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Determination of Stress Concentration Factor by Different Methods

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Determination of Stress Concentration Factor by Different Methods

Determination of Stress Concentration Factor by Different Methods

Stress Concentration Factor

Rabi Pathak,Nabin Giri

UG student(B.E Mechanical)

Nitte Meenakshi Institute of Technology Bangalore,India

Abstract In this modern day, FOS (Factor of Safety) considered is kept low to avoid unnecessary wastage of material due to which the degree of precision of stress value should be greater. The value of stress gets deviated from its precise value due to stress raisers (grooves notches etc.). So it is very important to analyze and find out the stress concentration factor. In this paper we discuss about the methods to determine the stress concentration factor.

INTRODUCTION

What is stress concentration and stress concentration factor?

A stress concentration often called as (stress raiser) is location where stress is concentrated. When a body of discontinuity is subject to different types of loading then more stress is concentrated at the discontinuities. The structure can fail due to such stress concentration, when concentration stress exceeds Material strength. Stress concentration factor Kt is a dimensionless factor that is used to qualify how concentrated the stress is in material. It is defined as the highest stress element to nominal stress in element.

Kt= Max/ Nom

The experimental, numerical and analytical studies were conducted to investigate the stress concentration around a U- notch grooved on a shaft.

The tension test was carried out on the grooved shaft using UTM (universal testing machine) and maximum stress is calculated using experimental approach and stress concentration factor was calculated

Kt= Max/ Nom Where,

Nom =Force/Original area

The stress concentration factor was calculated from the graph of stress concentration factor of grooved shaft in tension.We used the graph from DDHB VOLUME 1 by DR. K

Manmohan Thakur,Pramod K.C,Sujan Uprety

UG student(B.E Mechanical)

Nitte Meenakshi Institute of Technology Bangalore,India

METHODS TO DETERMINE STRESS CONCENTRATION FACTOR

NUMERICAL APPROACH

We can find the value of stress concentration factor by using the stress concentration factor graph for respective element , type of discontinuity and the loading condition (Axial,Bending or Torsional) .Here we have a solid shaft with a U-groove on the circumference and axial loading is given.Thus from the graph we have ;

Fig: Grooved shaft

Diameter of shaft(D)=25mm Diameter at the fillet part(d)=23mm Fillet radius(r)=1mm

Now,

LINGAIAH. Kt values are also calculated using Matlab coding and using software.

For Analytical approach FEA software ANSYS is used where shaft is loaded axially and maximum stress is calculated for calculation of Kt.

r/d = 1/23 = 0.0434

D/d = 25/23 = 1.086

Stress concentration factor for grooved shaft in tension graph

{From DDHB Vol-1 Page no:64 (By Dr. K. LINGAIAH)},

Stress Concentration factor is, Kt= 2.57

We obtained stress concentration factor from this method as 2.57.

EXPERIMENTAL APPROACH

Experimentally, the stress concentration factor can be found by using Universal Testing Machine, Instron test machine, polar scope system, etc. Although the most preferred method is by using the UTM. Though according to requirement of precision and availability of resource the other method also can be used.

OBJECTIVE

The purpose of the experiment is to determine the existence of the stress concentration in the vicinity of the geometric discontinuity (here U-groove) in a solid shaft.

MACHINES USED

  1. Universal testing machine

  2. Extensometer

  3. Mild steel rod

SPECIFICATION OF SOLID MILD STEEL ROD

  1. Material=Mild Steel

  2. Youngs Modulus=210GPA

  3. Initial diameter (D)=25mm

  4. Diameter of U-groove=25mm

  5. Maximum Load=1000N

    STEEL PROPERTIES

    • Standard grade for High Speed Steel, owing to its balanced composition has good toughness and cutting performance, hence many applications.

    • HSS containing Cobalt content is a high performance steel with good cutting capability and ensures high red hardness and tempering retention. It is particularly suitable in thermal stress situations and for intermittent cutting.

    • HSS containing high Molybdenum and Carbon has a wear resistance, high red hardness and good toughness.

    • HSS containing low Vanadium has a very good grind ability.

      RAW MATERIAL SPECIFICATION

      The material we used for determination of the stress concentration factor is mild steel which is also known as low carbon steel. It is the most generally used type of steel. This is also the cheapest of its category and its application is almost everywhere. Mild steel has relatively low tensile strength but it is very cheap and easy to form. The surface hardening can also be increased by carburizing

      Element

      Content(%)

      Carbon, C

      0.25-0.29

      Iron, Fe

      98

      Copper, Cu

      0.2

      Manganese, Mn

      1.03

      Phosphorous, P

      0.04

      Silicon, Si

      0.28

      Sulfur, S

      0.05

      CHEMICAL COMPOSITION OF RAW MATERIALS

      PROPERTIES

      METRIC

      IMPERIAL

      Density

      7.85 g/cm3

      0.284 lb/in3

      Tensile strength

      400-500 Mpa

      58000-79800 psi

      Yield strength

      250 Mpa

      36300

      Elongation at break in 200 mm

      20%

      20%

      Elongation at break in 50 mm

      23%

      23%

      Modulus of elasticity

      210 Gpa

      30450 ksi

      Poissons ratio

      0.3

      0.3

      Bulk modulus

      140 Gpa

      20300 ksi

      Shear modulus

      79.3 Gpa

      11500 ksi

      MECHANICAL PROPERTIES

      APPLICATONS

      High performance gear cutting hobs, shapers, milling cutters, Bevel tools, all kinds of highly stressed twist bits & taps, shaped shear blades, for working high strength materials, broaches, cutting tools for roughing and finishing such as helical bits, milling cutters of all types, taps, dies, spindles, reamers, thread cutting tools, drill bits, circular saw segments. Cold forming tools such as dies & punches for cold extrusion

      & fine cutting tools. MACHINE SPECIFICATION

      Type

      Stepped Cone Pulley Engine Lathe

      Power Supply

      3 Phase 440-440 Volts

      Motor

      1 HP 1440 RPM

      No. of Spindle Speeds (without back gear)

      04

      No. of Spindle Speeds (with back gear)

      08

      Size of Machine (L*B*H) in mm

      1400*450*1270

      Swing Over Bed

      165mm

      Max Centre Distance

      620mm

      LAHE MACHINE SPECIFICATON\

      CUTTER SPECIFICATION

      The material we used for cutter is high speed steel. The reason behind we use high speed steel for cutter was these material made cutters can be re sharpened repeatedly throughout the entire life of cutter, without changing the original cutting form.

      CUTTER MATERIAL SPECIFICATION

      It is a high performance steel with high hardness and good wear resistance properties. It is often used as a cutting tool as it can cut metal at very high rate than any other carbon too steel due to presence of tungsten and cobalt that gives it much higher strength.

      CHARACTERISTIC PROPERTIES

      • working hardness

      • excellent toughness

      • high wear resistance

      • high retention of hardness

Alloying Elements Present in HSS Properties

  1. Carbon: Forms carbides, increases wear resistance, is responsible for basic matrix hardness.

  2. Tungsten and molybdenum: Improved red hardness, retention of hardness and high temperature strength of matrix, forms special carbides of great hardness.

  3. Vanadium: Forms special carbides of supreme hardness, increase high temperature wear resistance

  4. Chromium: Promotes depth hardening, produces readily soluble carbides.

  5. Cobalt: Improves red hardness and retention of hardness of the matrix.

    LATHE MACHINE

    Used to cut groove in the mild steel specimen for conduction of required specimen.

    Capacity

    Up to 100KN

    1.

    Maximum force at full speed

    Compatible with full speed

    2.

    Width

    Preferably in the range of 1000- 1200mm

    3.

    Depth

    Preferably in the range of 500- 600mm

    4.

    Height

    Preferably in the range of 1600- 2000mm

    5.

    Total Crosshead travel

    Preferably in the range of 1200- 1400mm

    6.

    Total Vertical Test space

    Preferably in the range of 1200- 1400mm

    7.

    Mechanical test to be carried on UTM

    Must able to perform all mentioned test

    8.

    Accuracy

    ~0.5µm

    9.

    Repeatability

    ~0.25 µm

    10.

    Should be suitable for mechanical testing of following material:

    UTM must be suitable for testing all mentioned materials

    1. Tensile Test

    2. Compressive Test

    3. Ductility Test

    4. Fatigue/Cyclic Test

    5. Flexure/Bending

    6. Shear/Torsion Test

    • Stainless steel

    • Modified stainless steel such as SS 316, SS 316 LN

    • Polymers such FRP, Polyamides, Epoxy esters

    • Different Metals such as Copper, Aluminium alloys etc

    TESTING PROCEDURE

    1. Install the specimen in the UTM machine in tensile test setting

    2. Load the solid steel rod till 1000N

    3. Plot stress strain graph

    4. Find : max and min GRAPH

TECHNICAL SPECIFICATION OF UNIVERSAL TESTING MACHINE

Universal Testing Machine

Universal testing machine is used to conduct the tensile test experiment on the specimen. The specimen with given dimension is clamped between the jaws of UTM and tensile force was applied and results were calculated.

OUTPUT

From Graph,

Max=5.59MPA

Nom=Force/Original area (area before tensile test)

=2.41MPA

Stress concentration factor (K) =Max/Nom

=5.59/2.41

=2.32

RESULT

The stress concentration factor was obtained by a tensile test on mild Steel Specimen with a U-Groove of standard dimension and value was obtained as 2.32.

CONCLUSION

The value obtained from experimental approach is 2.32 and from theoretical approach (from graph and software), it is

    1. So the variation is (2.57-2.32)=0.25 This variation may be due to

      1. Improper gripping of specimen in UTM

      2. Improper taking of readings

      3. Manual errors of considering value in graph

CALCULATION OF STRESS CONCENTRATION FACTOR USING ANSYS FOR ANALYTICAL APPROACH

Ansys software is used for analytical approach. The shaft with groove is designed using Catia V5 software and imported in geometry section of Ansys static structural work bench then the shaft is meshed using triangular meshing of mesh size 0.5mm and mesh refinement is inserted in grooved area then axial force of 1000N is applied on both ends of shaft causing tension in shaft then solver is run and solution is obtained.

GEOMETRY

SHAFT WITH U-GROOVE

Material properties

Mild steel

Density

7850kg/m3

Young modulus

210GPA

Poissons ratio

0.3

Meshing information

Element size

0.5mm

Number of node

146261

Number of element

87648

Loading information

Tensile loading of 1000N in y direction

Boundary condition

Free at both ends

Maximum normal stress in Y direction is calculated;

Max=6.725MPA

Kt was calculated by dividing the max value by Nom

value

These things can be minimized by following ways:

  1. Choosing smaller mesh sizing as possible leads to more accuracy in result operation.

  2. Choosing element with more number of nodes leads to appropriate interpolation.

  3. By applying exact boundary conditions, we can get more accurate results.

USE OF SOFTWARE

Many different types of software are used to determine the stress concentration factor. Following software can be used directly used to directly determine the stress concentration factor:

  1. ALFAK SOFTWARE: A software created by faculty of Mechanical engineering Sarajevo. It is a very simple and easy to use software to determine stress concentration factor.

  2. PETERSONS METHOD: This method is explained in Walter Dr. Pikley publication from Virginia Engineering education.

  3. CATIA V5 AND ANSYS: These FEA software can be used to directly find the value and nature of stress concentration in complex geometries.

  4. STRESS CONCENTRATION FACTOR CALCULATOR: This is a basic and simple method of calculating the stress concentration factor. You just need to select the type of specimen and discontinuity, put the condition like load, diameter, moments, etc.

    Nom= 1000 ((4) ×232) =2.4068 Mpa

    The value of Kt from analytical approach i.e. using Ansys is 2.7944 deviates from actual value 2.575 due to following reason:

    1. Due to improper mesh sizing.

    2. Due to improper elemental shapes and number of nodes.

    3. Due to some in accuracy in boundary condition

    4. Due to consideration of less digits after decimal during calculations.

      CALCULATION OF STRESS CONCENTRATION FACTOR USING MATLAB

      1. Image display

        A figure of the case is displayed on the screen which will aids in better understanding of the parameters entered by the user and would avoid any confusion.Specimen.jpg is the name given to figure which has to be stored in Matlab folder.

        [P,map]=imread('specimen.jpg'); imagesc(P,[min(P(:)) max(P(:))]);colormap(map) axis off

        axis equal

      2. Prompt

        Tis section contains a set of codes that allow the user to input the various parameters that are necessary to compute the stress concentration factor shown in the figure by means of a dialog box that pops up when the program is run.

        prompt={'Enter the diameter of cylinder:D',…. 'Enter the depth of U notch:h',……

        ktheta1=k1+k2*y+k3*y^2+k4*y^3;

        disp('the value of stress concentration factor is:'); disp(ktheta1);

        elseif (x>2&x<=50) k1=0.833+2.069*sqrt(x)-0.009*x; k2=2.732-4.157*sqrt(x)+0.176*x; k3=-8.859+5.327*sqrt(x)-0.320*x; k4=6.294-3.239*sqrt(x)+0.154*x;

        ktheta1=k1+k2*y+k3*y^2+k4*y^3;

        disp('the value of stress concentration factor is:'); disp(ktheta1);

        else

        disp('This code is only valid for 0.1<=h/r<=50') end;

        'Enter the radius of curvature of U notch:r'}; dlg_title='Inputs';

        num_lines=1; answer=inputdlg(prompt,dlg_title,num_lines);

        D=str2double(answer(1,1)); h=str2double(answer(2,1)); r=str2double(answer(3,1));

      3. disp('This code is for elastic stress on a cylindrical section with two U notches')

      disp('under elastic stress and axial tension')

      disp('The output is the factor of stress concentration')

    5. Output

      The output, which is the factor of stress concentration is printed out on the screen.

      Reasons for variation in result of MATLAB with Theoretical value

      • Applied boundary condition may not be exact

d) Logic

This section contains the coding that is required to compute the stress concentration factor. A simple if-else statement is utilized to check the conditions if there is just one condition. If the input parameters do not satisfy the conditions, then This code is not valid for these parameters message is printed.

x=h/r; y=(2*h)/D;

if (x>=0.1&x<=2) k1=0.850+2.628*sqrt(x)-0.413*x; k2=-1.119-4.826*sqrt(x)+2.575*x; k3=3.563-0.514*sqrt(x)-2.402*x; k4=-2.294+2.713*sqrt(x)+0.240*x;

CREATE YOUR OWN SOFTWARE

You can make your own software as per your requirement in order to avoid expensive software. With a good knowledge of programming language, small software application can be developed. The initial approach can be to put the fundamental and relation in stress concentration topic in order to a program than process accordingly. In above method, we have used Matlab to solve a given specific problem. Similarly, that program can be expanded and can be converted into a simple software which can be used to find stress concentration.

Methods used

Values of (kt ) obtained

1. Numerical approach

2.575

2. Experimental approach

2.32

3. Analytical approach (FEA) analysis using

ANSYS (workbencp6)

2.7944

4. Using software (stress concentration factor

calculator )

2.57

5. Mat lab code

2.7999

RESULT COMPARISON

The variation in results calculated from different method and their reasons with remedies are explained above in each consecutive approaches.

BIBLIOGRAPHY

Pilkey, W. (1997), Peterson's Stress Concentration Factors, Wiley, New York.

ANSYS Workbench 16, ANSYS, Inc., www.ansys.com Machine Design Data Hand Book By Dr.K. Lingaiah . William J Palm III ,(2007) "A Concise Introduction to Matlab".

A Muminovic, I.Saric, and E.Mesic, CAD, Sarajevo:Faculty of Mechanical Engineering Sarajevo,2012.

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