Comparison of the Results from Analysis of Nonlinear ... · Jana Labudkova and Radim Cajka / Procedia Engineering 114 ( 2015 ) 522 – 529 523 difficult, and connected with the problematic

Procedia Engineering 114 ( 2015 ) 522 – 529

1877-7058 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).Peer-review under responsibility of INEGI - Institute of Science and Innovation in Mechanical and Industrial Engineeringdoi: 10.1016/j.proeng.2015.08.101

ScienceDirectAvailable online at www.sciencedirect.com

1st International Conference on Structural Integrity

Comparison of the Results from Analysis of Nonlinear Homogeneous and Nonlinear Inhomogeneous Half-Space

Jana Labudkovaa,*, Radim Cajkaa a Department of Structures, Faculty of Civil Engineering, VŠB – Technical University of Ostrava,

Ludvika Podeste 1875/17, 708 33 Ostrava-Poruba, Czech Republic

Abstract

Inhomogeneous half-space was used for the determination of the interaction of steel-fibre reinforced concrete foundation slab and subsoil in an analysis based on the finite element method. Inhomogeneous half-space aptly describes the stress-strain relationship in the subsoil. The soil is a heterogeneous substance. The Drucker-Prager material model was used for nonlinear analyses. Two sets of FEM analyses were performed. Nonlinear elastic isotropic homogeneous half-space was used for the first set of FEM analysis, and a nonlinear elastic isotropic inhomogeneous half-space was used for the second set.The parametric study shows a comparison of resulting deformations obtained by analysis using a homogeneous and an inhomogeneous half-space. © 2015 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of INEGI - Institute of Science and Innovation in Mechanical and Industrial Engineering.

Keywords: Foundation structure; soil – structure interaction; interaction models; 3D finite elements; FEM

1. Introduction

Physical-nonlinear behavior of the structure, the interaction of the upper building with foundation structure and the choice of interaction model are parameters influencing the accuracy of calculation of subsoil-structure interaction. Subsoil-structure interaction is also described in [1, 2, 3, 4]. Subsoil-structure interaction and its calculation is a subject of scientific research of many scientists and authors. However, the results of the various interaction models are different. Insufficient Theoretical basics and current state of the research of this issue still hinder to unambiguous solution of subsoil-structure interaction. Absence of appropriate calculation software also prevents the determination of an unambiguous solution. Low accuracy input data makes calculations of subsoil-structure interaction more

* Corresponding author. Tel.: +420-59-732-1925.

E-mail address: [email protected]

© 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).Peer-review under responsibility of INEGI - Institute of Science and Innovation in Mechanical and Industrial Engineering

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523 Jana Labudkova and Radim Cajka / Procedia Engineering 114 ( 2015 ) 522 – 529

difficult, and connected with the problematic description of the properties and behavior of foundation soil. Subsoil is composed of heterogeneous particles. The complex behavior of the soil is related to the complexity of the geological profile. Geological profile can be different in various parts of the area under the foundations. The use of combination of experimental measurements, laboratory tests or tests in-situ, and numerical modeling is the optimum approach to obtain reliable results. Such experimental measurements were carried out in the Czech Republic and abroad [5, 6].

The construction of the test equipment was conducted in the Faculty of Civil Engineering, VSB-TU Ostrava [7]. This equipment allows performing the experimental measurements of strain, stress monitoring and determination of stress-strain relations [7]. Process, results and conclusions of the performed loading tests are described in [8, 9, 10, 11].

Nomenclature

E0 modulus of elasticity on the surface of the model (E0 = E for z = 1), z z - coordinate (depth) m coefficient depending on Poisson ratio

2. Experimental measurements

Software ANSYS 15.0 was used for numerical modeling of the subsoil-structure interaction. A steel-fiber reinforced concrete slab sample was tested to determine which subsoil-structure interaction could be observed [10, 11, 12]. The steel-fiber reinforced concrete slab dimensions were 2000 x 2000 x 170 mm. The concrete class used was C25/30. The concrete was reinforced with fiber. The scattered reinforcement consisted of steel fibers type 3D DRAMIX 65 / 60B6 – 25kg.m-3.

Loess loam consistency F4 constitutes the upper layer of subsoil. Subsoil has a Poisson coefficient =0.35 and a modulus of deformability Edef,2 = 23.7MPa. Steel-fiber reinforced concrete slab was loaded applying pressure in the middle area of dimensions 200x200 mm. Loading was carried out in stages. Slab failed at load of 250kN by punching shear.

3. Nonlinear half-space

Structural and physical nonlinearity were used in the FEM analysis. Foundation-subsoil interaction is a contact task. The calculation is always performed by nonlinear analysis due to structural nonlinearities. This structural nonlinearity is induced by unilateral bond. This unilateral bond acts exclusively in the pressure. The nonlinear analysis requires an iterative solution. Physical nonlinearity is associated with material properties. Nonlinear material model was performed by application of Drucker-Prager model. Behavior of the subsoil can be described better due to the the Drucker-Prager model (Fig. 1).

Fig. 1. Drucker – Prager model.


In an nonlinear analysis, Hooke's law does is not applicable and the exceedance of conditions of plasticity leads to plastic deformations. Fig. 1 shows the elastic-plastic material behavior during uniaxial stress and it shows also a creation of plastic deformations.

Drucker-Prager model is derived from the condition of plasticity according to von Mises. Drucker-Prager model is used for cohesive materials with internal friction. Drucker-Prager model approximates to Mohr-Coulomb's plasticity condition (Fig. 1). So, the difference between the tensile and compressive strength is possible to be described (Fig. 1). Drucker-Prager model has a smooth curve of limit stress in contrast to Mohr-Coulomb's condition of plasticity. This is advantageous in terms of numerical calculations. The nonlinear model is defined by the modulus of elasticity E, Poisson coefficient , internal friction angle φ [o], soil cohesion c [kPa] and dilatancy angle ψ [o].

4. 3D numerical models of nonlinear homogeneous half-space and nonlinear inhomogeneous half-space

3D model of the subsoil can be created as a half-space. Half-space can be modeled discretely or as a continuum. Elastic homogeneous and isotropic body is easiest idealization of the half-space. Subsoil is heterogeneous material and its properties are different from idealization of linearly elastic isotropic and homogeneous material. This is the reason why calculated values of settlement and real measured settlement of building (or settlement Measured During the experiments) are different [13, 14]. If a 3D model of a linear homogeneous isotropic half-space is created using 3D elements, it is very difficult to correctly determine the size of the modeled area representing the subsoil, choose boundary conditions and determine the size of finite-element mesh. Inhomogeneous half-space describes behavior of heterogeneous materials better.

Inhomogeneous half-space was used for the analysis of the interaction of the steel fiber-reinforced concrete slab with the subsoil. Inhomogeneous half-space has different concentration of vertical stress than homogeneous half-space. Modulus of deformability increases with increasing depth of subsoil model. Finite element SHELL 181 (2D) was used to create the model of the steel-fiber reinforced concrete slab. Finite element SOLID 45 (3D) was used to create the subsoil model. The slab thickness was defined as a property of the SHELL 181. Slab thickness was 170 mm. Subsequently, the required material properties of the modeled slab and subsoil were assigned. Concrete with a modulus of elasticity of E=29.0GPa and a Poisson coefficient of =0.2 was identified as material No. 1. The subsoil model was divided into 30 layers. These layers had different material properties – Fig. 2 and equation (1). Inhomogeneity of subsoil model was taken into consideration by application of an increasing modulus of deformability Edef,2.

Fig. 2 Inhomogeneous subsoil model.

The subsoil was identified as a materials No. 2-31 with a Poisson's ratio of =0.35. Self-weight of the soil massif and self-weight of steel-fiber concrete slab were neglected. The thickness of one layer of subsoil model was 0.2m.


The modulus of deformability starts with value of E def,2 =23.7MPa at the surface of the model. The value increases gradually in the deeper layers according to the equation (1), [15]:

857,0235,0121

0 mwherezEE mdef

(1)

3D finite elements of subsoil model had dimensions 0.1x0.1x0.1m, and 2D finite elements of steel-fiber reinforced concrete slab had dimensions 0.1x0.1m. Both finite-element meshes defined as regular. The load was applied to the nodes in the loaded area. The size of the loaded area was 200x200mm. The load was approximately 250kN at the slab failure. This value was also used in the numerical model.

Transferring loading effects from the slab into the subsoil is mediated by establishing mutual contact and defining a contact area. The influence of friction between the slab and the subsoil was neglected.

Boundary conditions hindered the nodes shifts in the external walls of the modeled area representing the subsoil and shifts in the level of the lower base subsoil model. No boundary conditions hindered the nodes shifts in the level of the upper base subsoil model, which represents a terrain. Three variations of the boundary conditions were used to generate numerical models of the inhomogeneous half-space (Fig. 3). This allowed observing the influence of boundary conditions on the resulting vertical deformations.

Fig. 3. Variants of boundary conditions.

5. Comparison of settlements of nonlinear homogeneous half-space and nonlinear inhomogeneous half-space

Parametric study demonstrated that vertical deformations are most affected by changing depth of subsoil model [13, 14]. The main examined parameter in the parametric study with steel-fiber reinforced concrete slab on the subsoil modeled using an inhomogeneous half-space, was also changing depth. The above three variants of boundary conditions were used. Several depths of subsoil models and different types of boundary conditions have been used to create 12 models of inhomogeneous half-space and 12 models of homogeneous half-space. Dependence of deformation of the slab on the variable depth of subsoil models with the same ground area was investigated. Ground area was kept equal to 6.0x6.0m. Depth was increased using steps of 2.0m (2.0m, 4.0m, 6.0m, 8.0m).

Resulting vertical deformations on subsoil model, which were obtained within parametric study, are shown in Table 1, Table 2 and Table 3. The tables show a comparison of the calculated deformations of homogeneous subsoil model and inhomogeneous subsoil model. Tables rows show the difference of vertical deformations of homogeneous half-space and inhomogeneous for individual boundary conditions variations. Table columns show the increase in value of vertical deformations with increasing depth in comparison of both subsoil model types.

Table 1. Settlement of nonlinear homogeneous half-space and nonlinear inhomogeneous half-space, variant of boundary conditions A

De

pth Variant of boundary conditions A


NONLINEAR HOMOGENEOUS HALF-SPACE NONLINEAR INHOMOGENEOUS HALF-SPACE 2m

4m

6m

8m

Table 2. Settlement of nonlinear homogeneous half-space and nonlinear inhomogeneous half-space, variant of boundary conditions B

wmax is bigger about 34.8%

wmax is smaller about 15.9%



wmax is bigger about 16.8% wmax is bigger about 6.1%





Dep

th

Variant of boundary conditions B NONLINEAR HOMOGENEOUS HALF-SPACE NONLINEAR INHOMOGENEOUS HALF-SPACE

2m

4m

6m

8m










Table 3. Settlement of nonlinear homogeneous half-space and nonlinear inhomogeneous half-space, variant of boundary conditions C D

epth

Variant of boundary conditions C

NONLINEAR HOMOGENEOUS HALF-SPACE NONLINEAR INHOMOGENEOUS HALF-SPACE

2m

4m

6m

8m

wmax is smaller about 24,5%


wmax is bigger about 4,1% wmax is bigger about 1,4%






6. Conclusions

Inhomogeneous half-space provides smaller vertical deformations than homogeneous half-space. This is a consequence of the increasing modulus of deformability with the subsoil depth. The inhomogeneous half-space is not so heavily dependent on a randomly chosen geometric parameters of subsoil model as a homogenous half-space.

In this article, the depth of subsoil model parameter was observed. The difference between the smallest and the largest resulting vertical deformations in the middle of the slab for homogeneous subsoil model is 2.93mm. The difference between the smallest and the largest resulting vertical deformations in the middle of the slab for inhomogeneous subsoil model is only 0.67mm. This is more than three times smaller dispersion in resulting vertical deformations of geometrically identical models that differ only in homogeneity respectively. It can be concluded that inhomogeneous continuum provides more stable results less affected by the choice of the geometry and dimensions of the area representing the subsoil.

The maximum vertical deformation in the center of model of steel-fiber reinforced concrete slab calculated with inhomogeneous subsoil model with dimensions 6,0x6,0x6,0 and boundary conditions variant B, was 2.540mm. During the experiment, the measured deformation in the middle of the slab was 2.83mm. The calculated value was compared with the value measured during the experimental loading test and it is only about 11% smaller.

Acknowledgements

This outcome has been achieved with the financial support by Student Grant Competition VSB-TUO. Project registration number is SP2015/108.

References

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Comparison of the Results from Analysis of Nonlinear ... · Jana Labudkova and Radim Cajka / Procedia Engineering 114 ( 2015 ) 522 – 529 523 difficult, and connected with the problematic