Valentini Bernhard

Smart-Stress V1.00Copyright by Urich Christian and Valentini Bernhard

Calculation routine for stress, strain and elastic materialtensors, yield and failure functions of different yield andfailure hypothesis on TI89, TI89-Titanium, TI92+, V200

IContents

1 Introduction 1

2 Features 1

3 Requirements 1

4 Recommendations 1

5 Files 2

6 Installation 2

7 Starting the program 2

8 General Notes 2

9 Notes and warnings 3

10 Menu structure 4

11 File 411.1 New . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411.2 Save . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411.3 Open . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411.4 Exit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

12 Create 512.1 Stress tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

12.1.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 512.1.2 Input . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

12.2 Strain tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . 712.2.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 712.2.2 Input . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

12.3 Material tensor . . . . . . . . . . . . . . . . . . . . . . . . . . 812.3.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 812.3.2 Input . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

12.4 Yield hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . 1012.4.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 1012.4.2 Input . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

13 Edit 1413.1 List . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1413.2 Edit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1413.3 Delete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

II

14 Calculation 1514.1 Calculate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1514.2 Calculate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

15 Results 1615.1 Stress tabular . . . . . . . . . . . . . . . . . . . . . . . . . . . 1615.2 Mohrs circle . . . . . . . . . . . . . . . . . . . . . . . . . . . 1615.3 Strain tabular . . . . . . . . . . . . . . . . . . . . . . . . . . . 1715.4 Material tensor tabular . . . . . . . . . . . . . . . . . . . . . 1715.5 Yield hypothesis tabular . . . . . . . . . . . . . . . . . . . . . 17

16 Info 1816.1 Help . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1816.2 About . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

17 Developer 19

18 Thanks 19

19 History 20

1 INTRODUCTION 1

1 Introduction

Smart-Stress is a program for TI89, TI89-Titanium, TI92+ and V200 whichcalculates stress, strain and elastic material tensors, yield and failure func-tions of different yield and failure hypothesis. It provides graphic input/outputand a tabular view of the results.

2 Features

calculation of different elastic material parameters yield hypothesis: Tresca, von Mises,Mohr-Coulomb andDrucker-Prager

failure hypothesis: Rankine Mohrs circle graphic input via dialog boxes and graphical or tabular output of the result scroll and zoomable graphics scroll and zoomable tables calculated values:

hydrostatic and deviatory stress and strain tensor

principal stress and strain values and vectors

principal shear stress values and stress components normal tothem

yield and failure functions

3 Requirements

TI89, TI89-Titanium, TI92+ or V200 with AMS-Version 2.05 or higher 60kB free RAM Program HW3Patch to run the program on TI89-Titanium(You can download it from Kevin Koflers homepagehttp://kevinkofler.cjb.net.)

4 Recommendations

ProgramAuto Alpha-Lock Off from Kevin Kofler (http://kevinkofler.cjb.net)only for TI89 and TI89-Titanium

5 FILES 2

5 Files

Smart-Stress contains following files (for TI89/TI89-Titanium, TI92+, V200):

1. launcher:

smartstr.(89z, 9xz, v2z)2. dll:

sstress.(89y, 9xy, v2y)3. text file:

sstrhelp.(89t, 9xt, v2t)

6 Installation

Transfer all files of you calculator type via link cable to your calculator andarchive following files:

1. sstr.dll

2. sstrhelp.text

- thats all. All files must be in one folder.

When you have a TI89-Titanium install the program HW3Patch on yourcalculator.

When you have a TI89 or TI89-Titanium you can install the program AutoAlpha-Lock Off to avoid pressing the alpha button every time you makean input.

7 Starting the program

Write in the command line of the TI-application Home the expressionsmartstr().

8 General Notes

The handling of the program is made as easy as possible, so the input of alltensors and parameters can be done very quickly.

Negative numbers you have to input with the (-) sign next to the .at the bottom of the numeric block. You should never make an input with

9 NOTES AND WARNINGS 3

the sign minus -.

NO INPUT IS AUTOMATICALLY SET 0.

9 Notes and warnings

This program is distributed to help students of civil engineering and othertechnical fields, but WITHOUT ANY WARRANTY. (The authors make norepresentations or warranties about the suitability of the software, eitherexpress or implied. The authors are not liable for any damages suffered as aresult of using or distributing this software.) Every kind of commercial useis forbidden without the permission of the authors.

Certainly there are several bugs within the program. For this reason itsuseful to make a backup of your calculator before using it.

Wrong operation can lead to a complete crash of the calculators systemwhich can only be repaired with a reset (on+2nd+hand). The consequenceis that all data on your calculator which is not archived could be deleted.

Therefore you should be careful, especially at the start of using this pro-gram.

If you have comments, bug reports or anything else, email Valentini Bern-hard (bernhard.valentini@smart-programs.org) or visit the forum on our website http://www.smart-programs.org/.

10 MENU STRUCTURE 4

10 Menu structure

Table 1: Menu structure

11 File

11.1 New

Clears all data (stress, strain and elastic tensors, yield parameters and re-sults).

11.2 Save

The name of the savefile cant have more signs than eight.All savefiles have the ending .sstr.

11.3 Open

The program searches for all files with the ending .sstr.

11.4 Exit

Exits the program.

12 CREATE 5

12 Create

The available data is shown on the screen. So it should be easy to keep thesurvey of the input stress, strain and material tensors and the input yieldfunction.For the stress and strain tensor their state (three-dimensional or plane stressstate) is also output.

12.1 Stress tensor

12.1.1 Theory

Global system of coordinates (right hand system) with the positive direc-tions of the stress tensor:

Figure 1: Positive directions of the stress tensor

The symmetric stress tensor ij can be split up into hydrostatic hydand a deviatory part dev. 11 12 1321 22 2331 32 33

= m 0 00 m 0

0 0 m

hydrostatic

+

11 m 12 1321 22 m 2331 32 33 m

deviatory

(1)

or in index typing

ij = mij hydrostatic

+ sijdeviatory

, i, j = 1, 2, 3 (2)

12 CREATE 6

with

m =13(11 + 22 + 33) =

13I1 (3)

The unit of the stress tensor and its components is [N/mm2].

The principal stress components can be found by solving the eigenvalueproblem

|ij ij | = 0 (4)

3 + I1 2 I2 + I3 = 0 (5)

with the invariants of the stress tensor

I1 = 11 + 22 + 33,

I2 = 22 2332 33

+ 11 1331 33+ 11 1221 22

, (6)I3 =

11 12 1321 22 2331 32 33

.The principal stress components are sorted from biggest to smallest.

1 2 3 (7)

The eigenvector to each eigenvalue shows the direction of the principal stresscomponents in the global coordinate system.

The three principal shear stress components are

1 =2 32

, 2 = 3 12 , 3 = 1 22

. (8)Normally the planes with principal shear stress arent free from stress normalto these planes. So these values are

(1)nn =2 + 3

2, (2)nn =

3 + 12

, (3)nn =1 + 2

2. (9)

12.1.2 Input

You have to input the components of upper triangular stress tensor. Thelower one is symmetric to the upper one, so you share time during the input.The unit of the stress tensor is [N/mm2].

12 CREATE 7

The program itself scans the stress tensor whether its a three-dimensionalor a plane stress state. For example a plane stress state in the x1/x2-planeis specified by

33 = 31 = 32 = 0. (10)

12.2 Strain tensor

12.2.1 Theory

The positive directions of the strain tensor are the same as in the stresstensor, see section 1 on page 5.

The symmetric strain tensor ij can be split up into volumetric vol anda deviatory part dev. 11 12 1321 22 23

31 32 33

= m 0 00 m 0

0 0 m

volumetric

+

11 m 12 1321 22 m 2331 32 33 m

deviatory

(11)

or in index typing

ij = mij volumetric

+ eijdeviatory

, i, j = 1, 2, 3 (12)

with

m =13(11 + 22 + 33) =

13vol =

13I1 (13)

The unit of the strain tensor and its components is [-].

The principal strain components can be found by solving the eigenvalueproblem

|ij ij | = 0 (14)

3 + I12 I2+ I3 = 0 (15)

12 CREATE 8

with the invariants of the strain tensor

I1 = 11 + 22 + 33,

I2 = 22 2332 33

+ 11 1331 33+ 11 1221 22

, (16)I3 =

11 12 1321 22 2331 32 33

.The principal strain components are sorted from biggest to smallest.

1 2 3 (17)

The eigenvector to each eigenvalue shows the direction of the principal straincomponents in the global coordinate system.

12.2.2 Input

You have to input the components of upper triangular strain tensor. Thelower one is symmetric to the upper one, so you share time during the input.The unit of the strain tensor is [-].

The program itself scans the strain tensor whether its a three-dimensionalor a plane strain state. For example a plane strain state in the x1/x2-planeis specified by

33 = 31 = 32 = 0. (18)

12.3 Material tensor

12.3.1 Theory

From Hookes law

ij = Cijklkl (19)

we get the elastic or material stiffness tensor C with unit [N/mm2]. Thematerial stiffness tensor is symmetric with

Cijkl = Cjikl,Cijkl = Cijlk. (20)

12 CREATE 9

The inverse of the material stiffness tensor is also symmetric and calledmaterial compliance tensor

D = C1 (21)

with unit [mm2/N ].

The material tensor can be described with following material parameters:

1. Elastic modulus: E

2. Poisson ratio:

0 0.5 (22)

3. Shear modulus:

G =E

2(1 + )(23)

4. Compression modulus:

K =E

3(1 2) (24)

5. Lames material parameter:

=E

(1 + )(1 2) (25)

For isotropic materials Hookes law is

112233212223231

=

1E E E 0 0 0

1E E 0 0 0 E 1E 0 0 0

1G 0 0

1G 0

symm. 1G

D=C1

112233122331

(26)

or in index typing

ij =1 + E

ij Eijkk. (27)

12 CREATE 10

The inverse of this equation is

112233122331

=E(1 )

(1 + )(1 2)

1 1

1 0 0 01 1 0 0 0

1 0 0 0122(1) 0 0

122(1) 0

symm. 122(1)

C

112233212223231

(28)

or in index typing

ij =E

1 +

(ij +

1 2 kkij), (29)

ij = Kvolij + 2Geij , (30)ij = volij + 2Gij . (31)

12.3.2 Input

In the program you can decide which parameters you input for the isotropicmaterial tensors. Following combinations are available:

E [N/mm2] and [-] E [N/mm2] and G [N/mm2] G [N/mm2] and [-] G [N/mm2] and K [N/mm2] G [N/mm2] and [N/mm2]

12.4 Yield hypothesis

12.4.1 Theory

Four yield and one failure hypothesis have been implemented in the programto specify the mechanic criterion of the material in a specific point.

yield hypothesis: Tresca

von Mises

Mohr-Coulomb

Drucker-Prager

failure hypothesis:

12 CREATE 11

Rankine

1. Tresca yield hypothesis:

The maximal shear stress is decisive for yield in a specific point ofthe material.The Tresca yield hypothesis needs only one material parameter fy(yield stress).The yield function is

f() = max min fy. (32)

2. von Mises yield hypothesis:

The energy of distortion is decisive for yield in a specific point ofthe material.The von Mises yield hypothesis also needs only one material para-meter fy (yield stress).The yield function is

f() =ijij

23fy (33)

or

f() = okt 23y (34)

with

okt =13

(1 2)2 + (2 3)2 + (3 1)2, (35)

y =fy3. (36)

3. Mohr-Coulomb yield hypothesis:

The maximal shear stress is decisive for yield in a specific point ofthe material. But according to Coulombs law of friction the shearstress nt depends on the stress nn normal to this plane.

|nt| = f(nn) (37)The Mohr-Coulomb yield hypothesis needs two material parameterc (cohesion) and (friction angle).The yield function is

f() =maxft min

fc 1 (38)

12 CREATE 12

with

ft =2c cos1 + sin

, (39)

fc =2c cos1 sin. (40)

4. Drucker-Prager yield hypothesis:

This yield hypothesis is an extension to the von Mises yield hy-pothesis. The maximal octahedron shear stress is decisive for yieldin a specific point of the material. But according to Coulombs lawof friction the octahedron shear stress okt depends on the stress oktnormal to this plane.The Drucker-Prager yield hypothesis also needs two material pa-rameter (friction coefficient) and fy (yield stress).The yield function is

f() = okt + okt 23y (41)

or

f() =ijij +

3I1

2y (42)

with

okt =13(1 + 2 + 3) = m =

13I1 , (43)

okt =23y. (44)

5. Rankine failure hypothesis:

The maximal stress is decisive for failure in a specific point of thematerial.The Rankine failure hypothesis needs one material parameter for ten-sion ft (yield tension) and one for compression fc (yield compression).The yield or failure function is for tensile

f() = 1 ft, 1 > 0, (45)

and compression

f() = |3| fc, 3 < 0. (46)

12 CREATE 13

12.4.2 Input

First you have to choose which yield or failure hypothesis you want to use.Then you have to input the material parameters for the chosen hypothesis.

Tresca:fy [N/mm2]

von Mises:fy [N/mm2]

Mohr-Coulomb:c [N/mm2] and [rad]

Drucker-Prager:fy [N/mm2] and [-]

Rankine:ft [N/mm2] and fc [N/mm2]

13 EDIT 14

13 Edit

13.1 List

The input stress, strain and material tensor and yield parameters are listedthere.

The tabular outputs are scroll and zoomable.

Controls:

... scroll the table in one of these directions +/ . . . zoom the table (and font) 2nd . . . move to row number one Esc . . . cancel the table

13.2 Edit

You can edit the input stress, strain and material tensor and yield parame-ters with this routine.

13.3 Delete

You can delete all input data with this routine.

14 CALCULATION 15

14 Calculation

14.1 Calculate

When you have input the strain and material tensor this routine will calcu-late the stress tensor with Hookes law

ij = Cijklkl. (47)

14.2 Calculate

When you have input the stress and material tensor this routine will calcu-late the strain tensor with

ij = Dijklkl. (48)

15 RESULTS 16

15 Results

All tabular outputs are scroll and zoomable. The controls are described insection 13.1.

When you enter the graphic outputs you can scroll and zoom the draw-ing area. Only in the graphic and numeric output four color grayscale isturned on.

Controls:

. . . scroll in one of these directions +/ . . . zoom in and out 2nd . . . center the graphic to the area Esc . . . cancel the graphic mode

15.1 Stress tabular

The stress tensor components will be calculated like in section 12.1.1 shown.

Following results are output:

stress tensors: . . . stress tensor [N/mm2]

hyd . . . hydrostatic stress tensor [N/mm2]

dev . . . deviatory stress tensor [N/mm2]

1, 2, 3 . . . principal stresses and their eigenvectors [N/mm2] 1, 2, 3 . . . principal shear stresses [N/mm2] with(1), (2), (3) . . . stress components normal to the principal shearstress planes [N/mm2]

15.2 Mohrs circle

The Mohrs circles with its main components

1, 2, 3 . . . principal stresses [N/mm2] 1, 2, 3 . . . principal shear stresses [N/mm2]

is shown there.

15 RESULTS 17

15.3 Strain tabular

The strain tensor components will be calculated like in section 12.2.1 shown.

Following results are output:

strain tensors: . . . strain tensor [-]

vol . . . volumetric strain tensor [-]

dev . . . deviatory strain tensor [-]

1, 2, 3 . . . principal strains and their eigenvectors [-]

15.4 Material tensor tabular

The material tensor components will be calculated like in section 12.3.1shown.

Following results are output:

material tensors components: E . . . elastic modulus [N/mm2]

. . . poisson ratio [-]

G . . . shear modulus [N/mm2]

K . . . compression modulus [N/mm2]

. . . Lames material parameter [N/mm2]

material tensor: C . . . elastic or material stiffness tensor [N/mm2]

D . . . material compliance tensor [mm2/N ]

15.5 Yield hypothesis tabular

The yield function will be calculated like in section 12.4.1 shown.

Following results are output:

input yield hypothesis and parameters f() . . . yield function [N/mm2]

16 INFO 18

16 Info

16.1 Help

Shows a list of short cuts used in sections 15.1, 15.3 and 15.4.Also shows the control-keys for the graphic mode (section 15) and tables(section 13.1).

16.2 About

Prints some information about version and developer of the program andaddress of the homepage.

17 DEVELOPER 19

17 Developer

Valentini Bernhard (bernhard.valentini@smart-programs.org)

Web site: http://www.smart-programs.org/

18 Thanks

the TIGCC Team for making it possible to program in C (http://ticalc.ticalc.org) the TICT (TI-Chess Team) for their ExtGraph library (http://tict.ticalc.org) Christof Neuhauser for mathematic support

19 HISTORY 20

19 History

Smart-Section V1.00: 24.08.2005 - first release

new, save and load function added distinction between three-dimensional and plane stress andstrain state implemented Mohrs circle improved help file added

Smart-Section V0.10 Beta 2: 14.08.2005 - update

strain and material tensor implemented list, edit and delete functions implemented calculation routine for stress and strain implemented tabular output of strain and material tensor results imple-mented Mohrs circle improved manual added

Smart-Stress V0.10 Beta 1: 05.08.2005 - first beta release

stress tensor implemented five yield hypothesis implemented Mohrs circle implemented tabular output of stress and yield results implemented

REFERENCES 21

References

[Mang/Hofstetter (2000)] H. Mang and G. Hofstetter, Festigkeitslehre,Springer Verlag, Wien, 2000.

List of Figures

1 Positive directions of the stress tensor . . . . . . . . . . . . . 5

List of Tables

1 Menu structure . . . . . . . . . . . . . . . . . . . . . . . . . . 4

IntroductionFeaturesRequirementsRecommendationsFilesInstallationStarting the programGeneral NotesNotes and warningsMenu structureFileNewSaveOpenExit

CreateStress tensorTheoryInput

Strain tensorTheoryInput

Material tensorTheoryInput

Yield hypothesisTheoryInput

EditListEditDelete

CalculationCalculate Calculate

ResultsStress tabularMohr's circleStrain tabularMaterial tensor tabularYield hypothesis tabular

InfoHelpAbout

DeveloperThanksHistory