Stretching of a Prismatic Bar by its Own Weight

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1 APES documentation (revision date: 12.03.10) Stretching of a Prismatic Bar by its Own Weight. This sample analysis is provided in order to illustrate the correct specification of the gravitational acceleration g and of the material density ρ. As mentioned in the explanatory notes accompanying the program documentation, the exact values of g and ρ are not so important, provided that their product is equal to the weight density of the material. Consider a prismatic bar of length L equal to 12 and square cross-section of unit area. A schematic illustration of the bar is provided in Figure 1. z 1 1 bar assumed to be fixed at top E = 1.0e+06 L = 12 g = 500 x y Figure 1: Prismatic Bar Loaded by its Own Weight The exact solution for the stress and strain fields associated with this problem is [1] σ 33 = ρgz ; σ 11 = σ 22 = σ 12 = σ 13 = σ 23 = 0 (1) ε 33 = ρgz E ; ε 11 = ε 22 = ν ρgz E ; ε 12 = ε 13 = ε 23 = 0 (2)

2 where ρ is the mass density, g is the gravitational acceleration, E is the elastic modulus, and ν is Poisson s ratio. The exact displacement field depends upon the specific support conditions at z = L. The support must be such as to prevent translational and rotational motion of the bar. Typically the former constraint is realized by setting u = v = w = 0 at the point (x, y, z) = (0, 0, L) (i.e., at the centroid of the upper surface of the bar). To eliminate rotation of the bar about the axes through the centroid at z = 0, the following conditions are specified at the centroid: u z = v z = 0, v x = 0 (3) The resulting displacement field is u = ν ρgxz E v = ν ρgyz E w = ρg { z 2 + ν [ x 2 + y 2] L 2} (6) 2E The mesh used in the finite element analysis is shown in Figure 2. Although the displacement boundary conditions at node 41 (x, y, z) = (0, 0, L) can be specified exactly, the absence of rotational degrees of freedom precludes the constraints on slope (equation 3) from being realized. In order to obtain a well-posed problem, all nodal displacements must be fixed along the upper surface of the bar. The data file for this case is shown below. Note that the direction of the gravitational acceleration vector is specified as acting in the negative z-direction. The magnitude of this acceleration is set equal to one (1.0). As such, the material density is input equal to 500.0, thus assuring the product ρg to have the correct magnitude. ana tit "3-d stretching of a bar by its own weight" ana tit "mesh consisting of 16 H8P0 elements" analysis action analyze analysis description linear analysis idealization three-dimensional analysis temporal transient analysis type mechanical echo init off echo trans off echo warn off dim max material isotropic elastic 1 dim max nodes 45 dim max h8p0 16 gravity accel value 1.0 gravity accel history 0 gravity x1_angle value 90.0 (4) (5)

3 gravity x2_angle value 90.0 gravity x3_angle value 180.0 gravity angle history 0 fin sett material solid number 1 density 500.0 material elastic isotropic number 1 & description can be omitted modulus 1.0e+06 poissons 0.0 nodes line number 1 x1-0.5 x2-0.5 nodes line number 37 x1-0.5 x2-0.5 x3 12.0 inc 9 nodes line number 2 x2-0.5 nodes line number 38 x2-0.5 x3 12.0 inc 9 nodes line number 3 x1 0.5 x2-0.5 nodes line number 39 x1 0.5 x2-0.5 x3 12.0 inc 9 nodes line number 4 x1-0.5 nodes line number 40 x1-0.5 x3 12.0 inc 9 nodes line number 5 nodes line number 41 x3 12.0 inc 9 nodes line number 6 x1 0.5 nodes line number 42 x1 0.5 x3 12.0 inc 9 nodes line number 7 x1-0.5 x2 0.5 nodes line number 43 x1-0.5 x2 0.5 x3 12.0 inc 9 nodes line number 8 x2 0.5 nodes line number 44 x2 0.5 x3 12.0 inc 9 nodes line number 9 x1 0.5 x2 0.5 nodes line number 45 x1 0.5 x2 0.5 x3 12.0 inc 9 element irreducible type "h8p0" nodes 1 4 5 2 10 13 14 11 mat 1 & 1_add 1 1_inc 1 2_add 1 2_inc 3 3_add 3 3_inc 9 spe conc mec nod 41 1_disp 2_disp 3_disp spe conc mec nod 37:40 42:45 1_disp 2_disp 3_disp finish data solution time final 1.0 increments 1 output 1 finished loading From equilibrium, the exact distribution of σ 33 will always be predicted by the finite element analysis. The correct strain and displacement fields will, however, only be realized for the trivial case of ν = 0. Consequently, this value of ν was assumed. The problem was modelled and analyzed using the using the AP ES computer program. The results generated obtained are provided below.

4 z 39 38 42 37 41 40 44 43 30 45 34 21 36 25 12 27 16 18 3 7 x 6 9 8 y Figure 2: Eight-Node Hexahedral Finite Element Model Used in Analysis of Prismatic Bar Loaded by its Own Weight

5 3-d stretching of a bar by its own weight mesh consisting of 16 H8P0 elements 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 = 45 Max. no. of ISOTROPIC, LINEAR ELASTIC materials = 1 Max. no. of 8-node hexahedral (H8P0) elements = 16 = 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 = G R A V I T A T I O N A L I N F O R M A T I O N =

6 Initial value of gravitational acceleration, "g" = 1.000E+00 History function number associated with "g" = 0 Angle (in degrees) that "g" makes with the x1-axis = 9.000E+01 Angle (in degrees) that "g" makes with the x2-axis = 9.000E+01 Angle (in degrees) that "g" makes with the x3-axis = 1.800E+02 History function number associated w/these angles = 0 = 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. : Modulus of Elasticity = 1.000E+06 Poisson s ratio = 0.000E+00 Elastic bulk modulus of the solid phase = 0.000E+00 Material density of the solid phase = 5.000E+02 Combined bulk modulus for solid/fluid = 1.000E+10

7 = N O D A L C O O R D I N A T E S = node : 1 x1 = -5.000E-01 x2 = -5.000E-01 x3 = 0.000E+00 node : 2 x1 = 0.000E+00 x2 = -5.000E-01 x3 = 0.000E+00 node : 3 x1 = 5.000E-01 x2 = -5.000E-01 x3 = 0.000E+00 node : 4 x1 = -5.000E-01 x2 = 0.000E+00 x3 = 0.000E+00 node : 5 x1 = 0.000E+00 x2 = 0.000E+00 x3 = 0.000E+00 node : 6 x1 = 5.000E-01 x2 = 0.000E+00 x3 = 0.000E+00 node : 7 x1 = -5.000E-01 x2 = 5.000E-01 x3 = 0.000E+00 node : 8 x1 = 0.000E+00 x2 = 5.000E-01 x3 = 0.000E+00 node : 9 x1 = 5.000E-01 x2 = 5.000E-01 x3 = 0.000E+00 node : 10 x1 = -5.000E-01 x2 = -5.000E-01 x3 = 3.000E+00 node : 11 x1 = 0.000E+00 x2 = -5.000E-01 x3 = 3.000E+00 node : 12 x1 = 5.000E-01 x2 = -5.000E-01 x3 = 3.000E+00 node : 13 x1 = -5.000E-01 x2 = 0.000E+00 x3 = 3.000E+00 node : 14 x1 = 0.000E+00 x2 = 0.000E+00 x3 = 3.000E+00 node : 15 x1 = 5.000E-01 x2 = 0.000E+00 x3 = 3.000E+00 node : 16 x1 = -5.000E-01 x2 = 5.000E-01 x3 = 3.000E+00 node : 17 x1 = 0.000E+00 x2 = 5.000E-01 x3 = 3.000E+00 node : 18 x1 = 5.000E-01 x2 = 5.000E-01 x3 = 3.000E+00 node : 19 x1 = -5.000E-01 x2 = -5.000E-01 x3 = 6.000E+00 node : 20 x1 = 0.000E+00 x2 = -5.000E-01 x3 = 6.000E+00 node : 21 x1 = 5.000E-01 x2 = -5.000E-01 x3 = 6.000E+00 node : 22 x1 = -5.000E-01 x2 = 0.000E+00 x3 = 6.000E+00 node : 23 x1 = 0.000E+00 x2 = 0.000E+00 x3 = 6.000E+00 node : 24 x1 = 5.000E-01 x2 = 0.000E+00 x3 = 6.000E+00 node : 25 x1 = -5.000E-01 x2 = 5.000E-01 x3 = 6.000E+00 node : 26 x1 = 0.000E+00 x2 = 5.000E-01 x3 = 6.000E+00 node : 27 x1 = 5.000E-01 x2 = 5.000E-01 x3 = 6.000E+00 node : 28 x1 = -5.000E-01 x2 = -5.000E-01 x3 = 9.000E+00 node : 29 x1 = 0.000E+00 x2 = -5.000E-01 x3 = 9.000E+00 node : 30 x1 = 5.000E-01 x2 = -5.000E-01 x3 = 9.000E+00 node : 31 x1 = -5.000E-01 x2 = 0.000E+00 x3 = 9.000E+00 node : 32 x1 = 0.000E+00 x2 = 0.000E+00 x3 = 9.000E+00 node : 33 x1 = 5.000E-01 x2 = 0.000E+00 x3 = 9.000E+00 node : 34 x1 = -5.000E-01 x2 = 5.000E-01 x3 = 9.000E+00 node : 35 x1 = 0.000E+00 x2 = 5.000E-01 x3 = 9.000E+00 node : 36 x1 = 5.000E-01 x2 = 5.000E-01 x3 = 9.000E+00 node : 37 x1 = -5.000E-01 x2 = -5.000E-01 x3 = 1.200E+01 node : 38 x1 = 0.000E+00 x2 = -5.000E-01 x3 = 1.200E+01 node : 39 x1 = 5.000E-01 x2 = -5.000E-01 x3 = 1.200E+01 node : 40 x1 = -5.000E-01 x2 = 0.000E+00 x3 = 1.200E+01 node : 41 x1 = 0.000E+00 x2 = 0.000E+00 x3 = 1.200E+01 node : 42 x1 = 5.000E-01 x2 = 0.000E+00 x3 = 1.200E+01 node : 43 x1 = -5.000E-01 x2 = 5.000E-01 x3 = 1.200E+01 node : 44 x1 = 0.000E+00 x2 = 5.000E-01 x3 = 1.200E+01 node : 45 x1 = 5.000E-01 x2 = 5.000E-01 x3 = 1.200E+01

8 = E L E M E N T I N F O R M A T I O N = --> number: 1 (type: H8P0 ) (kind: IRREDUCIBLE ) 1 4 5 2 10 13 14 11 --> number: 2 (type: H8P0 ) (kind: IRREDUCIBLE ) 2 5 6 3 11 14 15 12 --> number: 3 (type: H8P0 ) (kind: IRREDUCIBLE ) 4 7 8 5 13 16 17 14 --> number: 4 (type: H8P0 ) (kind: IRREDUCIBLE ) 5 8 9 6 14 17 18 15 --> number: 5 (type: H8P0 ) (kind: IRREDUCIBLE ) 10 13 14 11 19 22 23 20 --> number: 6 (type: H8P0 ) (kind: IRREDUCIBLE )

11 14 15 12 20 23 24 21 --> number: 7 (type: H8P0 ) (kind: IRREDUCIBLE ) 13 16 17 14 22 25 26 23 --> number: 8 (type: H8P0 ) (kind: IRREDUCIBLE ) 14 17 18 15 23 26 27 24 --> number: 9 (type: H8P0 ) (kind: IRREDUCIBLE ) 19 22 23 20 28 31 32 29 --> number: 10 (type: H8P0 ) (kind: IRREDUCIBLE ) 20 23 24 21 29 32 33 30 --> number: 11 (type: H8P0 ) (kind: IRREDUCIBLE ) 22 25 26 23 31 34 35 32 --> number: 12 (type: H8P0 ) (kind: IRREDUCIBLE ) 9

10 23 26 27 24 32 35 36 33 --> number: 13 (type: H8P0 ) (kind: IRREDUCIBLE ) 28 31 32 29 37 40 41 38 --> number: 14 (type: H8P0 ) (kind: IRREDUCIBLE ) 29 32 33 30 38 41 42 39 --> number: 15 (type: H8P0 ) (kind: IRREDUCIBLE ) 31 34 35 32 40 43 44 41 --> number: 16 (type: H8P0 ) (kind: IRREDUCIBLE ) 32 35 36 33 41 44 45 42 = 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: ~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~

11 37 : ( x1 = -5.000E-01, x2 = -5.000E-01, x3 = 1.200E+01 ) displacement-1 = 0.000E+00 ; history no. = -2 displacement-2 = 0.000E+00 ; history no. = -2 displacement-3 = 0.000E+00 ; history no. = -2 38 : ( x1 = 0.000E+00, x2 = -5.000E-01, x3 = 1.200E+01 ) displacement-1 = 0.000E+00 ; history no. = -2 displacement-2 = 0.000E+00 ; history no. = -2 displacement-3 = 0.000E+00 ; history no. = -2 39 : ( x1 = 5.000E-01, x2 = -5.000E-01, x3 = 1.200E+01 ) displacement-1 = 0.000E+00 ; history no. = -2 displacement-2 = 0.000E+00 ; history no. = -2 displacement-3 = 0.000E+00 ; history no. = -2 40 : ( x1 = -5.000E-01, x2 = 0.000E+00, x3 = 1.200E+01 ) displacement-1 = 0.000E+00 ; history no. = -2 displacement-2 = 0.000E+00 ; history no. = -2 displacement-3 = 0.000E+00 ; history no. = -2 41 : ( x1 = 0.000E+00, x2 = 0.000E+00, x3 = 1.200E+01 ) displacement-1 = 0.000E+00 ; history no. = -2 displacement-2 = 0.000E+00 ; history no. = -2 displacement-3 = 0.000E+00 ; history no. = -2 42 : ( x1 = 5.000E-01, x2 = 0.000E+00, x3 = 1.200E+01 ) displacement-1 = 0.000E+00 ; history no. = -2 displacement-2 = 0.000E+00 ; history no. = -2 displacement-3 = 0.000E+00 ; history no. = -2 43 : ( x1 = -5.000E-01, x2 = 5.000E-01, x3 = 1.200E+01 ) displacement-1 = 0.000E+00 ; history no. = -2 displacement-2 = 0.000E+00 ; history no. = -2 displacement-3 = 0.000E+00 ; history no. = -2 44 : ( x1 = 0.000E+00, x2 = 5.000E-01, x3 = 1.200E+01 ) displacement-1 = 0.000E+00 ; history no. = -2 displacement-2 = 0.000E+00 ; history no. = -2 displacement-3 = 0.000E+00 ; history no. = -2 45 : ( x1 = 5.000E-01, x2 = 5.000E-01, x3 = 1.200E+01 ) displacement-1 = 0.000E+00 ; history no. = -2 displacement-2 = 0.000E+00 ; history no. = -2 displacement-3 = 0.000E+00 ; history no. = -2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ end of mathematical model data

12 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 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 = H8P0 ): @(x1 = -2.500E-01, x2 = -2.500E-01, x3 = 1.500E+00): eps_11 = -1.934E-19 ; eps_22 = -4.513E-19 ; eps_33 = 7.500E-04 gam_12 = 1.133E-18 ; gam_13 = 6.481E-11 ; gam_23 = 6.481E-11 sig_11 = -1.934E-13 ; sig_22 = -4.513E-13 ; sig_33 = 7.500E+02 sig_12 = 5.667E-13 ; sig_13 = 3.240E-05 ; sig_23 = 3.240E-05 --> element 2 ( type = H8P0 ): @(x1 = 2.500E-01, x2 = -2.500E-01, x3 = 1.500E+00): eps_11 = 3.467E-19 ; eps_22 = 2.715E-19 ; eps_33 = 7.500E-04 gam_12 = 1.705E-18 ; gam_13 = 6.481E-11 ; gam_23 = 6.481E-11 sig_11 = 3.467E-13 ; sig_22 = 2.715E-13 ; sig_33 = 7.500E+02 sig_12 = 8.524E-13 ; sig_13 = 3.240E-05 ; sig_23 = 3.240E-05 --> element 3 ( type = H8P0 ): @(x1 = -2.500E-01, x2 = 2.500E-01, x3 = 1.500E+00): eps_11 = 5.005E-19 ; eps_22 = -1.269E-18 ; eps_33 = 7.500E-04 gam_12 = 7.320E-19 ; gam_13 = 6.481E-11 ; gam_23 = 6.481E-11 sig_11 = 5.005E-13 ; sig_22 = -1.269E-12 ; sig_33 = 7.500E+02 sig_12 = 3.660E-13 ; sig_13 = 3.240E-05 ; sig_23 = 3.240E-05

13 --> element 4 ( type = H8P0 ): @(x1 = 2.500E-01, x2 = 2.500E-01, x3 = 1.500E+00): eps_11 = 1.758E-19 ; eps_22 = 9.043E-19 ; eps_33 = 7.500E-04 gam_12 = -5.569E-19 ; gam_13 = 6.481E-11 ; gam_23 = 6.481E-11 sig_11 = 1.758E-13 ; sig_22 = 9.043E-13 ; sig_33 = 7.500E+02 sig_12 = -2.785E-13 ; sig_13 = 3.240E-05 ; sig_23 = 3.240E-05 --> element 5 ( type = H8P0 ): @(x1 = -2.500E-01, x2 = -2.500E-01, x3 = 4.500E+00): eps_11 = -5.772E-20 ; eps_22 = 2.452E-20 ; eps_33 = 2.250E-03 gam_12 = 3.641E-19 ; gam_13 = 1.944E-10 ; gam_23 = 1.944E-10 sig_11 = -5.772E-14 ; sig_22 = 2.452E-14 ; sig_33 = 2.250E+03 sig_12 = 1.821E-13 ; sig_13 = 9.721E-05 ; sig_23 = 9.721E-05 --> element 6 ( type = H8P0 ): @(x1 = 2.500E-01, x2 = -2.500E-01, x3 = 4.500E+00): eps_11 = 2.678E-20 ; eps_22 = 1.357E-20 ; eps_33 = 2.250E-03 gam_12 = -1.382E-19 ; gam_13 = 1.944E-10 ; gam_23 = 1.944E-10 sig_11 = 2.678E-14 ; sig_22 = 1.357E-14 ; sig_33 = 2.250E+03 sig_12 = -6.911E-14 ; sig_13 = 9.721E-05 ; sig_23 = 9.721E-05 --> element 7 ( type = H8P0 ): @(x1 = -2.500E-01, x2 = 2.500E-01, x3 = 4.500E+00): eps_11 = -3.635E-20 ; eps_22 = 4.879E-19 ; eps_33 = 2.250E-03 gam_12 = -1.726E-19 ; gam_13 = 1.944E-10 ; gam_23 = 1.944E-10 sig_11 = -3.635E-14 ; sig_22 = 4.879E-13 ; sig_33 = 2.250E+03 sig_12 = -8.629E-14 ; sig_13 = 9.721E-05 ; sig_23 = 9.721E-05 --> element 8 ( type = H8P0 ): @(x1 = 2.500E-01, x2 = 2.500E-01, x3 = 4.500E+00):

14 eps_11 = 5.541E-19 ; eps_22 = 1.921E-19 ; eps_33 = 2.250E-03 gam_12 = -7.530E-19 ; gam_13 = 1.944E-10 ; gam_23 = 1.944E-10 sig_11 = 5.541E-13 ; sig_22 = 1.921E-13 ; sig_33 = 2.250E+03 sig_12 = -3.765E-13 ; sig_13 = 9.721E-05 ; sig_23 = 9.721E-05 --> element 9 ( type = H8P0 ): @(x1 = -2.500E-01, x2 = -2.500E-01, x3 = 7.500E+00): eps_11 = 1.526E-20 ; eps_22 = 6.359E-20 ; eps_33 = 3.750E-03 gam_12 = -2.302E-19 ; gam_13 = 3.240E-10 ; gam_23 = 3.240E-10 sig_11 = 1.526E-14 ; sig_22 = 6.359E-14 ; sig_33 = 3.750E+03 sig_12 = -1.151E-13 ; sig_13 = 1.620E-04 ; sig_23 = 1.620E-04 --> element 10 ( type = H8P0 ): @(x1 = 2.500E-01, x2 = -2.500E-01, x3 = 7.500E+00): eps_11 = -6.309E-20 ; eps_22 = -6.112E-20 ; eps_33 = 3.750E-03 gam_12 = -3.689E-19 ; gam_13 = 3.240E-10 ; gam_23 = 3.240E-10 sig_11 = -6.309E-14 ; sig_22 = -6.112E-14 ; sig_33 = 3.750E+03 sig_12 = -1.844E-13 ; sig_13 = 1.620E-04 ; sig_23 = 1.620E-04 --> element 11 ( type = H8P0 ): @(x1 = -2.500E-01, x2 = 2.500E-01, x3 = 7.500E+00): eps_11 = 2.983E-19 ; eps_22 = 1.065E-19 ; eps_33 = 3.750E-03 gam_12 = -5.552E-19 ; gam_13 = 3.240E-10 ; gam_23 = 3.240E-10 sig_11 = 2.983E-13 ; sig_22 = 1.065E-13 ; sig_33 = 3.750E+03 sig_12 = -2.776E-13 ; sig_13 = 1.620E-04 ; sig_23 = 1.620E-04 --> element 12 ( type = H8P0 ): @(x1 = 2.500E-01, x2 = 2.500E-01, x3 = 7.500E+00): eps_11 = 1.983E-19 ; eps_22 = -3.356E-19 ; eps_33 = 3.750E-03 gam_12 = -3.176E-20 ; gam_13 = 3.240E-10 ; gam_23 = 3.240E-10 sig_11 = 1.983E-13 ; sig_22 = -3.356E-13 ; sig_33 = 3.750E+03 sig_12 = -1.588E-14 ; sig_13 = 1.620E-04 ; sig_23 = 1.620E-04

15 --> element 13 ( type = H8P0 ): @(x1 = -2.500E-01, x2 = -2.500E-01, x3 = 1.050E+01): eps_11 = 3.037E-20 ; eps_22 = 1.262E-19 ; eps_33 = 5.250E-03 gam_12 = -3.039E-19 ; gam_13 = 4.537E-10 ; gam_23 = 4.537E-10 sig_11 = 3.037E-14 ; sig_22 = 1.262E-13 ; sig_33 = 5.250E+03 sig_12 = -1.520E-13 ; sig_13 = 2.268E-04 ; sig_23 = 2.268E-04 --> element 14 ( type = H8P0 ): @(x1 = 2.500E-01, x2 = -2.500E-01, x3 = 1.050E+01): eps_11 = 5.332E-20 ; eps_22 = 4.435E-20 ; eps_33 = 5.250E-03 gam_12 = 1.143E-19 ; gam_13 = 4.537E-10 ; gam_23 = 4.537E-10 sig_11 = 5.332E-14 ; sig_22 = 4.435E-14 ; sig_33 = 5.250E+03 sig_12 = 5.715E-14 ; sig_13 = 2.268E-04 ; sig_23 = 2.268E-04 --> element 15 ( type = H8P0 ): @(x1 = -2.500E-01, x2 = 2.500E-01, x3 = 1.050E+01): eps_11 = 2.885E-20 ; eps_22 = -2.030E-19 ; eps_33 = 5.250E-03 gam_12 = -2.427E-19 ; gam_13 = 4.537E-10 ; gam_23 = 4.537E-10 sig_11 = 2.885E-14 ; sig_22 = -2.030E-13 ; sig_33 = 5.250E+03 sig_12 = -1.213E-13 ; sig_13 = 2.268E-04 ; sig_23 = 2.268E-04 --> element 16 ( type = H8P0 ): @(x1 = 2.500E-01, x2 = 2.500E-01, x3 = 1.050E+01): eps_11 = 1.146E-19 ; eps_22 = -2.084E-19 ; eps_33 = 5.250E-03 gam_12 = -5.511E-19 ; gam_13 = 4.537E-10 ; gam_23 = 4.537E-10 sig_11 = 1.146E-13 ; sig_22 = -2.084E-13 ; sig_33 = 5.250E+03 sig_12 = -2.755E-13 ; sig_13 = 2.268E-04 ; sig_23 = 2.268E-04 At time 1.000E+00 (step no. 1):

16 = N O D A L Q U A N T I T I E S = node: 1 ( x1 = -5.000E-01, x2 = -5.000E-01, x3 = 0.000E+00 ) u_1 = -1.253E-07, u_2 = -1.253E-07, u_3 = -3.600E-02 node: 2 ( x1 = 0.000E+00, x2 = -5.000E-01, x3 = 0.000E+00 ) u_1 = -1.253E-07, u_2 = -1.253E-07, u_3 = -3.600E-02 node: 3 ( x1 = 5.000E-01, x2 = -5.000E-01, x3 = 0.000E+00 ) u_1 = -1.253E-07, u_2 = -1.253E-07, u_3 = -3.600E-02 node: 4 ( x1 = -5.000E-01, x2 = 0.000E+00, x3 = 0.000E+00 ) u_1 = -1.253E-07, u_2 = -1.253E-07, u_3 = -3.600E-02 node: 5 ( x1 = 0.000E+00, x2 = 0.000E+00, x3 = 0.000E+00 ) u_1 = -1.253E-07, u_2 = -1.253E-07, u_3 = -3.600E-02 node: 6 ( x1 = 5.000E-01, x2 = 0.000E+00, x3 = 0.000E+00 ) u_1 = -1.253E-07, u_2 = -1.253E-07, u_3 = -3.600E-02 node: 7 ( x1 = -5.000E-01, x2 = 5.000E-01, x3 = 0.000E+00 ) u_1 = -1.253E-07, u_2 = -1.253E-07, u_3 = -3.600E-02 node: 8 ( x1 = 0.000E+00, x2 = 5.000E-01, x3 = 0.000E+00 ) u_1 = -1.253E-07, u_2 = -1.253E-07, u_3 = -3.600E-02 node: 9 ( x1 = 5.000E-01, x2 = 5.000E-01, x3 = 0.000E+00 ) u_1 = -1.253E-07, u_2 = -1.253E-07, u_3 = -3.600E-02 node: 10 ( x1 = -5.000E-01, x2 = -5.000E-01, x3 = 3.000E+00 ) u_1 = -8.357E-08, u_2 = -8.357E-08, u_3 = -3.375E-02 node: 11 ( x1 = 0.000E+00, x2 = -5.000E-01, x3 = 3.000E+00 ) u_1 = -8.357E-08, u_2 = -8.357E-08, u_3 = -3.375E-02 node: 12 ( x1 = 5.000E-01, x2 = -5.000E-01, x3 = 3.000E+00 ) u_1 = -8.357E-08, u_2 = -8.357E-08, u_3 = -3.375E-02 node: 13 ( x1 = -5.000E-01, x2 = 0.000E+00, x3 = 3.000E+00 ) u_1 = -8.357E-08, u_2 = -8.357E-08, u_3 = -3.375E-02 node: 14 ( x1 = 0.000E+00, x2 = 0.000E+00, x3 = 3.000E+00 ) u_1 = -8.357E-08, u_2 = -8.357E-08, u_3 = -3.375E-02 node: 15 ( x1 = 5.000E-01, x2 = 0.000E+00, x3 = 3.000E+00 )

17 u_1 = -8.357E-08, u_2 = -8.357E-08, u_3 = -3.375E-02 node: 16 ( x1 = -5.000E-01, x2 = 5.000E-01, x3 = 3.000E+00 ) u_1 = -8.357E-08, u_2 = -8.357E-08, u_3 = -3.375E-02 node: 17 ( x1 = 0.000E+00, x2 = 5.000E-01, x3 = 3.000E+00 ) u_1 = -8.357E-08, u_2 = -8.357E-08, u_3 = -3.375E-02 node: 18 ( x1 = 5.000E-01, x2 = 5.000E-01, x3 = 3.000E+00 ) u_1 = -8.357E-08, u_2 = -8.357E-08, u_3 = -3.375E-02 node: 19 ( x1 = -5.000E-01, x2 = -5.000E-01, x3 = 6.000E+00 ) u_1 = -4.433E-08, u_2 = -4.433E-08, u_3 = -2.700E-02 node: 20 ( x1 = 0.000E+00, x2 = -5.000E-01, x3 = 6.000E+00 ) u_1 = -4.433E-08, u_2 = -4.433E-08, u_3 = -2.700E-02 node: 21 ( x1 = 5.000E-01, x2 = -5.000E-01, x3 = 6.000E+00 ) u_1 = -4.433E-08, u_2 = -4.433E-08, u_3 = -2.700E-02 node: 22 ( x1 = -5.000E-01, x2 = 0.000E+00, x3 = 6.000E+00 ) u_1 = -4.433E-08, u_2 = -4.433E-08, u_3 = -2.700E-02 node: 23 ( x1 = 0.000E+00, x2 = 0.000E+00, x3 = 6.000E+00 ) u_1 = -4.433E-08, u_2 = -4.433E-08, u_3 = -2.700E-02 node: 24 ( x1 = 5.000E-01, x2 = 0.000E+00, x3 = 6.000E+00 ) u_1 = -4.433E-08, u_2 = -4.433E-08, u_3 = -2.700E-02 node: 25 ( x1 = -5.000E-01, x2 = 5.000E-01, x3 = 6.000E+00 ) u_1 = -4.433E-08, u_2 = -4.433E-08, u_3 = -2.700E-02 node: 26 ( x1 = 0.000E+00, x2 = 5.000E-01, x3 = 6.000E+00 ) u_1 = -4.433E-08, u_2 = -4.433E-08, u_3 = -2.700E-02 node: 27 ( x1 = 5.000E-01, x2 = 5.000E-01, x3 = 6.000E+00 ) u_1 = -4.433E-08, u_2 = -4.433E-08, u_3 = -2.700E-02 node: 28 ( x1 = -5.000E-01, x2 = -5.000E-01, x3 = 9.000E+00 ) u_1 = -1.329E-08, u_2 = -1.329E-08, u_3 = -1.575E-02 node: 29 ( x1 = 0.000E+00, x2 = -5.000E-01, x3 = 9.000E+00 ) u_1 = -1.329E-08, u_2 = -1.329E-08, u_3 = -1.575E-02 node: 30 ( x1 = 5.000E-01, x2 = -5.000E-01, x3 = 9.000E+00 ) u_1 = -1.329E-08, u_2 = -1.329E-08, u_3 = -1.575E-02 node: 31 ( x1 = -5.000E-01, x2 = 0.000E+00, x3 = 9.000E+00 )

18 u_1 = -1.329E-08, u_2 = -1.329E-08, u_3 = -1.575E-02 node: 32 ( x1 = 0.000E+00, x2 = 0.000E+00, x3 = 9.000E+00 ) u_1 = -1.329E-08, u_2 = -1.329E-08, u_3 = -1.575E-02 node: 33 ( x1 = 5.000E-01, x2 = 0.000E+00, x3 = 9.000E+00 ) u_1 = -1.329E-08, u_2 = -1.329E-08, u_3 = -1.575E-02 node: 34 ( x1 = -5.000E-01, x2 = 5.000E-01, x3 = 9.000E+00 ) u_1 = -1.329E-08, u_2 = -1.329E-08, u_3 = -1.575E-02 node: 35 ( x1 = 0.000E+00, x2 = 5.000E-01, x3 = 9.000E+00 ) u_1 = -1.329E-08, u_2 = -1.329E-08, u_3 = -1.575E-02 node: 36 ( x1 = 5.000E-01, x2 = 5.000E-01, x3 = 9.000E+00 ) u_1 = -1.329E-08, u_2 = -1.329E-08, u_3 = -1.575E-02 node: 37 ( x1 = -5.000E-01, x2 = -5.000E-01, x3 = 1.200E+01 ) u_1 = -2.809E-31, u_2 = -2.809E-31, u_3 = -9.578E-24 node: 38 ( x1 = 0.000E+00, x2 = -5.000E-01, x3 = 1.200E+01 ) u_1 = -5.619E-31, u_2 = -5.619E-31, u_3 = -1.916E-23 node: 39 ( x1 = 5.000E-01, x2 = -5.000E-01, x3 = 1.200E+01 ) u_1 = -2.809E-31, u_2 = -2.809E-31, u_3 = -9.578E-24 node: 40 ( x1 = -5.000E-01, x2 = 0.000E+00, x3 = 1.200E+01 ) u_1 = -5.619E-31, u_2 = -5.619E-31, u_3 = -1.916E-23 node: 41 ( x1 = 0.000E+00, x2 = 0.000E+00, x3 = 1.200E+01 ) u_1 = -1.124E-30, u_2 = -1.124E-30, u_3 = -3.831E-23 node: 42 ( x1 = 5.000E-01, x2 = 0.000E+00, x3 = 1.200E+01 ) u_1 = -5.619E-31, u_2 = -5.619E-31, u_3 = -1.916E-23 node: 43 ( x1 = -5.000E-01, x2 = 5.000E-01, x3 = 1.200E+01 ) u_1 = -2.809E-31, u_2 = -2.809E-31, u_3 = -9.578E-24 node: 44 ( x1 = 0.000E+00, x2 = 5.000E-01, x3 = 1.200E+01 ) u_1 = -5.619E-31, u_2 = -5.619E-31, u_3 = -1.916E-23 node: 45 ( x1 = 5.000E-01, x2 = 5.000E-01, x3 = 1.200E+01 ) u_1 = -2.809E-31, u_2 = -2.809E-31, u_3 = -9.578E-24 apes -> end of analysis........

19 Table 1: Summary of Approximate Nodal Displacements in the Global z-direction Obtained using Eight-Node Hexahedral Elements z ŵ Nodes 0.0-3.600e-02 1 to 9 3.0-3.375e-02 10 to 18 6.0-2.700e-02 19 to 27 9.0-1.575e-02 28 to 36 12.0 0.0 37 to 45 Table 2: Summary of Approximate Element Normal Stresses in the Global z-direction Obtained using Eight-Node Hexahedral Elements z ˆσ (e) 33 Elements 1.50 750 1 to 4 4.50 2250 5 to 8 7.50 3750 9 to 12 10.5 5250 13 to 16 The results of the above analysis are next summarized. The first check on the accuracy of the results is the displacement field in the global z-direction. Specializing equation (6) for the problem under consideration gives w = 500 { z 2 (12) 2} = (2.50e 04)(z 2 144) (7) 2(1.0e + 06) The nodal values of w summarized in Table 1. These values agree exactly with the elasticity solution. It is also timely to note all of the values of ŵ located at a given value of z were the same. The second check of the numerical results involves σ 33. Specializing equation (1) for the values of ρ and g used in the analysis gives σ 33 = 500z (8) The values of ˆσ (e) 33 computed at the secondary quadrature points are summarized in Table 2. These values agree exactly with the elasticity solution.

20 Since the elasticity solution for w(x, y, z) is quadratic in x, y, and z (recall equation 6), a single quadratic Serendipity hexahedral element would be expected to give exact results. The analysis is thus repeated using the mesh shown in Figure 3 consisting of a single quadratic twenty-node hexahedral element. The data file for this case is shown below. ana tit "3-d stretching of a bar by its own weight" ana tit "mesh consisting of one H20P0 element" analysis action analyze analysis description linear analysis idealization three-dimensional analysis temporal transient analysis type mechanical echo init off echo trans off echo warn off dim max material isotropic elastic 1 dim max nodes 20 dim max h20p0 1 gravity accel value 1.0 gravity accel history 0 gravity x1_angle value 90.0 gravity x2_angle value 90.0 gravity x3_angle value 180.0 gravity angle history 0 fin sett material solid number 1 density 500.0 material elastic isotropic number 1 & description can be omitted modulus 1.0e+06 poissons 0.0 nodes line number 1 x1 0.5 x2 0.5 x3 12.0 nodes line number 2 x1-0.5 x2 0.5 x3 12.0 nodes line number 3 x1-0.5 x2-0.5 x3 12.0 nodes line number 4 x1 0.5 x2-0.5 x3 12.0 nodes line number 5 x1 0.5 x2 0.5 nodes line number 6 x1-0.5 x2 0.5 nodes line number 7 x1-0.5 x2-0.5 nodes line number 8 x1 0.5 x2-0.5 nodes line number 9 x2 0.5 x3 12.0 nodes line number 10 x1-0.5 x3 12.0 nodes line number 11 x2-0.5 x3 12.0 nodes line number 12 x1 0.5 x3 12.0 nodes line number 13 x2 0.5

nodes line number 14 x1-0.5 nodes line number 15 x2-0.5 nodes line number 16 x1 0.5 nodes line number 17 x1 0.5 x2 0.5 x3 6.0 nodes line number 18 x1-0.5 x2 0.5 x3 6.0 nodes line number 19 x1-0.5 x2-0.5 x3 6.0 nodes line number 20 x1 0.5 x2-0.5 x3 6.0 element irreducible type "h20p0" nodes 1 2 3 4 5 6 7 8 9 10 & 11 12 13 14 15 16 17 18 19 20 mat 1 spe conc mec nod 1:4 1_disp 2_disp 3_disp spe conc mec nod 9:12 1_disp 2_disp 3_disp finish data solution time final 1.0 increments 1 output 1 finished loading 21

22 z 11 3 10 4 2 12 1 9 19 20 18 17 15 7 14 8 6 x 16 5 13 y Figure 3: Twenty-Node Hexahedral Finite Element Model Used in Analysis of Prismatic Bar Loaded by its Own Weight The nodal values of w summarized in Table 3. As expected, these values agree exactly with the elasticity solution. Similar to the case of the eight-node element, all of the values of ŵ located at a given value of z were the same. The values of ˆσ (e) 33 computed at the secondary quadrature points are summarized in Table 4. Once again, these values agree exactly with the elasticity solution.

23 Table 3: Summary of Approximate Nodal Displacements in the Global z-direction Obtained using Twenty-Node Hexahedral Elements z ŵ Nodes 0.0-3.600e-02 5 to 8, 13 to 16 6.0-2.700e-02 17 to 20 12.0 0.0 1 to 4, 9 to 12 Table 4: Summary of Approximate Element Normal Stresses in the Global z-direction Obtained using Twenty-Node Hexahedral Elements z 2.536E+00 9.464E+00 ˆσ (e) 33 1.268E+03 4.732E+03

24 Finally, results identical to the above are also obtained if a single quadratic twenty-seven-node Lagrangian hexahedral element (Figure 4) is used. The data file for this case is shown below. z 11 3 10 4 25 2 12 1 9 24 19 22 20 27 18 21 17 23 15 7 14 8 26 6 x 16 5 13 y Figure 4: Twenty-Seven-Node Hexahedral Finite Element Model Used in Analysis of Prismatic Bar Loaded by its Own Weight ana tit "3-d stretching of a bar by its own weight" ana tit "mesh consisting of one H27P0 element" analysis action analyze analysis description linear analysis idealization three-dimensional analysis temporal transient analysis type mechanical echo init off echo trans off echo warn off

25 dim max material isotropic elastic 1 dim max nodes 27 dim max h27p0 1 gravity accel value 1.0 gravity accel history 0 gravity x1_angle value 90.0 gravity x2_angle value 90.0 gravity x3_angle value 180.0 gravity angle history 0 fin sett material solid number 1 density 500.0 material elastic isotropic number 1 & description can be omitted modulus 1.0e+06 poissons 0.0 nodes line number 1 x1 0.5 x2 0.5 x3 12.0 nodes line number 2 x1-0.5 x2 0.5 x3 12.0 nodes line number 3 x1-0.5 x2-0.5 x3 12.0 nodes line number 4 x1 0.5 x2-0.5 x3 12.0 nodes line number 5 x1 0.5 x2 0.5 nodes line number 6 x1-0.5 x2 0.5 nodes line number 7 x1-0.5 x2-0.5 nodes line number 8 x1 0.5 x2-0.5 nodes line number 9 x2 0.5 x3 12.0 nodes line number 10 x1-0.5 x3 12.0 nodes line number 11 x2-0.5 x3 12.0 nodes line number 12 x1 0.5 x3 12.0 nodes line number 13 x2 0.5 nodes line number 14 x1-0.5 nodes line number 15 x2-0.5 nodes line number 16 x1 0.5 nodes line number 17 x1 0.5 x2 0.5 x3 6.0 nodes line number 18 x1-0.5 x2 0.5 x3 6.0 nodes line number 19 x1-0.5 x2-0.5 x3 6.0 nodes line number 20 x1 0.5 x2-0.5 x3 6.0 element irreducible type "h27p0" nodes 1 2 3 4 5 6 7 8 9 10 & 11 12 13 14 15 16 17 18 19 20 & 21 22 23 24 25 26 27 mat 1 spe conc mec nod 1:4 1_disp 2_disp 3_disp spe conc mec nod 9:12 1_disp 2_disp 3_disp spe conc mec nod 25 1_disp 2_disp 3_disp finish data

26 solution time final 1.0 increments 1 output 1 finished loading

References [1] Timoshenko, S. P. and J. N. Goodier, Theory of Elasticity, Third Edition. New York: McGraw- Hill Book Co (1970). 27

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