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THE PHYSICALLY NONLINEAR CONTACT ANALYSIS OF CIRCULAR PLATE ON DEFORMABLE FOUNDATION BY ANSYS SOFTWARE

Valentinas Jankovski1, Valentinas Skaržauskas2

1, 2Dept of Structural Mechanics, Vilnius Gediminas Technical University, Saulėtekio al. 11, LT-10223 Vilnius, Lithuania

E-mail: 1JWMsoftcorp@gmail.com, 2Valentinas.Skarzauskas@vgtu.lt

Abstract. A circular reinforced concrete plate on deformable foundation is investigated in this paper. A foundation is composed of three layers of soil which are defined as isotropically nonlinear materials with variable physical proper-ties. The reinforced plate is treated as an elastic body. Dissymmetrical half-circular loading of linearly variable pres-sure is subjected on the plate. Generally the unified system “structure-foundation” is the contact analysis problem, which is modeled by ANSYS software. The plastic shear deformations are evaluated in solving of contact problem by the finite element method. Comparable analysis with treatment of physically linear system “structure-foundation” is performed. After linear analysis the comparison shows, that rigidity of the foundation generates bigger extreme stresses in the plate in context of the Huber-Mises criterion. However in nonlinear analysis, intensity of stresses is naturally decreasing when more flexible model of the foundation is used.

Keywords: circular plate, deformable foundation, contact problem, physical nonlinearity, finite element method, AN-SYS, plastic strains of soil, three dimensional model.

1. Introduction

The theory of bending plates on elastic foundation employs even some foundation models (Winkler, Paster-nak, Filonenko-Borodich, half space models) and is well developed and widely applied in practice (Bowles 1988; Celik et al. 1999; Gorbunov-Posadov et al. 1984; Mastro-janis 1986; Pavlik 1977; Silviera et al. 1999). However, the elastic theory solutions badly reflect an interaction between the plate and foundation in the near edge of the plate’s contact zone. More realistic and precise stresses and strains distributions are derived when selfweights of a structure and layered foundation with variable physical properties are evaluated (Krutinis et al. 2004; Regalado et al. 1992). However, nonlinearly deformable foundation model will be considered if material physical indices do not correspond to generalized Hooke‘s law (Colberg 1999; Hu et al. 1999; De Lima et al. 2001; Frank et al. 2005; Mučinis et al. 2009). The relation between stresses and strains becomes nonlinear in zones of large reactive stresses, also at edges of foundation where shear is char-acteristic, i.e. plastic strains emerge in soil (AFNOR 1994; Gorbunov-Posadov 1984). It is necessary to evalu-ate zones of plastic strains development bellow the struc-

ture especially for foundations of buildings, which are sensitive to settlements.

For thoroughly analysis and further rational design of the plate on deformable foundation, with evaluation of plastic shear strains and taking into account the settle-ments’ restrictions (Skaržauskas et al. 2009), can be used the concept of variable repeated load and adaptability principles (Atkočiūnas 1999). It would allow achieving more rational structural solution, especially for the circu-lar-like stamps and plates (Atkočiūnas et al. 2004–07), which would satisfy requirements of limit states (SNiP 1983–84; Frank et al. 2004).

2. A mathematical model of the system “structure-foundation”

The full system of equations (1) of symmetrical cir-cular plate on elastic foundation is composed of equilib-rium differential equations (a), geometrical differential equations (b), physical equations of the plate and founda-tion (c, d) and contact condition (e):

911

[ ]

[ ]

[ ]

( ) ( ) 0, ( )

( ) , ( )

, ( )

( ) ( ) ( ) 0, ( )

( ) ( ) 0. ( )

T

q r p r a

w r b

K c

D r p r s r d

w r s r e

⎧ + − =⎪⎪ + =⎪⎪

=⎨⎪ ⋅ − =⎪⎪ − =⎪⎩

M

κ 0

M κ

A

A

(1)

Here, [A] is the differential operator of equilibrium equa-

tions of the plate; [K] is the stiffness matrix of the plate; M is the vector of functions of plate’s bending moments; κ is the vector of curvatures functions of the plate; q(r) is a function of symmetrical external loading; p(r) is a function of reactive symmetrical pressure of foundation; w(r) is a function of deflections of the plate; s(r) is a function of foundation settlements; D(r) is a function of foundation flexibility. The described system of equations (1) includes such unknown functions: ( )rM r , ( )M r

θ,

( )p r , ( ) ( )w r s r= , ( )r rκ , ( )rθκ .

After some reorganizations of equations, we derive the equilibrium equation, the geometrical equation and the physical equations of the plate:

( )

0 0

1 1,

r rr

rd rM

M M qrdr prdrdr r rθ

+ − = − +∫ ∫ (2)

2

2,r

d w

drdw

rdrθ

κ

κ

−

= =

−

κ (3)

[ ]( )

( )

1

1

( ),

( )rr

r

M rK

M rθ

θθ

κ ν κ

κ ν κ

⋅ +

≡ = =

⋅ +

M κ

K

K (4)

where ( )

31

2112 1

E h

ν

=

−

K is a cylindrical stiffness of the

plate.

3. A circular plate on deformable foundation

The circular fairly flexible plate of reinforced con-crete subjected by asymmetrical loading together with layered deformable foundation (Fig 1) is analyzed as unified contact problem, taking into account the full sys-tem of equations (1). The depth of foundation is d = 3.0 m, radius of the plate R = 3.0 m and thickness t = 0.120 m, elasticity modulus of concrete E = 30 GPa, concrete density ρ = 2100 kg/m3 and Poisson‘s ratio ν = 0.3. The half-circular loading area is bound by internal Ri = 2.0 m and external Re = 2.5 m radii, maximum intensity of dis-tributed load is p1 = 6000 kN/m2, and p2 = 5000 kN/m2. The load intensity on the soil surface is calculated con-sidering to cutting depth and soil density by such an ex-pression 0 01p gdρ= . The elastic foundation of three

layers is bound by such a space: R0 = 6.0 m, z01 = 3.0 m, z02 = 6.0 m, z03 = 11.0 m. The physical-mechanical prop-erties of the foundation layers are described in Table 1.

Table 1. The physical-mechanical properties of layers of soil

Fine sand �

Sandy loam �

Clay loam �

Porosity ratio e 0.7 0.65 0.55

Density ρ0, kg/m3 1650 2050 2180

Strain modulus 0'E 38 17 45

Poisson‘s ratio ν0 0.30 0.33 0.33

Cohesion of soil c, kPa 10 31 30

Internal friction ratio ϕ, rad 35 24 28

±0.000

z

p

p

R0

Ri

ReR

t

E, ν, ρ

E , ν , ρ

p

d

0

1

2

01

z02

z03

01 01 01

E , ν , ρ02 02 02

E , ν , ρ03 03 03

1

2

3

Fig 1. Design schema of the structure

4. Physically nonlinear layered model of foundation

The dependency σ-ε of physically nonlinear material is defined for every layer under the plate in the stage of plastic shear strains development by soil test stamp. The foundation yield strain Ry (appoint the rise of plastic shear strains) and the foundation limit strength Ru (ap-point the rise of soil extrusion) are depicted in stresses axis. The values of these stresses must be known for foundations design, but in this paper the main criterion is considered as the development of soil plastic strains zones under the structure and settlements control, i.e. the stresses under the plate must not exceed the yield stress

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of foundation ( )H -M yf R≤σ , and the plate settlements

must not exceed the limit settlements lims s≤ . Sometimes

it is advisable to allow the formation of local plastic shear strains zones, herewith achieving the more rational struc-tural solution, which fully secures the requirements of limit states.

The foundation yield stress is calculated by such an expression (Šimkus 1984; Amšiejus et al. 2009)

( )ctan

ctan / 2max

y

z d cR d

π γ γ ϕγ

ϕ ϕ π

′+ + ⋅′= +

+ −, (5)

here, zmax is maximum depth of soil limit strength state distribution below the foundation; γ is unitary weight of soil below the foundation; γ ′ is unitary weight of soil

above the foundation; c is cohesion of soil below the foundation; ϕ is internal friction ratio of soil below the foundation [rad].

εε

E'

1

2

3

4 5

0.8*

R

Densificationstage of soil

Shear stage - the formation of plastic

shear strains

Extrusionstage of soil

δ

u

Ry , 0.25

Ry

y Fig. 2 The stamp test: idealized diagram of soil σ-ε dependency

The convenient approximation of this diagram

(Fig 2) is performed by Bezier curvature, which mathe-matical theory is presented here (Lengyel 2004).

σ

E'

PtApproximation of Bezier curvature

ε0.8*0.5

t =∞t = 0.25

Ry

Ry , 0.25

ε y εy , 0.25

0

Pt1 Pt2 Pt3

Fig 3. Approximation of plastic shear strains zone by 0.25 steps

Proposed Bezier approximation is realized numeri-cally by specially created computer program in MATLAB environment (Jankovski et al. 2008–10), and further em-ployed for every soil layer of the foundation. This pro-gram approximates plastic shear strains zone by 0.25 steps with respect to boundary values of Ry and Ry,0.25, on purpose to create the ANSYS Multilinear Kinematic Hardening (MKIN) curve (Fig 3). Finally approximated values of σ-ε dependency will be used for creation of physically nonlinear models of foundation layers in AN-SYS software (Fig 4). The physical model of isotropic material will be used for reason of idealization.

0

100

200

300

400

500

600

700

0 0.2 0.4 0.6 0.8ε , m/m

σ

, kPa

Layer 1

Layer 2

Layer 3

Fig 4. Nonlinear physical models MKIN of layers’ materials

5. Structural model generation in the ANSYS pre-processor

Structural model generation of the structure “plane-foundation” starts from creating the geometric discret model (GDM). Indeed, it is a planar 2D profile as axi-symmetric contour of the future foundation (Fig 5). Addi-tionally the profile is divided into separated areas A1–A6 which are composed of keypoints K* and lines L*. De-fined areas will be responsible for meshing control and densification which is described by a* parameters. Be-cause the foundation model is divided by regular three dimensional finite elements mesh, it is advisable to model the foundation as revolved body, and mesh it by desirable density and distribution quadrangle elements MESH200, which are specially intended for modeling by this way.

The loading of asymmetrical half-circular linearly variable pressure is subjected on the model of structure “plate-foundation”. The circular plate of reinforced con-crete is of the uniform thickness and meshed by plane quadrangle finite elements SHELL63. The plate is fairly flexible, therefore it is very likely that the contact with the foundation can be lost because of asymmetrical half-circular pressure. Therefore it is advisable to investigate such a case as contact problem, applying single-sided bonds modeled by CONTAC52 finite elements. The foundation model is composed of three soil layers of

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different physical-mechanical characteristics which are discussed early. These layers are modeled by 3D pris-matic finite elements SOLID45 (Gallagher 1975; Zien-kiewicz 1977; Образцов et al. 1985; Cook et al. 1989, 95; Wilson 2002; Barauskas et al. 2004).

The modeling schema (Fig 5) is used for the fully parametric batch and initial data ANSYS file formation which composed of the pre-processor, solution and post-processor stages. Since the system model is fully pa-rametrized, it allows modifying all the initial conditions, analyzing the results and discussing about a suitability of the design structure.

Such preparative and intermediate phases were de-rived: nonlinear physical-mechanical models of layered foundation (Fig 4); the preliminary rotational-planar pro-file of the foundation for generating 3D revolved solid of finite element model (Fig 6a); revolved solid foundation discretized by the prismatic finite elements (Fig 6b); the finite element model of the system “structure-foundation” (Fig 6c); boundary conditions of the medel’s surfaces; transfering the boundary conditions to the finite element model (Fig 6d).

K1 x

y

OK2 K3

K4 K5

K6 K7 K8

K9 K10 K11

K12K13

K14

L1

L2

L3

L4L5L6

L7

L8

L9

L10

L11

L12

L13

L14

L15

L16

L17

L18

L19

p

A1 A2

A3 A4

A5 A6

a3

a1 d1

c1

b1

a1

b1

c1

a4

a2

SHELL63

CONTAC52

MESH200+

SOLID45

FE Type Geometric Discret Model

K*L*

- keypoint- line

A*a*

- area- density of mesh

Fig 5. So-called “bottom up” modeling schema of circular plate on deformable foundation

a)

b)

c)

d)

Fig 6. Preparative and intermediate phases of model-ing: a) the planar profile discretized by the MESH200 finite elements; b) revolved solid foundation; c) the system “structure-foundation”; d) the final finite ele-ment model with baoundary conditions

6. The results obtained: comparative graphical interpretation

Performed comparative linear (L) and nonlinear (NL) static analysis of the circular plate on deformable foundation allowed visually to interpret such results by the ANSYS post-processor (Fig 7–Fig 13a).

Deformation character of soil shear surface and spread of strains in the layers of soil are formed along the plate perimeter (Fig 14).

Variation of maximum settlement of the structure node N1011 in the time history for both analysis cases (linear and nonlinear) are as shown in Fig 15.

Conclusions

Results of comparative analysis of the circular plate on deformable foundation shows, that in case of physical nonlinearity the stress-strain state in soil with plastic shear residual strains is more close to reality than in case of usual analysis of the structures on deformable founda-tion.

Increase of the plate settlements from 15.8 cm in linear analysis to 22.5 cm in nonlinear analysis is strongly accented for physical nonlinearity evaluation necessity. Deformation character of soil shear surface and spread of strains in the layers of soil are formed along the plate perimeter.

Stresses intensity in the foundation decreased from 0.8 MPa in linear analysis to 0.6 MPa in nonlinear analy-sis due to the development of soil strains.

Maximum stress intensities in the plate under the Huber-Mises criterion increased from 161 MPa to 182 MPa due to the increment of soil flexibility in nonlin-ear analysis.

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L

Fig 7. Foundation’s settlements sL ∈ [0; 0.158] m

NL

Fig 7a. Foundation’s settlements sNL ∈ [0; 0.225] m

L

Fig 8. Stress intensity contours by von Mises in the layered foundation fH-M, L(σ) ∈ [33.492; 807.281] kPa

NL

Fig 8a. Stress intensity contours by von Mises in the layered foundation fH-M, NL(σ) ∈ [20.048; 600.208] kPa

L

Fig 9. Foundation’s settlements in the section per node of the maximum settlement smax, L = 0.158 m

NL

Fig 9a. Foundation’s settlements in the section per node of the maximum settlement smax, NL = 0.225 m

915

L

Fig 10. Stress intensity contours by von Mises in the section (linear analysis) [Pa]

NL

Fig 10a. Stress intensity contours by von Mises in the section (nonlinear analysis) [Pa]

L

Fig 11. Deflections of the plate and foundation in the section [m]

NL

Fig 11a. Deflections of the plate and foundation in the section [m]

L

Fig 12. Plate’s deflections (linear analysis) [m]

NL

Fig 12a. Plate’s deflections (nonlinear analysis) [m]

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L

Fig 13. Stress intensity contours by von Mises in the plate fH-M, L(σ) ∈ [0.337; 161.0] MPa

NL

Fig 13a. Stress intensity contours by von Mises in the plate fH-M, NL(σ) ∈ [0.549; 182.0] MPa

NL

Fig 14. Distribution of plastic shear strains by von Mises criterion in soil fH-M, NL(ε) ∈ [0; 0.129] m/m

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0 0.2 0.4 0.6 0.8 1time , s

s10

11, m

Linear

Nonlinear

Fig 15. Settlements s1011(time) of foundation node N1011 in load history with and without evaluation of physical nonlinearity

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