ELEMENT IRREDUCIBLE TYPE H27P0 command · ELEMENT IRREDUCIBLE TYPE H27P0 command. ... speci c...

APES documentation (revision date March 16, 2011)

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ELEMENT IRREDUCIBLE TYPE H27P0 command.

Synopsis

The ELEMENT IRREDUCIBLE H27P0 command is used to describe all irreducible 27-node quadratic, hexahedral continuum elements that are to be used in mechanical analyses.

Syntax

The following syntax is used to describe a typical H27P0 irreducible hexahedral continuumelement:

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ELEment IRReducible TYPe H27P0 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 H27P0 is an irreducible, triquadratic, isoparametric Lagrangian hexahedral continuumelement [1]. The element

– Has six (6) faces.

– Contains eight (8) vertex nodes.

– Contains twelve (12) mid-side nodes.

– Contains seven (7) interior nodes.

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

– Possesses a total of eighty-one (81) displacement degrees of freedom.

• The numbering order of NODES associated with the H27P0 element is shown in Figure 1(see also Figure 2). For this element the numbers 1 to 27 must be input sequentially.

NOTE: Presently APES does not possess the ability to generate H27P0 elements. It isassumed that the analyst will use some stand alone pre-processing software to accomplish thistask. The resulting element and node data will then be translated to the format expected byAPES.

1

3

2

4

5

6

7

8

910

11

12

16

13

14

15

17

18

19

20

x1

x2

x3

21

22

23

24

25

26

27

Figure 1: Node Numbering Associated with a Typical Irreducible 27-Node Triquadratic (H27P0)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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161

4

5

8

12

17

20

21

15

11

2527

9

23

13

26

2418

2

3

6

14

19

7

2210

x1

x2

x3

Figure 2: Exploded View of Node Numbering Associated with a Typical Irreducible 27-Node Tri-quadratic (H27P0) Hexahedral Continuum Element

• 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 H27P0 elements corresponds to a 3 by 3by 3 Gauss-Legendre numerical integration scheme for the primary dependent variables (i.e.,nodal displacements) and a 2 by 2 by 2 Gauss-Legendre scheme for the secondary dependentvariables (i.e., strains and stresses). This is the default condition and requires no input usingthe INTCODE keyword.

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

INTCODE = 2: A 2 by 2 by 2 Gauss-Legendre numerical integration scheme is used for theprimary dependent variables (i.e., nodal displacements) and a 1-point Gauss-Legendrescheme for the secondary dependent variables (i.e., strains and stresses).

• 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.

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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.

• 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 not

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be used in conjunction with the H27P0 element. Under the default condition the volumetricstrain and the aforementioned ratio are not computed and printed.

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Example of Command Usage

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 3. 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 3: 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 APESformat. The input data file associated with the analysis is given below.

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

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ana tit "cantilever beam involving 5 x 2 x 1 mesh of H27P0 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 165

dim max h27p0 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 10 x1 0.000000000E+00 x2 -1.000000000E+00 x3 -5.000000000E-01

NODES LINE NUMBER 11 x1 0.000000000E+00 x2 -1.000000000E+00 x3 0.000000000E+00

NODES LINE NUMBER 12 x1 0.000000000E+00 x2 -1.000000000E+00 x3 5.000000000E-01







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!

ELEM IRREDUCIBLE TYPE H27P0 MAT 1 &

NODES 37 7 1 31 39 9 3 33 22 4 16 34 24 6 &

18 36 38 8 2 32 35 5 23 17 19 21 20


NODES 43 13 7 37 45 15 9 39 28 10 22 40 30 12 &

24 42 44 14 8 38 41 11 29 23 25 27 26


NODES 67 37 31 61 69 39 33 63 52 34 46 64 54 36 &

48 66 68 38 32 62 65 35 53 47 49 51 50


NODES 73 43 37 67 75 45 39 69 58 40 52 70 60 42 &

54 72 74 44 38 68 71 41 59 53 55 57 56


NODES 97 67 61 91 99 69 63 93 82 64 76 94 84 66 &

78 96 98 68 62 92 95 65 83 77 79 81 80


NODES 103 73 67 97 105 75 69 99 88 70 82 100 90 72 &

84 102 104 74 68 98 101 71 89 83 85 87 86


NODES 127 97 91 121 129 99 93 123 112 94 106 124 114 96 &

108 126 128 98 92 122 125 95 113 107 109 111 110


NODES 133 103 97 127 135 105 99 129 118 100 112 130 120 102 &

114 132 134 104 98 128 131 101 119 113 115 117 116


NODES 157 127 121 151 159 129 123 153 142 124 136 154 144 126 &

138 156 158 128 122 152 155 125 143 137 139 141 140


NODES 163 133 127 157 165 135 129 159 148 130 142 160 150 132 &

144 162 164 134 128 158 161 131 149 143 145 147 146

!

spe conc mec nod 1:15 1_dis 2_dis 3_dis

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spe conc mec nod 151:153:2 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 havebeen used so as to reduce the size of the output file file.

cantilever beam involving 5 x 2 x 1 mesh of H27P0 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 = 165

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

Max. no. of 27-node hexahedral (H27P0) 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

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--> 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

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

= 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

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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:

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

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 = 0.000E+00 )




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




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


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




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




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




11 : ( x1 = 0.000E+00, x2 = -1.000E+00, x3 = 0.000E+00 )




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




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




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




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




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

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

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


153 : ( 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 =

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

--> element 1 ( type = H27P0 ):

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

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

eps_11 = 2.302E-04 ; eps_22 = -7.659E-05 ; eps_33 = -7.645E-05

gam_12 = -4.264E-05 ; gam_13 = -1.055E-06 ; gam_23 = 3.034E-08

sig_11 = 6.647E+03 ; sig_22 = -4.324E+02 ; sig_33 = -4.293E+02

sig_12 = -4.920E+02 ; sig_13 = -1.217E+01 ; sig_23 = 3.501E-01

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

eps_11 = 2.281E-04 ; eps_22 = -4.617E-05 ; eps_33 = -4.680E-05

gam_12 = -1.818E-05 ; gam_13 = 1.533E-06 ; gam_23 = 7.283E-09

sig_11 = 7.602E+03 ; sig_22 = 1.273E+03 ; sig_33 = 1.259E+03

sig_12 = -2.098E+02 ; sig_13 = 1.769E+01 ; sig_23 = 8.403E-02

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

eps_11 = 8.538E-04 ; eps_22 = -1.743E-04 ; eps_33 = -1.751E-04

gam_12 = -2.515E-05 ; gam_13 = 6.056E-06 ; gam_23 = 2.266E-08

sig_11 = 2.843E+04 ; sig_22 = 4.707E+03 ; sig_33 = 4.688E+03

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sig_12 = -2.902E+02 ; sig_13 = 6.987E+01 ; sig_23 = 2.615E-01

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

eps_11 = 8.585E-04 ; eps_22 = -2.850E-04 ; eps_33 = -2.848E-04

gam_12 = -6.969E-07 ; gam_13 = -3.997E-06 ; gam_23 = -1.231E-08

sig_11 = 2.481E+04 ; sig_22 = -1.581E+03 ; sig_33 = -1.576E+03

sig_12 = -8.041E+00 ; sig_13 = -4.611E+01 ; sig_23 = -1.420E-01

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

eps_11 = 2.302E-04 ; eps_22 = -7.659E-05 ; eps_33 = -7.645E-05

gam_12 = -4.264E-05 ; gam_13 = 1.055E-06 ; gam_23 = -3.034E-08

sig_11 = 6.647E+03 ; sig_22 = -4.324E+02 ; sig_33 = -4.293E+02

sig_12 = -4.920E+02 ; sig_13 = 1.217E+01 ; sig_23 = -3.501E-01

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

eps_11 = 2.281E-04 ; eps_22 = -4.617E-05 ; eps_33 = -4.680E-05

gam_12 = -1.818E-05 ; gam_13 = -1.533E-06 ; gam_23 = -7.283E-09

sig_11 = 7.602E+03 ; sig_22 = 1.273E+03 ; sig_33 = 1.259E+03

sig_12 = -2.098E+02 ; sig_13 = -1.769E+01 ; sig_23 = -8.403E-02

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

eps_11 = 8.538E-04 ; eps_22 = -1.743E-04 ; eps_33 = -1.751E-04

gam_12 = -2.515E-05 ; gam_13 = -6.056E-06 ; gam_23 = -2.266E-08

sig_11 = 2.843E+04 ; sig_22 = 4.707E+03 ; sig_33 = 4.688E+03

sig_12 = -2.902E+02 ; sig_13 = -6.987E+01 ; sig_23 = -2.615E-01

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

eps_11 = 8.585E-04 ; eps_22 = -2.850E-04 ; eps_33 = -2.848E-04

gam_12 = -6.969E-07 ; gam_13 = 3.997E-06 ; gam_23 = 1.231E-08

sig_11 = 2.481E+04 ; sig_22 = -1.581E+03 ; sig_33 = -1.576E+03

sig_12 = -8.041E+00 ; sig_13 = 4.611E+01 ; sig_23 = 1.420E-01


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

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

eps_11 = -8.585E-04 ; eps_22 = 2.850E-04 ; eps_33 = 2.848E-04

gam_12 = -6.913E-07 ; gam_13 = 3.995E-06 ; gam_23 = -1.633E-08

sig_11 = -2.481E+04 ; sig_22 = 1.580E+03 ; sig_33 = 1.576E+03

sig_12 = -7.976E+00 ; sig_13 = 4.610E+01 ; sig_23 = -1.884E-01

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

eps_11 = -8.538E-04 ; eps_22 = 1.743E-04 ; eps_33 = 1.751E-04

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gam_12 = -2.515E-05 ; gam_13 = -6.055E-06 ; gam_23 = 2.489E-08

sig_11 = -2.843E+04 ; sig_22 = -4.707E+03 ; sig_33 = -4.688E+03

sig_12 = -2.902E+02 ; sig_13 = -6.986E+01 ; sig_23 = 2.872E-01

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

eps_11 = -2.281E-04 ; eps_22 = 4.617E-05 ; eps_33 = 4.680E-05

gam_12 = -1.818E-05 ; gam_13 = -1.533E-06 ; gam_23 = 6.695E-09

sig_11 = -7.602E+03 ; sig_22 = -1.273E+03 ; sig_33 = -1.259E+03

sig_12 = -2.098E+02 ; sig_13 = -1.769E+01 ; sig_23 = 7.726E-02

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

eps_11 = -2.302E-04 ; eps_22 = 7.659E-05 ; eps_33 = 7.645E-05

gam_12 = -4.264E-05 ; gam_13 = 1.054E-06 ; gam_23 = 3.136E-08

sig_11 = -6.647E+03 ; sig_22 = 4.324E+02 ; sig_33 = 4.293E+02

sig_12 = -4.920E+02 ; sig_13 = 1.217E+01 ; sig_23 = 3.618E-01

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

eps_11 = -8.585E-04 ; eps_22 = 2.850E-04 ; eps_33 = 2.848E-04

gam_12 = -6.913E-07 ; gam_13 = -3.995E-06 ; gam_23 = 1.633E-08

sig_11 = -2.481E+04 ; sig_22 = 1.580E+03 ; sig_33 = 1.576E+03

sig_12 = -7.976E+00 ; sig_13 = -4.610E+01 ; sig_23 = 1.884E-01

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

eps_11 = -8.538E-04 ; eps_22 = 1.743E-04 ; eps_33 = 1.751E-04

gam_12 = -2.515E-05 ; gam_13 = 6.055E-06 ; gam_23 = -2.489E-08

sig_11 = -2.843E+04 ; sig_22 = -4.707E+03 ; sig_33 = -4.688E+03

sig_12 = -2.902E+02 ; sig_13 = 6.986E+01 ; sig_23 = -2.872E-01

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

eps_11 = -2.281E-04 ; eps_22 = 4.617E-05 ; eps_33 = 4.680E-05

gam_12 = -1.818E-05 ; gam_13 = 1.533E-06 ; gam_23 = -6.695E-09

sig_11 = -7.602E+03 ; sig_22 = -1.273E+03 ; sig_33 = -1.259E+03

sig_12 = -2.098E+02 ; sig_13 = 1.769E+01 ; sig_23 = -7.726E-02

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

eps_11 = -2.302E-04 ; eps_22 = 7.659E-05 ; eps_33 = 7.645E-05

gam_12 = -4.264E-05 ; gam_13 = -1.054E-06 ; gam_23 = -3.136E-08

sig_11 = -6.647E+03 ; sig_22 = 4.324E+02 ; sig_33 = 4.293E+02

sig_12 = -4.920E+02 ; sig_13 = -1.217E+01 ; sig_23 = -3.618E-01


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

18 V. N. Kaliakin


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

eps_11 = 1.766E-04 ; eps_22 = -5.216E-05 ; eps_33 = -5.237E-05

gam_12 = -3.566E-05 ; gam_13 = -3.592E-07 ; gam_23 = -1.073E-08

sig_11 = 5.322E+03 ; sig_22 = 4.317E+01 ; sig_33 = 3.838E+01

sig_12 = -4.115E+02 ; sig_13 = -4.144E+00 ; sig_23 = -1.238E-01

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

eps_11 = 1.899E-04 ; eps_22 = -5.565E-05 ; eps_33 = -5.552E-05

gam_12 = -2.169E-05 ; gam_13 = 7.544E-07 ; gam_23 = -3.747E-09

sig_11 = 5.744E+03 ; sig_22 = 7.783E+01 ; sig_33 = 8.101E+01

sig_12 = -2.503E+02 ; sig_13 = 8.705E+00 ; sig_23 = -4.324E-02

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

eps_11 = 7.057E-04 ; eps_22 = -2.073E-04 ; eps_33 = -2.071E-04

gam_12 = -2.163E-05 ; gam_13 = 2.712E-06 ; gam_23 = -1.345E-08

sig_11 = 2.133E+04 ; sig_22 = 2.570E+02 ; sig_33 = 2.609E+02

sig_12 = -2.496E+02 ; sig_13 = 3.130E+01 ; sig_23 = -1.552E-01

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

eps_11 = 6.651E-04 ; eps_22 = -1.955E-04 ; eps_33 = -1.957E-04

gam_12 = -7.686E-06 ; gam_13 = -1.391E-06 ; gam_23 = 3.355E-08

sig_11 = 2.009E+04 ; sig_22 = 2.287E+02 ; sig_33 = 2.227E+02

sig_12 = -8.868E+01 ; sig_13 = -1.605E+01 ; sig_23 = 3.872E-01

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

eps_11 = 1.766E-04 ; eps_22 = -5.216E-05 ; eps_33 = -5.237E-05

gam_12 = -3.566E-05 ; gam_13 = 3.592E-07 ; gam_23 = 1.073E-08

sig_11 = 5.322E+03 ; sig_22 = 4.317E+01 ; sig_33 = 3.838E+01

sig_12 = -4.115E+02 ; sig_13 = 4.144E+00 ; sig_23 = 1.238E-01

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

eps_11 = 1.899E-04 ; eps_22 = -5.565E-05 ; eps_33 = -5.552E-05

gam_12 = -2.169E-05 ; gam_13 = -7.544E-07 ; gam_23 = 3.747E-09

sig_11 = 5.744E+03 ; sig_22 = 7.783E+01 ; sig_33 = 8.101E+01

sig_12 = -2.503E+02 ; sig_13 = -8.705E+00 ; sig_23 = 4.324E-02

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

eps_11 = 7.057E-04 ; eps_22 = -2.073E-04 ; eps_33 = -2.071E-04

gam_12 = -2.163E-05 ; gam_13 = -2.712E-06 ; gam_23 = 1.345E-08

sig_11 = 2.133E+04 ; sig_22 = 2.570E+02 ; sig_33 = 2.609E+02

sig_12 = -2.496E+02 ; sig_13 = -3.130E+01 ; sig_23 = 1.552E-01

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

eps_11 = 6.651E-04 ; eps_22 = -1.955E-04 ; eps_33 = -1.957E-04

gam_12 = -7.686E-06 ; gam_13 = 1.391E-06 ; gam_23 = -3.355E-08

19 V. N. Kaliakin


sig_11 = 2.009E+04 ; sig_22 = 2.287E+02 ; sig_33 = 2.227E+02

sig_12 = -8.868E+01 ; sig_13 = 1.605E+01 ; sig_23 = -3.872E-01


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

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

eps_11 = -6.651E-04 ; eps_22 = 1.955E-04 ; eps_33 = 1.957E-04

gam_12 = -7.656E-06 ; gam_13 = 1.385E-06 ; gam_23 = 1.168E-08

sig_11 = -2.009E+04 ; sig_22 = -2.292E+02 ; sig_33 = -2.226E+02

sig_12 = -8.834E+01 ; sig_13 = 1.598E+01 ; sig_23 = 1.348E-01

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

eps_11 = -7.057E-04 ; eps_22 = 2.073E-04 ; eps_33 = 2.071E-04

gam_12 = -2.165E-05 ; gam_13 = -2.709E-06 ; gam_23 = -4.162E-09

sig_11 = -2.133E+04 ; sig_22 = -2.568E+02 ; sig_33 = -2.610E+02

sig_12 = -2.498E+02 ; sig_13 = -3.126E+01 ; sig_23 = -4.802E-02

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

eps_11 = -1.899E-04 ; eps_22 = 5.566E-05 ; eps_33 = 5.551E-05

gam_12 = -2.169E-05 ; gam_13 = -7.529E-07 ; gam_23 = -6.045E-09

sig_11 = -5.744E+03 ; sig_22 = -7.781E+01 ; sig_33 = -8.113E+01

sig_12 = -2.503E+02 ; sig_13 = -8.688E+00 ; sig_23 = -6.975E-02

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

eps_11 = -1.766E-04 ; eps_22 = 5.215E-05 ; eps_33 = 5.238E-05

gam_12 = -3.566E-05 ; gam_13 = 3.561E-07 ; gam_23 = -5.437E-09

sig_11 = -5.322E+03 ; sig_22 = -4.333E+01 ; sig_33 = -3.810E+01

sig_12 = -4.115E+02 ; sig_13 = 4.108E+00 ; sig_23 = -6.273E-02

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

eps_11 = -6.651E-04 ; eps_22 = 1.955E-04 ; eps_33 = 1.957E-04

gam_12 = -7.656E-06 ; gam_13 = -1.385E-06 ; gam_23 = -1.168E-08

sig_11 = -2.009E+04 ; sig_22 = -2.292E+02 ; sig_33 = -2.226E+02

sig_12 = -8.834E+01 ; sig_13 = -1.598E+01 ; sig_23 = -1.348E-01

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

eps_11 = -7.057E-04 ; eps_22 = 2.073E-04 ; eps_33 = 2.071E-04

gam_12 = -2.165E-05 ; gam_13 = 2.709E-06 ; gam_23 = 4.162E-09

sig_11 = -2.133E+04 ; sig_22 = -2.568E+02 ; sig_33 = -2.610E+02

sig_12 = -2.498E+02 ; sig_13 = 3.126E+01 ; sig_23 = 4.802E-02

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

20 V. N. Kaliakin


eps_11 = -1.899E-04 ; eps_22 = 5.566E-05 ; eps_33 = 5.551E-05

gam_12 = -2.169E-05 ; gam_13 = 7.529E-07 ; gam_23 = 6.045E-09

sig_11 = -5.744E+03 ; sig_22 = -7.781E+01 ; sig_33 = -8.113E+01

sig_12 = -2.503E+02 ; sig_13 = 8.688E+00 ; sig_23 = 6.975E-02

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

eps_11 = -1.766E-04 ; eps_22 = 5.215E-05 ; eps_33 = 5.238E-05

gam_12 = -3.566E-05 ; gam_13 = -3.561E-07 ; gam_23 = 5.437E-09

sig_11 = -5.322E+03 ; sig_22 = -4.333E+01 ; sig_33 = -3.810E+01

sig_12 = -4.115E+02 ; sig_13 = -4.108E+00 ; sig_23 = 6.273E-02


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

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

eps_11 = 1.233E-04 ; eps_22 = -3.522E-05 ; eps_33 = -3.526E-05

gam_12 = -3.196E-05 ; gam_13 = -7.403E-08 ; gam_23 = -3.516E-08

sig_11 = 3.758E+03 ; sig_22 = 1.007E+02 ; sig_33 = 9.984E+01

sig_12 = -3.687E+02 ; sig_13 = -8.542E-01 ; sig_23 = -4.057E-01

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

eps_11 = 1.403E-04 ; eps_22 = -4.370E-05 ; eps_33 = -4.356E-05

gam_12 = -2.865E-05 ; gam_13 = 2.063E-07 ; gam_23 = 2.041E-08

sig_11 = 4.158E+03 ; sig_22 = -8.971E+01 ; sig_33 = -8.650E+01

sig_12 = -3.306E+02 ; sig_13 = 2.381E+00 ; sig_23 = 2.355E-01

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

eps_11 = 5.165E-04 ; eps_22 = -1.624E-04 ; eps_33 = -1.623E-04

gam_12 = -1.464E-05 ; gam_13 = 6.972E-07 ; gam_23 = -6.462E-08

sig_11 = 1.524E+04 ; sig_22 = -4.279E+02 ; sig_33 = -4.244E+02

sig_12 = -1.690E+02 ; sig_13 = 8.044E+00 ; sig_23 = -7.456E-01

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

eps_11 = 4.681E-04 ; eps_22 = -1.319E-04 ; eps_33 = -1.319E-04

gam_12 = -1.146E-05 ; gam_13 = -3.406E-07 ; gam_23 = 1.370E-07

sig_11 = 1.434E+04 ; sig_22 = 4.917E+02 ; sig_33 = 4.921E+02

sig_12 = -1.322E+02 ; sig_13 = -3.930E+00 ; sig_23 = 1.580E+00

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

eps_11 = 1.233E-04 ; eps_22 = -3.522E-05 ; eps_33 = -3.526E-05

gam_12 = -3.196E-05 ; gam_13 = 7.403E-08 ; gam_23 = 3.516E-08

sig_11 = 3.758E+03 ; sig_22 = 1.007E+02 ; sig_33 = 9.984E+01

sig_12 = -3.687E+02 ; sig_13 = 8.542E-01 ; sig_23 = 4.057E-01

21 V. N. Kaliakin


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

eps_11 = 1.403E-04 ; eps_22 = -4.370E-05 ; eps_33 = -4.356E-05

gam_12 = -2.865E-05 ; gam_13 = -2.063E-07 ; gam_23 = -2.041E-08

sig_11 = 4.158E+03 ; sig_22 = -8.971E+01 ; sig_33 = -8.650E+01

sig_12 = -3.306E+02 ; sig_13 = -2.381E+00 ; sig_23 = -2.355E-01

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

eps_11 = 5.165E-04 ; eps_22 = -1.624E-04 ; eps_33 = -1.623E-04

gam_12 = -1.464E-05 ; gam_13 = -6.972E-07 ; gam_23 = 6.462E-08

sig_11 = 1.524E+04 ; sig_22 = -4.279E+02 ; sig_33 = -4.244E+02

sig_12 = -1.690E+02 ; sig_13 = -8.044E+00 ; sig_23 = 7.456E-01

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

eps_11 = 4.681E-04 ; eps_22 = -1.319E-04 ; eps_33 = -1.319E-04

gam_12 = -1.146E-05 ; gam_13 = 3.406E-07 ; gam_23 = -1.370E-07

sig_11 = 1.434E+04 ; sig_22 = 4.917E+02 ; sig_33 = 4.921E+02

sig_12 = -1.322E+02 ; sig_13 = 3.930E+00 ; sig_23 = -1.580E+00


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

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

eps_11 = -4.681E-04 ; eps_22 = 1.318E-04 ; eps_33 = 1.320E-04

gam_12 = -1.132E-05 ; gam_13 = 3.165E-07 ; gam_23 = 9.822E-09

sig_11 = -1.434E+04 ; sig_22 = -4.947E+02 ; sig_33 = -4.909E+02

sig_12 = -1.306E+02 ; sig_13 = 3.652E+00 ; sig_23 = 1.133E-01

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

eps_11 = -5.165E-04 ; eps_22 = 1.625E-04 ; eps_33 = 1.622E-04

gam_12 = -1.471E-05 ; gam_13 = -6.852E-07 ; gam_23 = -1.117E-08

sig_11 = -1.524E+04 ; sig_22 = 4.291E+02 ; sig_33 = 4.239E+02

sig_12 = -1.697E+02 ; sig_13 = -7.906E+00 ; sig_23 = -1.289E-01

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

eps_11 = -1.404E-04 ; eps_22 = 4.373E-05 ; eps_33 = 4.354E-05

gam_12 = -2.866E-05 ; gam_13 = -2.005E-07 ; gam_23 = 7.694E-09

sig_11 = -4.158E+03 ; sig_22 = 9.040E+01 ; sig_33 = 8.595E+01

sig_12 = -3.307E+02 ; sig_13 = -2.313E+00 ; sig_23 = 8.877E-02

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

eps_11 = -1.232E-04 ; eps_22 = 3.513E-05 ; eps_33 = 3.531E-05

gam_12 = -3.193E-05 ; gam_13 = 6.236E-08 ; gam_23 = -5.357E-09

22 V. N. Kaliakin


sig_11 = -3.758E+03 ; sig_22 = -1.028E+02 ; sig_33 = -9.869E+01

sig_12 = -3.684E+02 ; sig_13 = 7.195E-01 ; sig_23 = -6.181E-02

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

eps_11 = -4.681E-04 ; eps_22 = 1.318E-04 ; eps_33 = 1.320E-04

gam_12 = -1.132E-05 ; gam_13 = -3.165E-07 ; gam_23 = -9.822E-09

sig_11 = -1.434E+04 ; sig_22 = -4.947E+02 ; sig_33 = -4.909E+02

sig_12 = -1.306E+02 ; sig_13 = -3.652E+00 ; sig_23 = -1.133E-01

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

eps_11 = -5.165E-04 ; eps_22 = 1.625E-04 ; eps_33 = 1.622E-04

gam_12 = -1.471E-05 ; gam_13 = 6.852E-07 ; gam_23 = 1.117E-08

sig_11 = -1.524E+04 ; sig_22 = 4.291E+02 ; sig_33 = 4.239E+02

sig_12 = -1.697E+02 ; sig_13 = 7.906E+00 ; sig_23 = 1.289E-01

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

eps_11 = -1.404E-04 ; eps_22 = 4.373E-05 ; eps_33 = 4.354E-05

gam_12 = -2.866E-05 ; gam_13 = 2.005E-07 ; gam_23 = -7.694E-09

sig_11 = -4.158E+03 ; sig_22 = 9.040E+01 ; sig_33 = 8.595E+01

sig_12 = -3.307E+02 ; sig_13 = 2.313E+00 ; sig_23 = -8.877E-02

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

eps_11 = -1.232E-04 ; eps_22 = 3.513E-05 ; eps_33 = 3.531E-05

gam_12 = -3.193E-05 ; gam_13 = -6.236E-08 ; gam_23 = 5.357E-09

sig_11 = -3.758E+03 ; sig_22 = -1.028E+02 ; sig_33 = -9.869E+01

sig_12 = -3.684E+02 ; sig_13 = -7.195E-01 ; sig_23 = 6.181E-02


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

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

eps_11 = 7.066E-05 ; eps_22 = -1.993E-05 ; eps_33 = -1.939E-05

gam_12 = -3.060E-05 ; gam_13 = 9.825E-09 ; gam_23 = -1.637E-07

sig_11 = 2.173E+03 ; sig_22 = 8.250E+01 ; sig_33 = 9.505E+01

sig_12 = -3.530E+02 ; sig_13 = 1.134E-01 ; sig_23 = -1.889E+00

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

eps_11 = 8.809E-05 ; eps_22 = -2.809E-05 ; eps_33 = -2.824E-05

gam_12 = -3.089E-05 ; gam_13 = 4.733E-08 ; gam_23 = 7.263E-08

sig_11 = 2.583E+03 ; sig_22 = -9.843E+01 ; sig_33 = -1.020E+02

sig_12 = -3.564E+02 ; sig_13 = 5.461E-01 ; sig_23 = 8.380E-01

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

23 V. N. Kaliakin


eps_11 = 3.206E-04 ; eps_22 = -1.047E-04 ; eps_33 = -1.051E-04

gam_12 = -1.229E-05 ; gam_13 = 9.476E-08 ; gam_23 = -3.213E-07

sig_11 = 9.317E+03 ; sig_22 = -4.982E+02 ; sig_33 = -5.063E+02

sig_12 = -1.418E+02 ; sig_13 = 1.093E+00 ; sig_23 = -3.708E+00

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

eps_11 = 2.716E-04 ; eps_22 = -7.393E-05 ; eps_33 = -7.290E-05

gam_12 = -1.304E-05 ; gam_13 = -1.041E-08 ; gam_23 = 7.563E-07

sig_11 = 8.428E+03 ; sig_22 = 4.535E+02 ; sig_33 = 4.775E+02

sig_12 = -1.504E+02 ; sig_13 = -1.201E-01 ; sig_23 = 8.727E+00

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

eps_11 = 7.066E-05 ; eps_22 = -1.993E-05 ; eps_33 = -1.939E-05

gam_12 = -3.060E-05 ; gam_13 = -9.825E-09 ; gam_23 = 1.637E-07

sig_11 = 2.173E+03 ; sig_22 = 8.250E+01 ; sig_33 = 9.505E+01

sig_12 = -3.530E+02 ; sig_13 = -1.134E-01 ; sig_23 = 1.889E+00

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

eps_11 = 8.809E-05 ; eps_22 = -2.809E-05 ; eps_33 = -2.824E-05

gam_12 = -3.089E-05 ; gam_13 = -4.733E-08 ; gam_23 = -7.263E-08

sig_11 = 2.583E+03 ; sig_22 = -9.843E+01 ; sig_33 = -1.020E+02

sig_12 = -3.564E+02 ; sig_13 = -5.461E-01 ; sig_23 = -8.380E-01

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

eps_11 = 3.206E-04 ; eps_22 = -1.047E-04 ; eps_33 = -1.051E-04

gam_12 = -1.229E-05 ; gam_13 = -9.476E-08 ; gam_23 = 3.213E-07

sig_11 = 9.317E+03 ; sig_22 = -4.982E+02 ; sig_33 = -5.063E+02

sig_12 = -1.418E+02 ; sig_13 = -1.093E+00 ; sig_23 = 3.708E+00

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

eps_11 = 2.716E-04 ; eps_22 = -7.393E-05 ; eps_33 = -7.290E-05

gam_12 = -1.304E-05 ; gam_13 = 1.041E-08 ; gam_23 = -7.563E-07

sig_11 = 8.428E+03 ; sig_22 = 4.535E+02 ; sig_33 = 4.775E+02

sig_12 = -1.504E+02 ; sig_13 = 1.201E-01 ; sig_23 = -8.727E+00


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

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

eps_11 = -2.715E-04 ; eps_22 = 7.321E-05 ; eps_33 = 7.330E-05

gam_12 = -1.254E-05 ; gam_13 = -6.246E-08 ; gam_23 = 1.460E-08

sig_11 = -8.429E+03 ; sig_22 = -4.739E+02 ; sig_33 = -4.719E+02

sig_12 = -1.447E+02 ; sig_13 = -7.207E-01 ; sig_23 = 1.685E-01

24 V. N. Kaliakin


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

eps_11 = -3.206E-04 ; eps_22 = 1.050E-04 ; eps_33 = 1.049E-04

gam_12 = -1.255E-05 ; gam_13 = -5.292E-08 ; gam_23 = -9.920E-09

sig_11 = -9.316E+03 ; sig_22 = 5.063E+02 ; sig_33 = 5.037E+02

sig_12 = -1.448E+02 ; sig_13 = -6.106E-01 ; sig_23 = -1.145E-01

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

eps_11 = -8.817E-05 ; eps_22 = 2.835E-05 ; eps_33 = 2.812E-05

gam_12 = -3.094E-05 ; gam_13 = -2.879E-08 ; gam_23 = 3.926E-10

sig_11 = -2.583E+03 ; sig_22 = 1.055E+02 ; sig_33 = 1.002E+02

sig_12 = -3.569E+02 ; sig_13 = -3.322E-01 ; sig_23 = 4.530E-03

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

eps_11 = -7.046E-05 ; eps_22 = 1.924E-05 ; eps_33 = 1.967E-05

gam_12 = -3.049E-05 ; gam_13 = -4.074E-08 ; gam_23 = 7.097E-09

sig_11 = -2.172E+03 ; sig_22 = -1.019E+02 ; sig_33 = -9.204E+01

sig_12 = -3.518E+02 ; sig_13 = -4.700E-01 ; sig_23 = 8.189E-02

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

eps_11 = -2.715E-04 ; eps_22 = 7.321E-05 ; eps_33 = 7.330E-05

gam_12 = -1.254E-05 ; gam_13 = 6.246E-08 ; gam_23 = -1.460E-08

sig_11 = -8.429E+03 ; sig_22 = -4.739E+02 ; sig_33 = -4.719E+02

sig_12 = -1.447E+02 ; sig_13 = 7.207E-01 ; sig_23 = -1.685E-01

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

eps_11 = -3.206E-04 ; eps_22 = 1.050E-04 ; eps_33 = 1.049E-04

gam_12 = -1.255E-05 ; gam_13 = 5.292E-08 ; gam_23 = 9.920E-09

sig_11 = -9.316E+03 ; sig_22 = 5.063E+02 ; sig_33 = 5.037E+02

sig_12 = -1.448E+02 ; sig_13 = 6.106E-01 ; sig_23 = 1.145E-01

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

eps_11 = -8.817E-05 ; eps_22 = 2.835E-05 ; eps_33 = 2.812E-05

gam_12 = -3.094E-05 ; gam_13 = 2.879E-08 ; gam_23 = -3.926E-10

sig_11 = -2.583E+03 ; sig_22 = 1.055E+02 ; sig_33 = 1.002E+02

sig_12 = -3.569E+02 ; sig_13 = 3.322E-01 ; sig_23 = -4.530E-03

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

eps_11 = -7.046E-05 ; eps_22 = 1.924E-05 ; eps_33 = 1.967E-05

gam_12 = -3.049E-05 ; gam_13 = 4.074E-08 ; gam_23 = -7.097E-09

sig_11 = -2.172E+03 ; sig_22 = -1.019E+02 ; sig_33 = -9.204E+01

sig_12 = -3.518E+02 ; sig_13 = 4.700E-01 ; sig_23 = -8.189E-02

25 V. N. Kaliakin



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

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

eps_11 = 1.950E-05 ; eps_22 = -8.207E-06 ; eps_33 = -4.722E-06

gam_12 = -3.092E-05 ; gam_13 = 3.133E-08 ; gam_23 = -9.550E-07

sig_11 = 5.639E+02 ; sig_22 = -7.558E+01 ; sig_33 = 4.851E+00

sig_12 = -3.567E+02 ; sig_13 = 3.615E-01 ; sig_23 = -1.102E+01

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

eps_11 = 3.512E-05 ; eps_22 = -1.071E-05 ; eps_33 = -1.217E-05

gam_12 = -3.157E-05 ; gam_13 = 4.908E-10 ; gam_23 = 3.963E-07

sig_11 = 1.022E+03 ; sig_22 = -3.513E+01 ; sig_33 = -6.885E+01

sig_12 = -3.643E+02 ; sig_13 = 5.663E-03 ; sig_23 = 4.572E+00

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

eps_11 = 1.238E-04 ; eps_22 = -4.170E-05 ; eps_33 = -4.439E-05

gam_12 = -1.145E-05 ; gam_13 = -6.324E-08 ; gam_23 = -1.848E-06

sig_11 = 3.511E+03 ; sig_22 = -3.091E+02 ; sig_33 = -3.712E+02

sig_12 = -1.321E+02 ; sig_13 = -7.297E-01 ; sig_23 = -2.133E+01

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

eps_11 = 7.951E-05 ; eps_22 = -2.645E-05 ; eps_33 = -2.003E-05

gam_12 = -1.252E-05 ; gam_13 = -2.325E-08 ; gam_23 = 4.407E-06

sig_11 = 2.407E+03 ; sig_22 = -3.878E+01 ; sig_33 = 1.096E+02

sig_12 = -1.445E+02 ; sig_13 = -2.682E-01 ; sig_23 = 5.085E+01

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

eps_11 = 1.950E-05 ; eps_22 = -8.207E-06 ; eps_33 = -4.722E-06

gam_12 = -3.092E-05 ; gam_13 = -3.133E-08 ; gam_23 = 9.550E-07

sig_11 = 5.639E+02 ; sig_22 = -7.558E+01 ; sig_33 = 4.851E+00

sig_12 = -3.567E+02 ; sig_13 = -3.615E-01 ; sig_23 = 1.102E+01

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

eps_11 = 3.512E-05 ; eps_22 = -1.071E-05 ; eps_33 = -1.217E-05

gam_12 = -3.157E-05 ; gam_13 = -4.908E-10 ; gam_23 = -3.963E-07

sig_11 = 1.022E+03 ; sig_22 = -3.513E+01 ; sig_33 = -6.885E+01

sig_12 = -3.643E+02 ; sig_13 = -5.663E-03 ; sig_23 = -4.572E+00

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

eps_11 = 1.238E-04 ; eps_22 = -4.170E-05 ; eps_33 = -4.439E-05

gam_12 = -1.145E-05 ; gam_13 = 6.324E-08 ; gam_23 = 1.848E-06

sig_11 = 3.511E+03 ; sig_22 = -3.091E+02 ; sig_33 = -3.712E+02

sig_12 = -1.321E+02 ; sig_13 = 7.297E-01 ; sig_23 = 2.133E+01

26 V. N. Kaliakin


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

eps_11 = 7.951E-05 ; eps_22 = -2.645E-05 ; eps_33 = -2.003E-05

gam_12 = -1.252E-05 ; gam_13 = 2.325E-08 ; gam_23 = -4.407E-06

sig_11 = 2.407E+03 ; sig_22 = -3.878E+01 ; sig_33 = 1.096E+02

sig_12 = -1.445E+02 ; sig_13 = 2.682E-01 ; sig_23 = -5.085E+01


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

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

eps_11 = -7.798E-05 ; eps_22 = 2.155E-05 ; eps_33 = 2.173E-05

gam_12 = -1.235E-05 ; gam_13 = -3.034E-08 ; gam_23 = 7.171E-08

sig_11 = -2.400E+03 ; sig_22 = -1.032E+02 ; sig_33 = -9.892E+01

sig_12 = -1.425E+02 ; sig_13 = -3.501E-01 ; sig_23 = 8.274E-01

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

eps_11 = -1.243E-04 ; eps_22 = 4.365E-05 ; eps_33 = 4.356E-05

gam_12 = -1.196E-05 ; gam_13 = 1.482E-07 ; gam_23 = -2.995E-08

sig_11 = -3.510E+03 ; sig_22 = 3.656E+02 ; sig_33 = 3.634E+02

sig_12 = -1.380E+02 ; sig_13 = 1.710E+00 ; sig_23 = -3.456E-01

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

eps_11 = -3.569E-05 ; eps_22 = 1.258E-05 ; eps_33 = 1.160E-05

gam_12 = -3.168E-05 ; gam_13 = 2.281E-08 ; gam_23 = -2.159E-08

sig_11 = -1.023E+03 ; sig_22 = 9.124E+01 ; sig_33 = 6.858E+01

sig_12 = -3.656E+02 ; sig_13 = 2.632E-01 ; sig_23 = -2.491E-01

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

eps_11 = -1.822E-05 ; eps_22 = 3.620E-06 ; eps_33 = 5.927E-06

gam_12 = -3.088E-05 ; gam_13 = -1.841E-08 ; gam_23 = 4.447E-08

sig_11 = -5.705E+02 ; sig_22 = -6.654E+01 ; sig_33 = -1.331E+01

sig_12 = -3.563E+02 ; sig_13 = -2.125E-01 ; sig_23 = 5.131E-01

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

eps_11 = -7.798E-05 ; eps_22 = 2.155E-05 ; eps_33 = 2.173E-05

gam_12 = -1.235E-05 ; gam_13 = 3.034E-08 ; gam_23 = -7.171E-08

sig_11 = -2.400E+03 ; sig_22 = -1.032E+02 ; sig_33 = -9.892E+01

sig_12 = -1.425E+02 ; sig_13 = 3.501E-01 ; sig_23 = -8.274E-01

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

eps_11 = -1.243E-04 ; eps_22 = 4.365E-05 ; eps_33 = 4.356E-05

gam_12 = -1.196E-05 ; gam_13 = -1.482E-07 ; gam_23 = 2.995E-08

sig_11 = -3.510E+03 ; sig_22 = 3.656E+02 ; sig_33 = 3.634E+02

27 V. N. Kaliakin


sig_12 = -1.380E+02 ; sig_13 = -1.710E+00 ; sig_23 = 3.456E-01

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

eps_11 = -3.569E-05 ; eps_22 = 1.258E-05 ; eps_33 = 1.160E-05

gam_12 = -3.168E-05 ; gam_13 = -2.281E-08 ; gam_23 = 2.159E-08

sig_11 = -1.023E+03 ; sig_22 = 9.124E+01 ; sig_33 = 6.858E+01

sig_12 = -3.656E+02 ; sig_13 = -2.632E-01 ; sig_23 = 2.491E-01

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

eps_11 = -1.822E-05 ; eps_22 = 3.620E-06 ; eps_33 = 5.927E-06

gam_12 = -3.088E-05 ; gam_13 = 1.841E-08 ; gam_23 = -4.447E-08

sig_11 = -5.705E+02 ; sig_22 = -6.654E+01 ; sig_33 = -1.331E+01

sig_12 = -3.563E+02 ; sig_13 = 2.125E-01 ; sig_23 = -5.131E-01

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

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

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

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

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

u_1 = 1.023E-24, u_2 = -1.656E-24, u_3 = 1.981E-24

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

u_1 = 1.352E-24, u_2 = -2.146E-24, u_3 = 1.429E-33

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

u_1 = 1.023E-24, u_2 = -1.656E-24, u_3 = -1.981E-24

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

u_1 = 4.826E-25, u_2 = 4.080E-25, u_3 = 1.018E-24

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

u_1 = 6.515E-25, u_2 = 5.700E-25, u_3 = 1.482E-33

node: 6 ( x1 = 0.000E+00, x2 = 1.000E+00, x3 = 5.000E-01 )

u_1 = 4.826E-25, u_2 = 4.080E-25, u_3 = -1.018E-24

node: 7 ( x1 = 0.000E+00, x2 = 0.000E+00, x3 = -5.000E-01 )

u_1 = -4.597E-31, u_2 = 6.567E-25, u_3 = -8.606E-30

28 V. N. Kaliakin


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

u_1 = -5.281E-30, u_2 = 8.497E-25, u_3 = 2.813E-33

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

u_1 = -4.530E-31, u_2 = 6.567E-25, u_3 = 8.599E-30

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

u_1 = -4.826E-25, u_2 = 4.080E-25, u_3 = -1.018E-24

node: 11 ( x1 = 0.000E+00, x2 = -1.000E+00, x3 = 0.000E+00 )

u_1 = -6.515E-25, u_2 = 5.700E-25, u_3 = 1.486E-33

node: 12 ( x1 = 0.000E+00, x2 = -1.000E+00, x3 = 5.000E-01 )

u_1 = -4.826E-25, u_2 = 4.080E-25, u_3 = 1.018E-24

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

u_1 = -1.023E-24, u_2 = -1.656E-24, u_3 = -1.981E-24

node: 14 ( x1 = 0.000E+00, x2 = -2.000E+00, x3 = 0.000E+00 )

u_1 = -1.352E-24, u_2 = -2.146E-24, u_3 = 1.433E-33

node: 15 ( x1 = 0.000E+00, x2 = -2.000E+00, x3 = 5.000E-01 )

u_1 = -1.023E-24, u_2 = -1.656E-24, u_3 = 1.981E-24

node: 16 ( x1 = 1.000E+01, x2 = 2.000E+00, x3 = -5.000E-01 )

u_1 = 1.084E-02, u_2 = -2.754E-02, u_3 = 1.887E-04

node: 17 ( x1 = 1.000E+01, x2 = 2.000E+00, x3 = 0.000E+00 )

u_1 = 1.084E-02, u_2 = -2.756E-02, u_3 = -1.399E-10

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

u_1 = 1.084E-02, u_2 = -2.754E-02, u_3 = -1.887E-04

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

u_1 = 5.408E-03, u_2 = -2.726E-02, u_3 = 9.423E-05

node: 20 ( x1 = 1.000E+01, x2 = 1.000E+00, x3 = 0.000E+00 )

u_1 = 5.407E-03, u_2 = -2.728E-02, u_3 = -1.416E-10

node: 21 ( x1 = 1.000E+01, x2 = 1.000E+00, x3 = 5.000E-01 )

u_1 = 5.408E-03, u_2 = -2.726E-02, u_3 = -9.423E-05

node: 22 ( x1 = 1.000E+01, x2 = 0.000E+00, x3 = -5.000E-01 )

u_1 = 3.494E-10, u_2 = -2.716E-02, u_3 = 5.056E-11

29 V. N. Kaliakin


node: 23 ( x1 = 1.000E+01, x2 = 0.000E+00, x3 = 0.000E+00 )

u_1 = 4.033E-10, u_2 = -2.719E-02, u_3 = -1.424E-10

node: 24 ( x1 = 1.000E+01, x2 = 0.000E+00, x3 = 5.000E-01 )

u_1 = 3.778E-10, u_2 = -2.716E-02, u_3 = -3.356E-10

node: 25 ( x1 = 1.000E+01, x2 = -1.000E+00, x3 = -5.000E-01 )

u_1 = -5.408E-03, u_2 = -2.726E-02, u_3 = -9.423E-05

node: 26 ( x1 = 1.000E+01, x2 = -1.000E+00, x3 = 0.000E+00 )

u_1 = -5.407E-03, u_2 = -2.728E-02, u_3 = -1.422E-10

node: 27 ( x1 = 1.000E+01, x2 = -1.000E+00, x3 = 5.000E-01 )

u_1 = -5.408E-03, u_2 = -2.726E-02, u_3 = 9.423E-05

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

u_1 = -1.084E-02, u_2 = -2.754E-02, u_3 = -1.887E-04

node: 29 ( x1 = 1.000E+01, x2 = -2.000E+00, x3 = 0.000E+00 )

u_1 = -1.084E-02, u_2 = -2.756E-02, u_3 = -1.411E-10

node: 30 ( x1 = 1.000E+01, x2 = -2.000E+00, x3 = 5.000E-01 )

u_1 = -1.084E-02, u_2 = -2.754E-02, u_3 = 1.887E-04

node: 31 ( x1 = 2.000E+01, x2 = 2.000E+00, x3 = -5.000E-01 )

u_1 = 2.172E-02, u_2 = -1.091E-01, u_3 = 1.200E-04

node: 32 ( x1 = 2.000E+01, x2 = 2.000E+00, x3 = 0.000E+00 )

u_1 = 2.172E-02, u_2 = -1.091E-01, u_3 = -5.645E-10

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

u_1 = 2.172E-02, u_2 = -1.091E-01, u_3 = -1.200E-04

node: 34 ( x1 = 2.000E+01, x2 = 1.000E+00, x3 = -5.000E-01 )

u_1 = 1.085E-02, u_2 = -1.089E-01, u_3 = 6.049E-05

node: 35 ( x1 = 2.000E+01, x2 = 1.000E+00, x3 = 0.000E+00 )

u_1 = 1.085E-02, u_2 = -1.089E-01, u_3 = -5.662E-10

node: 36 ( x1 = 2.000E+01, x2 = 1.000E+00, x3 = 5.000E-01 )

u_1 = 1.085E-02, u_2 = -1.089E-01, u_3 = -6.049E-05

node: 37 ( x1 = 2.000E+01, x2 = 0.000E+00, x3 = -5.000E-01 )

u_1 = 1.599E-09, u_2 = -1.088E-01, u_3 = -2.410E-09

30 V. N. Kaliakin


node: 38 ( x1 = 2.000E+01, x2 = 0.000E+00, x3 = 0.000E+00 )

u_1 = 1.390E-09, u_2 = -1.088E-01, u_3 = -5.674E-10

node: 39 ( x1 = 2.000E+01, x2 = 0.000E+00, x3 = 5.000E-01 )

u_1 = 1.656E-09, u_2 = -1.088E-01, u_3 = 1.275E-09

node: 40 ( x1 = 2.000E+01, x2 = -1.000E+00, x3 = -5.000E-01 )

u_1 = -1.085E-02, u_2 = -1.089E-01, u_3 = -6.049E-05

node: 41 ( x1 = 2.000E+01, x2 = -1.000E+00, x3 = 0.000E+00 )

u_1 = -1.085E-02, u_2 = -1.089E-01, u_3 = -5.679E-10

node: 42 ( x1 = 2.000E+01, x2 = -1.000E+00, x3 = 5.000E-01 )

u_1 = -1.085E-02, u_2 = -1.089E-01, u_3 = 6.049E-05

node: 43 ( x1 = 2.000E+01, x2 = -2.000E+00, x3 = -5.000E-01 )

u_1 = -2.172E-02, u_2 = -1.091E-01, u_3 = -1.200E-04

node: 44 ( x1 = 2.000E+01, x2 = -2.000E+00, x3 = 0.000E+00 )

u_1 = -2.172E-02, u_2 = -1.091E-01, u_3 = -5.677E-10

node: 45 ( x1 = 2.000E+01, x2 = -2.000E+00, x3 = 5.000E-01 )

u_1 = -2.172E-02, u_2 = -1.091E-01, u_3 = 1.200E-04

node: 46 ( x1 = 3.000E+01, x2 = 2.000E+00, x3 = -5.000E-01 )

u_1 = 3.062E-02, u_2 = -2.403E-01, u_3 = 1.347E-04

node: 47 ( x1 = 3.000E+01, x2 = 2.000E+00, x3 = 0.000E+00 )

u_1 = 3.062E-02, u_2 = -2.403E-01, u_3 = -1.238E-09

node: 48 ( x1 = 3.000E+01, x2 = 2.000E+00, x3 = 5.000E-01 )

u_1 = 3.062E-02, u_2 = -2.403E-01, u_3 = -1.347E-04

node: 49 ( x1 = 3.000E+01, x2 = 1.000E+00, x3 = -5.000E-01 )

u_1 = 1.531E-02, u_2 = -2.401E-01, u_3 = 6.709E-05

node: 50 ( x1 = 3.000E+01, x2 = 1.000E+00, x3 = 0.000E+00 )

u_1 = 1.531E-02, u_2 = -2.401E-01, u_3 = -1.240E-09

node: 51 ( x1 = 3.000E+01, x2 = 1.000E+00, x3 = 5.000E-01 )

u_1 = 1.531E-02, u_2 = -2.401E-01, u_3 = -6.709E-05

node: 52 ( x1 = 3.000E+01, x2 = 0.000E+00, x3 = -5.000E-01 )

u_1 = -1.532E-09, u_2 = -2.401E-01, u_3 = 3.711E-10

31 V. N. Kaliakin


node: 53 ( x1 = 3.000E+01, x2 = 0.000E+00, x3 = 0.000E+00 )

u_1 = -1.278E-09, u_2 = -2.401E-01, u_3 = -1.242E-09

node: 54 ( x1 = 3.000E+01, x2 = 0.000E+00, x3 = 5.000E-01 )

u_1 = -1.454E-09, u_2 = -2.401E-01, u_3 = -2.855E-09

node: 55 ( x1 = 3.000E+01, x2 = -1.000E+00, x3 = -5.000E-01 )

u_1 = -1.531E-02, u_2 = -2.401E-01, u_3 = -6.709E-05

node: 56 ( x1 = 3.000E+01, x2 = -1.000E+00, x3 = 0.000E+00 )

u_1 = -1.531E-02, u_2 = -2.401E-01, u_3 = -1.243E-09

node: 57 ( x1 = 3.000E+01, x2 = -1.000E+00, x3 = 5.000E-01 )

u_1 = -1.531E-02, u_2 = -2.401E-01, u_3 = 6.709E-05

node: 58 ( x1 = 3.000E+01, x2 = -2.000E+00, x3 = -5.000E-01 )

u_1 = -3.062E-02, u_2 = -2.403E-01, u_3 = -1.347E-04

node: 59 ( x1 = 3.000E+01, x2 = -2.000E+00, x3 = 0.000E+00 )

u_1 = -3.062E-02, u_2 = -2.403E-01, u_3 = -1.243E-09

node: 60 ( x1 = 3.000E+01, x2 = -2.000E+00, x3 = 5.000E-01 )

u_1 = -3.062E-02, u_2 = -2.403E-01, u_3 = 1.347E-04

node: 61 ( x1 = 4.000E+01, x2 = 2.000E+00, x3 = -5.000E-01 )

u_1 = 3.911E-02, u_2 = -4.150E-01, u_3 = 1.077E-04

node: 62 ( x1 = 4.000E+01, x2 = 2.000E+00, x3 = 0.000E+00 )

u_1 = 3.912E-02, u_2 = -4.150E-01, u_3 = -2.126E-09

node: 63 ( x1 = 4.000E+01, x2 = 2.000E+00, x3 = 5.000E-01 )

u_1 = 3.911E-02, u_2 = -4.150E-01, u_3 = -1.077E-04

node: 64 ( x1 = 4.000E+01, x2 = 1.000E+00, x3 = -5.000E-01 )

u_1 = 1.953E-02, u_2 = -4.149E-01, u_3 = 5.414E-05

node: 65 ( x1 = 4.000E+01, x2 = 1.000E+00, x3 = 0.000E+00 )

u_1 = 1.953E-02, u_2 = -4.149E-01, u_3 = -2.128E-09

node: 66 ( x1 = 4.000E+01, x2 = 1.000E+00, x3 = 5.000E-01 )

u_1 = 1.953E-02, u_2 = -4.149E-01, u_3 = -5.415E-05

node: 67 ( x1 = 4.000E+01, x2 = 0.000E+00, x3 = -5.000E-01 )

u_1 = 1.460E-08, u_2 = -4.148E-01, u_3 = -1.436E-08

32 V. N. Kaliakin


node: 68 ( x1 = 4.000E+01, x2 = 0.000E+00, x3 = 0.000E+00 )

u_1 = 1.368E-08, u_2 = -4.148E-01, u_3 = -2.130E-09

node: 69 ( x1 = 4.000E+01, x2 = 0.000E+00, x3 = 5.000E-01 )

u_1 = 1.470E-08, u_2 = -4.148E-01, u_3 = 1.010E-08

node: 70 ( x1 = 4.000E+01, x2 = -1.000E+00, x3 = -5.000E-01 )

u_1 = -1.953E-02, u_2 = -4.149E-01, u_3 = -5.416E-05

node: 71 ( x1 = 4.000E+01, x2 = -1.000E+00, x3 = 0.000E+00 )

u_1 = -1.953E-02, u_2 = -4.149E-01, u_3 = -2.132E-09

node: 72 ( x1 = 4.000E+01, x2 = -1.000E+00, x3 = 5.000E-01 )

u_1 = -1.953E-02, u_2 = -4.149E-01, u_3 = 5.415E-05

node: 73 ( x1 = 4.000E+01, x2 = -2.000E+00, x3 = -5.000E-01 )

u_1 = -3.911E-02, u_2 = -4.150E-01, u_3 = -1.077E-04

node: 74 ( x1 = 4.000E+01, x2 = -2.000E+00, x3 = 0.000E+00 )

u_1 = -3.912E-02, u_2 = -4.150E-01, u_3 = -2.133E-09

node: 75 ( x1 = 4.000E+01, x2 = -2.000E+00, x3 = 5.000E-01 )

u_1 = -3.911E-02, u_2 = -4.150E-01, u_3 = 1.077E-04

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

u_1 = 4.559E-02, u_2 = -6.272E-01, u_3 = 9.432E-05

node: 77 ( x1 = 5.000E+01, x2 = 2.000E+00, x3 = 0.000E+00 )

u_1 = 4.559E-02, u_2 = -6.272E-01, u_3 = -3.192E-09

node: 78 ( x1 = 5.000E+01, x2 = 2.000E+00, x3 = 5.000E-01 )

u_1 = 4.559E-02, u_2 = -6.272E-01, u_3 = -9.432E-05

node: 79 ( x1 = 5.000E+01, x2 = 1.000E+00, x3 = -5.000E-01 )

u_1 = 2.280E-02, u_2 = -6.271E-01, u_3 = 4.707E-05

node: 80 ( x1 = 5.000E+01, x2 = 1.000E+00, x3 = 0.000E+00 )

u_1 = 2.280E-02, u_2 = -6.271E-01, u_3 = -3.195E-09

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

u_1 = 2.280E-02, u_2 = -6.271E-01, u_3 = -4.707E-05

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

u_1 = -2.193E-08, u_2 = -6.270E-01, u_3 = 7.247E-09

33 V. N. Kaliakin


node: 83 ( x1 = 5.000E+01, x2 = 0.000E+00, x3 = 0.000E+00 )

u_1 = -2.102E-08, u_2 = -6.270E-01, u_3 = -3.198E-09

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

u_1 = -2.181E-08, u_2 = -6.270E-01, u_3 = -1.364E-08

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

u_1 = -2.280E-02, u_2 = -6.271E-01, u_3 = -4.707E-05

node: 86 ( x1 = 5.000E+01, x2 = -1.000E+00, x3 = 0.000E+00 )

u_1 = -2.280E-02, u_2 = -6.271E-01, u_3 = -3.200E-09

node: 87 ( x1 = 5.000E+01, x2 = -1.000E+00, x3 = 5.000E-01 )

u_1 = -2.280E-02, u_2 = -6.271E-01, u_3 = 4.706E-05

node: 88 ( x1 = 5.000E+01, x2 = -2.000E+00, x3 = -5.000E-01 )

u_1 = -4.559E-02, u_2 = -6.272E-01, u_3 = -9.430E-05

node: 89 ( x1 = 5.000E+01, x2 = -2.000E+00, x3 = 0.000E+00 )

u_1 = -4.559E-02, u_2 = -6.272E-01, u_3 = -3.201E-09

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

u_1 = -4.559E-02, u_2 = -6.272E-01, u_3 = 9.429E-05

node: 91 ( x1 = 6.000E+01, x2 = 2.000E+00, x3 = -5.000E-01 )

u_1 = 5.160E-02, u_2 = -8.706E-01, u_3 = 7.446E-05

node: 92 ( x1 = 6.000E+01, x2 = 2.000E+00, x3 = 0.000E+00 )

u_1 = 5.160E-02, u_2 = -8.706E-01, u_3 = -4.400E-09

node: 93 ( x1 = 6.000E+01, x2 = 2.000E+00, x3 = 5.000E-01 )

u_1 = 5.160E-02, u_2 = -8.706E-01, u_3 = -7.447E-05

node: 94 ( x1 = 6.000E+01, x2 = 1.000E+00, x3 = -5.000E-01 )

u_1 = 2.577E-02, u_2 = -8.705E-01, u_3 = 3.732E-05

node: 95 ( x1 = 6.000E+01, x2 = 1.000E+00, x3 = 0.000E+00 )

u_1 = 2.577E-02, u_2 = -8.705E-01, u_3 = -4.404E-09

node: 96 ( x1 = 6.000E+01, x2 = 1.000E+00, x3 = 5.000E-01 )

u_1 = 2.577E-02, u_2 = -8.705E-01, u_3 = -3.733E-05

node: 97 ( x1 = 6.000E+01, x2 = 0.000E+00, x3 = -5.000E-01 )

u_1 = 1.074E-07, u_2 = -8.704E-01, u_3 = -7.921E-08

34 V. N. Kaliakin


node: 98 ( x1 = 6.000E+01, x2 = 0.000E+00, x3 = 0.000E+00 )

u_1 = 1.048E-07, u_2 = -8.704E-01, u_3 = -4.407E-09

node: 99 ( x1 = 6.000E+01, x2 = 0.000E+00, x3 = 5.000E-01 )

u_1 = 1.076E-07, u_2 = -8.704E-01, u_3 = 7.039E-08

node: 100 ( x1 = 6.000E+01, x2 = -1.000E+00, x3 = -5.000E-01 )

u_1 = -2.577E-02, u_2 = -8.705E-01, u_3 = -3.739E-05

node: 101 ( x1 = 6.000E+01, x2 = -1.000E+00, x3 = 0.000E+00 )

u_1 = -2.577E-02, u_2 = -8.705E-01, u_3 = -4.410E-09

node: 102 ( x1 = 6.000E+01, x2 = -1.000E+00, x3 = 5.000E-01 )

u_1 = -2.577E-02, u_2 = -8.705E-01, u_3 = 3.738E-05

node: 103 ( x1 = 6.000E+01, x2 = -2.000E+00, x3 = -5.000E-01 )

u_1 = -5.160E-02, u_2 = -8.706E-01, u_3 = -7.467E-05

node: 104 ( x1 = 6.000E+01, x2 = -2.000E+00, x3 = 0.000E+00 )

u_1 = -5.160E-02, u_2 = -8.706E-01, u_3 = -4.412E-09

node: 105 ( x1 = 6.000E+01, x2 = -2.000E+00, x3 = 5.000E-01 )

u_1 = -5.160E-02, u_2 = -8.706E-01, u_3 = 7.466E-05

node: 106 ( x1 = 7.000E+01, x2 = 2.000E+00, x3 = -5.000E-01 )

u_1 = 5.559E-02, u_2 = -1.139E+00, u_3 = 5.618E-05

node: 107 ( x1 = 7.000E+01, x2 = 2.000E+00, x3 = 0.000E+00 )

u_1 = 5.559E-02, u_2 = -1.139E+00, u_3 = -5.714E-09

node: 108 ( x1 = 7.000E+01, x2 = 2.000E+00, x3 = 5.000E-01 )

u_1 = 5.559E-02, u_2 = -1.139E+00, u_3 = -5.620E-05

node: 109 ( x1 = 7.000E+01, x2 = 1.000E+00, x3 = -5.000E-01 )

u_1 = 2.780E-02, u_2 = -1.139E+00, u_3 = 2.808E-05

node: 110 ( x1 = 7.000E+01, x2 = 1.000E+00, x3 = 0.000E+00 )

u_1 = 2.780E-02, u_2 = -1.139E+00, u_3 = -5.718E-09

node: 111 ( x1 = 7.000E+01, x2 = 1.000E+00, x3 = 5.000E-01 )

u_1 = 2.780E-02, u_2 = -1.139E+00, u_3 = -2.809E-05

node: 112 ( x1 = 7.000E+01, x2 = 0.000E+00, x3 = -5.000E-01 )

u_1 = -1.970E-07, u_2 = -1.139E+00, u_3 = 5.790E-08

35 V. N. Kaliakin


node: 113 ( x1 = 7.000E+01, x2 = 0.000E+00, x3 = 0.000E+00 )

u_1 = -1.947E-07, u_2 = -1.139E+00, u_3 = -5.722E-09

node: 114 ( x1 = 7.000E+01, x2 = 0.000E+00, x3 = 5.000E-01 )

u_1 = -1.969E-07, u_2 = -1.139E+00, u_3 = -6.934E-08

node: 115 ( x1 = 7.000E+01, x2 = -1.000E+00, x3 = -5.000E-01 )

u_1 = -2.780E-02, u_2 = -1.139E+00, u_3 = -2.804E-05

node: 116 ( x1 = 7.000E+01, x2 = -1.000E+00, x3 = 0.000E+00 )

u_1 = -2.780E-02, u_2 = -1.139E+00, u_3 = -5.726E-09

node: 117 ( x1 = 7.000E+01, x2 = -1.000E+00, x3 = 5.000E-01 )

u_1 = -2.780E-02, u_2 = -1.139E+00, u_3 = 2.803E-05

node: 118 ( x1 = 7.000E+01, x2 = -2.000E+00, x3 = -5.000E-01 )

u_1 = -5.559E-02, u_2 = -1.139E+00, u_3 = -5.602E-05

node: 119 ( x1 = 7.000E+01, x2 = -2.000E+00, x3 = 0.000E+00 )

u_1 = -5.559E-02, u_2 = -1.139E+00, u_3 = -5.729E-09

node: 120 ( x1 = 7.000E+01, x2 = -2.000E+00, x3 = 5.000E-01 )

u_1 = -5.559E-02, u_2 = -1.139E+00, u_3 = 5.601E-05

node: 121 ( x1 = 8.000E+01, x2 = 2.000E+00, x3 = -5.000E-01 )

u_1 = 5.911E-02, u_2 = -1.426E+00, u_3 = 3.913E-05

node: 122 ( x1 = 8.000E+01, x2 = 2.000E+00, x3 = 0.000E+00 )

u_1 = 5.911E-02, u_2 = -1.426E+00, u_3 = -7.097E-09

node: 123 ( x1 = 8.000E+01, x2 = 2.000E+00, x3 = 5.000E-01 )

u_1 = 5.911E-02, u_2 = -1.426E+00, u_3 = -3.915E-05

node: 124 ( x1 = 8.000E+01, x2 = 1.000E+00, x3 = -5.000E-01 )

u_1 = 2.952E-02, u_2 = -1.426E+00, u_3 = 1.959E-05

node: 125 ( x1 = 8.000E+01, x2 = 1.000E+00, x3 = 0.000E+00 )

u_1 = 2.952E-02, u_2 = -1.426E+00, u_3 = -7.102E-09

node: 126 ( x1 = 8.000E+01, x2 = 1.000E+00, x3 = 5.000E-01 )

u_1 = 2.952E-02, u_2 = -1.426E+00, u_3 = -1.961E-05

node: 127 ( x1 = 8.000E+01, x2 = 0.000E+00, x3 = -5.000E-01 )

u_1 = 6.871E-07, u_2 = -1.426E+00, u_3 = -4.351E-07

36 V. N. Kaliakin


node: 128 ( x1 = 8.000E+01, x2 = 0.000E+00, x3 = 0.000E+00 )

u_1 = 6.886E-07, u_2 = -1.426E+00, u_3 = -7.106E-09

node: 129 ( x1 = 8.000E+01, x2 = 0.000E+00, x3 = 5.000E-01 )

u_1 = 6.872E-07, u_2 = -1.426E+00, u_3 = 4.209E-07

node: 130 ( x1 = 8.000E+01, x2 = -1.000E+00, x3 = -5.000E-01 )

u_1 = -2.952E-02, u_2 = -1.426E+00, u_3 = -1.989E-05

node: 131 ( x1 = 8.000E+01, x2 = -1.000E+00, x3 = 0.000E+00 )

u_1 = -2.952E-02, u_2 = -1.426E+00, u_3 = -7.111E-09

node: 132 ( x1 = 8.000E+01, x2 = -1.000E+00, x3 = 5.000E-01 )

u_1 = -2.952E-02, u_2 = -1.426E+00, u_3 = 1.987E-05

node: 133 ( x1 = 8.000E+01, x2 = -2.000E+00, x3 = -5.000E-01 )

u_1 = -5.911E-02, u_2 = -1.426E+00, u_3 = -4.030E-05

node: 134 ( x1 = 8.000E+01, x2 = -2.000E+00, x3 = 0.000E+00 )

u_1 = -5.911E-02, u_2 = -1.426E+00, u_3 = -7.115E-09

node: 135 ( x1 = 8.000E+01, x2 = -2.000E+00, x3 = 5.000E-01 )

u_1 = -5.911E-02, u_2 = -1.426E+00, u_3 = 4.029E-05

node: 136 ( x1 = 9.000E+01, x2 = 2.000E+00, x3 = -5.000E-01 )

u_1 = 6.061E-02, u_2 = -1.726E+00, u_3 = 1.778E-05

node: 137 ( x1 = 9.000E+01, x2 = 2.000E+00, x3 = 0.000E+00 )

u_1 = 6.061E-02, u_2 = -1.726E+00, u_3 = -8.523E-09

node: 138 ( x1 = 9.000E+01, x2 = 2.000E+00, x3 = 5.000E-01 )

u_1 = 6.061E-02, u_2 = -1.726E+00, u_3 = -1.780E-05

node: 139 ( x1 = 9.000E+01, x2 = 1.000E+00, x3 = -5.000E-01 )

u_1 = 3.031E-02, u_2 = -1.726E+00, u_3 = 8.845E-06

node: 140 ( x1 = 9.000E+01, x2 = 1.000E+00, x3 = 0.000E+00 )

u_1 = 3.031E-02, u_2 = -1.726E+00, u_3 = -8.528E-09

node: 141 ( x1 = 9.000E+01, x2 = 1.000E+00, x3 = 5.000E-01 )

u_1 = 3.031E-02, u_2 = -1.726E+00, u_3 = -8.862E-06

node: 142 ( x1 = 9.000E+01, x2 = 0.000E+00, x3 = -5.000E-01 )

u_1 = -1.543E-06, u_2 = -1.726E+00, u_3 = 3.537E-07

37 V. N. Kaliakin


node: 143 ( x1 = 9.000E+01, x2 = 0.000E+00, x3 = 0.000E+00 )

u_1 = -1.545E-06, u_2 = -1.726E+00, u_3 = -8.533E-09

node: 144 ( x1 = 9.000E+01, x2 = 0.000E+00, x3 = 5.000E-01 )

u_1 = -1.543E-06, u_2 = -1.726E+00, u_3 = -3.708E-07

node: 145 ( x1 = 9.000E+01, x2 = -1.000E+00, x3 = -5.000E-01 )

u_1 = -3.031E-02, u_2 = -1.726E+00, u_3 = -8.633E-06

node: 146 ( x1 = 9.000E+01, x2 = -1.000E+00, x3 = 0.000E+00 )

u_1 = -3.031E-02, u_2 = -1.726E+00, u_3 = -8.538E-09

node: 147 ( x1 = 9.000E+01, x2 = -1.000E+00, x3 = 5.000E-01 )

u_1 = -3.031E-02, u_2 = -1.726E+00, u_3 = 8.616E-06

node: 148 ( x1 = 9.000E+01, x2 = -2.000E+00, x3 = -5.000E-01 )

u_1 = -6.062E-02, u_2 = -1.726E+00, u_3 = -1.680E-05

node: 149 ( x1 = 9.000E+01, x2 = -2.000E+00, x3 = 0.000E+00 )

u_1 = -6.062E-02, u_2 = -1.726E+00, u_3 = -8.543E-09

node: 150 ( x1 = 9.000E+01, x2 = -2.000E+00, x3 = 5.000E-01 )

u_1 = -6.062E-02, u_2 = -1.726E+00, u_3 = 1.678E-05

node: 151 ( x1 = 1.000E+02, x2 = 2.000E+00, x3 = -5.000E-01 )

u_1 = 6.169E-02, u_2 = -2.032E+00, u_3 = 1.195E-05

node: 152 ( x1 = 1.000E+02, x2 = 2.000E+00, x3 = 0.000E+00 )

u_1 = 6.169E-02, u_2 = -2.032E+00, u_3 = -9.965E-09

node: 153 ( x1 = 1.000E+02, x2 = 2.000E+00, x3 = 5.000E-01 )

u_1 = 6.169E-02, u_2 = -2.032E+00, u_3 = -1.197E-05

node: 154 ( x1 = 1.000E+02, x2 = 1.000E+00, x3 = -5.000E-01 )

u_1 = 3.081E-02, u_2 = -2.032E+00, u_3 = 6.178E-06

node: 155 ( x1 = 1.000E+02, x2 = 1.000E+00, x3 = 0.000E+00 )

u_1 = 3.081E-02, u_2 = -2.032E+00, u_3 = -9.970E-09

node: 156 ( x1 = 1.000E+02, x2 = 1.000E+00, x3 = 5.000E-01 )

u_1 = 3.081E-02, u_2 = -2.032E+00, u_3 = -6.198E-06

node: 157 ( x1 = 1.000E+02, x2 = 0.000E+00, x3 = -5.000E-01 )

u_1 = 4.069E-06, u_2 = -2.032E+00, u_3 = -2.246E-06

38 V. N. Kaliakin


node: 158 ( x1 = 1.000E+02, x2 = 0.000E+00, x3 = 0.000E+00 )

u_1 = 4.154E-06, u_2 = -2.032E+00, u_3 = -9.975E-09

node: 159 ( x1 = 1.000E+02, x2 = 0.000E+00, x3 = 5.000E-01 )

u_1 = 4.069E-06, u_2 = -2.032E+00, u_3 = 2.226E-06

node: 160 ( x1 = 1.000E+02, x2 = -1.000E+00, x3 = -5.000E-01 )

u_1 = -3.080E-02, u_2 = -2.032E+00, u_3 = -7.179E-06

node: 161 ( x1 = 1.000E+02, x2 = -1.000E+00, x3 = 0.000E+00 )

u_1 = -3.080E-02, u_2 = -2.032E+00, u_3 = -9.981E-09

node: 162 ( x1 = 1.000E+02, x2 = -1.000E+00, x3 = 5.000E-01 )

u_1 = -3.080E-02, u_2 = -2.032E+00, u_3 = 7.159E-06

node: 163 ( x1 = 1.000E+02, x2 = -2.000E+00, x3 = -5.000E-01 )

u_1 = -6.168E-02, u_2 = -2.032E+00, u_3 = -1.788E-05

node: 164 ( x1 = 1.000E+02, x2 = -2.000E+00, x3 = 0.000E+00 )

u_1 = -6.168E-02, u_2 = -2.032E+00, u_3 = -9.986E-09

node: 165 ( x1 = 1.000E+02, x2 = -2.000E+00, x3 = 5.000E-01 )

u_1 = -6.168E-02, u_2 = -2.032E+00, u_3 = 1.786E-05

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

As evident from the above results, the transverse displacement of the beam tip (nodes 151 to165) is equal to -2.032 inches, which is approximately 2.4% in error. The magnitude of the maximumaxial stress, obtained at quadrature points in elements 1 or 2 closest to (x = 0), is 2.843E+04, whichis approximately 24% in error.

Refining to a 10 by 4 by 1 mesh, the transverse displacement of the beam tip increases to -2.063E+00, which is approximately 1.0% in error. The magnitude of the maximum axial stress,obtained at quadrature points closest to (x = 0), is 3.329E+04, which is approximately 11% inerror.

39 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).

40

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