Finite Element Modelling for Elastic Plastic Stress ... Finite element elastic-plastic analysis

Finite Element Modelling for Elastic Plastic Stress ... Finite element elastic-plastic analysis
Finite Element Modelling for Elastic Plastic Stress ... Finite element elastic-plastic analysis
Finite Element Modelling for Elastic Plastic Stress ... Finite element elastic-plastic analysis
Finite Element Modelling for Elastic Plastic Stress ... Finite element elastic-plastic analysis
Finite Element Modelling for Elastic Plastic Stress ... Finite element elastic-plastic analysis
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Transcript of Finite Element Modelling for Elastic Plastic Stress ... Finite element elastic-plastic analysis

  • International Conference on Challenges and Opportunities in Mechanical Engineering, Industrial Engineering and Management Studies 188

    (ICCOMIM - 2012), 11-13 July, 2012

    ISBN 978-93-82338-03-1 | 2012 Bonfring

    Abstract--- The focus of this paper is on the use of ANSYS software for elastic-plastic stress analysis. NAFEMS

    (International association for the Engineering analysis community) India is proposing a benchmark for nonlinear

    finite element analysis: Material non-linearity. The objective of this study is to present accurate target solutions to

    this test problem. The Finite Element Model is developed using ANSYS. It is validated using another NAFEMS

    Benchmark namely a thick walled cylinder under internal pressure for which target solutions are available in the

    NAFEMS document [1]. Using this Finite Element Model, Target solutions to the test problem are graphically

    presented and discussed. A number of agencies in India and worldwide will be contributing target solutions to the

    same problem. This will be consolidated before NAFEMS issuing the Benchmark.

    Keywords--- Elastic Plastic Stress Analysis, Non-linear Finite Element Analysis, Material nonlinearity,

    Rectangular Plates with Circular cut-out, Linearly Varying In-Plane Tensile Load

    I. INTRODUCTION

    HE use of thin plates is very common in many engineering applications, such as offshore platforms, ship decks

    and hulls, box sections of bridge girders and aircraft industries. There is often a need for cut-out in the plates for

    services, inspection and maintenance, etc. The presence of these holes changes the stress distribution and cause

    reduction in its strength and buckling characteristics. Analysis of such systems was mainly carried out for axial

    compressive forces resulting in instability [2] and elasto-plastic buckling in the system [3, 4], axial tensile loading to

    study on Stress Concentration Factors [5] and non-linear behaviour subjected to shear loading [6]. While in this case

    the analysis is carried out in axial tensile loading resulting in Materials to exhibit nonlinearities as loads and

    deformations increase.

    When ductile metals are loaded beyond elastic range, the initial linear stress response will give way to a

    complicated nonlinear response, characterized by a much-reduced modulus and different stress behaviour along load

    and unloading path. Finite element elastic-plastic analysis is no longer linear, but a set of nonlinear equations that

    needs to be solved iteratively. Typically, we divide the applied load into small increments so as to have a better

    numerical performance.

    The Finite Element Model is developed using ANSYS software, a pioneer in the discipline of nonlinear analysis.

    ANSYS has both penalty-based and Lagrangian multiplier based mixed u-P formulations. Lagrange multiplier based

    formulation is available in the 180-series solid elements, and is meant for nearly incompressible elasto-plastic,

    nearly incompressible hyperelastic and fully incompressible hyperelastic materials [7]. ANSYS employs the

    "Newton-Raphson" approach to solve nonlinear problems. The "top" level consists of the load steps that we define

    explicitly over a "time" span. Within each load step, we can direct the program to perform several solutions

    (substeps or time steps) to apply the load gradually. At each substep, the program will perform a number of

    equilibrium iterations to obtain a converged solution.

    Bhavesh Govind Naik, Department of Mechanical Engineering, Dayananda Sagar College of Engineering, Bangalore.

    Shivashankar R. Srivatsa, Department of Mechanical Engineering, Dayananda Sagar College of Engineering, Bangalore.

    PAPER ID: MED32

    Finite Element Modelling for Elastic Plastic

    Stress Analysis: Development, Validation and

    Case Study Bhavesh Govind Naik and Shivashankar R. Srivatsa

    T

  • International Conference on Challenges and Opportunities in Mechanical Engineering, Industrial Engineering and Management Studies 189

    (ICCOMIM - 2012), 11-13 July, 2012

    ISBN 978-93-82338-03-1 | 2012 Bonfring

    II. NAFEMS BENCHMARK

    The Finite element modelling for elastic plastic stress analysis using ANSYS software is validated using

    NAFEMS Benchmark namely a thick walled cylinder under internal pressure for which target solutions are available

    in the NAFEMS document [1].

    A thick walled cylinder of internal radius and external radius is subjected to a

    uniform internal pressure (fig 1). Exploiting Symmetry, one octant of the cylindrical tube has been modelled. The

    sector is discretised using Plane 183 element (8-noded quadrilateral) in ANSYS. A typical mesh generated is shown

    in fig 2. Axisymmetric boundary conditions are enforced along lines AB and CD. The internal pressure is applied

    along AC. The material properties of the mild steel cylinder are listed in table 1. Bilinear uniaxial stress strain curve

    for an elastic perfectly plastic material used in the analysis is shown in fig 3.

    Figure 1: Thick Walled Cylinder

    Subjected to Internal Pressure

    Figure 2: Finite Element Model of

    an Octant of Cylinder

    Figure 3: Bilinear Uniaxial Stress

    Strain Curve for an Elastic Perfectly

    Plastic Material

    Figure 4: Stress Distribution for a Cylinder under Internal Pressure

    A maximum pressure of 160 N/mm2 is applied in steps. The predicted hoop stress (yy) and Radial stress (xx)

    along the x-axis are plotted in fig 4. These results arefound to closely match with the target solutions reported in the

    NAFEMS document [1].

    To study the effect of Tangent modulus on the behavior of the cylinder, computations are performed with

    Tangent Modulus =1250 N/mm2.The internal pressure is applied in load steps of 80, 100, 120, 140, 160 and 165

    N/mm2. The radial, hoop and von Mises stresses are plotted in fig 5.

    -1

    -0.8

    -0.6

    -0.4

    -0.2

    0

    0.2

    0.4

    0.6

    0.8

    1

    0.5 0.6 0.7 0.8 0.9 1

    Case 1 = 80 N/mm2

    Case 2 = 100 N/mm2

    Case 3 = 120 N/mm2

    Case 4 = 140 N/mm2

    Case 5 = 160 N/mm2

    P / P max = 0.5

    P / P max = 0.625

    P / P max = 0.75

    P / P max = 0.875

    P / P max = 1.0

    xx/o

    yy/o

    xx/o

    yy/o

    Radius r/b

    max

  • International Conference on Challenges and Opportunities in Mechanical Engineering, Industrial Engineering and Management Studies 190

    (ICCOMIM - 2012), 11-13 July, 2012

    ISBN 978-93-82338-03-1 | 2012 Bonfring

    Figure 5: Plot of Radial, Hoop and Von Mises Stress from Inner Radius to outer Radius for Various Load Steps of

    80,100,120,140,160 and 165 N/mm2.

    Comparison of figure 4 and figure 5 shows the significant influence of the tangent modulus on the stress

    distribution.

    III. NAFEMS INDIA PROPOSED BENCHMARK

    A rectangular plate of uniform thickness with a central circular hole as shown in fig 6 is fixed at one end and

    subjected to maximum axial tensile stress of 272 N/mm2 at other end. The material properties of the rectangular

    plate are listed in table 2. Bilinear uniaxial stress strain curve for an elastic plastic material used in the analysis is

    shown in fig 7. The objective is to perform elastic-plastic stress analysis and to report maximum equivalent plastic

    strain and stress and their locations.

    Figure 6: Proposed Bencmmark Problem Figure 7: Bilinear Isotropic Hardening Material for

    Proposed NAFEMS Benchmark

    Table 2: Rectangular Plate Material Properties: No. of Plane 183 Elements = 1144

    Elastic Modulus, E 200000 N/mm2

    Poisson's ratio, 0.3

    Yield Stress, Y 400 N/mm2

    Tangent Modulus, ET 1250 N/mm2

    Radial

    Hoop

    von Mises

    P =80 N/mm2

    P =165 N/mm2 P =160 N/mm2

    P =120 N/mm2

    P =140 N/mm2

    P =100 N/mm2

  • International Conference on Challenges and Opportunities in Mechanical Engineering, Industrial Engineering and Management Studies 191

    (ICCOMIM - 2012), 11-13 July, 2012

    ISBN 978-93-82338-03-1 | 2012 Bonfring

    Figure 8: FE Mesh of Proposed Benchmark

    Considering symmetry, one half of the problem has been modelled. Plane183 elements (8-noded quadrilateral)

    that deal with both small and large strains, with a variety of material options including elasto-plasticity, have been

    used to create FE model. Plasticity is defined by yield stress and the elastic-plastic tangent modulus. A typical finite

    element model is shown in fig 8. One end is rigidly fixed with all DOF and other end is subjected to axial load,

    Symmetric boundary condition is applied across horizontal centre line.

    Initial Pressure of 110 N/mm2 initiates the plate to yield. Load is then increased in steps to 130, 150, 170, 190,

    210, 230, 250 & 272 N/mm2. The variation of von Mises equivalent stress and the equivalent plastic strain along the

    hole boundary are presented in figures 9 and 10.

    Figure 9: Rectangular Plate with Hole stress Distribution

    0

    100

    200

    300

    400

    500

    600

    0 5 10 15 20 25 30 35 40

    vom

    Mis

    es s

    tres

    s

    Node path on circumference of hole

    P= 110 MPa

    P=150 MPa

    P=190 MPa

    P=230 MPa

    P=250 MPa

    P=272 MPa

  • International Conference on Challenges and Opportunities in Mechanical Engi