The following syntax is used to describe a typical irreducible continuum element:

Transcription

1 ELEMENT IRREDUCIBLE T7P0 command.. Synopsis The ELEMENT IRREDUCIBLE T7P0 command is used to describe all irreducible 7-node enhanced quadratic triangular continuum elements that are to be used in mechanical analyses. Syntax The following syntax is used to describe a typical irreducible continuum element: Explanatory Notes ELEment IRReducible TYPe T7P0 NODes #:#:# (MATerial #) (INItial #) (THIckness #.#) (INTcode #) (CONstruction #) (EXCavation #) (DONT PRINT Results) (DONT PRINT STRAins) (DONT PRINT STREsses) (PRINT PRIN STRAins) (PRINT PRIN STREsses) (PRINT VOLUMETRIC STRAIN).. The T7P0 is an irreducible, enhanced quadratic, isoparametric triangular continuum element [1]. The element Contains three (3) vertex nodes. Contains three (3) mid-side nodes. Contains one (1) mid-side node. Has two (2) displacements degrees of freedom at each node. Possesses a total of fourteen (14) displacement degrees of freedom. The numbering order of NODES associated with T6P0 elements, which must be specified sequentially from 1 to 7, is shown in Figure 1. NOTE: Presently APES does not possess the ability to generate T7P0 elements. It is assumed that the analyst will thus use some stand-alone pre-processing software to accomplish this task. The resulting element and node data will then be translated to the format expected by APES. 1 V. N. Kaliakin

2 x 2 1 x Figure 1: Node Numbering Associated with a Typical Irreducible 7-Node (T7P0) Triangular Element The MATERIAL keyword is used to specify the number of the material idealization associated 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). The THICKNESS keyword is used to specify the material thickness assumed for the element. Over a given element, the thickness is assumed to be constant. The default THICKNESS value is equal to one (1.0). For AXISYMMETRIC and PLANE STRAIN idealizations (see discussion of the ANALYSIS IDEALIZATION command), the THICKNESS must be equal to 1.0. For such idealizations, specified values different from 1.0 are ignored and the proper value is used. The value specified in conjunction with the INTCODE keyword describes the order of numerical integration scheme to be used in developing the element equations for the element. The commonly used numerical integration rule for T7P0 elements corresponds to a 6- point numerical integration scheme (degree of precision equal to 4) for the primary dependent variables (i.e., nodal displacements) and a 4-point scheme (degree 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 integer values are associated with this keyword: INTCODE = 31: a 3-point numerical integration scheme (degree of precision equal to 2) is used to compute the primary dependent variables (i.e., nodal displacements) and a 1- point scheme (degree of precision equal to 1) is used to compute the secondary dependent variables (i.e., strains and stresses). INTCODE = 63: a 6-point numerical integration scheme (degree of precision equal to 4) is used to compute the primary dependent variables (i.e., nodal displacements). A 3- point scheme (degree of precision equal to 2) is used is used to compute the secondary dependent variables (i.e., strains and stresses). 2 V. N. Kaliakin

3 INTCODE = 64: a 6-point numerical integration scheme (degree of precision equal to 4) is used to compute the primary dependent variables (i.e., nodal displacements). A 4- point scheme (degree of precision equal to 3) is used is used to compute the secondary dependent variables (i.e., strains and stresses). This is equivalent to the aforementioned default setting. The incremental CONSTRUCTION and EXCAVATION numbers represent the time increment 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 the beginning of the analysis. Since this is the default condition, no input is required in such a case. The condition of no excavation is likewise the default. The purpose of the PRINT commands is to eliminate unnecessary output generated by APES. More precisely, if the time history of strains and/or stresses is desired only for a select few elements, this option greatly speeds program output and facilitates inspection of results by the user. Information associated with the elements specified in this section will be printed for every solution (time) step. If generation is performed using this ELEMENT IRREDUCIBLE command, then all the elements generated will be affected in a like manner by the above print control commands. Specification of the keyword DONT PRINT Results indicates that the analyst does not desire to see output of secondary dependent variables (i.e., strains and stresses) for this element. Specification of the DONT PRINT STRAINS keyword indicates that element strains are not to be printed. Under the default condition both strains are printed. Specification of the keyword DONT PRINT STRESSES indicates that stresses are not to be printed. Under the default condition stresses are printed. The PRINT PRIN STRAINS keyword indicates that principal strains are to be computed and printed for the element. Under the default condition these quantities are not computed and printed. The PRINT PRIN STRESSES keyword indicates that principal stresses are to be computed and printed for the element. Under the default condition these quantities are not computed and printed. The keyword PRINT VOLUMETRIC STRAIN causes the volumetric strain to be computed and printed for the element. In addition, the ratio of the absolute value of the volumetric strain to the absolute value of the minimum non-zero normal strain in the element is printed. That is, ε vol min (ε 11, ε 22, ε 33 ) ; min (ε 11, ε 22, ε 33 ) 0 This ratio is instructive in the assessment of mixed and mixed/penalty elements used to simulate material response in the incompressible limit. As such, this keyword would likely not 3 V. N. Kaliakin

4 be used in conjunction with the T7P0 element. Under the default condition the volumetric strain and the aforementioned ratio are not computed and printed. 4 V. N. Kaliakin

5 Example of Command Usage Element Performance in Simple Patch Test Consider a simple eight-element mesh of T7P0 elements. Although the solution domain is square with a side dimension of 2.0, the middle node is purposely not placed at the centroid of the solution domain Ω. As such, the elements are mildly distorted. A distributed traction of 20.0, acting in the positive x 2 -coordinate direction, is applied along the top boundary of Ω. Along the right boundary, a distributed traction of 10.0, acting in the positive x 1 -coordinate direction, is applied. The material is characterized using the isotropic elastic constitutive model with elastic modulus equal to 1000 and a Poisson s ratio equal to The input data associated with this problem is given next. ana tit "patch test C involving 8 quadratic T7P0 triangles" ana tit " sig_11 = 20.0 ; sig_22 = 10.0 ; sig_33 = sig_12 = 0.0" analysis type mech analysis idealization plane_stress analysis temp transient echo init off echo memory off echo grav off echo warn off integration time parameter 0.50 dim max material isotropic elastic 1 dim max nodes 33 dim max t7p0 8 finished settings mat elastic isotropic number 1 desc " test 1 " mod = poisson 0.30 nodes line number 1 x1 0.0 x2 0.0 nodes line number 5 x x2 0.0 incr 1 nodes line number 6 x1 0.0 x2 2.5 nodes line number 10 x x2 2.5 incr 1 nodes line number 11 x1 0.0 x2 4.8 nodes line number 15 x x2 5.2 incr 1 nodes line number 16 x1 0.0 x2 7.5 nodes line number 20 x x2 7.5 incr 1 5 V. N. Kaliakin

6 nodes line number 21 x1 0.0 x nodes line number 25 x x incr 1 nodes line number 26 x E+00 x E+00 nodes line number 27 x E+00 x E+00 nodes line number 28 x E+00 x E+00 nodes line number 29 x E+00 x E+00 nodes line number 30 x E+00 x E+00 nodes line number 31 x E+00 x E+00 nodes line number 32 x E+00 x E+00 nodes line number 33 x E+00 x E+00 element irreducible type "t7p0" nodes mat 1 element irreducible type "t7p0" nodes mat 1 element irreducible type "t7p0" nodes mat 1 element irreducible type "t7p0" nodes mat 1 element irreducible type "t7p0" nodes mat 1 element irreducible type "t7p0" nodes mat 1 element irreducible type "t7p0" nodes mat 1 element irreducible type "t7p0" nodes mat 1 spec line quad mech node_b 21 node_end 25 1_incr 2 2_incr 1 1_hist 0 2_hist 0 & np_begin 10.0 np_end 10.0 spec line quad mech node_b 25 node_end 5 1_incr -10 2_incr -5 1_hist 0 2_hist 0 & np_begin 20.0 np_end 20.0 specification conc mech nodes 1:5 1_for 1_val 0.0 1_hist 0 2_dis 2_val 0.0 specification conc mech nodes 1:21:5 1_dis 1_val 0.0 2_for 2_val 0.0 2_hist 0 finished data solution time final 1.0 increments 1 output 1:5:1 finished load Using the above data in conjunction with the APES computer program, the results shown below are obtained. For clarity, the header that is printed at the top of the file is omitted from this file. patch test C involving 8 quadratic T7P0 triangles sig_11 = 20.0 ; sig_22 = 10.0 ; sig_33 = sig_12 = V. N. Kaliakin

7 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 = 33 Max. no. of ISOTROPIC, LINEAR ELASTIC materials = 1 Max. no. of 7-node triangular (T7P0) elements = 8 = 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 --> TWO-DIMENSIONAL solution domain assumed (PLANE STRESS idealization) --> Nodal coordinates will NOT be updated --> solver type used: SKYLINE --> storage type: SYMMETRIC --> "Isoparametric" mesh generation scheme used = 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" = 5.000E-01 7 V. N. Kaliakin

8 = 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. : test 1 Modulus of Elasticity = 1.000E+03 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 A L C O O R D I N A T E S = node : 1 x1 = 0.000E+00 x2 = 0.000E+00 node : 2 x1 = 2.500E+00 x2 = 0.000E+00 node : 3 x1 = 5.000E+00 x2 = 0.000E+00 node : 4 x1 = 7.500E+00 x2 = 0.000E+00 node : 5 x1 = 1.000E+01 x2 = 0.000E+00 node : 6 x1 = 0.000E+00 x2 = 2.500E+00 8 V. N. Kaliakin

9 node : 7 x1 = 2.500E+00 x2 = 2.500E+00 node : 8 x1 = 5.000E+00 x2 = 2.500E+00 node : 9 x1 = 7.500E+00 x2 = 2.500E+00 node : 10 x1 = 1.000E+01 x2 = 2.500E+00 node : 11 x1 = 0.000E+00 x2 = 4.800E+00 node : 12 x1 = 2.500E+00 x2 = 4.900E+00 node : 13 x1 = 5.000E+00 x2 = 5.000E+00 node : 14 x1 = 7.500E+00 x2 = 5.100E+00 node : 15 x1 = 1.000E+01 x2 = 5.200E+00 node : 16 x1 = 0.000E+00 x2 = 7.500E+00 node : 17 x1 = 2.500E+00 x2 = 7.500E+00 node : 18 x1 = 5.000E+00 x2 = 7.500E+00 node : 19 x1 = 7.500E+00 x2 = 7.500E+00 node : 20 x1 = 1.000E+01 x2 = 7.500E+00 node : 21 x1 = 0.000E+00 x2 = 1.000E+01 node : 22 x1 = 2.500E+00 x2 = 1.000E+01 node : 23 x1 = 5.000E+00 x2 = 1.000E+01 node : 24 x1 = 7.500E+00 x2 = 1.000E+01 node : 25 x1 = 1.000E+01 x2 = 1.000E+01 node : 26 x1 = 1.667E+00 x2 = 3.267E+00 node : 27 x1 = 3.333E+00 x2 = 1.667E+00 node : 28 x1 = 6.667E+00 x2 = 1.667E+00 node : 29 x1 = 8.333E+00 x2 = 3.400E+00 node : 30 x1 = 1.667E+00 x2 = 6.600E+00 node : 31 x1 = 3.333E+00 x2 = 8.333E+00 node : 32 x1 = 6.667E+00 x2 = 8.333E+00 node : 33 x1 = 8.333E+00 x2 = 6.733E+00 = E L E M E N T I N F O R M A T I O N = --> number: 1 (type: T7P0 ) (kind: IRREDUCIBLE ) nodes: integration rule for primary variables: 6-point formula for triangles integration rule for secondary variables: 4-point formula for triangles material no. = 1 material type: isotropic linear elastic thickness = 1.000E > number: 2 (type: T7P0 ) (kind: IRREDUCIBLE ) nodes: V. N. Kaliakin

10 integration rule for primary variables: 6-point formula for triangles integration rule for secondary variables: 4-point formula for triangles material no. = 1 material type: isotropic linear elastic thickness = 1.000E > number: 3 (type: T7P0 ) (kind: IRREDUCIBLE ) nodes: integration rule for primary variables: 6-point formula for triangles integration rule for secondary variables: 4-point formula for triangles material no. = 1 material type: isotropic linear elastic thickness = 1.000E > number: 4 (type: T7P0 ) (kind: IRREDUCIBLE ) nodes: integration rule for primary variables: 6-point formula for triangles integration rule for secondary variables: 4-point formula for triangles material no. = 1 material type: isotropic linear elastic thickness = 1.000E > number: 5 (type: T7P0 ) (kind: IRREDUCIBLE ) nodes: integration rule for primary variables: 6-point formula for triangles integration rule for secondary variables: 4-point formula for triangles material no. = 1 material type: isotropic linear elastic thickness = 1.000E > number: 6 (type: T7P0 ) (kind: IRREDUCIBLE ) nodes: integration rule for primary variables: 6-point formula for triangles integration rule for secondary variables: 4-point formula for triangles material no. = 1 material type: isotropic linear elastic thickness = 1.000E > number: 7 (type: T7P0 ) (kind: IRREDUCIBLE ) nodes: V. N. Kaliakin

11 integration rule for primary variables: 6-point formula for triangles integration rule for secondary variables: 4-point formula for triangles material no. = 1 material type: isotropic linear elastic thickness = 1.000E > number: 8 (type: T7P0 ) (kind: IRREDUCIBLE ) nodes: integration rule for primary variables: 6-point formula for triangles integration rule for secondary variables: 4-point formula for triangles material no. = 1 material type: isotropic linear elastic thickness = 1.000E = 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 = 0.000E+00 ) displacement-1 = 0.000E+00 ; history no. = -2 displacement-2 = 0.000E+00 ; history no. = -2 2 : ( x1 = 2.500E+00, x2 = 0.000E+00 ) force-1 = 0.000E+00 ; history no. = 0 displacement-2 = 0.000E+00 ; history no. = -2 3 : ( x1 = 5.000E+00, x2 = 0.000E+00 ) force-1 = 0.000E+00 ; history no. = 0 displacement-2 = 0.000E+00 ; history no. = -2 4 : ( x1 = 7.500E+00, x2 = 0.000E+00 ) force-1 = 0.000E+00 ; history no. = 0 displacement-2 = 0.000E+00 ; history no. = -2 5 : ( x1 = 1.000E+01, x2 = 0.000E+00 ) force-1 = 1.600E+01 ; history no. = 0 displacement-2 = 0.000E+00 ; history no. = V. N. Kaliakin

12 6 : ( x1 = 0.000E+00, x2 = 2.500E+00 ) displacement-1 = 0.000E+00 ; history no. = -2 force-2 = 0.000E+00 ; history no. = 0 10 : ( x1 = 1.000E+01, x2 = 2.500E+00 ) force-1 = 6.933E+01 ; history no. = 0 force-2 = 0.000E+00 ; history no. = 0 11 : ( x1 = 0.000E+00, x2 = 4.800E+00 ) displacement-1 = 0.000E+00 ; history no. = -2 force-2 = 0.000E+00 ; history no. = 0 15 : ( x1 = 1.000E+01, x2 = 5.200E+00 ) force-1 = 3.333E+01 ; history no. = 0 force-2 = 0.000E+00 ; history no. = 0 16 : ( x1 = 0.000E+00, x2 = 7.500E+00 ) displacement-1 = 0.000E+00 ; history no. = -2 force-2 = 0.000E+00 ; history no. = 0 20 : ( x1 = 1.000E+01, x2 = 7.500E+00 ) force-1 = 6.400E+01 ; history no. = 0 force-2 = 0.000E+00 ; history no. = 0 21 : ( x1 = 0.000E+00, x2 = 1.000E+01 ) displacement-1 = 0.000E+00 ; history no. = -2 force-2 = 8.333E+00 ; history no. = 0 22 : ( x1 = 2.500E+00, x2 = 1.000E+01 ) force-1 = 0.000E+00 ; history no. = 0 force-2 = 3.333E+01 ; history no. = 0 23 : ( x1 = 5.000E+00, x2 = 1.000E+01 ) force-1 = 0.000E+00 ; history no. = 0 force-2 = 1.667E+01 ; history no. = 0 24 : ( x1 = 7.500E+00, x2 = 1.000E+01 ) force-1 = 0.000E+00 ; history no. = 0 force-2 = 3.333E+01 ; history no. = 0 25 : ( x1 = 1.000E+01, x2 = 1.000E+01 ) force-1 = 1.733E+01 ; history no. = 0 force-2 = 8.333E+00 ; history no. = 0 12 V. N. Kaliakin

13 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 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 = T7P0 = 1.667E+00, x2 = 3.267E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = E-03 ; gam_12 = 1.885E-10 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = = 1.000E+00, x2 = 1.979E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = E-03 ; gam_12 = E-12 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = = 3.000E+00, x2 = 3.947E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = E-03 ; gam_12 = 2.740E-10 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = = 1.000E+00, x2 = 3.899E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = E-03 ; gam_12 = 3.026E-10 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = 1.164E-07 --> element 2 ( type = T7P0 = 3.333E+00, x2 = 1.667E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = E-03 ; gam_12 = E-10 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = E V. N. Kaliakin

14 @(x1 = 2.000E+00, x2 = 1.000E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = E-03 ; gam_12 = E-10 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = = 4.000E+00, x2 = 1.000E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = E-03 ; gam_12 = E-10 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = = 4.000E+00, x2 = 3.000E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = E-03 ; gam_12 = E-10 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = E-07 --> element 3 ( type = T7P0 = 6.667E+00, x2 = 1.667E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = E-03 ; gam_12 = 2.106E-11 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = = 6.000E+00, x2 = 1.000E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = E-03 ; gam_12 = 1.530E-10 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = = 8.000E+00, x2 = 1.000E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = E-03 ; gam_12 = 8.722E-12 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = = 6.000E+00, x2 = 3.000E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = E-03 ; gam_12 = E-11 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = E-08 --> element 4 ( type = T7P0 = 8.333E+00, x2 = 3.400E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = E-03 ; gam_12 = E-10 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = = 9.000E+00, x2 = 2.021E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = E-03 ; gam_12 = E-10 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = E V. N. Kaliakin

15 @(x1 = 9.000E+00, x2 = 4.101E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = E-03 ; gam_12 = E-11 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = = 7.000E+00, x2 = 4.053E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = E-03 ; gam_12 = E-10 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = E-07 --> element 5 ( type = T7P0 = 1.667E+00, x2 = 6.600E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = E-03 ; gam_12 = E-10 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = = 1.000E+00, x2 = 5.899E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = E-03 ; gam_12 = 3.822E-12 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = = 3.000E+00, x2 = 5.947E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = E-03 ; gam_12 = E-10 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = = 1.000E+00, x2 = 7.979E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = E-03 ; gam_12 = E-10 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = E-07 --> element 6 ( type = T7P0 = 3.333E+00, x2 = 8.333E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = E-03 ; gam_12 = 9.784E-11 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = = 4.000E+00, x2 = 7.000E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = E-03 ; gam_12 = E-11 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = = 4.000E+00, x2 = 9.000E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = E-03 ; gam_12 = 1.525E V. N. Kaliakin

16 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = = 2.000E+00, x2 = 9.000E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = E-03 ; gam_12 = 2.352E-10 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = 9.045E-08 --> element 7 ( type = T7P0 = 6.667E+00, x2 = 8.333E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = E-03 ; gam_12 = E-10 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = = 6.000E+00, x2 = 7.000E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = E-03 ; gam_12 = E-10 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = = 8.000E+00, x2 = 9.000E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = E-03 ; gam_12 = E-10 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = = 6.000E+00, x2 = 9.000E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = E-03 ; gam_12 = E-11 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = E-08 --> element 8 ( type = T7P0 = 8.333E+00, x2 = 6.733E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = E-03 ; gam_12 = 1.308E-10 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = = 7.000E+00, x2 = 6.053E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = E-03 ; gam_12 = 1.669E-10 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = = 9.000E+00, x2 = 6.101E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = E-03 ; gam_12 = E-11 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = = 9.000E+00, x2 = 8.021E+00): 16 V. N. Kaliakin

17 eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = E-03 ; gam_12 = 2.422E-10 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = 9.317E-08 = N O D A L Q U A N T I T I E S = node: 1 ( x1 = 0.000E+00, x2 = 0.000E+00 ) u_1 = 9.340E-22, u_2 = 2.047E-22 node: 2 ( x1 = 2.500E+00, x2 = 0.000E+00 ) u_1 = 4.250E-02, u_2 = 1.308E-22 node: 3 ( x1 = 5.000E+00, x2 = 0.000E+00 ) u_1 = 8.500E-02, u_2 = 3.210E-22 node: 4 ( x1 = 7.500E+00, x2 = 0.000E+00 ) u_1 = 1.275E-01, u_2 = 1.308E-22 node: 5 ( x1 = 1.000E+01, x2 = 0.000E+00 ) u_1 = 1.700E-01, u_2 = 4.397E-22 node: 6 ( x1 = 0.000E+00, x2 = 2.500E+00 ) u_1 = 2.597E-22, u_2 = 1.000E-02 node: 7 ( x1 = 2.500E+00, x2 = 2.500E+00 ) u_1 = 4.250E-02, u_2 = 1.000E-02 node: 8 ( x1 = 5.000E+00, x2 = 2.500E+00 ) u_1 = 8.500E-02, u_2 = 1.000E-02 node: 9 ( x1 = 7.500E+00, x2 = 2.500E+00 ) u_1 = 1.275E-01, u_2 = 1.000E-02 node: 10 ( x1 = 1.000E+01, x2 = 2.500E+00 ) u_1 = 1.700E-01, u_2 = 1.000E-02 node: 11 ( x1 = 0.000E+00, x2 = 4.800E+00 ) u_1 = 6.483E-22, u_2 = 1.920E-02 node: 12 ( x1 = 2.500E+00, x2 = 4.900E+00 ) 17 V. N. Kaliakin

18 u_1 = 4.250E-02, u_2 = 1.960E-02 node: 13 ( x1 = 5.000E+00, x2 = 5.000E+00 ) u_1 = 8.500E-02, u_2 = 2.000E-02 node: 14 ( x1 = 7.500E+00, x2 = 5.100E+00 ) u_1 = 1.275E-01, u_2 = 2.040E-02 node: 15 ( x1 = 1.000E+01, x2 = 5.200E+00 ) u_1 = 1.700E-01, u_2 = 2.080E-02 node: 16 ( x1 = 0.000E+00, x2 = 7.500E+00 ) u_1 = 2.630E-22, u_2 = 3.000E-02 node: 17 ( x1 = 2.500E+00, x2 = 7.500E+00 ) u_1 = 4.250E-02, u_2 = 3.000E-02 node: 18 ( x1 = 5.000E+00, x2 = 7.500E+00 ) u_1 = 8.500E-02, u_2 = 3.000E-02 node: 19 ( x1 = 7.500E+00, x2 = 7.500E+00 ) u_1 = 1.275E-01, u_2 = 3.000E-02 node: 20 ( x1 = 1.000E+01, x2 = 7.500E+00 ) u_1 = 1.700E-01, u_2 = 3.000E-02 node: 21 ( x1 = 0.000E+00, x2 = 1.000E+01 ) u_1 = 8.230E-22, u_2 = 4.000E-02 node: 22 ( x1 = 2.500E+00, x2 = 1.000E+01 ) u_1 = 4.250E-02, u_2 = 4.000E-02 node: 23 ( x1 = 5.000E+00, x2 = 1.000E+01 ) u_1 = 8.500E-02, u_2 = 4.000E-02 node: 24 ( x1 = 7.500E+00, x2 = 1.000E+01 ) u_1 = 1.275E-01, u_2 = 4.000E-02 node: 25 ( x1 = 1.000E+01, x2 = 1.000E+01 ) u_1 = 1.700E-01, u_2 = 4.000E-02 node: 26 ( x1 = 1.667E+00, x2 = 3.267E+00 ) u_1 = 2.834E-02, u_2 = 1.307E-02 node: 27 ( x1 = 3.333E+00, x2 = 1.667E+00 ) 18 V. N. Kaliakin

19 u_1 = 5.666E-02, u_2 = 6.668E-03 node: 28 ( x1 = 6.667E+00, x2 = 1.667E+00 ) u_1 = 1.133E-01, u_2 = 6.668E-03 node: 29 ( x1 = 8.333E+00, x2 = 3.400E+00 ) u_1 = 1.417E-01, u_2 = 1.360E-02 node: 30 ( x1 = 1.667E+00, x2 = 6.600E+00 ) u_1 = 2.834E-02, u_2 = 2.640E-02 node: 31 ( x1 = 3.333E+00, x2 = 8.333E+00 ) u_1 = 5.666E-02, u_2 = 3.333E-02 node: 32 ( x1 = 6.667E+00, x2 = 8.333E+00 ) u_1 = 1.133E-01, u_2 = 3.333E-02 node: 33 ( x1 = 8.333E+00, x2 = 6.733E+00 ) u_1 = 1.417E-01, u_2 = 2.693E-02 apes -> end of analysis V. N. Kaliakin

20 Bibliography [1] Kaliakin, V. N., Approximate Solution Techniques, Numerical Modeling and Finite Element Methods. New York: Marcel Dekker, Inc. (2001). 20