Analysis of Planar Truss

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Horace Fox
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1 Analysis of Planar Truss Although the APES computer program is not a specific matrix structural code, it can none the less be used to analyze simple structures. In this example, the following statically determinate, planar truss is analyzed. 20 kn 10 kn b d f h 12 kn j 6 m a c e 6 kn 6 kn g i 8 m 8 m 8 m 8 m The mathematical model used has the following appearance. 20 kn 10 kn kn kn 6 kn A copy of the input data file used is shown below. This is followed by the associated output file. 1

2 ana title "statically determinate truss analyzed using L2P0 bar elements" analysis type mechanical anal ideal plane_stress anal temp transient analysis description linear dim max material isotropic elastic 1 dim max nodes 10 dim max l2p0 17 fin sett mat elastic isotropic number 1 & desc "hypothetical material for bar elements" & modulus 3.0e+09 nodes line number 1 nodes line number 9 x incr 2 nodes line number 2 x2 6.0 nodes line number 10 x x2 6.0 incr 2 element bar_mech type "l2p0" nodes 1 2 mat 1 area 1.0 element bar_mech type "l2p0" nodes 2 4 mat 1 area 1.0 element bar_mech type "l2p0" nodes 1 4 mat 1 area 1.0 element bar_mech type "l2p0" nodes 1 3 mat 1 area 1.0 element bar_mech type "l2p0" nodes 3 4 mat 1 area 1.0 element bar_mech type "l2p0" nodes 4 6 mat 1 area 1.0 element bar_mech type "l2p0" nodes 3 6 mat 1 area 1.0 element bar_mech type "l2p0" nodes 3 5 mat 1 area 1.0 element bar_mech type "l2p0" nodes 5 6 mat 1 area 1.0 element bar_mech type "l2p0" nodes 6 8 mat 1 area 1.0 element bar_mech type "l2p0" nodes 6 7 mat 1 area 1.0 element bar_mech type "l2p0" nodes 5 7 mat 1 area 1.0 element bar_mech type "l2p0" nodes 7 8 mat 1 area 1.0 element bar_mech type "l2p0" nodes 8 10 mat 1 area 1.0 element bar_mech type "l2p0" nodes 8 9 mat 1 area 1.0 element bar_mech type "l2p0" nodes 7 9 mat 1 area 1.0 element bar_mech type "l2p0" nodes 9 10 mat 1 area 1.0 specification conc mech nodes 1 1_disp specification conc mech nodes 2 1_disp 2_disp specification conc mech nodes 3 2_hist 0 2_forc 2_value -6.0 specification conc mech nodes 6 2_hist 0 2_forc 2_value specification conc mech nodes 7 2_hist 0 2_forc 2_value -6.0 specification conc mech nodes 10 1_hist 0 1_forc 1_value 12.0 & 2_hist 0 2_forc 2_value finish data solution time final 1.0 increments 1 output 1:10:1 finished loading 2

3 apes : version 3.10 : DATE OF ANALYSIS :: day:20 month:03 year:98 ANALYSIS INITIATED AT TIME :: 11:37:04 statically determinate truss analyzed using L2P0 bar elements WARNING: no INTEGRATION commands specified <<< 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 = 10 Max. no. of ISOTROPIC, LINEAR ELASTIC materials = 1 Max. no. of 2-node line (L2P0) elements = 17 = 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 3

4 -- "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" = 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 = NOTE: no gravity specifications have been made ; the following DEFAULT values are thus assumed: Initial value of gravitational acceleration, "g" = 0.000E+00 History function number associated with "g" = -2 Angle (in degrees) that "g" makes with the x1-axis = E+01 Angle (in degrees) that "g" makes with the x2-axis = 1.800E+02 Angle (in degrees) that "g" makes with the x3-axis = E+01 History function number associated w/these angles = -2 = 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 = I N I T I A L S T A T E I N F O R M A T I O N = <<< NONE 4

5 = 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. : hypothetical material for bar elements Modulus of Elasticity = 3.000E+09 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 = 0.000E+00 x2 = 6.000E+00 node : 3 x1 = 8.000E+00 x2 = 0.000E+00 node : 4 x1 = 8.000E+00 x2 = 6.000E+00 node : 5 x1 = 1.600E+01 x2 = 0.000E+00 node : 6 x1 = 1.600E+01 x2 = 6.000E+00 node : 7 x1 = 2.400E+01 x2 = 0.000E+00 node : 8 x1 = 2.400E+01 x2 = 6.000E+00 node : 9 x1 = 3.200E+01 x2 = 0.000E+00 node : 10 x1 = 3.200E+01 x2 = 6.000E+00 = E L E M E N T I N F O R M A T I O N = -- number : 1 (type : L2P0 ) (kind : BAR_MECH ) ~~~~~~ nodes : number : 2 (type : L2P0 ) (kind : BAR_MECH ) ~~~~~~ nodes : 2 4 5

6 -- number : 3 (type : L2P0 ) (kind : BAR_MECH ) ~~~~~~ nodes : number : 4 (type : L2P0 ) (kind : BAR_MECH ) ~~~~~~ nodes : number : 5 (type : L2P0 ) (kind : BAR_MECH ) ~~~~~~ nodes : number : 6 (type : L2P0 ) (kind : BAR_MECH ) 6

7 ~~~~~~ nodes : number : 7 (type : L2P0 ) (kind : BAR_MECH ) ~~~~~~ nodes : number : 8 (type : L2P0 ) (kind : BAR_MECH ) ~~~~~~ nodes : number : 9 (type : L2P0 ) (kind : BAR_MECH ) ~~~~~~ nodes : 5 6 7

8 -- number : 10 (type : L2P0 ) (kind : BAR_MECH ) ~~~~~~ nodes : number : 11 (type : L2P0 ) (kind : BAR_MECH ) ~~~~~~ nodes : number : 12 (type : L2P0 ) (kind : BAR_MECH ) ~~~~~~ nodes : number : 13 (type : L2P0 ) (kind : BAR_MECH ) ~~~~~~ nodes : 7 8 8

9 -- number : 14 (type : L2P0 ) (kind : BAR_MECH ) ~~~~~~ nodes : number : 15 (type : L2P0 ) (kind : BAR_MECH ) ~~~~~~ nodes : number : 16 (type : L2P0 ) (kind : BAR_MECH ) ~~~~~~ nodes : number : 17 (type : L2P0 ) (kind : BAR_MECH ) ~~~~~~ nodes :

10 = N O D E P O I N T S P E C I F I C A T I O N S = Node Number s p e c i f i c a t i o n: ~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~ 1 : 2 : 3 : 6 : 7 : 10 : displacement-1 = 0.000E+00 ; history no. = -2 force-2 = 0.000E+00 ; history no. = -2 displacement-1 = 0.000E+00 ; history no. = -2 displacement-2 = 0.000E+00 ; history no. = -2 force-1 = 0.000E+00 ; history no. = -2 force-2 = E+00 ; history no. = 0 force-1 = 0.000E+00 ; history no. = -2 force-2 = E+01 ; history no. = 0 force-1 = 0.000E+00 ; history no. = -2 force-2 = E+00 ; history no. = 0 force-1 = 1.200E+01 ; history no. = 0 force-2 = E+01 ; history no. = 0 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 = L2P0 ) = 0.000E+00, x2 = 3.000E+00, x3 = 0.000E+00) : eps_11 = 1.400E-08 ; sig_11 = 4.200E+01 ; axial force = 4.200E element 2 ( type = L2P0 ) = 4.000E+00, x2 = 6.000E+00, x3 = 0.000E+00) : eps_11 = 5.022E-08 ; sig_11 = 1.507E+02 ; axial force = 1.507E+02 10

11 -- element 3 ( type = L2P0 ) = 4.000E+00, x2 = 3.000E+00, x3 = 0.000E+00) : eps_11 = E-08 ; sig_11 = E+01 ; axial force = E element 4 ( type = L2P0 ) = 4.000E+00, x2 = 0.000E+00, x3 = 0.000E+00) : eps_11 = E-08 ; sig_11 = E+01 ; axial force = E element 5 ( type = L2P0 ) = 8.000E+00, x2 = 3.000E+00, x3 = 0.000E+00) : eps_11 = 1.400E-08 ; sig_11 = 4.200E+01 ; axial force = 4.200E element 6 ( type = L2P0 ) = 1.200E+01, x2 = 6.000E+00, x3 = 0.000E+00) : eps_11 = 3.156E-08 ; sig_11 = 9.467E+01 ; axial force = 9.467E element 7 ( type = L2P0 ) = 1.200E+01, x2 = 3.000E+00, x3 = 0.000E+00) : eps_11 = E-08 ; sig_11 = E+01 ; axial force = E element 8 ( type = L2P0 ) = 1.200E+01, x2 = 0.000E+00, x3 = 0.000E+00) : eps_11 = E-08 ; sig_11 = E+01 ; axial force = E element 9 ( type = L2P0 ) = 1.600E+01, x2 = 3.000E+00, x3 = 0.000E+00) : eps_11 = E-15 ; sig_11 = E-06 ; axial force = E element 10 ( type = L2P0 ) = 2.000E+01, x2 = 6.000E+00, x3 = 0.000E+00) : eps_11 = 8.444E-09 ; sig_11 = 2.533E+01 ; axial force = 2.533E element 11 ( type = L2P0 ) = 2.000E+01, x2 = 3.000E+00, x3 = 0.000E+00) : eps_11 = 8.889E-09 ; sig_11 = 2.667E+01 ; axial force = 2.667E+01 11

12 -- element 12 ( type = L2P0 ) = 2.000E+01, x2 = 0.000E+00, x3 = 0.000E+00) : eps_11 = E-08 ; sig_11 = E+01 ; axial force = E element 13 ( type = L2P0 ) = 2.400E+01, x2 = 3.000E+00, x3 = 0.000E+00) : eps_11 = E-09 ; sig_11 = E+01 ; axial force = E element 14 ( type = L2P0 ) = 2.800E+01, x2 = 6.000E+00, x3 = 0.000E+00) : eps_11 = 4.000E-09 ; sig_11 = 1.200E+01 ; axial force = 1.200E element 15 ( type = L2P0 ) = 2.800E+01, x2 = 3.000E+00, x3 = 0.000E+00) : eps_11 = 5.556E-09 ; sig_11 = 1.667E+01 ; axial force = 1.667E element 16 ( type = L2P0 ) = 2.800E+01, x2 = 0.000E+00, x3 = 0.000E+00) : eps_11 = E-09 ; sig_11 = E+01 ; axial force = E element 17 ( type = L2P0 ) = 3.200E+01, x2 = 3.000E+00, x3 = 0.000E+00) : eps_11 = E-09 ; sig_11 = E+01 ; axial force = E+01 = 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 = E-18, u_2 = E-08 node : 2 ( x1 = 0.000E+00, x2 = 6.000E+00 ) u_1 = 1.507E-18, u_2 = E-28 node : 3 ( x1 = 8.000E+00, x2 = 0.000E+00 ) u_1 = E-07, u_2 = E-06 node : 4 ( x1 = 8.000E+00, x2 = 6.000E+00 ) u_1 = 4.018E-07, u_2 = E-06 12

13 node : 5 ( x1 = 1.600E+01, x2 = 0.000E+00 ) u_1 = E-07, u_2 = E-06 node : 6 ( x1 = 1.600E+01, x2 = 6.000E+00 ) u_1 = 6.542E-07, u_2 = E-06 node : 7 ( x1 = 2.400E+01, x2 = 0.000E+00 ) u_1 = E-07, u_2 = E-06 node : 8 ( x1 = 2.400E+01, x2 = 6.000E+00 ) u_1 = 7.218E-07, u_2 = E-06 node : 9 ( x1 = 3.200E+01, x2 = 0.000E+00 ) u_1 = E-07, u_2 = E-06 node : 10 ( x1 = 3.200E+01, x2 = 6.000E+00 ) u_1 = 7.538E-07, u_2 = E-06 apes - end of analysis

Stretching of a Prismatic Bar by its Own Weight

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