ELEMENT IRREDUCIBLE TYPE H20P0 command · ELEMENT IRREDUCIBLE TYPE H20P0 command. ... speci c problem considered is the bending of a cantilever beam loaded by a ... is generated using

APES documentation (revision date March 14, 2011)

.

ELEMENT IRREDUCIBLE TYPE H20P0 command.

Synopsis

The ELEMENT IRREDUCIBLE H20P0 command is used to describe all irreducible 20-node quadratic, hexahedral continuum elements that are to be used in mechanical analyses.

Syntax

The following syntax is used to describe a typical H20P0 irreducible hexahedral continuumelement:

.

ELEment IRReducible TYPe H20P0 NODes #:#:#(MATerial #) (INItial #) (INTcode #)

(CONstruction #) (EXCavation #)(DONT PRINT Results)

(DONT PRINT STRAins) (DONT PRINT STREsses)(PRINT AVG STRAins) (PRINT AVG STREsses)

(PRINT PRIN STRAins) (PRINT PRIN STREsses)(PRINT VOLUMETRIC STRAIN) (PRINT AVG VOLumetric strain)

.

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Explanatory Notes

• The H20P0 is an irreducible, quadratic, isoparametric “Serendipity” hexahedral continuumelement [1]. The element

– Has six (6) faces.

– Contains eight (8) vertex nodes.

– Contains twelve (12) mid-side nodes.

– Has three (3) displacements degrees of freedom at each node.

– Possesses a total of sixty (60) displacement degrees of freedom.

• The numbering order of NODES associated with H20P0 elements, which must be specifiedsequentially from 1 to 20, is shown in Figure 1.

NOTE: Presently APES does not possess the ability to generate H20P0 elements. It isassumed that the analyst will thus use some stand-alone pre-processing software to accomplishthis task. The resulting element and node data will then be translated to the format expectedby APES.

9

16

3

2

1

4

5

6

7

8

10

11

12

13

14

15

17

18

19

20

x1

x2

x3

Figure 1: Node Numbering Associated with a Typical Irreducible 20-Node Quadratic (H20P0)Hexahedral Continuum Element

• The MATERIAL keyword is used to specify the number of the material idealization associ-ated with the element. The default values for the MATERIAL number is one (1).

• The INITIAL keyword is used to specify the initial state number associated with the element.The default value for the INITIAL is zero (0).

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• The value specified in conjunction with the INTCODE keyword describes the order of nu-merical integration scheme to be used in developing the element equations for the element.

The “commonly” used numerical integration rule for H8P0 elements corresponds to a 2 by2 by 2 Gauss-Legendre quadrature scheme (degree of precision equal to 3) for the primarydependent variables (i.e., nodal displacements) and a 1-point Gauss-Legendre scheme (degreeof precision equal to 1) for the secondary dependent variables (i.e., strains and stresses). Thisis the default condition and requires no input using the INTCODE keyword. If a quadratureorder different from the default condition is desired, the following integer values are associatedwith this keyword:

The “commonly” used numerical integration rule for H20P0 elements corresponds to a 3 by3 by 3 Gauss-Legendre quadrature scheme (degree of precision equal to 5) for the primarydependent variables (i.e., nodal displacements) and a 2 by 2 by 2 Gauss-Legendre scheme (de-gree of precision equal to 3) for the secondary dependent variables (i.e., strains and stresses).This is the default condition and requires no input using the INTCODE keyword.

If a quadrature order different from the default condition is desired, the following integervalues are associated with this keyword:

INTCODE = 32: a 3 by 3 by 3 Gauss-Legendre quadrature scheme (degree of precisionequal to 5) is used to compute the primary dependent variables (i.e., nodal displacements)and a 2 by 2 by 2 Gauss-Legendre scheme (degree of precision equal to 3) is used tocompute the secondary dependent variables (i.e., strains and stresses). This is equivalentto the aforementioned default setting.

INTCODE = 21: A 2 by 2 by 2 Gauss-Legendre quadrature scheme (degree of precisionequal to 3) is used for the primary dependent variables (i.e., nodal displacements) and a1-point Gauss-Legendre scheme for the secondary dependent variables (i.e., strains andstresses).

• The incremental CONSTRUCTION and EXCAVATION numbers represent the timeincrement in which the material in this element(s) is added to or removed from the model.A CONSTRUCTION number equal to zero corresponds to a material in existence at thebeginning of the analysis. Since this is the default condition, no input is required in such acase. The condition of no excavation is likewise the default.

• The purpose of the PRINT commands is to eliminate unnecessary output generated byAPES. More precisely, if the time history of strains and/or stresses is desired only for aselect few elements, this option greatly speeds program output and facilitates inspection ofresults by the user. Information associated with the elements specified in this section willbe printed for every solution (time) step. If generation is performed using this ELEMENTIRREDUCIBLE command, then all the elements generated will be affected in a like mannerby the above print control commands.

• Specification of the keyword DONT PRINT Results indicates that the analyst does notdesire to see output of secondary dependent variables (i.e., strains and stresses) for this ele-ment.

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• Specification of the DONT PRINT STRAINS keyword indicates that element strains arenot to be printed. Under the default condition both strains are printed.

• Specification of the keyword DONT PRINT STRESSES indicates that stresses are not tobe printed. Under the default condition stresses are printed.

• The PRINT PRIN STRAINS keyword indicates that principal strains are to be computedand printed for the element. Under the default condition these quantities are not computedand printed.

• The PRINT PRIN STRESSES keyword indicates that principal stresses are to be com-puted and printed for the element. Under the default condition these quantities are notcomputed and printed.

• The PRINT AVG STRAINS keyword indicates that average strains (averaged over thesecondary quadrature points) are to be computed and printed for the element. Under thedefault condition average strains are not computed and printed.

• The PRINT AVG STRESSES keyword indicates that average stresses (averaged over thesecondary quadrature points) are to be computed and printed for the element. Under thedefault condition average stresses are not computed and printed.

• The keyword PRINT VOLUMETRIC STRAIN causes the volumetric strain to be com-puted and printed for the element. In addition, the ratio of the absolute value of the volumetricstrain to the absolute value of the minimum non-zero normal strain in the element is printed.That is,

|εvol||min (ε11, ε22, ε33)|

; min (ε11, ε22, ε33) 6= 0

This ratio is instructive in the assessment of mixed and mixed/penalty elements used tosimulate material response in the incompressible limit. As such, this keyword would likely notbe used in conjunction with the H20P0 element. Under the default condition the volumetricstrain and the aforementioned ratio are not computed and printed.

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Example of Command Usage

Simulation of Three-Dimensional Beam Bending

A three-dimensional simulation of beam bending is investigated in this sample analysis. Thespecific problem considered is the bending of a cantilever beam loaded by a concentrated forceapplied at its free end in the manner shown in Figure 2. The beam has a span (L) of 100 inches,a depth (d) of 4 inches, and a unit thickness (b). The beam is loaded by a concentrated force ofP = 1, 000 applied at its free end. The material is idealized as being isotropic linear elastic. Theelastic modulus (E) is equal to 30 x 106, and a Poisson’s ratio (ν) equal to 0.30 is assumed.1

x

y

z

P

L

d

b

Figure 2: Physical Problem Analyzed: Cantilever Beam Subjected to Concentrated Load Appliedat Free End

Since the span-to-depth ratio for the beam is relatively large (25:1), the contribution of sheardeformations to the overall transverse displacement will be quite small. As such, the Bernoulli-Eulersolution for the transverse displacement will be accurate. In particular, the transverse displacementof the free end (vmax) is PL3/3EI. Noting that the moment of inertia for bending about the z-axisis equal to I = (1)(4)3/12 = 5.333, it follows that vmax is equal to -2.083 inches.

In accordance with the Bernoulli-Euler beam theory, the axial stress varies linearly with thetransverse coordinate (y); viz., σ11 = −My/I. The largest axial stresses occur at the fixed end(x = 0), where the bending moment is M = PL. Thus, σ11(0) = −PLy/I = −18, 750y. Themaximum axial stress occurs at outer fibers (y = ±2.0). The magnitude of the maximum axialstress is thus σ11 max = σ11(0,±2) = 37, 500 psi.

A mesh 5 elements (in the x-direction), by 4 elements (in the y-direction), by 1 element (in thez-direction) is generated using the PATRAN program [3] and is subsequently translated to APES

1This example is taken from Logan [2], who analyzed the beam using the Algor computer program.

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format. The input data file associated with the analysis is given below.

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ana tit "cantilever beam involving 10 x 4 x 1 mesh of H20P0 elements"

ana tit "Example 12.2 from Logan Algor/FE book (1997)"

ana tit "5 (in x-dir) by 2 (in y-dir) by 1 (in z-dir) mesh"

ana tit "PATRAN generated"

!

anal act analyze

anal type mechanical

anal description linear

anal idealization three-dimensional

anal temporal transient

!

echo init off

echo grav off

echo elements off

echo nodes off

echo warn off

!

dim max material isotropic elastic 1

dim max nodes 108

dim max h20p0 10

!

FINISHED SETTINGS

!

mat elastic isotropic number 1 desc "sample isotropic material" &

modulus 30.0e+06 poissons 0.30

!

NODES LINE NUMBER 1 x1 0.000000000E+00 x2 2.000000000E+00 x3 -5.000000000E-01

NODES LINE NUMBER 2 x1 0.000000000E+00 x2 2.000000000E+00 x3 0.000000000E+00

NODES LINE NUMBER 3 x1 0.000000000E+00 x2 2.000000000E+00 x3 5.000000000E-01






NODES LINE NUMBER 9 x1 0.000000000E+00 x2 -1.000000000E+00 x3 -5.000000000E-01

NODES LINE NUMBER 10 x1 0.000000000E+00 x2 -1.000000000E+00 x3 5.000000000E-01


NODES LINE NUMBER 12 x1 0.000000000E+00 x2 -2.000000000E+00 x3 0.000000000E+00







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!

ELEM IRREDUCIBLE TYPE H20P0 MAT 1 &

NODES 1 3 8 6 20 22 27 25 2 5 &

7 4 21 24 26 23 14 15 17 16


NODES 6 8 13 11 25 27 32 30 7 10 &

12 9 26 29 31 28 16 17 19 18


NODES 20 22 27 25 39 41 46 44 21 24 &

26 23 40 43 45 42 33 34 36 35


NODES 25 27 32 30 44 46 51 49 26 29 &

31 28 45 48 50 47 35 36 38 37


NODES 39 41 46 44 58 60 65 63 40 43 &

45 42 59 62 64 61 52 53 55 54


NODES 44 46 51 49 63 65 70 68 45 48 &

50 47 64 67 69 66 54 55 57 56


NODES 58 60 65 63 77 79 84 82 59 62 &

64 61 78 81 83 80 71 72 74 73


NODES 63 65 70 68 82 84 89 87 64 67 &

69 66 83 86 88 85 73 74 76 75


NODES 77 79 84 82 96 98 103 101 78 81 &

83 80 97 100 102 99 90 91 93 92


NODES 82 84 89 87 101 103 108 106 83 86 &

88 85 102 105 107 104 92 93 95 94

!

spe conc mec nod 1:13:1 1_dis 2_dis 3_dis

spe conc mec nod 96:98:2 2_forc 2_value -500.0 2_hist 0

!

finish data

!

solution time final 1.0 increments 1 output 1:10:1

!

finished loading

The results shown below are obtained using the above data in conjunction with the APEScomputer program. For clarity, the “header” that is printed at the top of the file is omitted fromthis file. Also note that the ECHO ELEMENTS OFF and ECHO NODES OFF commands have

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been used so as to reduce the size of the output file file.

cantilever beam involving 10 x 4 x 1 mesh of H20P0 elements

Example 12.2 from Logan Algor/FE book (1997)

5 (in x-dir) by 2 (in y-dir) by 1 (in z-dir) mesh

PATRAN generated

======================================================================

| D Y N A M I C S T O R A G E A L L O C A T I O N |

======================================================================

Largest NODE number which can used in the mesh = 108

Max. no. of ISOTROPIC, LINEAR ELASTIC materials = 1

Max. no. of 20-node hexahedral (H20P0) elements = 10

======================================================================

= G E N E R A L A N A L Y S I S I N F O R M A T I O N =

======================================================================

--> MECHANICAL analysis shall be performed

--> Fluid flow is NOT accounted for in the analysis

--> Thermal effects are NOT accounted for in analysis

--> THREE-DIMENSIONAL solution domain assumed

--> Nodal coordinates will NOT be updated

--> solver type used: SKYLINE

--> storage type: SYMMETRIC

--> "Isoparametric" scheme used for native mesh generation (if applicable)

======================================================================

= I N T E G R A T I O N O P T I O N S =

======================================================================

In approximating time derivatives, the value of "THETA" = 6.667E-01

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======================================================================

= N O N L I N E A R A N A L Y S I S I N F O R M A T I O N =

======================================================================

--> LINEAR analysis

======================================================================

= H I S T O R Y F U N C T I O N I N F O R M A T I O N =

======================================================================

<<< NONE >>>

======================================================================

= M A T E R I A L I D E A L I Z A T I O N S =

======================================================================

--> Material number: 1

~~~~~~~~~~~~~~~

type : isotropic linear elastic

info. : sample isotropic material

Modulus of Elasticity = 3.000E+07

Poisson’s ratio = 3.000E-01

Elastic bulk modulus of the solid phase = 0.000E+00

Material density of the solid phase = 0.000E+00

Combined bulk modulus for solid/fluid = 0.000E+00

======================================================================

= N O D E P O I N T S P E C I F I C A T I O N S =

======================================================================

Node ( c o o r d i n a t e s )

Number s p e c i f i c a t i o n:

~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~

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1 : ( x1 = 0.000E+00, x2 = 2.000E+00, x3 = -5.000E-01 )

displacement-1 = 0.000E+00 ; history no. = -2



2 : ( x1 = 0.000E+00, x2 = 2.000E+00, x3 = 0.000E+00 )




3 : ( x1 = 0.000E+00, x2 = 2.000E+00, x3 = 5.000E-01 )




4 : ( x1 = 0.000E+00, x2 = 1.000E+00, x3 = -5.000E-01 )




5 : ( x1 = 0.000E+00, x2 = 1.000E+00, x3 = 5.000E-01 )




6 : ( x1 = 0.000E+00, x2 = 0.000E+00, x3 = -5.000E-01 )




7 : ( x1 = 0.000E+00, x2 = 0.000E+00, x3 = 0.000E+00 )




8 : ( x1 = 0.000E+00, x2 = 0.000E+00, x3 = 5.000E-01 )




9 : ( x1 = 0.000E+00, x2 = -1.000E+00, x3 = -5.000E-01 )




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10 : ( x1 = 0.000E+00, x2 = -1.000E+00, x3 = 5.000E-01 )




11 : ( x1 = 0.000E+00, x2 = -2.000E+00, x3 = -5.000E-01 )




12 : ( x1 = 0.000E+00, x2 = -2.000E+00, x3 = 0.000E+00 )




13 : ( x1 = 0.000E+00, x2 = -2.000E+00, x3 = 5.000E-01 )




96 : ( x1 = 1.000E+02, x2 = 2.000E+00, x3 = -5.000E-01 )

force-1 = 0.000E+00 ; history no. = -2

force-2 = -5.000E+02 ; history no. = 0


98 : ( x1 = 1.000E+02, x2 = 2.000E+00, x3 = 5.000E-01 )


force-2 = -5.000E+02 ; history no. = 0


~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

end of mathematical model data

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

At time 1.000E+00 (step no. 1): NO iteration was required

======================================================================

= E L E M E N T S T R A I N S & S T R E S S E S =

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======================================================================

--> element 1 ( type = H20P0 ):

.................................

@(x1 = 4.226E+00, x2 = 1.577E+00, x3 = -2.887E-01):

eps_11 = 8.457E-04 ; eps_22 = -1.304E-04 ; eps_33 = -1.841E-04

gam_12 = -3.107E-05 ; gam_13 = 1.003E-05 ; gam_23 = 2.172E-05

sig_11 = 2.871E+04 ; sig_22 = 6.186E+03 ; sig_33 = 4.946E+03

sig_12 = -3.585E+02 ; sig_13 = 1.157E+02 ; sig_23 = 2.507E+02

@(x1 = 4.226E+00, x2 = 1.577E+00, x3 = 2.887E-01):

eps_11 = 8.457E-04 ; eps_22 = -1.304E-04 ; eps_33 = -1.841E-04

gam_12 = -3.107E-05 ; gam_13 = -1.003E-05 ; gam_23 = -2.172E-05

sig_11 = 2.871E+04 ; sig_22 = 6.186E+03 ; sig_33 = 4.946E+03

sig_12 = -3.585E+02 ; sig_13 = -1.157E+02 ; sig_23 = -2.507E+02

@(x1 = 4.226E+00, x2 = 4.226E-01, x3 = 2.887E-01):

eps_11 = 2.264E-04 ; eps_22 = -8.247E-05 ; eps_33 = -4.933E-05

gam_12 = -1.226E-05 ; gam_13 = -2.686E-06 ; gam_23 = -2.172E-05

sig_11 = 6.862E+03 ; sig_22 = -2.657E+02 ; sig_33 = 4.990E+02

sig_12 = -1.415E+02 ; sig_13 = -3.100E+01 ; sig_23 = -2.506E+02

@(x1 = 4.226E+00, x2 = 4.226E-01, x3 = -2.887E-01):

eps_11 = 2.264E-04 ; eps_22 = -8.247E-05 ; eps_33 = -4.933E-05

gam_12 = -1.226E-05 ; gam_13 = 2.686E-06 ; gam_23 = 2.172E-05

sig_11 = 6.862E+03 ; sig_22 = -2.657E+02 ; sig_33 = 4.990E+02

sig_12 = -1.415E+02 ; sig_13 = 3.100E+01 ; sig_23 = 2.506E+02

@(x1 = 1.577E+01, x2 = 1.577E+00, x3 = -2.887E-01):

eps_11 = 8.519E-04 ; eps_22 = -2.675E-04 ; eps_33 = -2.870E-04

gam_12 = -9.031E-06 ; gam_13 = -4.259E-06 ; gam_23 = 7.851E-06

sig_11 = 2.481E+04 ; sig_22 = -1.026E+03 ; sig_33 = -1.476E+03

sig_12 = -1.042E+02 ; sig_13 = -4.914E+01 ; sig_23 = 9.059E+01

@(x1 = 1.577E+01, x2 = 1.577E+00, x3 = 2.887E-01):

eps_11 = 8.519E-04 ; eps_22 = -2.675E-04 ; eps_33 = -2.870E-04

gam_12 = -9.031E-06 ; gam_13 = 4.259E-06 ; gam_23 = -7.851E-06

sig_11 = 2.481E+04 ; sig_22 = -1.026E+03 ; sig_33 = -1.476E+03

sig_12 = -1.042E+02 ; sig_13 = 4.914E+01 ; sig_23 = -9.059E+01

@(x1 = 1.577E+01, x2 = 4.226E-01, x3 = 2.887E-01):

eps_11 = 2.281E-04 ; eps_22 = -8.875E-05 ; eps_33 = -7.687E-05

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gam_12 = -3.431E-05 ; gam_13 = 1.141E-06 ; gam_23 = -7.830E-06

sig_11 = 6.343E+03 ; sig_22 = -9.677E+02 ; sig_33 = -6.935E+02

sig_12 = -3.958E+02 ; sig_13 = 1.317E+01 ; sig_23 = -9.034E+01

@(x1 = 1.577E+01, x2 = 4.226E-01, x3 = -2.887E-01):

eps_11 = 2.281E-04 ; eps_22 = -8.875E-05 ; eps_33 = -7.687E-05

gam_12 = -3.431E-05 ; gam_13 = -1.141E-06 ; gam_23 = 7.830E-06

sig_11 = 6.343E+03 ; sig_22 = -9.677E+02 ; sig_33 = -6.935E+02

sig_12 = -3.958E+02 ; sig_13 = -1.317E+01 ; sig_23 = 9.034E+01


.................................

@(x1 = 4.226E+00, x2 = -4.226E-01, x3 = -2.887E-01):

eps_11 = -2.264E-04 ; eps_22 = 8.247E-05 ; eps_33 = 4.933E-05

gam_12 = -1.226E-05 ; gam_13 = -2.686E-06 ; gam_23 = 2.172E-05

sig_11 = -6.862E+03 ; sig_22 = 2.657E+02 ; sig_33 = -4.991E+02

sig_12 = -1.415E+02 ; sig_13 = -3.099E+01 ; sig_23 = 2.506E+02

@(x1 = 4.226E+00, x2 = -4.226E-01, x3 = 2.887E-01):

eps_11 = -2.264E-04 ; eps_22 = 8.247E-05 ; eps_33 = 4.933E-05

gam_12 = -1.226E-05 ; gam_13 = 2.686E-06 ; gam_23 = -2.172E-05

sig_11 = -6.862E+03 ; sig_22 = 2.657E+02 ; sig_33 = -4.991E+02

sig_12 = -1.415E+02 ; sig_13 = 3.099E+01 ; sig_23 = -2.506E+02

@(x1 = 4.226E+00, x2 = -1.577E+00, x3 = 2.887E-01):

eps_11 = -8.457E-04 ; eps_22 = 1.304E-04 ; eps_33 = 1.841E-04

gam_12 = -3.107E-05 ; gam_13 = 1.003E-05 ; gam_23 = -2.172E-05

sig_11 = -2.871E+04 ; sig_22 = -6.186E+03 ; sig_33 = -4.946E+03

sig_12 = -3.585E+02 ; sig_13 = 1.157E+02 ; sig_23 = -2.506E+02

@(x1 = 4.226E+00, x2 = -1.577E+00, x3 = -2.887E-01):

eps_11 = -8.457E-04 ; eps_22 = 1.304E-04 ; eps_33 = 1.841E-04

gam_12 = -3.107E-05 ; gam_13 = -1.003E-05 ; gam_23 = 2.172E-05

sig_11 = -2.871E+04 ; sig_22 = -6.186E+03 ; sig_33 = -4.946E+03

sig_12 = -3.585E+02 ; sig_13 = -1.157E+02 ; sig_23 = 2.506E+02

@(x1 = 1.577E+01, x2 = -4.226E-01, x3 = -2.887E-01):

eps_11 = -2.281E-04 ; eps_22 = 8.875E-05 ; eps_33 = 7.687E-05

gam_12 = -3.430E-05 ; gam_13 = 1.142E-06 ; gam_23 = 7.823E-06

sig_11 = -6.343E+03 ; sig_22 = 9.676E+02 ; sig_33 = 6.935E+02

sig_12 = -3.958E+02 ; sig_13 = 1.318E+01 ; sig_23 = 9.026E+01

16 V. N. Kaliakin


@(x1 = 1.577E+01, x2 = -4.226E-01, x3 = 2.887E-01):

eps_11 = -2.281E-04 ; eps_22 = 8.875E-05 ; eps_33 = 7.687E-05

gam_12 = -3.430E-05 ; gam_13 = -1.142E-06 ; gam_23 = -7.823E-06

sig_11 = -6.343E+03 ; sig_22 = 9.676E+02 ; sig_33 = 6.935E+02

sig_12 = -3.958E+02 ; sig_13 = -1.318E+01 ; sig_23 = -9.026E+01

@(x1 = 1.577E+01, x2 = -1.577E+00, x3 = 2.887E-01):

eps_11 = -8.519E-04 ; eps_22 = 2.675E-04 ; eps_33 = 2.870E-04

gam_12 = -9.031E-06 ; gam_13 = -4.259E-06 ; gam_23 = -7.833E-06

sig_11 = -2.481E+04 ; sig_22 = 1.026E+03 ; sig_33 = 1.476E+03

sig_12 = -1.042E+02 ; sig_13 = -4.914E+01 ; sig_23 = -9.038E+01

@(x1 = 1.577E+01, x2 = -1.577E+00, x3 = -2.887E-01):

eps_11 = -8.519E-04 ; eps_22 = 2.675E-04 ; eps_33 = 2.870E-04

gam_12 = -9.031E-06 ; gam_13 = 4.259E-06 ; gam_23 = 7.833E-06

sig_11 = -2.481E+04 ; sig_22 = 1.026E+03 ; sig_33 = 1.476E+03

sig_12 = -1.042E+02 ; sig_13 = 4.914E+01 ; sig_23 = 9.038E+01


.................................

@(x1 = 2.423E+01, x2 = 1.577E+00, x3 = -2.887E-01):

eps_11 = 7.090E-04 ; eps_22 = -2.354E-04 ; eps_33 = -2.009E-04

gam_12 = -1.256E-05 ; gam_13 = 1.754E-06 ; gam_23 = -1.401E-05

sig_11 = 2.108E+04 ; sig_22 = -7.120E+02 ; sig_33 = 8.331E+01

sig_12 = -1.449E+02 ; sig_13 = 2.024E+01 ; sig_23 = -1.616E+02

@(x1 = 2.423E+01, x2 = 1.577E+00, x3 = 2.887E-01):

eps_11 = 7.090E-04 ; eps_22 = -2.354E-04 ; eps_33 = -2.009E-04

gam_12 = -1.256E-05 ; gam_13 = -1.754E-06 ; gam_23 = 1.401E-05

sig_11 = 2.108E+04 ; sig_22 = -7.120E+02 ; sig_33 = 8.331E+01

sig_12 = -1.449E+02 ; sig_13 = -2.024E+01 ; sig_23 = 1.616E+02

@(x1 = 2.423E+01, x2 = 4.226E-01, x3 = 2.887E-01):

eps_11 = 1.900E-04 ; eps_22 = -3.229E-05 ; eps_33 = -5.381E-05

gam_12 = -3.077E-05 ; gam_13 = -4.707E-07 ; gam_23 = 1.401E-05

sig_11 = 6.184E+03 ; sig_22 = 1.054E+03 ; sig_33 = 5.568E+02

sig_12 = -3.550E+02 ; sig_13 = -5.431E+00 ; sig_23 = 1.616E+02

@(x1 = 2.423E+01, x2 = 4.226E-01, x3 = -2.887E-01):

eps_11 = 1.900E-04 ; eps_22 = -3.229E-05 ; eps_33 = -5.381E-05

gam_12 = -3.077E-05 ; gam_13 = 4.707E-07 ; gam_23 = -1.401E-05

sig_11 = 6.184E+03 ; sig_22 = 1.054E+03 ; sig_33 = 5.568E+02

17 V. N. Kaliakin


sig_12 = -3.550E+02 ; sig_13 = 5.431E+00 ; sig_23 = -1.616E+02

@(x1 = 3.577E+01, x2 = 1.577E+00, x3 = -2.887E-01):

eps_11 = 6.676E-04 ; eps_22 = -1.935E-04 ; eps_33 = -1.969E-04

gam_12 = -1.124E-05 ; gam_13 = -6.093E-07 ; gam_23 = 1.292E-06

sig_11 = 2.020E+04 ; sig_22 = 3.303E+02 ; sig_33 = 2.525E+02

sig_12 = -1.296E+02 ; sig_13 = -7.031E+00 ; sig_23 = 1.491E+01

@(x1 = 3.577E+01, x2 = 1.577E+00, x3 = 2.887E-01):

eps_11 = 6.676E-04 ; eps_22 = -1.935E-04 ; eps_33 = -1.969E-04

gam_12 = -1.124E-05 ; gam_13 = 6.093E-07 ; gam_23 = -1.292E-06

sig_11 = 2.020E+04 ; sig_22 = 3.303E+02 ; sig_33 = 2.525E+02

sig_12 = -1.296E+02 ; sig_13 = 7.031E+00 ; sig_23 = -1.491E+01

@(x1 = 3.577E+01, x2 = 4.226E-01, x3 = 2.887E-01):

eps_11 = 1.789E-04 ; eps_22 = -5.488E-05 ; eps_33 = -5.275E-05

gam_12 = -3.212E-05 ; gam_13 = 1.642E-07 ; gam_23 = -1.358E-06

sig_11 = 5.363E+03 ; sig_22 = -3.257E+01 ; sig_33 = 1.651E+01

sig_12 = -3.706E+02 ; sig_13 = 1.895E+00 ; sig_23 = -1.567E+01

@(x1 = 3.577E+01, x2 = 4.226E-01, x3 = -2.887E-01):

eps_11 = 1.789E-04 ; eps_22 = -5.488E-05 ; eps_33 = -5.275E-05

gam_12 = -3.212E-05 ; gam_13 = -1.642E-07 ; gam_23 = 1.358E-06

sig_11 = 5.363E+03 ; sig_22 = -3.257E+01 ; sig_33 = 1.651E+01

sig_12 = -3.706E+02 ; sig_13 = -1.895E+00 ; sig_23 = 1.567E+01


.................................

@(x1 = 2.423E+01, x2 = -4.226E-01, x3 = -2.887E-01):

eps_11 = -1.900E-04 ; eps_22 = 3.229E-05 ; eps_33 = 5.381E-05

gam_12 = -3.077E-05 ; gam_13 = -4.704E-07 ; gam_23 = -1.401E-05

sig_11 = -6.184E+03 ; sig_22 = -1.054E+03 ; sig_33 = -5.570E+02

sig_12 = -3.551E+02 ; sig_13 = -5.428E+00 ; sig_23 = -1.616E+02

@(x1 = 2.423E+01, x2 = -4.226E-01, x3 = 2.887E-01):

eps_11 = -1.900E-04 ; eps_22 = 3.229E-05 ; eps_33 = 5.381E-05

gam_12 = -3.077E-05 ; gam_13 = 4.704E-07 ; gam_23 = 1.401E-05

sig_11 = -6.184E+03 ; sig_22 = -1.054E+03 ; sig_33 = -5.570E+02

sig_12 = -3.551E+02 ; sig_13 = 5.428E+00 ; sig_23 = 1.616E+02

@(x1 = 2.423E+01, x2 = -1.577E+00, x3 = 2.887E-01):

eps_11 = -7.090E-04 ; eps_22 = 2.354E-04 ; eps_33 = 2.009E-04

18 V. N. Kaliakin


gam_12 = -1.257E-05 ; gam_13 = 1.753E-06 ; gam_23 = 1.400E-05

sig_11 = -2.108E+04 ; sig_22 = 7.121E+02 ; sig_33 = -8.337E+01

sig_12 = -1.450E+02 ; sig_13 = 2.023E+01 ; sig_23 = 1.616E+02

@(x1 = 2.423E+01, x2 = -1.577E+00, x3 = -2.887E-01):

eps_11 = -7.090E-04 ; eps_22 = 2.354E-04 ; eps_33 = 2.009E-04

gam_12 = -1.257E-05 ; gam_13 = -1.753E-06 ; gam_23 = -1.400E-05

sig_11 = -2.108E+04 ; sig_22 = 7.121E+02 ; sig_33 = -8.337E+01

sig_12 = -1.450E+02 ; sig_13 = -2.023E+01 ; sig_23 = -1.616E+02

@(x1 = 3.577E+01, x2 = -4.226E-01, x3 = -2.887E-01):

eps_11 = -1.789E-04 ; eps_22 = 5.486E-05 ; eps_33 = 5.277E-05

gam_12 = -3.211E-05 ; gam_13 = 1.603E-07 ; gam_23 = 1.383E-06

sig_11 = -5.363E+03 ; sig_22 = 3.229E+01 ; sig_33 = -1.591E+01

sig_12 = -3.705E+02 ; sig_13 = 1.850E+00 ; sig_23 = 1.596E+01

@(x1 = 3.577E+01, x2 = -4.226E-01, x3 = 2.887E-01):

eps_11 = -1.789E-04 ; eps_22 = 5.486E-05 ; eps_33 = 5.277E-05

gam_12 = -3.211E-05 ; gam_13 = -1.603E-07 ; gam_23 = -1.383E-06

sig_11 = -5.363E+03 ; sig_22 = 3.229E+01 ; sig_33 = -1.591E+01

sig_12 = -3.705E+02 ; sig_13 = -1.850E+00 ; sig_23 = -1.596E+01

@(x1 = 3.577E+01, x2 = -1.577E+00, x3 = 2.887E-01):

eps_11 = -6.676E-04 ; eps_22 = 1.935E-04 ; eps_33 = 1.969E-04

gam_12 = -1.120E-05 ; gam_13 = -6.067E-07 ; gam_23 = -1.358E-06

sig_11 = -2.020E+04 ; sig_22 = -3.308E+02 ; sig_33 = -2.526E+02

sig_12 = -1.292E+02 ; sig_13 = -7.000E+00 ; sig_23 = -1.567E+01

@(x1 = 3.577E+01, x2 = -1.577E+00, x3 = -2.887E-01):

eps_11 = -6.676E-04 ; eps_22 = 1.935E-04 ; eps_33 = 1.969E-04

gam_12 = -1.120E-05 ; gam_13 = 6.067E-07 ; gam_23 = 1.358E-06

sig_11 = -2.020E+04 ; sig_22 = -3.308E+02 ; sig_33 = -2.526E+02

sig_12 = -1.292E+02 ; sig_13 = 7.000E+00 ; sig_23 = 1.567E+01


.................................

@(x1 = 4.423E+01, x2 = 1.577E+00, x3 = -2.887E-01):

eps_11 = 5.142E-04 ; eps_22 = -1.573E-04 ; eps_33 = -1.631E-04

gam_12 = -1.264E-05 ; gam_13 = 1.207E-07 ; gam_23 = 2.352E-06

sig_11 = 1.522E+04 ; sig_22 = -2.791E+02 ; sig_33 = -4.128E+02

sig_12 = -1.459E+02 ; sig_13 = 1.393E+00 ; sig_23 = 2.714E+01

19 V. N. Kaliakin


@(x1 = 4.423E+01, x2 = 1.577E+00, x3 = 2.887E-01):

eps_11 = 5.142E-04 ; eps_22 = -1.573E-04 ; eps_33 = -1.631E-04

gam_12 = -1.264E-05 ; gam_13 = -1.207E-07 ; gam_23 = -2.352E-06

sig_11 = 1.522E+04 ; sig_22 = -2.791E+02 ; sig_33 = -4.128E+02

sig_12 = -1.459E+02 ; sig_13 = -1.393E+00 ; sig_23 = -2.714E+01

@(x1 = 4.423E+01, x2 = 4.226E-01, x3 = 2.887E-01):

eps_11 = 1.378E-04 ; eps_22 = -4.734E-05 ; eps_33 = -4.374E-05

gam_12 = -3.065E-05 ; gam_13 = -3.419E-08 ; gam_23 = -2.355E-06

sig_11 = 3.987E+03 ; sig_22 = -2.845E+02 ; sig_33 = -2.015E+02

sig_12 = -3.537E+02 ; sig_13 = -3.945E-01 ; sig_23 = -2.717E+01

@(x1 = 4.423E+01, x2 = 4.226E-01, x3 = -2.887E-01):

eps_11 = 1.378E-04 ; eps_22 = -4.734E-05 ; eps_33 = -4.374E-05

gam_12 = -3.065E-05 ; gam_13 = 3.419E-08 ; gam_23 = 2.355E-06

sig_11 = 3.987E+03 ; sig_22 = -2.845E+02 ; sig_33 = -2.015E+02

sig_12 = -3.537E+02 ; sig_13 = 3.945E-01 ; sig_23 = 2.717E+01

@(x1 = 5.577E+01, x2 = 1.577E+00, x3 = -2.887E-01):

eps_11 = 4.691E-04 ; eps_22 = -1.317E-04 ; eps_33 = -1.318E-04

gam_12 = -1.223E-05 ; gam_13 = -1.733E-07 ; gam_23 = 3.884E-07

sig_11 = 1.438E+04 ; sig_22 = 5.181E+02 ; sig_33 = 5.160E+02

sig_12 = -1.411E+02 ; sig_13 = -2.000E+00 ; sig_23 = 4.482E+00

@(x1 = 5.577E+01, x2 = 1.577E+00, x3 = 2.887E-01):

eps_11 = 4.691E-04 ; eps_22 = -1.317E-04 ; eps_33 = -1.318E-04

gam_12 = -1.223E-05 ; gam_13 = 1.733E-07 ; gam_23 = -3.884E-07

sig_11 = 1.438E+04 ; sig_22 = 5.181E+02 ; sig_33 = 5.160E+02

sig_12 = -1.411E+02 ; sig_13 = 2.000E+00 ; sig_23 = -4.482E+00

@(x1 = 5.577E+01, x2 = 4.226E-01, x3 = 2.887E-01):

eps_11 = 1.257E-04 ; eps_22 = -3.554E-05 ; eps_33 = -3.534E-05

gam_12 = -3.119E-05 ; gam_13 = 4.462E-08 ; gam_23 = -1.486E-07

sig_11 = 3.849E+03 ; sig_22 = 1.287E+02 ; sig_33 = 1.333E+02

sig_12 = -3.599E+02 ; sig_13 = 5.148E-01 ; sig_23 = -1.715E+00

@(x1 = 5.577E+01, x2 = 4.226E-01, x3 = -2.887E-01):

eps_11 = 1.257E-04 ; eps_22 = -3.554E-05 ; eps_33 = -3.534E-05

gam_12 = -3.119E-05 ; gam_13 = -4.462E-08 ; gam_23 = 1.486E-07

sig_11 = 3.849E+03 ; sig_22 = 1.287E+02 ; sig_33 = 1.333E+02

sig_12 = -3.599E+02 ; sig_13 = -5.148E-01 ; sig_23 = 1.715E+00


20 V. N. Kaliakin


.................................

@(x1 = 4.423E+01, x2 = -4.226E-01, x3 = -2.887E-01):

eps_11 = -1.378E-04 ; eps_22 = 4.738E-05 ; eps_33 = 4.371E-05

gam_12 = -3.068E-05 ; gam_13 = -3.048E-08 ; gam_23 = 2.356E-06

sig_11 = -3.987E+03 ; sig_22 = 2.854E+02 ; sig_33 = 2.008E+02

sig_12 = -3.540E+02 ; sig_13 = -3.517E-01 ; sig_23 = 2.718E+01

@(x1 = 4.423E+01, x2 = -4.226E-01, x3 = 2.887E-01):

eps_11 = -1.378E-04 ; eps_22 = 4.738E-05 ; eps_33 = 4.371E-05

gam_12 = -3.068E-05 ; gam_13 = 3.048E-08 ; gam_23 = -2.356E-06

sig_11 = -3.987E+03 ; sig_22 = 2.854E+02 ; sig_33 = 2.008E+02

sig_12 = -3.540E+02 ; sig_13 = 3.517E-01 ; sig_23 = -2.718E+01

@(x1 = 4.423E+01, x2 = -1.577E+00, x3 = 2.887E-01):

eps_11 = -5.142E-04 ; eps_22 = 1.574E-04 ; eps_33 = 1.631E-04

gam_12 = -1.269E-05 ; gam_13 = 1.187E-07 ; gam_23 = -2.352E-06

sig_11 = -1.522E+04 ; sig_22 = 2.802E+02 ; sig_33 = 4.125E+02

sig_12 = -1.464E+02 ; sig_13 = 1.370E+00 ; sig_23 = -2.714E+01

@(x1 = 4.423E+01, x2 = -1.577E+00, x3 = -2.887E-01):

eps_11 = -5.142E-04 ; eps_22 = 1.574E-04 ; eps_33 = 1.631E-04

gam_12 = -1.269E-05 ; gam_13 = -1.187E-07 ; gam_23 = 2.352E-06

sig_11 = -1.522E+04 ; sig_22 = 2.802E+02 ; sig_33 = 4.125E+02

sig_12 = -1.464E+02 ; sig_13 = -1.370E+00 ; sig_23 = 2.714E+01

@(x1 = 5.577E+01, x2 = -4.226E-01, x3 = -2.887E-01):

eps_11 = -1.257E-04 ; eps_22 = 3.545E-05 ; eps_33 = 3.537E-05

gam_12 = -3.107E-05 ; gam_13 = 4.826E-08 ; gam_23 = 5.560E-08

sig_11 = -3.849E+03 ; sig_22 = -1.311E+02 ; sig_33 = -1.329E+02

sig_12 = -3.585E+02 ; sig_13 = 5.568E-01 ; sig_23 = 6.415E-01

@(x1 = 5.577E+01, x2 = -4.226E-01, x3 = 2.887E-01):

eps_11 = -1.257E-04 ; eps_22 = 3.545E-05 ; eps_33 = 3.537E-05

gam_12 = -3.107E-05 ; gam_13 = -4.826E-08 ; gam_23 = -5.560E-08

sig_11 = -3.849E+03 ; sig_22 = -1.311E+02 ; sig_33 = -1.329E+02

sig_12 = -3.585E+02 ; sig_13 = -5.568E-01 ; sig_23 = -6.415E-01

@(x1 = 5.577E+01, x2 = -1.577E+00, x3 = 2.887E-01):

eps_11 = -4.691E-04 ; eps_22 = 1.316E-04 ; eps_33 = 1.319E-04

gam_12 = -1.218E-05 ; gam_13 = -1.752E-07 ; gam_23 = -1.393E-07

sig_11 = -1.438E+04 ; sig_22 = -5.210E+02 ; sig_33 = -5.141E+02

sig_12 = -1.405E+02 ; sig_13 = -2.022E+00 ; sig_23 = -1.607E+00

@(x1 = 5.577E+01, x2 = -1.577E+00, x3 = -2.887E-01):

21 V. N. Kaliakin


eps_11 = -4.691E-04 ; eps_22 = 1.316E-04 ; eps_33 = 1.319E-04

gam_12 = -1.218E-05 ; gam_13 = 1.752E-07 ; gam_23 = 1.393E-07

sig_11 = -1.438E+04 ; sig_22 = -5.210E+02 ; sig_33 = -5.141E+02

sig_12 = -1.405E+02 ; sig_13 = 2.022E+00 ; sig_23 = 1.607E+00


.................................

@(x1 = 6.423E+01, x2 = 1.577E+00, x3 = -2.887E-01):

eps_11 = 3.192E-04 ; eps_22 = -1.065E-04 ; eps_33 = -1.045E-04

gam_12 = -1.215E-05 ; gam_13 = 1.764E-08 ; gam_23 = -9.438E-07

sig_11 = 9.238E+03 ; sig_22 = -5.871E+02 ; sig_33 = -5.406E+02

sig_12 = -1.402E+02 ; sig_13 = 2.035E-01 ; sig_23 = -1.089E+01

@(x1 = 6.423E+01, x2 = 1.577E+00, x3 = 2.887E-01):

eps_11 = 3.192E-04 ; eps_22 = -1.065E-04 ; eps_33 = -1.045E-04

gam_12 = -1.215E-05 ; gam_13 = -1.764E-08 ; gam_23 = 9.438E-07

sig_11 = 9.238E+03 ; sig_22 = -5.871E+02 ; sig_33 = -5.406E+02

sig_12 = -1.402E+02 ; sig_13 = -2.035E-01 ; sig_23 = 1.089E+01

@(x1 = 6.423E+01, x2 = 4.226E-01, x3 = 2.887E-01):

eps_11 = 8.551E-05 ; eps_22 = -2.631E-05 ; eps_33 = -2.806E-05

gam_12 = -3.103E-05 ; gam_13 = -6.371E-09 ; gam_23 = 9.432E-07

sig_11 = 2.512E+03 ; sig_22 = -6.795E+01 ; sig_33 = -1.084E+02

sig_12 = -3.580E+02 ; sig_13 = -7.351E-02 ; sig_23 = 1.088E+01

@(x1 = 6.423E+01, x2 = 4.226E-01, x3 = -2.887E-01):

eps_11 = 8.551E-05 ; eps_22 = -2.631E-05 ; eps_33 = -2.806E-05

gam_12 = -3.103E-05 ; gam_13 = 6.371E-09 ; gam_23 = -9.432E-07

sig_11 = 2.512E+03 ; sig_22 = -6.795E+01 ; sig_33 = -1.084E+02

sig_12 = -3.580E+02 ; sig_13 = 7.351E-02 ; sig_23 = -1.088E+01

@(x1 = 7.577E+01, x2 = 1.577E+00, x3 = -2.887E-01):

eps_11 = 2.731E-04 ; eps_22 = -7.222E-05 ; eps_33 = -7.360E-05

gam_12 = -1.306E-05 ; gam_13 = 2.747E-07 ; gam_23 = -2.804E-07

sig_11 = 8.505E+03 ; sig_22 = 5.362E+02 ; sig_33 = 5.045E+02

sig_12 = -1.507E+02 ; sig_13 = 3.170E+00 ; sig_23 = -3.235E+00

@(x1 = 7.577E+01, x2 = 1.577E+00, x3 = 2.887E-01):

eps_11 = 2.731E-04 ; eps_22 = -7.222E-05 ; eps_33 = -7.360E-05

gam_12 = -1.306E-05 ; gam_13 = -2.747E-07 ; gam_23 = 2.804E-07

sig_11 = 8.505E+03 ; sig_22 = 5.362E+02 ; sig_33 = 5.045E+02

sig_12 = -1.507E+02 ; sig_13 = -3.170E+00 ; sig_23 = 3.235E+00

22 V. N. Kaliakin


@(x1 = 7.577E+01, x2 = 4.226E-01, x3 = 2.887E-01):

eps_11 = 7.326E-05 ; eps_22 = -2.166E-05 ; eps_33 = -1.944E-05

gam_12 = -3.058E-05 ; gam_13 = -4.348E-08 ; gam_23 = -6.197E-07

sig_11 = 2.247E+03 ; sig_22 = 5.677E+01 ; sig_33 = 1.082E+02

sig_12 = -3.529E+02 ; sig_13 = -5.017E-01 ; sig_23 = -7.150E+00

@(x1 = 7.577E+01, x2 = 4.226E-01, x3 = -2.887E-01):

eps_11 = 7.326E-05 ; eps_22 = -2.166E-05 ; eps_33 = -1.944E-05

gam_12 = -3.058E-05 ; gam_13 = 4.348E-08 ; gam_23 = 6.197E-07

sig_11 = 2.247E+03 ; sig_22 = 5.677E+01 ; sig_33 = 1.082E+02

sig_12 = -3.529E+02 ; sig_13 = 5.017E-01 ; sig_23 = 7.150E+00


.................................

@(x1 = 6.423E+01, x2 = -4.226E-01, x3 = -2.887E-01):

eps_11 = -8.559E-05 ; eps_22 = 2.657E-05 ; eps_33 = 2.791E-05

gam_12 = -3.113E-05 ; gam_13 = 5.783E-09 ; gam_23 = -9.453E-07

sig_11 = -2.513E+03 ; sig_22 = 7.498E+01 ; sig_33 = 1.059E+02

sig_12 = -3.592E+02 ; sig_13 = 6.672E-02 ; sig_23 = -1.091E+01

@(x1 = 6.423E+01, x2 = -4.226E-01, x3 = 2.887E-01):

eps_11 = -8.559E-05 ; eps_22 = 2.657E-05 ; eps_33 = 2.791E-05

gam_12 = -3.113E-05 ; gam_13 = -5.783E-09 ; gam_23 = 9.453E-07

sig_11 = -2.513E+03 ; sig_22 = 7.498E+01 ; sig_33 = 1.059E+02

sig_12 = -3.592E+02 ; sig_13 = -6.672E-02 ; sig_23 = 1.091E+01

@(x1 = 6.423E+01, x2 = -1.577E+00, x3 = 2.887E-01):

eps_11 = -3.193E-04 ; eps_22 = 1.068E-04 ; eps_33 = 1.044E-04

gam_12 = -1.235E-05 ; gam_13 = 1.430E-09 ; gam_23 = 9.445E-07

sig_11 = -9.237E+03 ; sig_22 = 5.949E+02 ; sig_33 = 5.388E+02

sig_12 = -1.425E+02 ; sig_13 = 1.650E-02 ; sig_23 = 1.090E+01

@(x1 = 6.423E+01, x2 = -1.577E+00, x3 = -2.887E-01):

eps_11 = -3.193E-04 ; eps_22 = 1.068E-04 ; eps_33 = 1.044E-04

gam_12 = -1.235E-05 ; gam_13 = -1.430E-09 ; gam_23 = -9.445E-07

sig_11 = -9.237E+03 ; sig_22 = 5.949E+02 ; sig_33 = 5.388E+02

sig_12 = -1.425E+02 ; sig_13 = -1.650E-02 ; sig_23 = -1.090E+01

@(x1 = 7.577E+01, x2 = -4.226E-01, x3 = -2.887E-01):

eps_11 = -7.307E-05 ; eps_22 = 2.092E-05 ; eps_33 = 1.988E-05

gam_12 = -3.059E-05 ; gam_13 = -1.176E-07 ; gam_23 = 9.673E-07

23 V. N. Kaliakin


sig_11 = -2.245E+03 ; sig_22 = -7.593E+01 ; sig_33 = -9.987E+01

sig_12 = -3.530E+02 ; sig_13 = -1.357E+00 ; sig_23 = 1.116E+01

@(x1 = 7.577E+01, x2 = -4.226E-01, x3 = 2.887E-01):

eps_11 = -7.307E-05 ; eps_22 = 2.092E-05 ; eps_33 = 1.988E-05

gam_12 = -3.059E-05 ; gam_13 = 1.176E-07 ; gam_23 = -9.673E-07

sig_11 = -2.245E+03 ; sig_22 = -7.593E+01 ; sig_33 = -9.987E+01

sig_12 = -3.530E+02 ; sig_13 = 1.357E+00 ; sig_23 = -1.116E+01

@(x1 = 7.577E+01, x2 = -1.577E+00, x3 = 2.887E-01):

eps_11 = -2.730E-04 ; eps_22 = 7.157E-05 ; eps_33 = 7.383E-05

gam_12 = -1.243E-05 ; gam_13 = 3.433E-07 ; gam_23 = -6.547E-07

sig_11 = -8.507E+03 ; sig_22 = -5.564E+02 ; sig_33 = -5.042E+02

sig_12 = -1.435E+02 ; sig_13 = 3.961E+00 ; sig_23 = -7.554E+00

@(x1 = 7.577E+01, x2 = -1.577E+00, x3 = -2.887E-01):

eps_11 = -2.730E-04 ; eps_22 = 7.157E-05 ; eps_33 = 7.383E-05

gam_12 = -1.243E-05 ; gam_13 = -3.433E-07 ; gam_23 = 6.547E-07

sig_11 = -8.507E+03 ; sig_22 = -5.564E+02 ; sig_33 = -5.042E+02

sig_12 = -1.435E+02 ; sig_13 = -3.961E+00 ; sig_23 = 7.554E+00


.................................

@(x1 = 8.423E+01, x2 = 1.577E+00, x3 = -2.887E-01):

eps_11 = 1.225E-04 ; eps_22 = -4.296E-05 ; eps_33 = -4.377E-05

gam_12 = -1.146E-05 ; gam_13 = -3.467E-07 ; gam_23 = -4.822E-07

sig_11 = 3.446E+03 ; sig_22 = -3.726E+02 ; sig_33 = -3.912E+02

sig_12 = -1.322E+02 ; sig_13 = -4.001E+00 ; sig_23 = -5.564E+00

@(x1 = 8.423E+01, x2 = 1.577E+00, x3 = 2.887E-01):

eps_11 = 1.225E-04 ; eps_22 = -4.296E-05 ; eps_33 = -4.377E-05

gam_12 = -1.146E-05 ; gam_13 = 3.467E-07 ; gam_23 = 4.822E-07

sig_11 = 3.446E+03 ; sig_22 = -3.726E+02 ; sig_33 = -3.912E+02

sig_12 = -1.322E+02 ; sig_13 = 4.001E+00 ; sig_23 = 5.564E+00

@(x1 = 8.423E+01, x2 = 4.226E-01, x3 = 2.887E-01):

eps_11 = 3.256E-05 ; eps_22 = -9.224E-06 ; eps_33 = -1.197E-05

gam_12 = -3.155E-05 ; gam_13 = 5.057E-08 ; gam_23 = 4.871E-07

sig_11 = 9.479E+02 ; sig_22 = -1.624E+01 ; sig_33 = -7.964E+01

sig_12 = -3.640E+02 ; sig_13 = 5.835E-01 ; sig_23 = 5.620E+00

@(x1 = 8.423E+01, x2 = 4.226E-01, x3 = -2.887E-01):

24 V. N. Kaliakin


eps_11 = 3.256E-05 ; eps_22 = -9.224E-06 ; eps_33 = -1.197E-05

gam_12 = -3.155E-05 ; gam_13 = -5.057E-08 ; gam_23 = -4.871E-07

sig_11 = 9.479E+02 ; sig_22 = -1.624E+01 ; sig_33 = -7.964E+01

sig_12 = -3.640E+02 ; sig_13 = -5.835E-01 ; sig_23 = -5.620E+00

@(x1 = 9.577E+01, x2 = 1.577E+00, x3 = -2.887E-01):

eps_11 = 8.142E-05 ; eps_22 = -2.662E-05 ; eps_33 = -2.056E-05

gam_12 = -1.252E-05 ; gam_13 = 6.541E-08 ; gam_23 = 3.802E-06

sig_11 = 2.471E+03 ; sig_22 = -2.179E+01 ; sig_33 = 1.182E+02

sig_12 = -1.445E+02 ; sig_13 = 7.548E-01 ; sig_23 = 4.387E+01

@(x1 = 9.577E+01, x2 = 1.577E+00, x3 = 2.887E-01):

eps_11 = 8.142E-05 ; eps_22 = -2.662E-05 ; eps_33 = -2.056E-05

gam_12 = -1.252E-05 ; gam_13 = -6.541E-08 ; gam_23 = -3.802E-06

sig_11 = 2.471E+03 ; sig_22 = -2.179E+01 ; sig_33 = 1.182E+02

sig_12 = -1.445E+02 ; sig_13 = -7.548E-01 ; sig_23 = -4.387E+01

@(x1 = 9.577E+01, x2 = 4.226E-01, x3 = 2.887E-01):

eps_11 = 2.221E-05 ; eps_22 = -9.140E-06 ; eps_33 = -5.539E-06

gam_12 = -3.093E-05 ; gam_13 = -2.048E-08 ; gam_23 = -4.254E-07

sig_11 = 6.428E+02 ; sig_22 = -8.063E+01 ; sig_33 = 2.461E+00

sig_12 = -3.569E+02 ; sig_13 = -2.363E-01 ; sig_23 = -4.909E+00

@(x1 = 9.577E+01, x2 = 4.226E-01, x3 = -2.887E-01):

eps_11 = 2.221E-05 ; eps_22 = -9.140E-06 ; eps_33 = -5.539E-06

gam_12 = -3.093E-05 ; gam_13 = 2.048E-08 ; gam_23 = 4.254E-07

sig_11 = 6.428E+02 ; sig_22 = -8.063E+01 ; sig_33 = 2.461E+00

sig_12 = -3.569E+02 ; sig_13 = 2.363E-01 ; sig_23 = 4.909E+00


.................................

@(x1 = 8.423E+01, x2 = -4.226E-01, x3 = -2.887E-01):

eps_11 = -3.313E-05 ; eps_22 = 1.115E-05 ; eps_33 = 1.128E-05

gam_12 = -3.153E-05 ; gam_13 = 1.380E-07 ; gam_23 = -4.865E-07

sig_11 = -9.495E+02 ; sig_22 = 7.226E+01 ; sig_33 = 7.536E+01

sig_12 = -3.638E+02 ; sig_13 = 1.592E+00 ; sig_23 = -5.614E+00

@(x1 = 8.423E+01, x2 = -4.226E-01, x3 = 2.887E-01):

eps_11 = -3.313E-05 ; eps_22 = 1.115E-05 ; eps_33 = 1.128E-05

gam_12 = -3.153E-05 ; gam_13 = -1.380E-07 ; gam_23 = 4.865E-07

sig_11 = -9.495E+02 ; sig_22 = 7.226E+01 ; sig_33 = 7.536E+01

sig_12 = -3.638E+02 ; sig_13 = -1.592E+00 ; sig_23 = 5.614E+00

25 V. N. Kaliakin


@(x1 = 8.423E+01, x2 = -1.577E+00, x3 = 2.887E-01):

eps_11 = -1.230E-04 ; eps_22 = 4.487E-05 ; eps_33 = 4.306E-05

gam_12 = -1.213E-05 ; gam_13 = -4.187E-07 ; gam_23 = 4.851E-07

sig_11 = -3.444E+03 ; sig_22 = 4.291E+02 ; sig_33 = 3.874E+02

sig_12 = -1.400E+02 ; sig_13 = -4.831E+00 ; sig_23 = 5.598E+00

@(x1 = 8.423E+01, x2 = -1.577E+00, x3 = -2.887E-01):

eps_11 = -1.230E-04 ; eps_22 = 4.487E-05 ; eps_33 = 4.306E-05

gam_12 = -1.213E-05 ; gam_13 = 4.187E-07 ; gam_23 = -4.851E-07

sig_11 = -3.444E+03 ; sig_22 = 4.291E+02 ; sig_33 = 3.874E+02

sig_12 = -1.400E+02 ; sig_13 = 4.831E+00 ; sig_23 = -5.598E+00

@(x1 = 9.577E+01, x2 = -4.226E-01, x3 = -2.887E-01):

eps_11 = -2.092E-05 ; eps_22 = 4.606E-06 ; eps_33 = 6.620E-06

gam_12 = -3.091E-05 ; gam_13 = -3.968E-08 ; gam_23 = -8.571E-07

sig_11 = -6.507E+02 ; sig_22 = -6.153E+01 ; sig_33 = -1.506E+01

sig_12 = -3.567E+02 ; sig_13 = -4.578E-01 ; sig_23 = -9.890E+00

@(x1 = 9.577E+01, x2 = -4.226E-01, x3 = 2.887E-01):

eps_11 = -2.092E-05 ; eps_22 = 4.606E-06 ; eps_33 = 6.620E-06

gam_12 = -3.091E-05 ; gam_13 = 3.968E-08 ; gam_23 = 8.571E-07

sig_11 = -6.507E+02 ; sig_22 = -6.153E+01 ; sig_33 = -1.506E+01

sig_12 = -3.567E+02 ; sig_13 = 4.578E-01 ; sig_23 = 9.890E+00

@(x1 = 9.577E+01, x2 = -1.577E+00, x3 = 2.887E-01):

eps_11 = -7.988E-05 ; eps_22 = 2.166E-05 ; eps_33 = 2.239E-05

gam_12 = -1.230E-05 ; gam_13 = 9.723E-08 ; gam_23 = -3.158E-07

sig_11 = -2.464E+03 ; sig_22 = -1.203E+02 ; sig_33 = -1.034E+02

sig_12 = -1.419E+02 ; sig_13 = 1.122E+00 ; sig_23 = -3.644E+00

@(x1 = 9.577E+01, x2 = -1.577E+00, x3 = -2.887E-01):

eps_11 = -7.988E-05 ; eps_22 = 2.166E-05 ; eps_33 = 2.239E-05

gam_12 = -1.230E-05 ; gam_13 = -9.723E-08 ; gam_23 = 3.158E-07

sig_11 = -2.464E+03 ; sig_22 = -1.203E+02 ; sig_33 = -1.034E+02

sig_12 = -1.419E+02 ; sig_13 = -1.122E+00 ; sig_23 = 3.644E+00

At time 1.000E+00 (step no. 1):

======================================================================

= N O D A L Q U A N T I T I E S =

26 V. N. Kaliakin


======================================================================

node: 1 ( x1 = 0.000E+00, x2 = 2.000E+00, x3 = -5.000E-01 )

u_1 = -2.784E-26, u_2 = -9.951E-25, u_3 = 4.908E-25

node: 2 ( x1 = 0.000E+00, x2 = 2.000E+00, x3 = 0.000E+00 )

u_1 = 5.123E-25, u_2 = -5.594E-25, u_3 = 8.974E-35

node: 3 ( x1 = 0.000E+00, x2 = 2.000E+00, x3 = 5.000E-01 )

u_1 = -2.784E-26, u_2 = -9.951E-25, u_3 = -4.908E-25

node: 4 ( x1 = 0.000E+00, x2 = 1.000E+00, x3 = -5.000E-01 )

u_1 = 6.938E-25, u_2 = 9.063E-25, u_3 = 7.187E-25

node: 5 ( x1 = 0.000E+00, x2 = 1.000E+00, x3 = 5.000E-01 )

u_1 = 6.938E-25, u_2 = 9.063E-25, u_3 = -7.187E-25

node: 6 ( x1 = 0.000E+00, x2 = 0.000E+00, x3 = -5.000E-01 )

u_1 = -4.346E-30, u_2 = -1.090E-24, u_3 = -2.008E-30

node: 7 ( x1 = 0.000E+00, x2 = 0.000E+00, x3 = 0.000E+00 )

u_1 = -3.590E-31, u_2 = 5.198E-25, u_3 = 1.773E-34

node: 8 ( x1 = 0.000E+00, x2 = 0.000E+00, x3 = 5.000E-01 )

u_1 = -4.346E-30, u_2 = -1.090E-24, u_3 = 2.007E-30

node: 9 ( x1 = 0.000E+00, x2 = -1.000E+00, x3 = -5.000E-01 )

u_1 = -6.938E-25, u_2 = 9.063E-25, u_3 = -7.187E-25

node: 10 ( x1 = 0.000E+00, x2 = -1.000E+00, x3 = 5.000E-01 )

u_1 = -6.938E-25, u_2 = 9.063E-25, u_3 = 7.187E-25

node: 11 ( x1 = 0.000E+00, x2 = -2.000E+00, x3 = -5.000E-01 )

u_1 = 2.783E-26, u_2 = -9.951E-25, u_3 = -4.908E-25

node: 12 ( x1 = 0.000E+00, x2 = -2.000E+00, x3 = 0.000E+00 )

u_1 = -5.123E-25, u_2 = -5.594E-25, u_3 = 8.777E-35

node: 13 ( x1 = 0.000E+00, x2 = -2.000E+00, x3 = 5.000E-01 )

u_1 = 2.783E-26, u_2 = -9.951E-25, u_3 = 4.908E-25

node: 14 ( x1 = 1.000E+01, x2 = 2.000E+00, x3 = -5.000E-01 )

u_1 = 1.073E-02, u_2 = -2.723E-02, u_3 = 1.958E-04

27 V. N. Kaliakin


node: 15 ( x1 = 1.000E+01, x2 = 2.000E+00, x3 = 5.000E-01 )

u_1 = 1.073E-02, u_2 = -2.723E-02, u_3 = -1.958E-04

node: 16 ( x1 = 1.000E+01, x2 = 0.000E+00, x3 = -5.000E-01 )

u_1 = 2.489E-10, u_2 = -2.686E-02, u_3 = 1.461E-10

node: 17 ( x1 = 1.000E+01, x2 = 0.000E+00, x3 = 5.000E-01 )

u_1 = 2.506E-10, u_2 = -2.686E-02, u_3 = -1.631E-10

node: 18 ( x1 = 1.000E+01, x2 = -2.000E+00, x3 = -5.000E-01 )

u_1 = -1.073E-02, u_2 = -2.723E-02, u_3 = -1.958E-04

node: 19 ( x1 = 1.000E+01, x2 = -2.000E+00, x3 = 5.000E-01 )

u_1 = -1.073E-02, u_2 = -2.723E-02, u_3 = 1.958E-04

node: 20 ( x1 = 2.000E+01, x2 = 2.000E+00, x3 = -5.000E-01 )

u_1 = 2.153E-02, u_2 = -1.080E-01, u_3 = 1.130E-04

node: 21 ( x1 = 2.000E+01, x2 = 2.000E+00, x3 = 0.000E+00 )

u_1 = 2.153E-02, u_2 = -1.081E-01, u_3 = -3.446E-11

node: 22 ( x1 = 2.000E+01, x2 = 2.000E+00, x3 = 5.000E-01 )

u_1 = 2.153E-02, u_2 = -1.080E-01, u_3 = -1.130E-04

node: 23 ( x1 = 2.000E+01, x2 = 1.000E+00, x3 = -5.000E-01 )

u_1 = 1.076E-02, u_2 = -1.078E-01, u_3 = 5.647E-05

node: 24 ( x1 = 2.000E+01, x2 = 1.000E+00, x3 = 5.000E-01 )

u_1 = 1.076E-02, u_2 = -1.078E-01, u_3 = -5.647E-05

node: 25 ( x1 = 2.000E+01, x2 = 0.000E+00, x3 = -5.000E-01 )

u_1 = 5.274E-09, u_2 = -1.078E-01, u_3 = -2.112E-09

node: 26 ( x1 = 2.000E+01, x2 = 0.000E+00, x3 = 0.000E+00 )

u_1 = 5.484E-09, u_2 = -1.078E-01, u_3 = -3.410E-11

node: 27 ( x1 = 2.000E+01, x2 = 0.000E+00, x3 = 5.000E-01 )

u_1 = 5.277E-09, u_2 = -1.078E-01, u_3 = 2.043E-09

node: 28 ( x1 = 2.000E+01, x2 = -1.000E+00, x3 = -5.000E-01 )

u_1 = -1.076E-02, u_2 = -1.078E-01, u_3 = -5.647E-05

node: 29 ( x1 = 2.000E+01, x2 = -1.000E+00, x3 = 5.000E-01 )

u_1 = -1.076E-02, u_2 = -1.078E-01, u_3 = 5.647E-05

28 V. N. Kaliakin


node: 30 ( x1 = 2.000E+01, x2 = -2.000E+00, x3 = -5.000E-01 )

u_1 = -2.153E-02, u_2 = -1.080E-01, u_3 = -1.130E-04

node: 31 ( x1 = 2.000E+01, x2 = -2.000E+00, x3 = 0.000E+00 )

u_1 = -2.153E-02, u_2 = -1.081E-01, u_3 = -3.347E-11

node: 32 ( x1 = 2.000E+01, x2 = -2.000E+00, x3 = 5.000E-01 )

u_1 = -2.153E-02, u_2 = -1.080E-01, u_3 = 1.130E-04

node: 33 ( x1 = 3.000E+01, x2 = 2.000E+00, x3 = -5.000E-01 )

u_1 = 3.049E-02, u_2 = -2.385E-01, u_3 = 1.338E-04

node: 34 ( x1 = 3.000E+01, x2 = 2.000E+00, x3 = 5.000E-01 )

u_1 = 3.049E-02, u_2 = -2.385E-01, u_3 = -1.338E-04

node: 35 ( x1 = 3.000E+01, x2 = 0.000E+00, x3 = -5.000E-01 )

u_1 = -5.319E-09, u_2 = -2.382E-01, u_3 = 1.734E-09

node: 36 ( x1 = 3.000E+01, x2 = 0.000E+00, x3 = 5.000E-01 )

u_1 = -5.313E-09, u_2 = -2.382E-01, u_3 = -1.901E-09

node: 37 ( x1 = 3.000E+01, x2 = -2.000E+00, x3 = -5.000E-01 )

u_1 = -3.049E-02, u_2 = -2.385E-01, u_3 = -1.338E-04

node: 38 ( x1 = 3.000E+01, x2 = -2.000E+00, x3 = 5.000E-01 )

u_1 = -3.049E-02, u_2 = -2.385E-01, u_3 = 1.338E-04

node: 39 ( x1 = 4.000E+01, x2 = 2.000E+00, x3 = -5.000E-01 )

u_1 = 3.899E-02, u_2 = -4.126E-01, u_3 = 1.086E-04

node: 40 ( x1 = 4.000E+01, x2 = 2.000E+00, x3 = 0.000E+00 )

u_1 = 3.899E-02, u_2 = -4.126E-01, u_3 = -1.644E-10

node: 41 ( x1 = 4.000E+01, x2 = 2.000E+00, x3 = 5.000E-01 )

u_1 = 3.899E-02, u_2 = -4.126E-01, u_3 = -1.086E-04

node: 42 ( x1 = 4.000E+01, x2 = 1.000E+00, x3 = -5.000E-01 )

u_1 = 1.949E-02, u_2 = -4.124E-01, u_3 = 5.428E-05

node: 43 ( x1 = 4.000E+01, x2 = 1.000E+00, x3 = 5.000E-01 )

u_1 = 1.949E-02, u_2 = -4.124E-01, u_3 = -5.428E-05

node: 44 ( x1 = 4.000E+01, x2 = 0.000E+00, x3 = -5.000E-01 )

u_1 = 2.140E-08, u_2 = -4.123E-01, u_3 = -9.388E-09

29 V. N. Kaliakin


node: 45 ( x1 = 4.000E+01, x2 = 0.000E+00, x3 = 0.000E+00 )

u_1 = 2.063E-08, u_2 = -4.123E-01, u_3 = -1.634E-10

node: 46 ( x1 = 4.000E+01, x2 = 0.000E+00, x3 = 5.000E-01 )

u_1 = 2.141E-08, u_2 = -4.123E-01, u_3 = 9.061E-09

node: 47 ( x1 = 4.000E+01, x2 = -1.000E+00, x3 = -5.000E-01 )

u_1 = -1.949E-02, u_2 = -4.124E-01, u_3 = -5.431E-05

node: 48 ( x1 = 4.000E+01, x2 = -1.000E+00, x3 = 5.000E-01 )

u_1 = -1.949E-02, u_2 = -4.124E-01, u_3 = 5.431E-05

node: 49 ( x1 = 4.000E+01, x2 = -2.000E+00, x3 = -5.000E-01 )

u_1 = -3.899E-02, u_2 = -4.126E-01, u_3 = -1.086E-04

node: 50 ( x1 = 4.000E+01, x2 = -2.000E+00, x3 = 0.000E+00 )

u_1 = -3.899E-02, u_2 = -4.126E-01, u_3 = -1.620E-10

node: 51 ( x1 = 4.000E+01, x2 = -2.000E+00, x3 = 5.000E-01 )

u_1 = -3.899E-02, u_2 = -4.126E-01, u_3 = 1.086E-04

node: 52 ( x1 = 5.000E+01, x2 = 2.000E+00, x3 = -5.000E-01 )

u_1 = 4.547E-02, u_2 = -6.241E-01, u_3 = 9.453E-05

node: 53 ( x1 = 5.000E+01, x2 = 2.000E+00, x3 = 5.000E-01 )

u_1 = 4.547E-02, u_2 = -6.241E-01, u_3 = -9.453E-05

node: 54 ( x1 = 5.000E+01, x2 = 0.000E+00, x3 = -5.000E-01 )

u_1 = -2.570E-08, u_2 = -6.239E-01, u_3 = 7.366E-09

node: 55 ( x1 = 5.000E+01, x2 = 0.000E+00, x3 = 5.000E-01 )

u_1 = -2.568E-08, u_2 = -6.239E-01, u_3 = -7.928E-09

node: 56 ( x1 = 5.000E+01, x2 = -2.000E+00, x3 = -5.000E-01 )

u_1 = -4.547E-02, u_2 = -6.241E-01, u_3 = -9.454E-05

node: 57 ( x1 = 5.000E+01, x2 = -2.000E+00, x3 = 5.000E-01 )

u_1 = -4.547E-02, u_2 = -6.241E-01, u_3 = 9.454E-05

node: 58 ( x1 = 6.000E+01, x2 = 2.000E+00, x3 = -5.000E-01 )

u_1 = 5.145E-02, u_2 = -8.668E-01, u_3 = 7.412E-05

node: 59 ( x1 = 6.000E+01, x2 = 2.000E+00, x3 = 0.000E+00 )

u_1 = 5.145E-02, u_2 = -8.668E-01, u_3 = -4.450E-10

30 V. N. Kaliakin


node: 60 ( x1 = 6.000E+01, x2 = 2.000E+00, x3 = 5.000E-01 )

u_1 = 5.145E-02, u_2 = -8.668E-01, u_3 = -7.412E-05

node: 61 ( x1 = 6.000E+01, x2 = 1.000E+00, x3 = -5.000E-01 )

u_1 = 2.572E-02, u_2 = -8.667E-01, u_3 = 3.709E-05

node: 62 ( x1 = 6.000E+01, x2 = 1.000E+00, x3 = 5.000E-01 )

u_1 = 2.572E-02, u_2 = -8.667E-01, u_3 = -3.709E-05

node: 63 ( x1 = 6.000E+01, x2 = 0.000E+00, x3 = -5.000E-01 )

u_1 = 1.624E-07, u_2 = -8.666E-01, u_3 = -5.370E-08

node: 64 ( x1 = 6.000E+01, x2 = 0.000E+00, x3 = 0.000E+00 )

u_1 = 1.655E-07, u_2 = -8.667E-01, u_3 = -4.430E-10

node: 65 ( x1 = 6.000E+01, x2 = 0.000E+00, x3 = 5.000E-01 )

u_1 = 1.624E-07, u_2 = -8.666E-01, u_3 = 5.282E-08

node: 66 ( x1 = 6.000E+01, x2 = -1.000E+00, x3 = -5.000E-01 )

u_1 = -2.572E-02, u_2 = -8.667E-01, u_3 = -3.717E-05

node: 67 ( x1 = 6.000E+01, x2 = -1.000E+00, x3 = 5.000E-01 )

u_1 = -2.572E-02, u_2 = -8.667E-01, u_3 = 3.717E-05

node: 68 ( x1 = 6.000E+01, x2 = -2.000E+00, x3 = -5.000E-01 )

u_1 = -5.145E-02, u_2 = -8.668E-01, u_3 = -7.428E-05

node: 69 ( x1 = 6.000E+01, x2 = -2.000E+00, x3 = 0.000E+00 )

u_1 = -5.145E-02, u_2 = -8.668E-01, u_3 = -4.404E-10

node: 70 ( x1 = 6.000E+01, x2 = -2.000E+00, x3 = 5.000E-01 )

u_1 = -5.145E-02, u_2 = -8.668E-01, u_3 = 7.428E-05

node: 71 ( x1 = 7.000E+01, x2 = 2.000E+00, x3 = -5.000E-01 )

u_1 = 5.546E-02, u_2 = -1.134E+00, u_3 = 5.617E-05

node: 72 ( x1 = 7.000E+01, x2 = 2.000E+00, x3 = 5.000E-01 )

u_1 = 5.546E-02, u_2 = -1.134E+00, u_3 = -5.618E-05

node: 73 ( x1 = 7.000E+01, x2 = 0.000E+00, x3 = -5.000E-01 )

u_1 = -2.460E-07, u_2 = -1.134E+00, u_3 = 4.292E-08

node: 74 ( x1 = 7.000E+01, x2 = 0.000E+00, x3 = 5.000E-01 )

u_1 = -2.460E-07, u_2 = -1.134E+00, u_3 = -4.422E-08

31 V. N. Kaliakin


node: 75 ( x1 = 7.000E+01, x2 = -2.000E+00, x3 = -5.000E-01 )

u_1 = -5.546E-02, u_2 = -1.134E+00, u_3 = -5.600E-05

node: 76 ( x1 = 7.000E+01, x2 = -2.000E+00, x3 = 5.000E-01 )

u_1 = -5.546E-02, u_2 = -1.134E+00, u_3 = 5.600E-05

node: 77 ( x1 = 8.000E+01, x2 = 2.000E+00, x3 = -5.000E-01 )

u_1 = 5.896E-02, u_2 = -1.421E+00, u_3 = 4.045E-05

node: 78 ( x1 = 8.000E+01, x2 = 2.000E+00, x3 = 0.000E+00 )

u_1 = 5.896E-02, u_2 = -1.421E+00, u_3 = -9.135E-10

node: 79 ( x1 = 8.000E+01, x2 = 2.000E+00, x3 = 5.000E-01 )

u_1 = 5.896E-02, u_2 = -1.421E+00, u_3 = -4.045E-05

node: 80 ( x1 = 8.000E+01, x2 = 1.000E+00, x3 = -5.000E-01 )

u_1 = 2.947E-02, u_2 = -1.421E+00, u_3 = 1.986E-05

node: 81 ( x1 = 8.000E+01, x2 = 1.000E+00, x3 = 5.000E-01 )

u_1 = 2.947E-02, u_2 = -1.421E+00, u_3 = -1.987E-05

node: 82 ( x1 = 8.000E+01, x2 = 0.000E+00, x3 = -5.000E-01 )

u_1 = 7.111E-07, u_2 = -1.421E+00, u_3 = -2.468E-07

node: 83 ( x1 = 8.000E+01, x2 = 0.000E+00, x3 = 0.000E+00 )

u_1 = 7.010E-07, u_2 = -1.421E+00, u_3 = -9.096E-10

node: 84 ( x1 = 8.000E+01, x2 = 0.000E+00, x3 = 5.000E-01 )

u_1 = 7.111E-07, u_2 = -1.421E+00, u_3 = 2.450E-07

node: 85 ( x1 = 8.000E+01, x2 = -1.000E+00, x3 = -5.000E-01 )

u_1 = -2.947E-02, u_2 = -1.421E+00, u_3 = -2.045E-05

node: 86 ( x1 = 8.000E+01, x2 = -1.000E+00, x3 = 5.000E-01 )

u_1 = -2.947E-02, u_2 = -1.421E+00, u_3 = 2.045E-05

node: 87 ( x1 = 8.000E+01, x2 = -2.000E+00, x3 = -5.000E-01 )

u_1 = -5.896E-02, u_2 = -1.421E+00, u_3 = -4.070E-05

node: 88 ( x1 = 8.000E+01, x2 = -2.000E+00, x3 = 0.000E+00 )

u_1 = -5.896E-02, u_2 = -1.421E+00, u_3 = -9.053E-10

node: 89 ( x1 = 8.000E+01, x2 = -2.000E+00, x3 = 5.000E-01 )

u_1 = -5.896E-02, u_2 = -1.421E+00, u_3 = 4.070E-05

32 V. N. Kaliakin


node: 90 ( x1 = 9.000E+01, x2 = 2.000E+00, x3 = -5.000E-01 )

u_1 = 6.048E-02, u_2 = -1.720E+00, u_3 = 1.655E-05

node: 91 ( x1 = 9.000E+01, x2 = 2.000E+00, x3 = 5.000E-01 )

u_1 = 6.048E-02, u_2 = -1.720E+00, u_3 = -1.655E-05

node: 92 ( x1 = 9.000E+01, x2 = 0.000E+00, x3 = -5.000E-01 )

u_1 = -1.522E-06, u_2 = -1.720E+00, u_3 = 2.007E-07

node: 93 ( x1 = 9.000E+01, x2 = 0.000E+00, x3 = 5.000E-01 )

u_1 = -1.522E-06, u_2 = -1.720E+00, u_3 = -2.031E-07

node: 94 ( x1 = 9.000E+01, x2 = -2.000E+00, x3 = -5.000E-01 )

u_1 = -6.048E-02, u_2 = -1.720E+00, u_3 = -1.646E-05

node: 95 ( x1 = 9.000E+01, x2 = -2.000E+00, x3 = 5.000E-01 )

u_1 = -6.048E-02, u_2 = -1.720E+00, u_3 = 1.646E-05

node: 96 ( x1 = 1.000E+02, x2 = 2.000E+00, x3 = -5.000E-01 )

u_1 = 6.155E-02, u_2 = -2.025E+00, u_3 = 1.430E-05

node: 97 ( x1 = 1.000E+02, x2 = 2.000E+00, x3 = 0.000E+00 )

u_1 = 6.155E-02, u_2 = -2.025E+00, u_3 = -1.521E-09

node: 98 ( x1 = 1.000E+02, x2 = 2.000E+00, x3 = 5.000E-01 )

u_1 = 6.155E-02, u_2 = -2.025E+00, u_3 = -1.431E-05

node: 99 ( x1 = 1.000E+02, x2 = 1.000E+00, x3 = -5.000E-01 )

u_1 = 3.077E-02, u_2 = -2.025E+00, u_3 = 7.390E-06

node: 100 ( x1 = 1.000E+02, x2 = 1.000E+00, x3 = 5.000E-01 )

u_1 = 3.077E-02, u_2 = -2.025E+00, u_3 = -7.393E-06

node: 101 ( x1 = 1.000E+02, x2 = 0.000E+00, x3 = -5.000E-01 )

u_1 = 4.020E-06, u_2 = -2.025E+00, u_3 = -1.226E-06

node: 102 ( x1 = 1.000E+02, x2 = 0.000E+00, x3 = 0.000E+00 )

u_1 = 4.064E-06, u_2 = -2.025E+00, u_3 = -1.516E-09

node: 103 ( x1 = 1.000E+02, x2 = 0.000E+00, x3 = 5.000E-01 )

u_1 = 4.020E-06, u_2 = -2.025E+00, u_3 = 1.223E-06

node: 104 ( x1 = 1.000E+02, x2 = -1.000E+00, x3 = -5.000E-01 )

u_1 = -3.076E-02, u_2 = -2.025E+00, u_3 = -9.512E-06

33 V. N. Kaliakin


node: 105 ( x1 = 1.000E+02, x2 = -1.000E+00, x3 = 5.000E-01 )

u_1 = -3.076E-02, u_2 = -2.025E+00, u_3 = 9.509E-06

node: 106 ( x1 = 1.000E+02, x2 = -2.000E+00, x3 = -5.000E-01 )

u_1 = -6.154E-02, u_2 = -2.025E+00, u_3 = -1.767E-05

node: 107 ( x1 = 1.000E+02, x2 = -2.000E+00, x3 = 0.000E+00 )

u_1 = -6.154E-02, u_2 = -2.025E+00, u_3 = -1.511E-09

node: 108 ( x1 = 1.000E+02, x2 = -2.000E+00, x3 = 5.000E-01 )

u_1 = -6.154E-02, u_2 = -2.025E+00, u_3 = 1.767E-05

apes -> end of analysis . . . . . . . .

As evident from the above results, the transverse displacement of the beam tip (nodes 98 to 108)is equal to -2.025 inches, which is approximately 2.8% in error. The magnitude of the maximumaxial stress, obtained at quadrature points in elements 1 or 2 closest to (x = 0), is 2.481E+04,which is approximately 34% in error.

34 V. N. Kaliakin

Bibliography

[1] Kaliakin, V. N., Approximate Solution Techniques, Numerical Modeling and Finite ElementMethods. New York: Marcel Dekker, Inc. (2001).

[2] Logan, D. L., A First Course in the Finite Element Method Using AlgorTM . Boston, MA: PWSPublishing Co. (1997).

[3] User Documentation for MSC Patran, Version 2005r2, MSC Software Corporation, Santa Ana,CA (2005).

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