000006, 0.114 105, 2., 0.400000006, 0.115 106, 0., 0.5, 0.116 107, 0.100000001, 0.5, 0.117 108, 0.200000003, 0.5, 0.118 109, 0.300000012, 0.5, 0.119 110, 0.400000006, 0.5, 0.120 111, 0.5, 0.5, 0.121 112, 0.600000024, 0.5, 0.122 113, 0.699999988, 0.5, 0.123 114, 0.800000012, 0.5, 0.124 115, 0.899999976, 0.5, 0.125 116, 1., 0.5, 0.126 117, 1.10000002, 0.5, 0.127 118, 1.20000005, 0.5, 0.128 119, 1.29999995, 0.5, 0.129 120, 1.39999998, 0.5, 0.130 121, 1.5, 0.5, 0.131 122, 1.60000002, 0.5, 0.132 123, 1.70000005, 0.5, 0.133 124, 1.79999995, 0.5, 0.134 125, 1.89999998, 0.5, 0.135 126, 2., 0.5, 0.136 127, 0., 0.600000024, 0.137 128, 0.100000001, 0.600000024, 0.138 129, 0.200000003, 0.600000024, 0.


of 97

139 130, 0.300000012, 0.600000024, 0.140 131, 0.400000006, 0.600000024, 0.141 132, 0.5, 0.600000024, 0.142 133, 0.600000024, 0.600000024, 0.143 134, 0.699999988, 0.600000024, 0.144 135, 0.800000012, 0.600000024, 0.145 136, 0.899999976, 0.600000024, 0.146 137, 1., 0.600000024, 0.147 138, 1.10000002, 0.600000024, 0.148 139, 1.20000005, 0.600000024, 0.149 140, 1.29999995, 0.600000024, 0.150 141, 1.39999998, 0.600000024, 0.151 142, 1.5, 0.600000024, 0.152 143, 1.60000002, 0.600000024, 0.153 144, 1.70000005, 0.600000024, 0.154 145, 1.79999995, 0.600000024, 0.155 146, 1.89999998, 0.600000024, 0.156 147, 2., 0.600000024, 0.157 148, 0., 0.699999988, 0.158 149, 0.100000001, 0.699999988, 0.159 150, 0.200000003, 0.699999988, 0.160 151, 0.300000012, 0.699999988, 0.161 152, 0.400000006, 0.699999988, 0.162 153, 0.5, 0.699999988, 0.163 154, 0.600000024, 0.699999988, 0.164 155, 0.699999988, 0.699999988, 0.165 156, 0.800000012, 0.699999988, 0.166 157, 0.899999976, 0.699999988, 0.167 158, 1., 0.699999988, 0.168 159, 1.10000002, 0.699999988, 0.169 160, 1.20000005, 0.699999988, 0.170 161, 1.29999995, 0.699999988, 0.171 162, 1.39999998, 0.699999988, 0.172 163, 1.5, 0.699999988, 0.173 164, 1.60000002, 0.699999988, 0.174 165, 1.70000005, 0.699999988, 0.175 166, 1.79999995, 0.699999988, 0.176 167, 1.89999998, 0.699999988, 0.177 168, 2., 0.699999988, 0.178 169, 0., 0.800000012, 0.179 170, 0.100000001, 0.800000012, 0.180 171, 0.200000003, 0.800000012, 0.181 172, 0.300000012, 0.800000012, 0.182 173, 0.400000006, 0.800000012, 0.183 174, 0.5, 0.800000012, 0.184 175, 0.600000024, 0.800000012, 0.185 176, 0.699999988, 0.800000012, 0.186 177, 0.800000012, 0.800000012, 0.187 178, 0.899999976, 0.800000012, 0.188 179, 1., 0.800000012, 0.189 180, 1.10000002, 0.800000012, 0.190 181, 1.20000005, 0.800000012, 0.191 182, 1.29999995, 0.800000012, 0.192 183, 1.39999998, 0.800000012, 0.193 184, 1.5, 0.800000012, 0.194 185, 1.60000002, 0.800000012, 0.195 186, 1.70000005, 0.800000012, 0.196 187, 1.79999995, 0.800000012, 0.197 188, 1.89999998, 0.800000012, 0.198 189, 2., 0.800000012, 0.199 190, 0., 0.899999976, 0.200 191, 0.100000001, 0.899999976, 0.201 192, 0.200000003, 0.899999976, 0.202 193, 0.300000012, 0.899999976, 0.203 194, 0.400000006, 0.899999976, 0.204 195, 0.5, 0.899999976, 0.205 196, 0.600000024, 0.899999976, 0.206 197, 0.699999988, 0.899999976, 0.207 198, 0.800000012, 0.899999976, 0.


of 97

208 199, 0.899999976, 0.899999976, 0.209 200, 1., 0.899999976, 0.210 201, 1.10000002, 0.899999976, 0.211 202, 1.20000005, 0.899999976, 0.212 203, 1.29999995, 0.899999976, 0.213 204, 1.39999998, 0.899999976, 0.214 205, 1.5, 0.899999976, 0.215 206, 1.60000002, 0.899999976, 0.216 207, 1.70000005, 0.899999976, 0.217 208, 1.79999995, 0.899999976, 0.218 209, 1.89999998, 0.899999976, 0.219 210, 2., 0.899999976, 0.220 211, 0., 1., 0.221 212, 0.100000001, 1., 0.222 213, 0.200000003, 1., 0.223 214, 0.300000012, 1., 0.224 215, 0.400000006, 1., 0.225 216, 0.5, 1., 0.226 217, 0.600000024, 1., 0.227 218, 0.699999988, 1., 0.228 219, 0.800000012, 1., 0.229 220, 0.899999976, 1., 0.230 221, 1., 1., 0.231 222, 1.10000002, 1., 0.232 223, 1.20000005, 1., 0.233 224, 1.29999995, 1., 0.234 225, 1.39999998, 1., 0.235 226, 1.5, 1., 0.236 227, 1.60000002, 1., 0.237 228, 1.70000005, 1., 0.238 229, 1.79999995, 1., 0.239 230, 1.89999998, 1., 0.240 231, 2., 1., 0.241 232, 0., 1.10000002, 0.242 233, 0.100000001, 1.10000002, 0.243 234, 0.200000003, 1.10000002, 0.244 235, 0.300000012, 1.10000002, 0.245 236, 0.400000006, 1.10000002, 0.246 237, 0.5, 1.10000002, 0.247 238, 0.600000024, 1.10000002, 0.248 239, 0.699999988, 1.10000002, 0.249 240, 0.800000012, 1.10000002, 0.250 241, 0.899999976, 1.10000002, 0.251 242, 1., 1.10000002, 0.252 243, 1.10000002, 1.10000002, 0.253 244, 1.20000005, 1.10000002, 0.254 245, 1.29999995, 1.10000002, 0.255 246, 1.39999998, 1.10000002, 0.256 247, 1.5, 1.10000002, 0.257 248, 1.60000002, 1.10000002, 0.258 249, 1.70000005, 1.10000002, 0.259 250, 1.79999995, 1.10000002, 0.260 251, 1.89999998, 1.10000002, 0.261 252, 2., 1.10000002, 0.262 253, 0., 1.20000005, 0.263 254, 0.100000001, 1.20000005, 0.264 255, 0.200000003, 1.20000005, 0.265 256, 0.300000012, 1.20000005, 0.266 257, 0.400000006, 1.20000005, 0.267 258, 0.5, 1.20000005, 0.268 259, 0.600000024, 1.20000005, 0.269 260, 0.699999988, 1.20000005, 0.270 261, 0.800000012, 1.20000005, 0.271 262, 0.899999976, 1.20000005, 0.272 263, 1., 1.20000005, 0.273 264, 1.10000002, 1.20000005, 0.274 265, 1.20000005, 1.20000005, 0.275 266, 1.29999995, 1.20000005, 0.276 267, 1.39999998, 1.20000005, 0.


of 97

277 268, 1.5, 1.20000005, 0.278 269, 1.60000002, 1.20000005, 0.279 270, 1.70000005, 1.20000005, 0.280 271, 1.79999995, 1.20000005, 0.281 272, 1.89999998, 1.20000005, 0.282 273, 2., 1.20000005, 0.283 274, 0., 1.29999995, 0.284 275, 0.100000001, 1.29999995, 0.285 276, 0.200000003, 1.29999995, 0.286 277, 0.300000012, 1.29999995, 0.287 278, 0.400000006, 1.29999995, 0.288 279, 0.5, 1.29999995, 0.289 280, 0.600000024, 1.29999995, 0.290 281, 0.699999988, 1.29999995, 0.291 282, 0.800000012, 1.29999995, 0.292 283, 0.899999976, 1.29999995, 0.293 284, 1., 1.29999995, 0.294 285, 1.10000002, 1.29999995, 0.295 286, 1.20000005, 1.29999995, 0.296 287, 1.29999995, 1.29999995, 0.297 288, 1.39999998, 1.29999995, 0.298 289, 1.5, 1.29999995, 0.299 290, 1.60000002, 1.29999995, 0.300 291, 1.70000005, 1.29999995, 0.301 292, 1.79999995, 1.29999995, 0.302 293, 1.89999998, 1.29999995, 0.303 294, 2., 1.29999995, 0.304 295, 0., 1.39999998, 0.305 296, 0.100000001, 1.39999998, 0.306 297, 0.200000003, 1.39999998, 0.307 298, 0.300000012, 1.39999998, 0.308 299, 0.400000006, 1.39999998, 0.309 300, 0.5, 1.39999998, 0.310 301, 0.600000024, 1.39999998, 0.311 302, 0.699999988, 1.39999998, 0.312 303, 0.800000012, 1.39999998, 0.313 304, 0.899999976, 1.39999998, 0.314 305, 1., 1.39999998, 0.315 306, 1.10000002, 1.39999998, 0.316 307, 1.20000005, 1.39999998, 0.317 308, 1.29999995, 1.39999998, 0.318 309, 1.39999998, 1.39999998, 0.319 310, 1.5, 1.39999998, 0.320 311, 1.60000002, 1.39999998, 0.321 312, 1.70000005, 1.39999998, 0.322 313, 1.79999995, 1.39999998, 0.323 314, 1.89999998, 1.39999998, 0.324 315, 2., 1.39999998, 0.325 316, 0., 1.5, 0.326 317, 0.100000001, 1.5, 0.327 318, 0.200000003, 1.5, 0.328 319, 0.300000012, 1.5, 0.329 320, 0.400000006, 1.5, 0.330 321, 0.5, 1.5, 0.331 322, 0.600000024, 1.5, 0.332 323, 0.699999988, 1.5, 0.333 324, 0.800000012, 1.5, 0.334 325, 0.899999976, 1.5, 0.335 326, 1., 1.5, 0.336 327, 1.10000002, 1.5, 0.337 328, 1.20000005, 1.5, 0.338 329, 1.29999995, 1.5, 0.339 330, 1.39999998, 1.5, 0.340 331, 1.5, 1.5, 0.341 332, 1.60000002, 1.5, 0.342 333, 1.70000005, 1.5, 0.343 334, 1.79999995, 1.5, 0.344 335, 1.89999998, 1.5, 0.345 336, 2., 1.5, 0.


of 97

346 337, 0., 1.60000002, 0.347 338, 0.100000001, 1.60000002, 0.348 339, 0.200000003, 1.60000002, 0.349 340, 0.300000012, 1.60000002, 0.350 341, 0.400000006, 1.60000002, 0.351 342, 0.5, 1.60000002, 0.352 343, 0.600000024, 1.60000002, 0.353 344, 0.699999988, 1.60000002, 0.354 345, 0.800000012, 1.60000002, 0.355 346, 0.899999976, 1.60000002, 0.356 347, 1., 1.60000002, 0.357 348, 1.10000002, 1.60000002, 0.358 349, 1.20000005, 1.60000002, 0.359 350, 1.29999995, 1.60000002, 0.360 351, 1.39999998, 1.60000002, 0.361 352, 1.5, 1.60000002, 0.362 353, 1.60000002, 1.60000002, 0.363 354, 1.70000005, 1.60000002, 0.364 355, 1.79999995, 1.60000002, 0.365 356, 1.89999998, 1.60000002, 0.366 357, 2., 1.60000002, 0.367 358, 0., 1.70000005, 0.368 359, 0.100000001, 1.70000005, 0.369 360, 0.200000003, 1.70000005, 0.370 361, 0.300000012, 1.70000005, 0.371 362, 0.400000006, 1.70000005, 0.372 363, 0.5, 1.70000005, 0.373 364, 0.600000024, 1.70000005, 0.374 365, 0.699999988, 1.70000005, 0.375 366, 0.800000012, 1.70000005, 0.376 367, 0.899999976, 1.70000005, 0.377 368, 1., 1.70000005, 0.378 369, 1.10000002, 1.70000005, 0.379 370, 1.20000005, 1.70000005, 0.380 371, 1.29999995, 1.70000005, 0.381 372, 1.39999998, 1.70000005, 0.382 373, 1.5, 1.70000005, 0.383 374, 1.60000002, 1.70000005, 0.384 375, 1.70000005, 1.70000005, 0.385 376, 1.79999995, 1.70000005, 0.386 377, 1.89999998, 1.70000005, 0.387 378, 2., 1.70000005, 0.388 379, 0., 1.79999995, 0.389 380, 0.100000001, 1.79999995, 0.390 381, 0.200000003, 1.79999995, 0.391 382, 0.300000012, 1.79999995, 0.392 383, 0.400000006, 1.79999995, 0.393 384, 0.5, 1.79999995, 0.394 385, 0.600000024, 1.79999995, 0.395 386, 0.699999988, 1.79999995, 0.396 387, 0.800000012, 1.79999995, 0.397 388, 0.899999976, 1.79999995, 0.398 389, 1., 1.79999995, 0.399 390, 1.10000002, 1.79999995, 0.400 391, 1.20000005, 1.79999995, 0.401 392, 1.29999995, 1.79999995, 0.402 393, 1.39999998, 1.79999995, 0.403 394, 1.5, 1.79999995, 0.404 395, 1.60000002, 1.79999995, 0.405 396, 1.70000005, 1.79999995, 0.406 397, 1.79999995, 1.79999995, 0.407 398, 1.89999998, 1.79999995, 0.408 399, 2., 1.79999995, 0.409 400, 0., 1.89999998, 0.410 401, 0.100000001, 1.89999998, 0.411 402, 0.200000003, 1.89999998, 0.412 403, 0.300000012, 1.89999998, 0.413 404, 0.400000006, 1.89999998, 0.414 405, 0.5, 1.89999998, 0.


of 97

415 406, 0.600000024, 1.89999998, 0.416 407, 0.699999988, 1.89999998, 0.417 408, 0.800000012, 1.89999998, 0.418 409, 0.899999976, 1.89999998, 0.419 410, 1., 1.89999998, 0.420 411, 1.10000002, 1.89999998, 0.421 412, 1.20000005, 1.89999998, 0.422 413, 1.29999995, 1.89999998, 0.423 414, 1.39999998, 1.89999998, 0.424 415, 1.5, 1.89999998, 0.425 416, 1.60000002, 1.89999998, 0.426 417, 1.70000005, 1.89999998, 0.427 418, 1.79999995, 1.89999998, 0.428 419, 1.89999998, 1.89999998, 0.429 420, 2., 1.89999998, 0.430 421, 0., 2., 0.431 422, 0.100000001, 2., 0.432 423, 0.200000003, 2., 0.433 424, 0.300000012, 2., 0.434 425, 0.400000006, 2., 0.435 426, 0.5, 2., 0.436 427, 0.600000024, 2., 0.437 428, 0.699999988, 2., 0.438 429, 0.800000012, 2., 0.439 430, 0.899999976, 2., 0.440 431, 1., 2., 0.441 432, 1.10000002, 2., 0.442 433, 1.20000005, 2., 0.443 434, 1.29999995, 2., 0.444 435, 1.39999998, 2., 0.445 436, 1.5, 2., 0.446 437, 1.60000002, 2., 0.447 438, 1.70000005, 2., 0.448 439, 1.79999995, 2., 0.449 440, 1.89999998, 2., 0.450 441, 2., 2., 0.451 ******************************************************452 ******************************************************453 ******************************************************454 *USER ELEMENT, TYPE=U1, NODES=4, COORDINATES=3, VAR=40, 455 INTEGRATION=4456 3,4,5457 *ELEMENT,TYPE=U1,ELSET=SOLID458 1, 1, 2, 23, 22459 2, 2, 3, 24, 23460 3, 3, 4, 25, 24461 4, 4, 5, 26, 25462 5, 5, 6, 27, 26463 6, 6, 7, 28, 27464 7, 7, 8, 29, 28465 8, 8, 9, 30, 29466 9, 9, 10, 31, 30467 10, 10, 11, 32, 31468 11, 11, 12, 33, 32469 12, 12, 13, 34, 33470 13, 13, 14, 35, 34471 14, 14, 15, 36, 35472 15, 15, 16, 37, 36473 16, 16, 17, 38, 37474 17, 17, 18, 39, 38475 18, 18, 19, 40, 39476 19, 19, 20, 41, 40477 20, 20, 21, 42, 41478 21, 22, 23, 44, 43479 22, 23, 24, 45, 44480 23, 24, 25, 46, 45481 24, 25, 26, 47, 46482 25, 26, 27, 48, 47483 26, 27, 28, 49, 48


of 97

484 27, 28, 29, 50, 49485 28, 29, 30, 51, 50486 29, 30, 31, 52, 51487 30, 31, 32, 53, 52488 31, 32, 33, 54, 53489 32, 33, 34, 55, 54490 33, 34, 35, 56, 55491 34, 35, 36, 57, 56492 35, 36, 37, 58, 57493 36, 37, 38, 59, 58494 37, 38, 39, 60, 59495 38, 39, 40, 61, 60496 39, 40, 41, 62, 61497 40, 41, 42, 63, 62498 41, 43, 44, 65, 64499 42, 44, 45, 66, 65500 43, 45, 46, 67, 66501 44, 46, 47, 68, 67502 45, 47, 48, 69, 68503 46, 48, 49, 70, 69504 47, 49, 50, 71, 70505 48, 50, 51, 72, 71506 49, 51, 52, 73, 72507 50, 52, 53, 74, 73508 51, 53, 54, 75, 74509 52, 54, 55, 76, 75510 53, 55, 56, 77, 76511 54, 56, 57, 78, 77512 55, 57, 58, 79, 78513 56, 58, 59, 80, 79514 57, 59, 60, 81, 80515 58, 60, 61, 82, 81516 59, 61, 62, 83, 82517 60, 62, 63, 84, 83518 61, 64, 65, 86, 85519 62, 65, 66, 87, 86520 63, 66, 67, 88, 87521 64, 67, 68, 89, 88522 65, 68, 69, 90, 89523 66, 69, 70, 91, 90524 67, 70, 71, 92, 91525 68, 71, 72, 93, 92526 69, 72, 73, 94, 93527 70, 73, 74, 95, 94528 71, 74, 75, 96, 95529 72, 75, 76, 97, 96530 73, 76, 77, 98, 97531 74, 77, 78, 99, 98532 75, 78, 79, 100, 99533 76, 79, 80, 101, 100534 77, 80, 81, 102, 101535 78, 81, 82, 103, 102536 79, 82, 83, 104, 103537 80, 83, 84, 105, 104538 81, 85, 86, 107, 106539 82, 86, 87, 108, 107540 83, 87, 88, 109, 108541 84, 88, 89, 110, 109542 85, 89, 90, 111, 110543 86, 90, 91, 112, 111544 87, 91, 92, 113, 112545 88, 92, 93, 114, 113546 89, 93, 94, 115, 114547 90, 94, 95, 116, 115548 91, 95, 96, 117, 116549 92, 96, 97, 118, 117550 93, 97, 98, 119, 118551 94, 98, 99, 120, 119552 95, 99, 100, 121, 120


of 97

553 96, 100, 101, 122, 121554 97, 101, 102, 123, 122555 98, 102, 103, 124, 123556 99, 103, 104, 125, 124557 100, 104, 105, 126, 125558 101, 106, 107, 128, 127559 102, 107, 108, 129, 128560 103, 108, 109, 130, 129561 104, 109, 110, 131, 130562 105, 110, 111, 132, 131563 106, 111, 112, 133, 132564 107, 112, 113, 134, 133565 108, 113, 114, 135, 134566 109, 114, 115, 136, 135567 110, 115, 116, 137, 136568 111, 116, 117, 138, 137569 112, 117, 118, 139, 138570 113, 118, 119, 140, 139571 114, 119, 120, 141, 140572 115, 120, 121, 142, 141573 116, 121, 122, 143, 142574 117, 122, 123, 144, 143575 118, 123, 124, 145, 144576 119, 124, 125, 146, 145577 120, 125, 126, 147, 146578 121, 127, 128, 149, 148579 122, 128, 129, 150, 149580 123, 129, 130, 151, 150581 124, 130, 131, 152, 151582 125, 131, 132, 153, 152583 126, 132, 133, 154, 153584 127, 133, 134, 155, 154585 128, 134, 135, 156, 155586 129, 135, 136, 157, 156587 130, 136, 137, 158, 157588 131, 137, 138, 159, 158589 132, 138, 139, 160, 159590 133, 139, 140, 161, 160591 134, 140, 141, 162, 161592 135, 141, 142, 163, 162593 136, 142, 143, 164, 163594 137, 143, 144, 165, 164595 138, 144, 145, 166, 165596 139, 145, 146, 167, 166597 140, 146, 147, 168, 167598 141, 148, 149, 170, 169599 142, 149, 150, 171, 170600 143, 150, 151, 172, 171601 144, 151, 152, 173, 172602 145, 152, 153, 174, 173603 146, 153, 154, 175, 174604 147, 154, 155, 176, 175605 148, 155, 156, 177, 176606 149, 156, 157, 178, 177607 150, 157, 158, 179, 178608 151, 158, 159, 180, 179609 152, 159, 160, 181, 180610 153, 160, 161, 182, 181611 154, 161, 162, 183, 182612 155, 162, 163, 184, 183613 156, 163, 164, 185, 184614 157, 164, 165, 186, 185615 158, 165, 166, 187, 186616 159, 166, 167, 188, 187617 160, 167, 168, 189, 188618 161, 169, 170, 191, 190619 162, 170, 171, 192, 191620 163, 171, 172, 193, 192621 164, 172, 173, 194, 193


of 97

622 165, 173, 174, 195, 194623 166, 174, 175, 196, 195624 167, 175, 176, 197, 196625 168, 176, 177, 198, 197626 169, 177, 178, 199, 198627 170, 178, 179, 200, 199628 171, 179, 180, 201, 200629 172, 180, 181, 202, 201630 173, 181, 182, 203, 202631 174, 182, 183, 204, 203632 175, 183, 184, 205, 204633 176, 184, 185, 206, 205634 177, 185, 186, 207, 206635 178, 186, 187, 208, 207636 179, 187, 188, 209, 208637 180, 188, 189, 210, 209638 181, 190, 191, 212, 211639 182, 191, 192, 213, 212640 183, 192, 193, 214, 213641 184, 193, 194, 215, 214642 185, 194, 195, 216, 215643 186, 195, 196, 217, 216644 187, 196, 197, 218, 217645 188, 197, 198, 219, 218646 189, 198, 199, 220, 219647 190, 199, 200, 221, 220648 191, 200, 201, 222, 221649 192, 201, 202, 223, 222650 193, 202, 203, 224, 223651 194, 203, 204, 225, 224652 195, 204, 205, 226, 225653 196, 205, 206, 227, 226654 197, 206, 207, 228, 227655 198, 207, 208, 229, 228656 199, 208, 209, 230, 229657 200, 209, 210, 231, 230658 201, 211, 212, 233, 232659 202, 212, 213, 234, 233660 203, 213, 214, 235, 234661 204, 214, 215, 236, 235662 205, 215, 216, 237, 236663 206, 216, 217, 238, 237664 207, 217, 218, 239, 238665 208, 218, 219, 240, 239666 209, 219, 220, 241, 240667 210, 220, 221, 242, 241668 211, 221, 222, 243, 242669 212, 222, 223, 244, 243670 213, 223, 224, 245, 244671 214, 224, 225, 246, 245672 215, 225, 226, 247, 246673 216, 226, 227, 248, 247674 217, 227, 228, 249, 248675 218, 228, 229, 250, 249676 219, 229, 230, 251, 250677 220, 230, 231, 252, 251678 221, 232, 233, 254, 253679 222, 233, 234, 255, 254680 223, 234, 235, 256, 255681 224, 235, 236, 257, 256682 225, 236, 237, 258, 257683 226, 237, 238, 259, 258684 227, 238, 239, 260, 259685 228, 239, 240, 261, 260686 229, 240, 241, 262, 261687 230, 241, 242, 263, 262688 231, 242, 243, 264, 263689 232, 243, 244, 265, 264690 233, 244, 245, 266, 265


of 97

691 234, 245, 246, 267, 266692 235, 246, 247, 268, 267693 236, 247, 248, 269, 268694 237, 248, 249, 270, 269695 238, 249, 250, 271, 270696 239, 250, 251, 272, 271697 240, 251, 252, 273, 272698 241, 253, 254, 275, 274699 242, 254, 255, 276, 275700 243, 255, 256, 277, 276701 244, 256, 257, 278, 277702 245, 257, 258, 279, 278703 246, 258, 259, 280, 279704 247, 259, 260, 281, 280705 248, 260, 261, 282, 281706 249, 261, 262, 283, 282707 250, 262, 263, 284, 283708 251, 263, 264, 285, 284709 252, 264, 265, 286, 285710 253, 265, 266, 287, 286711 254, 266, 267, 288, 287712 255, 267, 268, 289, 288713 256, 268, 269, 290, 289714 257, 269, 270, 291, 290715 258, 270, 271, 292, 291716 259, 271, 272, 293, 292717 260, 272, 273, 294, 293718 261, 274, 275, 296, 295719 262, 275, 276, 297, 296720 263, 276, 277, 298, 297721 264, 277, 278, 299, 298722 265, 278, 279, 300, 299723 266, 279, 280, 301, 300724 267, 280, 281, 302, 301725 268, 281, 282, 303, 302726 269, 282, 283, 304, 303727 270, 283, 284, 305, 304728 271, 284, 285, 306, 305729 272, 285, 286, 307, 306730 273, 286, 287, 308, 307731 274, 287, 288, 309, 308732 275, 288, 289, 310, 309733 276, 289, 290, 311, 310734 277, 290, 291, 312, 311735 278, 291, 292, 313, 312736 279, 292, 293, 314, 313737 280, 293, 294, 315, 314738 281, 295, 296, 317, 316739 282, 296, 297, 318, 317740 283, 297, 298, 319, 318741 284, 298, 299, 320, 319742 285, 299, 300, 321, 320743 286, 300, 301, 322, 321744 287, 301, 302, 323, 322745 288, 302, 303, 324, 323746 289, 303, 304, 325, 324747 290, 304, 305, 326, 325748 291, 305, 306, 327, 326749 292, 306, 307, 328, 327750 293, 307, 308, 329, 328751 294, 308, 309, 330, 329752 295, 309, 310, 331, 330753 296, 310, 311, 332, 331754 297, 311, 312, 333, 332755 298, 312, 313, 334, 333756 299, 313, 314, 335, 334757 300, 314, 315, 336, 335758 301, 316, 317, 338, 337759 302, 317, 318, 339, 338


of 97

760 303, 318, 319, 340, 339761 304, 319, 320, 341, 340762 305, 320, 321, 342, 341763 306, 321, 322, 343, 342764 307, 322, 323, 344, 343765 308, 323, 324, 345, 344766 309, 324, 325, 346, 345767 310, 325, 326, 347, 346768 311, 326, 327, 348, 347769 312, 327, 328, 349, 348770 313, 328, 329, 350, 349771 314, 329, 330, 351, 350772 315, 330, 331, 352, 351773 316, 331, 332, 353, 352774 317, 332, 333, 354, 353775 318, 333, 334, 355, 354776 319, 334, 335, 356, 355777 320, 335, 336, 357, 356778 321, 337, 338, 359, 358779 322, 338, 339, 360, 359780 323, 339, 340, 361, 360781 324, 340, 341, 362, 361782 325, 341, 342, 363, 362783 326, 342, 343, 364, 363784 327, 343, 344, 365, 364785 328, 344, 345, 366, 365786 329, 345, 346, 367, 366787 330, 346, 347, 368, 367788 331, 347, 348, 369, 368789 332, 348, 349, 370, 369790 333, 349, 350, 371, 370791 334, 350, 351, 372, 371792 335, 351, 352, 373, 372793 336, 352, 353, 374, 373794 337, 353, 354, 375, 374795 338, 354, 355, 376, 375796 339, 355, 356, 377, 376797 340, 356, 357, 378, 377798 341, 358, 359, 380, 379799 342, 359, 360, 381, 380800 343, 360, 361, 382, 381801 344, 361, 362, 383, 382802 345, 362, 363, 384, 383803 346, 363, 364, 385, 384804 347, 364, 365, 386, 385805 348, 365, 366, 387, 386806 349, 366, 367, 388, 387807 350, 367, 368, 389, 388808 351, 368, 369, 390, 389809 352, 369, 370, 391, 390810 353, 370, 371, 392, 391811 354, 371, 372, 393, 392812 355, 372, 373, 394, 393813 356, 373, 374, 395, 394814 357, 374, 375, 396, 395815 358, 375, 376, 397, 396816 359, 376, 377, 398, 397817 360, 377, 378, 399, 398818 361, 379, 380, 401, 400819 362, 380, 381, 402, 401820 363, 381, 382, 403, 402821 364, 382, 383, 404, 403822 365, 383, 384, 405, 404823 366, 384, 385, 406, 405824 367, 385, 386, 407, 406825 368, 386, 387, 408, 407826 369, 387, 388, 409, 408827 370, 388, 389, 410, 409828 371, 389, 390, 411, 410


of 97

829 372, 390, 391, 412, 411830 373, 391, 392, 413, 412831 374, 392, 393, 414, 413832 375, 393, 394, 415, 414833 376, 394, 395, 416, 415834 377, 395, 396, 417, 416835 378, 396, 397, 418, 417836 379, 397, 398, 419, 418837 380, 398, 399, 420, 419838 381, 400, 401, 422, 421839 382, 401, 402, 423, 422840 383, 402, 403, 424, 423841 384, 403, 404, 425, 424842 385, 404, 405, 426, 425843 386, 405, 406, 427, 426844 387, 406, 407, 428, 427845 388, 407, 408, 429, 428846 389, 408, 409, 430, 429847 390, 409, 410, 431, 430848 391, 410, 411, 432, 431849 392, 411, 412, 433, 432850 393, 412, 413, 434, 433851 394, 413, 414, 435, 434852 395, 414, 415, 436, 435853 396, 415, 416, 437, 436854 397, 416, 417, 438, 437855 398, 417, 418, 439, 438856 399, 418, 419, 440, 439857 400, 419, 420, 441, 440858 *UEL PROPERTY, ELSET=SOLID859 **************************************860 *End Part861 ** 862 **863 ** ASSEMBLY864 **865 *Assembly, name=Assembly866 ** 867 *Instance, name=Plate-1, part=Plate868 *End Instance869 ** 870 *Nset, nset=all_elements, instance=Plate-1, generate871 1, 441, 1872 *Elset, elset=all_elements, instance=Plate-1, generate873 1, 400, 1874 *Nset, nset=all_nodes, instance=Plate-1, generate875 1, 441, 1876 *Nset, nset=encastre_nodes, instance=Plate-1877 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16878 17, 18, 19, 20, 21, 22, 42, 43, 63, 64, 84, 85, 105, 106, 126, 127879 147, 148, 168, 169, 189, 190, 210, 211, 231, 232, 252, 253, 273, 274, 294, 295880 315, 316, 336, 337, 357, 358, 378, 379, 399, 400, 420, 421, 422, 423, 424, 425881 426, 427, 428, 429, 430, 431, 432, 433, 434, 435, 436, 437, 438, 439, 440, 441882 *End Assembly883 ** 884 ** 885 ** BOUNDARY CONDITIONS886 ** 887 ** Name: fix_nodes Type: Symmetry/Antisymmetry/Encastre888 *Boundary889 encastre_nodes, ENCASTRE890 ** ----------------------------------------------------------------891 ** 892 ** STEP: loading893 ** 894 *Step, name=loading, nlgeom=NO, inc=100000895 *Static, direct896 0.1, 1., 897 **


of 97

898 ** LOADS899 ** 900 ** Name: Load-1 Type: Concentrated force901 *Cload902 all_nodes, 3, -0.01903 ** 904 ** OUTPUT REQUESTS905 ** 906 *Restart, write, frequency=0907 ** 908 ** FIELD OUTPUT: F-Output-2909 ** 910 *Output, field911 *Node Output912 CF, RF, RM, RT, TF, VF913 *Element Output, directions=YES914 BF, CENTMAG, CENTRIFMAG, CORIOMAG, ESF1, GRAV, HP, NFORC, NFORCSO, P, ROTAMAG, SF,



of 97

1 *Heading2 ** Job name: abaqus_4_elements_lift_up Model name: Model-13 ** Generated by: Abaqus/CAE 2016.HF24 *Preprint, echo=NO, model=NO, history=NO, contact=NO5 **6 ** PARTS7 **8 *Part, name=Plate9 *Node

10 1, 0., 0., 0.11 2, 2., 0., 0.12 3, 0., 2., 0.13 4, 2., 2., 0.14 *Element, type=S415 1, 1, 2, 4, 316 *Nset, nset=_PickedSet2, internal, generate17 1, 4, 118 *Elset, elset=_PickedSet2, internal19 1,20 ** Section: Section21 *Shell Section, elset=_PickedSet2, material=Steel22 0.1, 523 *End Part24 ** 25 **26 ** ASSEMBLY27 **28 *Assembly, name=Assembly29 ** 30 *Instance, name=Plate-1, part=Plate31 *End Instance32 ** 33 *Nset, nset=all_elements, instance=Plate-1, generate34 1, 4, 135 *Elset, elset=all_elements, instance=Plate-136 1,37 *Nset, nset=_PickedSet13, internal, instance=Plate-138 4,39 *Nset, nset=_PickedSet14, internal, instance=Plate-1, generate40 1, 3, 141 *End Assembly42 ** 43 ** MATERIALS44 ** 45 *Material, name=Steel46 *Elastic47 2e+08, 0.348 ** 49 ** BOUNDARY CONDITIONS50 ** 51 ** Name: fix_nodes Type: Symmetry/Antisymmetry/Encastre52 *Boundary53 _PickedSet14, ENCASTRE54 ** ----------------------------------------------------------------55 ** 56 ** STEP: loading57 ** 58 *Step, name=loading, nlgeom=NO, inc=10000059 *Static, direct60 0.1, 1., 61 *****************************************************62 *ELEMENT MATRIX

OUTPUT,ELSET=all_elements,STIFF=YES,MASS=YES,DLOAD=YES,OUTPUTFILE=USER,FILENAME=Dump63 *****************************************************64 ** BOUNDARY CONDITIONS65 ** 66 ** Name: lift_up Type: Displacement/Rotation67 *Boundary68 _PickedSet13, 3, 3, 0.1


Abaqus job input file for default S4 element

of 97

69 _PickedSet13, 4, 470 _PickedSet13, 5, 571 ** 72 ** OUTPUT REQUESTS73 ** 74 *Restart, write, frequency=075 ** 76 ** FIELD OUTPUT: F-Output-277 ** 78 *Output, field79 *Node Output80 CF, RF, RM, RT, TF, VF81 *Element Output, directions=YES82 BF, CENTMAG, CENTRIFMAG, CORIOMAG, ESF1, GRAV, HP, NFORC, NFORCSO, P, ROTAMAG, SF,

TRNOR, TRSHR83 ** 84 ** FIELD OUTPUT: F-Output-185 ** 86 *Output, field, variable=PRESELECT87 ** 88 ** HISTORY OUTPUT: H-Output-289 ** 90 *Output, history91 *Element Output92 IRF1, IRF2, IRF3, IRM1, IRM2, IRM393 ** 94 ** HISTORY OUTPUT: H-Output-195 ** 96 *Output, history, variable=PRESELECT97 *End Step98 *STEP99 *MATRIX GENERATE, STIFFNESS

100 *MATRIX OUTPUT, STIFFNESS, FORMAT=COORDINATE101 *END STEP102


of 97

1 %................................................................2 3 % MATLAB codes for Finite Element Analysis4 % problem19.m5 % Mindlin plate in bending6 % antonio ferreira 20087 8 % clear memory9 clc

10 clear *;11 clear all;12 colordef white;13 format long;14 close all;15 16 % materials17 E = 200000000; poisson = 0.30; kapa=5/6;18 thickness=0.1;19 I=thickness^3/12;20 21 % matrix C22 % bending part23 C_bending=I*E/(1-poisson^2)*...24 [1 poisson 0;poisson 1 0;0 0 (1-poisson)/2];25 % shear part26 C_shear=kapa*thickness*E/2/(1+poisson)*eye(2);27 28 % load29 P = -1;30 31 %Mesh generation32 L = 2;33 numberElementsX=20;34 numberElementsY=20;35 numberElements=numberElementsX*numberElementsY;36 %37 [nodeCoordinates, elementNodes] = ...38 rectangularMesh(L,L,numberElementsX,numberElementsY);39 xx=nodeCoordinates(:,1);40 yy=nodeCoordinates(:,2);41 42 %drawingMesh(nodeCoordinates,elementNodes,'Q4','k-');43 %axis off44 numberNodes=size(xx,1);45 46 % GDof: global number of degrees of freedom47 GDof=3*numberNodes;48 49 % computation of the system stiffness matrix and force vector50 [stiffness]=...51 formStiffnessMatrixMindlinQ4(GDof,numberElements,...52 elementNodes,numberNodes,nodeCoordinates,C_shear,...53 C_bending,thickness,I);54 55 56 [force]=...57 formForceVectorMindlinQ4(GDof,numberElements,...58 elementNodes,numberNodes,nodeCoordinates,P);59 60 % % boundary conditions 61 [prescribedDof,activeDof]=...62 EssentialBC('cccc',GDof,xx,yy,nodeCoordinates,numberNodes);63 64 % solution65 displacements=solution(GDof,prescribedDof,stiffness,force);66 67 % displacements68 disp('Displacements')69 jj=1:GDof; format


Modified Matlab main routine for multiple elements

of 97

70 f=[jj; displacements'];71 fprintf('node U\n')72 fprintf('%3d %12.8f\n',f)73 74 % deformed shape75 figure76 plot3(xx,yy,displacements(1:numberNodes),'.')77 xlabel('X-axis') % x-axis label78 ylabel('Y-axis') % y-axis label79 zlabel('Z-axis') % y-axis label80 format long81 D1=E*thickness^3/12/(1-poisson^2);82 min(displacements(1:numberNodes))


of 97

1 *Heading2 ** Job name: push_center_Down Model name: Model-13 ** Generated by: Abaqus/CAE 2016.HF24 *Preprint, echo=NO, model=NO, history=NO, contact=NO5 **6 ** PARTS7 **8 *Part, name=Plate9 *Node

10 1, 0., 0., 0.11 2, 0.100000001, 0., 0.12 3, 0.200000003, 0., 0.13 4, 0.300000012, 0., 0.14 5, 0.400000006, 0., 0.15 6, 0.5, 0., 0.16 7, 0.600000024, 0., 0.17 8, 0.699999988, 0., 0.18 9, 0.800000012, 0., 0.19 10, 0.899999976, 0., 0.20 11, 1., 0., 0.21 12, 1.10000002, 0., 0.22 13, 1.20000005, 0., 0.23 14, 1.29999995, 0., 0.24 15, 1.39999998, 0., 0.25 16, 1.5, 0., 0.26 17, 1.60000002, 0., 0.27 18, 1.70000005, 0., 0.28 19, 1.79999995, 0., 0.29 20, 1.89999998, 0., 0.30 21, 2., 0., 0.31 22, 0., 0.100000001, 0.32 23, 0.100000001, 0.100000001, 0.33 24, 0.200000003, 0.100000001, 0.34 25, 0.300000012, 0.100000001, 0.35 26, 0.400000006, 0.100000001, 0.36 27, 0.5, 0.100000001, 0.37 28, 0.600000024, 0.100000001, 0.38 29, 0.699999988, 0.100000001, 0.39 30, 0.800000012, 0.100000001, 0.40 31, 0.899999976, 0.100000001, 0.41 32, 1., 0.100000001, 0.42 33, 1.10000002, 0.100000001, 0.43 34, 1.20000005, 0.100000001, 0.44 35, 1.29999995, 0.100000001, 0.45 36, 1.39999998, 0.100000001, 0.46 37, 1.5, 0.100000001, 0.47 38, 1.60000002, 0.100000001, 0.48 39, 1.70000005, 0.100000001, 0.49 40, 1.79999995, 0.100000001, 0.50 41, 1.89999998, 0.100000001, 0.51 42, 2., 0.100000001, 0.52 43, 0., 0.200000003, 0.53 44, 0.100000001, 0.200000003, 0.54 45, 0.200000003, 0.200000003, 0.55 46, 0.300000012, 0.200000003, 0.56 47, 0.400000006, 0.200000003, 0.57 48, 0.5, 0.200000003, 0.58 49, 0.600000024, 0.200000003, 0.59 50, 0.699999988, 0.200000003, 0.60 51, 0.800000012, 0.200000003, 0.61 52, 0.899999976, 0.200000003, 0.62 53, 1., 0.200000003, 0.63 54, 1.10000002, 0.200000003, 0.64 55, 1.20000005, 0.200000003, 0.65 56, 1.29999995, 0.200000003, 0.66 57, 1.39999998, 0.200000003, 0.67 58, 1.5, 0.200000003, 0.68 59, 1.60000002, 0.200000003, 0.69 60, 1.70000005, 0.200000003, 0.


Abaqus job input file for multiple default S4 elements

of 97

70 61, 1.79999995, 0.200000003, 0.71 62, 1.89999998, 0.200000003, 0.72 63, 2., 0.200000003, 0.73 64, 0., 0.300000012, 0.74 65, 0.100000001, 0.300000012, 0.75 66, 0.200000003, 0.300000012, 0.76 67, 0.300000012, 0.300000012, 0.77 68, 0.400000006, 0.300000012, 0.78 69, 0.5, 0.300000012, 0.79 70, 0.600000024, 0.300000012, 0.80 71, 0.699999988, 0.300000012, 0.81 72, 0.800000012, 0.300000012, 0.82 73, 0.899999976, 0.300000012, 0.83 74, 1., 0.300000012, 0.84 75, 1.10000002, 0.300000012, 0.85 76, 1.20000005, 0.300000012, 0.86 77, 1.29999995, 0.300000012, 0.87 78, 1.39999998, 0.300000012, 0.88 79, 1.5, 0.300000012, 0.89 80, 1.60000002, 0.300000012, 0.90 81, 1.70000005, 0.300000012, 0.91 82, 1.79999995, 0.300000012, 0.92 83, 1.89999998, 0.300000012, 0.93 84, 2., 0.300000012, 0.94 85, 0., 0.400000006, 0.95 86, 0.100000001, 0.400000006, 0.96 87, 0.200000003, 0.400000006, 0.97 88, 0.300000012, 0.400000006, 0.98 89, 0.400000006, 0.400000006, 0.99 90, 0.5, 0.400000006, 0.

100 91, 0.600000024, 0.400000006, 0.101 92, 0.699999988, 0.400000006, 0.102 93, 0.800000012, 0.400000006, 0.103 94, 0.899999976, 0.400000006, 0.104 95, 1., 0.400000006, 0.105 96, 1.10000002, 0.400000006, 0.106 97, 1.20000005, 0.400000006, 0.107 98, 1.29999995, 0.400000006, 0.108 99, 1.39999998, 0.400000006, 0.109 100, 1.5, 0.400000006, 0.110 101, 1.60000002, 0.400000006, 0.111 102, 1.70000005, 0.400000006, 0.112 103, 1.79999995, 0.400000006, 0.113 104, 1.89999998, 0.400000006, 0.114 105, 2., 0.400000006, 0.115 106, 0., 0.5, 0.116 107, 0.100000001, 0.5, 0.117 108, 0.200000003, 0.5, 0.118 109, 0.300000012, 0.5, 0.119 110, 0.400000006, 0.5, 0.120 111, 0.5, 0.5, 0.121 112, 0.600000024, 0.5, 0.122 113, 0.699999988, 0.5, 0.123 114, 0.800000012, 0.5, 0.124 115, 0.899999976, 0.5, 0.125 116, 1., 0.5, 0.126 117, 1.10000002, 0.5, 0.127 118, 1.20000005, 0.5, 0.128 119, 1.29999995, 0.5, 0.129 120, 1.39999998, 0.5, 0.130 121, 1.5, 0.5, 0.131 122, 1.60000002, 0.5, 0.132 123, 1.70000005, 0.5, 0.133 124, 1.79999995, 0.5, 0.134 125, 1.89999998, 0.5, 0.135 126, 2., 0.5, 0.136 127, 0., 0.600000024, 0.137 128, 0.100000001, 0.600000024, 0.138 129, 0.200000003, 0.600000024, 0.


of 97

139 130, 0.300000012, 0.600000024, 0.140 131, 0.400000006, 0.600000024, 0.141 132, 0.5, 0.600000024, 0.142 133, 0.600000024, 0.600000024, 0.143 134, 0.699999988, 0.600000024, 0.144 135, 0.800000012, 0.600000024, 0.145 136, 0.899999976, 0.600000024, 0.146 137, 1., 0.600000024, 0.147 138, 1.10000002, 0.600000024, 0.148 139, 1.20000005, 0.600000024, 0.149 140, 1.29999995, 0.600000024, 0.150 141, 1.39999998, 0.600000024, 0.151 142, 1.5, 0.600000024, 0.152 143, 1.60000002, 0.600000024, 0.153 144, 1.70000005, 0.600000024, 0.154 145, 1.79999995, 0.600000024, 0.155 146, 1.89999998, 0.600000024, 0.156 147, 2., 0.600000024, 0.157 148, 0., 0.699999988, 0.158 149, 0.100000001, 0.699999988, 0.159 150, 0.200000003, 0.699999988, 0.160 151, 0.300000012, 0.699999988, 0.161 152, 0.400000006, 0.699999988, 0.162 153, 0.5, 0.699999988, 0.163 154, 0.600000024, 0.699999988, 0.164 155, 0.699999988, 0.699999988, 0.165 156, 0.800000012, 0.699999988, 0.166 157, 0.899999976, 0.699999988, 0.167 158, 1., 0.699999988, 0.168 159, 1.10000002, 0.699999988, 0.169 160, 1.20000005, 0.699999988, 0.170 161, 1.29999995, 0.699999988, 0.171 162, 1.39999998, 0.699999988, 0.172 163, 1.5, 0.699999988, 0.173 164, 1.60000002, 0.699999988, 0.174 165, 1.70000005, 0.699999988, 0.175 166, 1.79999995, 0.699999988, 0.176 167, 1.89999998, 0.699999988, 0.177 168, 2., 0.699999988, 0.178 169, 0., 0.800000012, 0.179 170, 0.100000001, 0.800000012, 0.180 171, 0.200000003, 0.800000012, 0.181 172, 0.300000012, 0.800000012, 0.182 173, 0.400000006, 0.800000012, 0.183 174, 0.5, 0.800000012, 0.184 175, 0.600000024, 0.800000012, 0.185 176, 0.699999988, 0.800000012, 0.186 177, 0.800000012, 0.800000012, 0.187 178, 0.899999976, 0.800000012, 0.188 179, 1., 0.800000012, 0.189 180, 1.10000002, 0.800000012, 0.190 181, 1.20000005, 0.800000012, 0.191 182, 1.29999995, 0.800000012, 0.192 183, 1.39999998, 0.800000012, 0.193 184, 1.5, 0.800000012, 0.194 185, 1.60000002, 0.800000012, 0.195 186, 1.70000005, 0.800000012, 0.196 187, 1.79999995, 0.800000012, 0.197 188, 1.89999998, 0.800000012, 0.198 189, 2., 0.800000012, 0.199 190, 0., 0.899999976, 0.200 191, 0.100000001, 0.899999976, 0.201 192, 0.200000003, 0.899999976, 0.202 193, 0.300000012, 0.899999976, 0.203 194, 0.400000006, 0.899999976, 0.204 195, 0.5, 0.899999976, 0.205 196, 0.600000024, 0.899999976, 0.206 197, 0.699999988, 0.899999976, 0.207 198, 0.800000012, 0.899999976, 0.


of 97

208 199, 0.899999976, 0.899999976, 0.209 200, 1., 0.899999976, 0.210 201, 1.10000002, 0.899999976, 0.211 202, 1.20000005, 0.899999976, 0.212 203, 1.29999995, 0.899999976, 0.213 204, 1.39999998, 0.899999976, 0.214 205, 1.5, 0.899999976, 0.215 206, 1.60000002, 0.899999976, 0.216 207, 1.70000005, 0.899999976, 0.217 208, 1.79999995, 0.899999976, 0.218 209, 1.89999998, 0.899999976, 0.219 210, 2., 0.899999976, 0.220 211, 0., 1., 0.221 212, 0.100000001, 1., 0.222 213, 0.200000003, 1., 0.223 214, 0.300000012, 1., 0.224 215, 0.400000006, 1., 0.225 216, 0.5, 1., 0.226 217, 0.600000024, 1., 0.227 218, 0.699999988, 1., 0.228 219, 0.800000012, 1., 0.229 220, 0.899999976, 1., 0.230 221, 1., 1., 0.231 222, 1.10000002, 1., 0.232 223, 1.20000005, 1., 0.233 224, 1.29999995, 1., 0.234 225, 1.39999998, 1., 0.235 226, 1.5, 1., 0.236 227, 1.60000002, 1., 0.237 228, 1.70000005, 1., 0.238 229, 1.79999995, 1., 0.239 230, 1.89999998, 1., 0.240 231, 2., 1., 0.241 232, 0., 1.10000002, 0.242 233, 0.100000001, 1.10000002, 0.243 234, 0.200000003, 1.10000002, 0.244 235, 0.300000012, 1.10000002, 0.245 236, 0.400000006, 1.10000002, 0.246 237, 0.5, 1.10000002, 0.247 238, 0.600000024, 1.10000002, 0.248 239, 0.699999988, 1.10000002, 0.249 240, 0.800000012, 1.10000002, 0.250 241, 0.899999976, 1.10000002, 0.251 242, 1., 1.10000002, 0.252 243, 1.10000002, 1.10000002, 0.253 244, 1.20000005, 1.10000002, 0.254 245, 1.29999995, 1.10000002, 0.255 246, 1.39999998, 1.10000002, 0.256 247, 1.5, 1.10000002, 0.257 248, 1.60000002, 1.10000002, 0.258 249, 1.70000005, 1.10000002, 0.259 250, 1.79999995, 1.10000002, 0.260 251, 1.89999998, 1.10000002, 0.261 252, 2., 1.10000002, 0.262 253, 0., 1.20000005, 0.263 254, 0.100000001, 1.20000005, 0.264 255, 0.200000003, 1.20000005, 0.265 256, 0.300000012, 1.20000005, 0.266 257, 0.400000006, 1.20000005, 0.267 258, 0.5, 1.20000005, 0.268 259, 0.600000024, 1.20000005, 0.269 260, 0.699999988, 1.20000005, 0.270 261, 0.800000012, 1.20000005, 0.271 262, 0.899999976, 1.20000005, 0.272 263, 1., 1.20000005, 0.273 264, 1.10000002, 1.20000005, 0.274 265, 1.20000005, 1.20000005, 0.275 266, 1.29999995, 1.20000005, 0.276 267, 1.39999998, 1.20000005, 0.


of 97

277 268, 1.5, 1.20000005, 0.278 269, 1.60000002, 1.20000005, 0.279 270, 1.70000005, 1.20000005, 0.280 271, 1.79999995, 1.20000005, 0.281 272, 1.89999998, 1.20000005, 0.282 273, 2., 1.20000005, 0.283 274, 0., 1.29999995, 0.284 275, 0.100000001, 1.29999995, 0.285 276, 0.200000003, 1.29999995, 0.286 277, 0.300000012, 1.29999995, 0.287 278, 0.400000006, 1.29999995, 0.288 279, 0.5, 1.29999995, 0.289 280, 0.600000024, 1.29999995, 0.290 281, 0.699999988, 1.29999995, 0.291 282, 0.800000012, 1.29999995, 0.292 283, 0.899999976, 1.29999995, 0.293 284, 1., 1.29999995, 0.294 285, 1.10000002, 1.29999995, 0.295 286, 1.20000005, 1.29999995, 0.296 287, 1.29999995, 1.29999995, 0.297 288, 1.39999998, 1.29999995, 0.298 289, 1.5, 1.29999995, 0.299 290, 1.60000002, 1.29999995, 0.300 291, 1.70000005, 1.29999995, 0.301 292, 1.79999995, 1.29999995, 0.302 293, 1.89999998, 1.29999995, 0.303 294, 2., 1.29999995, 0.304 295, 0., 1.39999998, 0.305 296, 0.100000001, 1.39999998, 0.306 297, 0.200000003, 1.39999998, 0.307 298, 0.300000012, 1.39999998, 0.308 299, 0.400000006, 1.39999998, 0.309 300, 0.5, 1.39999998, 0.310 301, 0.600000024, 1.39999998, 0.311 302, 0.699999988, 1.39999998, 0.312 303, 0.800000012, 1.39999998, 0.313 304, 0.899999976, 1.39999998, 0.314 305, 1., 1.39999998, 0.315 306, 1.10000002, 1.39999998, 0.316 307, 1.20000005, 1.39999998, 0.317 308, 1.29999995, 1.39999998, 0.318 309, 1.39999998, 1.39999998, 0.319 310, 1.5, 1.39999998, 0.320 311, 1.60000002, 1.39999998, 0.321 312, 1.70000005, 1.39999998, 0.322 313, 1.79999995, 1.39999998, 0.323 314, 1.89999998, 1.39999998, 0.324 315, 2., 1.39999998, 0.325 316, 0., 1.5, 0.326 317, 0.100000001, 1.5, 0.327 318, 0.200000003, 1.5, 0.328 319, 0.300000012, 1.5, 0.329 320, 0.400000006, 1.5, 0.330 321, 0.5, 1.5, 0.331 322, 0.600000024, 1.5, 0.332 323, 0.699999988, 1.5, 0.333 324, 0.800000012, 1.5, 0.334 325, 0.899999976, 1.5, 0.335 326, 1., 1.5, 0.336 327, 1.10000002, 1.5, 0.337 328, 1.20000005, 1.5, 0.338 329, 1.29999995, 1.5, 0.339 330, 1.39999998, 1.5, 0.340 331, 1.5, 1.5, 0.341 332, 1.60000002, 1.5, 0.342 333, 1.70000005, 1.5, 0.343 334, 1.79999995, 1.5, 0.344 335, 1.89999998, 1.5, 0.345 336, 2., 1.5, 0.


of 97

346 337, 0., 1.60000002, 0.347 338, 0.100000001, 1.60000002, 0.348 339, 0.200000003, 1.60000002, 0.349 340, 0.300000012, 1.60000002, 0.350 341, 0.400000006, 1.60000002, 0.351 342, 0.5, 1.60000002, 0.352 343, 0.600000024, 1.60000002, 0.353 344, 0.699999988, 1.60000002, 0.354 345, 0.800000012, 1.60000002, 0.355 346, 0.899999976, 1.60000002, 0.356 347, 1., 1.60000002, 0.357 348, 1.10000002, 1.60000002, 0.358 349, 1.20000005, 1.60000002, 0.359 350, 1.29999995, 1.60000002, 0.360 351, 1.39999998, 1.60000002, 0.361 352, 1.5, 1.60000002, 0.362 353, 1.60000002, 1.60000002, 0.363 354, 1.70000005, 1.60000002, 0.364 355, 1.79999995, 1.60000002, 0.365 356, 1.89999998, 1.60000002, 0.366 357, 2., 1.60000002, 0.367 358, 0., 1.70000005, 0.368 359, 0.100000001, 1.70000005, 0.369 360, 0.200000003, 1.70000005, 0.370 361, 0.300000012, 1.70000005, 0.371 362, 0.400000006, 1.70000005, 0.372 363, 0.5, 1.70000005, 0.373 364, 0.600000024, 1.70000005, 0.374 365, 0.699999988, 1.70000005, 0.375 366, 0.800000012, 1.70000005, 0.376 367, 0.899999976, 1.70000005, 0.377 368, 1., 1.70000005, 0.378 369, 1.10000002, 1.70000005, 0.379 370, 1.20000005, 1.70000005, 0.380 371, 1.29999995, 1.70000005, 0.381 372, 1.39999998, 1.70000005, 0.382 373, 1.5, 1.70000005, 0.383 374, 1.60000002, 1.70000005, 0.384 375, 1.70000005, 1.70000005, 0.385 376, 1.79999995, 1.70000005, 0.386 377, 1.89999998, 1.70000005, 0.387 378, 2., 1.70000005, 0.388 379, 0., 1.79999995, 0.389 380, 0.100000001, 1.79999995, 0.390 381, 0.200000003, 1.79999995, 0.391 382, 0.300000012, 1.79999995, 0.392 383, 0.400000006, 1.79999995, 0.393 384, 0.5, 1.79999995, 0.394 385, 0.600000024, 1.79999995, 0.395 386, 0.699999988, 1.79999995, 0.396 387, 0.800000012, 1.79999995, 0.397 388, 0.899999976, 1.79999995, 0.398 389, 1., 1.79999995, 0.399 390, 1.10000002, 1.79999995, 0.400 391, 1.20000005, 1.79999995, 0.401 392, 1.29999995, 1.79999995, 0.402 393, 1.39999998, 1.79999995, 0.403 394, 1.5, 1.79999995, 0.404 395, 1.60000002, 1.79999995, 0.405 396, 1.70000005, 1.79999995, 0.406 397, 1.79999995, 1.79999995, 0.407 398, 1.89999998, 1.79999995, 0.408 399, 2., 1.79999995, 0.409 400, 0., 1.89999998, 0.410 401, 0.100000001, 1.89999998, 0.411 402, 0.200000003, 1.89999998, 0.412 403, 0.300000012, 1.89999998, 0.413 404, 0.400000006, 1.89999998, 0.414 405, 0.5, 1.89999998, 0.


of 97

415 406, 0.600000024, 1.89999998, 0.416 407, 0.699999988, 1.89999998, 0.417 408, 0.800000012, 1.89999998, 0.418 409, 0.899999976, 1.89999998, 0.419 410, 1., 1.89999998, 0.420 411, 1.10000002, 1.89999998, 0.421 412, 1.20000005, 1.89999998, 0.422 413, 1.29999995, 1.89999998, 0.423 414, 1.39999998, 1.89999998, 0.424 415, 1.5, 1.89999998, 0.425 416, 1.60000002, 1.89999998, 0.426 417, 1.70000005, 1.89999998, 0.427 418, 1.79999995, 1.89999998, 0.428 419, 1.89999998, 1.89999998, 0.429 420, 2., 1.89999998, 0.430 421, 0., 2., 0.431 422, 0.100000001, 2., 0.432 423, 0.200000003, 2., 0.433 424, 0.300000012, 2., 0.434 425, 0.400000006, 2., 0.435 426, 0.5, 2., 0.436 427, 0.600000024, 2., 0.437 428, 0.699999988, 2., 0.438 429, 0.800000012, 2., 0.439 430, 0.899999976, 2., 0.440 431, 1., 2., 0.441 432, 1.10000002, 2., 0.442 433, 1.20000005, 2., 0.443 434, 1.29999995, 2., 0.444 435, 1.39999998, 2., 0.445 436, 1.5, 2., 0.446 437, 1.60000002, 2., 0.447 438, 1.70000005, 2., 0.448 439, 1.79999995, 2., 0.449 440, 1.89999998, 2., 0.450 441, 2., 2., 0.451 *Element, type=S4R452 1, 1, 2, 23, 22453 2, 2, 3, 24, 23454 3, 3, 4, 25, 24455 4, 4, 5, 26, 25456 5, 5, 6, 27, 26457 6, 6, 7, 28, 27458 7, 7, 8, 29, 28459 8, 8, 9, 30, 29460 9, 9, 10, 31, 30461 10, 10, 11, 32, 31462 11, 11, 12, 33, 32463 12, 12, 13, 34, 33464 13, 13, 14, 35, 34465 14, 14, 15, 36, 35466 15, 15, 16, 37, 36467 16, 16, 17, 38, 37468 17, 17, 18, 39, 38469 18, 18, 19, 40, 39470 19, 19, 20, 41, 40471 20, 20, 21, 42, 41472 21, 22, 23, 44, 43473 22, 23, 24, 45, 44474 23, 24, 25, 46, 45475 24, 25, 26, 47, 46476 25, 26, 27, 48, 47477 26, 27, 28, 49, 48478 27, 28, 29, 50, 49479 28, 29, 30, 51, 50480 29, 30, 31, 52, 51481 30, 31, 32, 53, 52482 31, 32, 33, 54, 53483 32, 33, 34, 55, 54


of 97

484 33, 34, 35, 56, 55485 34, 35, 36, 57, 56486 35, 36, 37, 58, 57487 36, 37, 38, 59, 58488 37, 38, 39, 60, 59489 38, 39, 40, 61, 60490 39, 40, 41, 62, 61491 40, 41, 42, 63, 62492 41, 43, 44, 65, 64493 42, 44, 45, 66, 65494 43, 45, 46, 67, 66495 44, 46, 47, 68, 67496 45, 47, 48, 69, 68497 46, 48, 49, 70, 69498 47, 49, 50, 71, 70499 48, 50, 51, 72, 71500 49, 51, 52, 73, 72501 50, 52, 53, 74, 73502 51, 53, 54, 75, 74503 52, 54, 55, 76, 75504 53, 55, 56, 77, 76505 54, 56, 57, 78, 77506 55, 57, 58, 79, 78507 56, 58, 59, 80, 79508 57, 59, 60, 81, 80509 58, 60, 61, 82, 81510 59, 61, 62, 83, 82511 60, 62, 63, 84, 83512 61, 64, 65, 86, 85513 62, 65, 66, 87, 86514 63, 66, 67, 88, 87515 64, 67, 68, 89, 88516 65, 68, 69, 90, 89517 66, 69, 70, 91, 90518 67, 70, 71, 92, 91519 68, 71, 72, 93, 92520 69, 72, 73, 94, 93521 70, 73, 74, 95, 94522 71, 74, 75, 96, 95523 72, 75, 76, 97, 96524 73, 76, 77, 98, 97525 74, 77, 78, 99, 98526 75, 78, 79, 100, 99527 76, 79, 80, 101, 100528 77, 80, 81, 102, 101529 78, 81, 82, 103, 102530 79, 82, 83, 104, 103531 80, 83, 84, 105, 104532 81, 85, 86, 107, 106533 82, 86, 87, 108, 107534 83, 87, 88, 109, 108535 84, 88, 89, 110, 109536 85, 89, 90, 111, 110537 86, 90, 91, 112, 111538 87, 91, 92, 113, 112539 88, 92, 93, 114, 113540 89, 93, 94, 115, 114541 90, 94, 95, 116, 115542 91, 95, 96, 117, 116543 92, 96, 97, 118, 117544 93, 97, 98, 119, 118545 94, 98, 99, 120, 119546 95, 99, 100, 121, 120547 96, 100, 101, 122, 121548 97, 101, 102, 123, 122549 98, 102, 103, 124, 123550 99, 103, 104, 125, 124551 100, 104, 105, 126, 125552 101, 106, 107, 128, 127


of 97

553 102, 107, 108, 129, 128554 103, 108, 109, 130, 129555 104, 109, 110, 131, 130556 105, 110, 111, 132, 131557 106, 111, 112, 133, 132558 107, 112, 113, 134, 133559 108, 113, 114, 135, 134560 109, 114, 115, 136, 135561 110, 115, 116, 137, 136562 111, 116, 117, 138, 137563 112, 117, 118, 139, 138564 113, 118, 119, 140, 139565 114, 119, 120, 141, 140566 115, 120, 121, 142, 141567 116, 121, 122, 143, 142568 117, 122, 123, 144, 143569 118, 123, 124, 145, 144570 119, 124, 125, 146, 145571 120, 125, 126, 147, 146572 121, 127, 128, 149, 148573 122, 128, 129, 150, 149574 123, 129, 130, 151, 150575 124, 130, 131, 152, 151576 125, 131, 132, 153, 152577 126, 132, 133, 154, 153578 127, 133, 134, 155, 154579 128, 134, 135, 156, 155580 129, 135, 136, 157, 156581 130, 136, 137, 158, 157582 131, 137, 138, 159, 158583 132, 138, 139, 160, 159584 133, 139, 140, 161, 160585 134, 140, 141, 162, 161586 135, 141, 142, 163, 162587 136, 142, 143, 164, 163588 137, 143, 144, 165, 164589 138, 144, 145, 166, 165590 139, 145, 146, 167, 166591 140, 146, 147, 168, 167592 141, 148, 149, 170, 169593 142, 149, 150, 171, 170594 143, 150, 151, 172, 171595 144, 151, 152, 173, 172596 145, 152, 153, 174, 173597 146, 153, 154, 175, 174598 147, 154, 155, 176, 175599 148, 155, 156, 177, 176600 149, 156, 157, 178, 177601 150, 157, 158, 179, 178602 151, 158, 159, 180, 179603 152, 159, 160, 181, 180604 153, 160, 161, 182, 181605 154, 161, 162, 183, 182606 155, 162, 163, 184, 183607 156, 163, 164, 185, 184608 157, 164, 165, 186, 185609 158, 165, 166, 187, 186610 159, 166, 167, 188, 187611 160, 167, 168, 189, 188612 161, 169, 170, 191, 190613 162, 170, 171, 192, 191614 163, 171, 172, 193, 192615 164, 172, 173, 194, 193616 165, 173, 174, 195, 194617 166, 174, 175, 196, 195618 167, 175, 176, 197, 196619 168, 176, 177, 198, 197620 169, 177, 178, 199, 198621 170, 178, 179, 200, 199


of 97

622 171, 179, 180, 201, 200623 172, 180, 181, 202, 201624 173, 181, 182, 203, 202625 174, 182, 183, 204, 203626 175, 183, 184, 205, 204627 176, 184, 185, 206, 205628 177, 185, 186, 207, 206629 178, 186, 187, 208, 207630 179, 187, 188, 209, 208631 180, 188, 189, 210, 209632 181, 190, 191, 212, 211633 182, 191, 192, 213, 212634 183, 192, 193, 214, 213635 184, 193, 194, 215, 214636 185, 194, 195, 216, 215637 186, 195, 196, 217, 216638 187, 196, 197, 218, 217639 188, 197, 198, 219, 218640 189, 198, 199, 220, 219641 190, 199, 200, 221, 220642 191, 200, 201, 222, 221643 192, 201, 202, 223, 222644 193, 202, 203, 224, 223645 194, 203, 204, 225, 224646 195, 204, 205, 226, 225647 196, 205, 206, 227, 226648 197, 206, 207, 228, 227649 198, 207, 208, 229, 228650 199, 208, 209, 230, 229651 200, 209, 210, 231, 230652 201, 211, 212, 233, 232653 202, 212, 213, 234, 233654 203, 213, 214, 235, 234655 204, 214, 215, 236, 235656 205, 215, 216, 237, 236657 206, 216, 217, 238, 237658 207, 217, 218, 239, 238659 208, 218, 219, 240, 239660 209, 219, 220, 241, 240661 210, 220, 221, 242, 241662 211, 221, 222, 243, 242663 212, 222, 223, 244, 243664 213, 223, 224, 245, 244665 214, 224, 225, 246, 245666 215, 225, 226, 247, 246667 216, 226, 227, 248, 247668 217, 227, 228, 249, 248669 218, 228, 229, 250, 249670 219, 229, 230, 251, 250671 220, 230, 231, 252, 251672 221, 232, 233, 254, 253673 222, 233, 234, 255, 254674 223, 234, 235, 256, 255675 224, 235, 236, 257, 256676 225, 236, 237, 258, 257677 226, 237, 238, 259, 258678 227, 238, 239, 260, 259679 228, 239, 240, 261, 260680 229, 240, 241, 262, 261681 230, 241, 242, 263, 262682 231, 242, 243, 264, 263683 232, 243, 244, 265, 264684 233, 244, 245, 266, 265685 234, 245, 246, 267, 266686 235, 246, 247, 268, 267687 236, 247, 248, 269, 268688 237, 248, 249, 270, 269689 238, 249, 250, 271, 270690 239, 250, 251, 272, 271


of 97

691 240, 251, 252, 273, 272692 241, 253, 254, 275, 274693 242, 254, 255, 276, 275694 243, 255, 256, 277, 276695 244, 256, 257, 278, 277696 245, 257, 258, 279, 278697 246, 258, 259, 280, 279698 247, 259, 260, 281, 280699 248, 260, 261, 282, 281700 249, 261, 262, 283, 282701 250, 262, 263, 284, 283702 251, 263, 264, 285, 284703 252, 264, 265, 286, 285704 253, 265, 266, 287, 286705 254, 266, 267, 288, 287706 255, 267, 268, 289, 288707 256, 268, 269, 290, 289708 257, 269, 270, 291, 290709 258, 270, 271, 292, 291710 259, 271, 272, 293, 292711 260, 272, 273, 294, 293712 261, 274, 275, 296, 295713 262, 275, 276, 297, 296714 263, 276, 277, 298, 297715 264, 277, 278, 299, 298716 265, 278, 279, 300, 299717 266, 279, 280, 301, 300718 267, 280, 281, 302, 301719 268, 281, 282, 303, 302720 269, 282, 283, 304, 303721 270, 283, 284, 305, 304722 271, 284, 285, 306, 305723 272, 285, 286, 307, 306724 273, 286, 287, 308, 307725 274, 287, 288, 309, 308726 275, 288, 289, 310, 309727 276, 289, 290, 311, 310728 277, 290, 291, 312, 311729 278, 291, 292, 313, 312730 279, 292, 293, 314, 313731 280, 293, 294, 315, 314732 281, 295, 296, 317, 316733 282, 296, 297, 318, 317734 283, 297, 298, 319, 318735 284, 298, 299, 320, 319736 285, 299, 300, 321, 320737 286, 300, 301, 322, 321738 287, 301, 302, 323, 322739 288, 302, 303, 324, 323740 289, 303, 304, 325, 324741 290, 304, 305, 326, 325742 291, 305, 306, 327, 326743 292, 306, 307, 328, 327744 293, 307, 308, 329, 328745 294, 308, 309, 330, 329746 295, 309, 310, 331, 330747 296, 310, 311, 332, 331748 297, 311, 312, 333, 332749 298, 312, 313, 334, 333750 299, 313, 314, 335, 334751 300, 314, 315, 336, 335752 301, 316, 317, 338, 337753 302, 317, 318, 339, 338754 303, 318, 319, 340, 339755 304, 319, 320, 341, 340756 305, 320, 321, 342, 341757 306, 321, 322, 343, 342758 307, 322, 323, 344, 343759 308, 323, 324, 345, 344


of 97

760 309, 324, 325, 346, 345761 310, 325, 326, 347, 346762 311, 326, 327, 348, 347763 312, 327, 328, 349, 348764 313, 328, 329, 350, 349765 314, 329, 330, 351, 350766 315, 330, 331, 352, 351767 316, 331, 332, 353, 352768 317, 332, 333, 354, 353769 318, 333, 334, 355, 354770 319, 334, 335, 356, 355771 320, 335, 336, 357, 356772 321, 337, 338, 359, 358773 322, 338, 339, 360, 359774 323, 339, 340, 361, 360775 324, 340, 341, 362, 361776 325, 341, 342, 363, 362777 326, 342, 343, 364, 363778 327, 343, 344, 365, 364779 328, 344, 345, 366, 365780 329, 345, 346, 367, 366781 330, 346, 347, 368, 367782 331, 347, 348, 369, 368783 332, 348, 349, 370, 369784 333, 349, 350, 371, 370785 334, 350, 351, 372, 371786 335, 351, 352, 373, 372787 336, 352, 353, 374, 373788 337, 353, 354, 375, 374789 338, 354, 355, 376, 375790 339, 355, 356, 377, 376791 340, 356, 357, 378, 377792 341, 358, 359, 380, 379793 342, 359, 360, 381, 380794 343, 360, 361, 382, 381795 344, 361, 362, 383, 382796 345, 362, 363, 384, 383797 346, 363, 364, 385, 384798 347, 364, 365, 386, 385799 348, 365, 366, 387, 386800 349, 366, 367, 388, 387801 350, 367, 368, 389, 388802 351, 368, 369, 390, 389803 352, 369, 370, 391, 390804 353, 370, 371, 392, 391805 354, 371, 372, 393, 392806 355, 372, 373, 394, 393807 356, 373, 374, 395, 394808 357, 374, 375, 396, 395809 358, 375, 376, 397, 396810 359, 376, 377, 398, 397811 360, 377, 378, 399, 398812 361, 379, 380, 401, 400813 362, 380, 381, 402, 401814 363, 381, 382, 403, 402815 364, 382, 383, 404, 403816 365, 383, 384, 405, 404817 366, 384, 385, 406, 405818 367, 385, 386, 407, 406819 368, 386, 387, 408, 407820 369, 387, 388, 409, 408821 370, 388, 389, 410, 409822 371, 389, 390, 411, 410823 372, 390, 391, 412, 411824 373, 391, 392, 413, 412825 374, 392, 393, 414, 413826 375, 393, 394, 415, 414827 376, 394, 395, 416, 415828 377, 395, 396, 417, 416


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829 378, 396, 397, 418, 417830 379, 397, 398, 419, 418831 380, 398, 399, 420, 419832 381, 400, 401, 422, 421833 382, 401, 402, 423, 422834 383, 402, 403, 424, 423835 384, 403, 404, 425, 424836 385, 404, 405, 426, 425837 386, 405, 406, 427, 426838 387, 406, 407, 428, 427839 388, 407, 408, 429, 428840 389, 408, 409, 430, 429841 390, 409, 410, 431, 430842 391, 410, 411, 432, 431843 392, 411, 412, 433, 432844 393, 412, 413, 434, 433845 394, 413, 414, 435, 434846 395, 414, 415, 436, 435847 396, 415, 416, 437, 436848 397, 416, 417, 438, 437849 398, 417, 418, 439, 438850 399, 418, 419, 440, 439851 400, 419, 420, 441, 440852 *Nset, nset=_PickedSet2, internal, generate853 1, 441, 1854 *Elset, elset=_PickedSet2, internal, generate855 1, 400, 1856 ** Section: Section857 *Shell Section, elset=_PickedSet2, material=Steel858 0.1, 5859 *End Part860 ** 861 **862 ** ASSEMBLY863 **864 *Assembly, name=Assembly865 ** 866 *Instance, name=Plate-1, part=Plate867 *End Instance868 ** 869 *Nset, nset=all_elements, instance=Plate-1, generate870 1, 441, 1871 *Elset, elset=all_elements, instance=Plate-1, generate872 1, 400, 1873 *Nset, nset=encastre_nodes, instance=Plate-1874 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16875 17, 18, 19, 20, 21, 22, 42, 43, 63, 64, 84, 85, 105, 106, 126, 127876 147, 148, 168, 169, 189, 190, 210, 211, 231, 232, 252, 253, 273, 274, 294, 295877 315, 316, 336, 337, 357, 358, 378, 379, 399, 400, 420, 421, 422, 423, 424, 425878 426, 427, 428, 429, 430, 431, 432, 433, 434, 435, 436, 437, 438, 439, 440, 441879 *Elset, elset=__PickedSurf13_SPOS, internal, instance=Plate-1, generate880 1, 400, 1881 *Surface, type=ELEMENT, name=_PickedSurf13, internal882 __PickedSurf13_SPOS, SPOS883 *End Assembly884 ** 885 ** MATERIALS886 ** 887 *Material, name=Steel888 *Elastic889 2e+08, 0.3890 ** 891 ** BOUNDARY CONDITIONS892 ** 893 ** Name: fix_nodes Type: Symmetry/Antisymmetry/Encastre894 *Boundary895 encastre_nodes, ENCASTRE896 ** ----------------------------------------------------------------897 **


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898 ** STEP: loading899 ** 900 *Step, name=loading, nlgeom=NO, inc=100000901 *Static902 0.0001, 1., 1e-08, 0.01903 ** 904 ** LOADS905 ** 906 ** Name: Load-1 Type: Pressure907 *Dsload908 _PickedSurf13, P, 1.909 ** 910 ** OUTPUT REQUESTS911 ** 912 *Restart, write, frequency=0913 ** 914 ** FIELD OUTPUT: F-Output-2915 ** 916 *Output, field917 *Node Output918 CF, RF, RM, RT, TF, VF919 *Element Output, directions=YES920 BF, CENTMAG, CENTRIFMAG, CORIOMAG, ESF1, GRAV, HP, NFORC, NFORCSO, P, ROTAMAG, SF,



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mindlin plate theory and abaqus uel implementationweb.mit.edu/npdhye/www/padhye-uel.pdfmindlin plate...

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