Verification Examples

Transcription

1 Verification Examples 018

2 AxisVM X4 Verification Examples Linear static... 4 Supported bar with concentrated loads Thermally loaded bar structure Continously supported beam with point loads External prestressed beam Periodically supported infinite membrane wall with constant distributed load Clamped beam examination with plane stress elements Clamped thin square plate Simply supported XLAM plate Plate with fixed support and constant distributed load.... Annular plate All edges simply supported plate with partial distributed load Clamped plate with linear distributed load Hemisphere displacement Nonlinear static D beam structure Plate with fixed end and bending moment Plastic material Clamped beam with plastic material under cyclic loading Clamped beam with symmetric nonlinear material model Clamped beam with asymmetric nonlinear material model... 4 Clamped beam with only compression material model Dynamic Deep simply supported beam Clamped thin rhombic plate Cantilevered thin square plate Cantilevered tapered membrane Flat grillages Stability Simply supported beam Simply supported beam Design N-M interaction curve of cross-section (EN :004) RC beam deflection according to EC, EN : RC one-way slab deflection according to EC, EN : RC two-way slab deflection according to EC, EN : Nonlinear analysis of RC columns according to EC, EN : Axially loaded RC column check according to EC, EN : Axially loaded RC column check according to EC, EN : Required steel reinforcement of RC plate according to EC, EN : Interaction check of simply supported beam under biaxial bending (EN ) Interaction check of simply supported beam under normal force, bending and shear force Buckling resistance of simply supported beam (EN ) Buckling resistance of simply supported beam (EN ) Buckling of a hollow cross-section beam (EN ) Lateral torsional buckling of a beam (EN ) Interaction check of beam in section class 4 (EN , EN ) Fire design of steel elements Unprotected column under axial compression (EN ) Fire design of steel elements Unrestrained beam (EN ) Fire design of steel elements Unrestrained beam-column (EN ) Fire design of steel elements Beam-column with restrained lateral displacements (EN ) Earth-quake design using response-spectrum method Design of an XLAM shell (EC5)

3 AxisVM X4 Verification Examples 3 Appendix A Clamped beam with symmetrical nonlinear material model Theoretical background... 1 Clamped beam with asymmetrical nonlinear material model Theoretical background Clamped beam with only compression nonlinear material model Theoretical background... 15

4 AxisVM X4 Verification Examples 4 Linear static

5 AxisVM X4 Verification Examples 5 Software Release Number: X4r1 Date: Tested by: InterCAD File name: beam1.axs Thema Analysis Type Geometry Supported bar with concentrated loads. Linear analysis. Side view Section Area = 100,0 cm (10 10) Loads Boundary Conditions Material Properties Element types Mesh Axial direction forces P1 = -00 kn, P = 100 kn, P3 = -40 kn Fix ends, at R1 and R5. E = 0000 kn / cm = 0,3 Beam element Target Results R1, R5 support forces Theory AxisVM % R 1 [kn] -,00 -,00 0,00 R 5 [kn] 118,00 118,00 0,00

6 AxisVM X4 Verification Examples 6 Software Release Number: X4r1 Date: Tested by: InterCAD File name: beam.axs Thema Analysis Type Geometry Thermally loaded bar structure. Linear analysis. z x Sections: Steel: AS = x 10-4 m (D=cm) Copper: AC = x 10-4 m (D=cm) Side view Loads Boundary Conditions Material Properties Element types Target Results P = -1 kn (Point load) Temperature rise of 10 C in the structure after assembly. The upper end of bars are fixed. Nodal DOF: Frame X-Z plane Steel: ES = 0700 kn / cm, = 0,3, S = 1, x 10-5 C -1 Copper: EC = kn / cm, = 0,3, C = 1,7 x 10-5 C -1 Beam element Smax in the three bars. Theory AxisVM % Steel S max [MPa] 3,84 3,848 0,10 Cooper S max [MPa] 7,185 7,199 0,19

7 AxisVM X4 Verification Examples 7 Software Release Number: X4r1 Date: Tested by: InterCAD File name: beam3.axs Thema Analysis Type Geometry Continously supported beam with point loads. Linear analysis. Side view (Section width = 1,00 m, height1 = 0,30 m, height = 0,60 m) Loads Boundary Conditions Material Properties Element types Target Results P1= -300 kn, P= -150 kn, P3= -800 kn, P4= -450 kn Elastic supported. From A to D is Kz = 5000 kn/m/m. From D to F is Kz = kn/m/m. Nodal DOF: Frame X-Z plane E = 3000 kn/cm = 0,3 Rib element: Three node beam element. Shear deformation is taken into account. ez, My, Vz, Rz Diagram ez Diagram My Results

8 AxisVM X4 Verification Examples 8 Diagram Vz Diagram Rz Reference AxisVM e [%] e A [m] e B [m] e C [m] e D [m] e E [m] e F [m] Reference AxisVM e [%] M A [KNm] M B [KNm] M C [KNm] M D [KNm] M E [KNm] M F [KNm]

9 AxisVM X4 Verification Examples 9 Results Reference AxisVM e [%] V A [KN] V B [KN] V C [KN] V D [KN] V E [KN] V F [KN] Reference AxisVM e [%] R A [KN/m ] R B [KN/m ] R C [KN/m ] R D [KN/m ] R E [KN/m ] R F [KN/m ]

10 AxisVM X4 Verification Examples 10 Software Release Number: X4r1 Date: Tested by: InterCAD File name: beam4.axs Thema Analysis Type Geometry External prestressed beam. Linear analysis. Side view Loads Boundary Conditions Material Properties p = -50 kn /m distributed load Length change = -6,5E-3 at beam 5-6 ey = ez = = 0 at node 1 ex = ey = ez = 0 at node 4 E =,1E11 N / m Beam 1-5, 5-6, 6-4 A = 4,5E-3 m Iz= 0,E-5 m 4 Truss -5, 3-6 A = 3,48E-3 m Iz= 0,E-5 m 4 Beam 1-4 A = 1,516E- m Iz=,174E-4 m 4 Mesh Beam 1-4: division into N segment: N = 1 Element types Rib element: Three node beam element, 1-5, 5-6, 6-4, 1-4 (shear deformation is taken into account) Truss element -5, 3-6 Target NX at beam 1-4 My,max at beam -3 ez at node

11 AxisVM X4 Verification Examples 11 Results Diagram ez ROBOT V6 AxisVM % N x [kn] 584,56 584,81 0,04 M y [knm] 49,6 49,60 0,68 e z [mm] -0,541-0,5469 0,89

12 AxisVM X4 Verification Examples 1 Software Release Number: X4r1 Date: Tested by: InterCAD File name: plane1.axs Thema Analysis Type Geometry Periodically supported infinite membrane wall with constant distributed load. Linear analysis. Loads p = 00 kn / m Side view (thickness = 0,0 cm) Boundary Conditions Material Properties Element types Mesh vertical support at every 4,0 m support length is 0,4 m (Rz = 1E+3) Symmetry edges Nodal DOF: (C C f C C C) E = 880 kn / cm = 0,16 Parabolic quadrilateral membrane (plane stress) Target Sxx at 1-10 nodes (1-5 at middle, 6-10 at support)

13 AxisVM X4 Verification Examples 13 Results Node Analytical [kn/cm ] AxisVM [kn/cm ] % 1 0,1313 0,1310-0,3 0,0399 0,0395-1,00 3-0,0093-0,0095,15 4-0,041-0,0413 0,4 5-0,1073-0,1070-0,8 6-0,9317-0,9166-1,6 7 0,0401 0,046 6,3 8 0,0465 0,0469 0,86 9 0,0538 0,0537-0, ,149 0,145-0,3 Reference: Dr. Bölcskey Elemér Dr. Orosz Árpád: Vasbeton szerkezetek Faltartók, Lemezek, Tárolók

14 AxisVM X4 Verification Examples 14 Software Release Number: X4r1 Date: Tested by: InterCAD File name: plane.axs Thema Analysis Type Geometry Clamped beam examination with plane stress elements. Linear analysis. Side view Loads Boundary Conditions Material Properties Element types Mesh p = -5 kn/m Both ends built-in. Line support component stiffness: 1E+10. Symmetry edge Nodal DOF: (C C f C C C) E = 880 kn / cm = 0 Parabolic quadrilateral membrane (plane stress) Side view

15 AxisVM X4 Verification Examples 15 Target Results xy, max at section C Diagram xy Diagram xy at section C

16 AxisVM X4 Verification Examples 16 V 65,65 kn ( from beam theory) S ' y 0, m 3 b 0,5 m I y 0, m 4 xy V S b I ' y y 65,650, ,5 0,50, kn/ m AxisVM result xy = 786,8 kn / m Difference = -0,09 % AxisVM result V nxy 65, 33 kn Difference = -0,45 %

17 AxisVM X4 Verification Examples 17 Software Release Number: X4r1 Date: Tested by: InterCAD File name: plate1.axs Thema Analysis Type Geometry Clamped thin square plate. Linear analysis. Top view (thickness = 5,0 cm) Loads Boundary Conditions Material Properties Element types Mesh P = -10 kn (at the middle of the plate) ex = ey = ez = fix = fiy = fiz = 0 along all edges Nodal DOF: Plate in X-Y plane E = 0000 kn / cm = 0,3 Plate element (Parabolic quadrilateral, heterosis type) Target Displacement of middle of the plate

18 Displacements AxisVM X4 Verification Examples 18 Results Displacements Mode Mesh Book 1 Timoshenko AxisVM Diff 1 [%] Diff [%] 1 x 0,40 0,40 4,48 10,53 4x4 0,416 0,369-11,30 -,89 3 8x8 0,394 0,38 0,381-3,30 0,6 4 1x1 0,387 0,383-1,03 0, x16 0,385 0,383-0,5 0,79 References: 1.) The Finite Element Method (Fourth Edition) Volume. /O.C. Zienkiewicz and R.L. Taylor/ McGraw-Hill Book Company 1991 London.) Result of analytical solution of Timoshenko Convergency 15,00 10,00 5,00 0, Diff1 [%] Diff [%] -5,00-10,00-15,00 Mesh density

19 AxisVM X4 Verification Examples 19 Software Release Number: X4r1 Date: Tested by: InterCAD File name: XLAM_Example_1.axs Thema Analysis Type Geometry Simply supported XLAM plate. Linear analysis. Top view (x-y plane) Loads P = -100 kn concentrated force acting at point (x = 9.0 m, y =.5 m) PZ = kn/m uniform load Boundary ex = ey = ez = fiy = fiz = 0, fix = free along top and bottom edges Conditions ex = ey = ez = fix = fiz = 0, fiy = free along left and right edges Nodal DOF at the remaining nodes: Plate in X-Y plane Material Material quality equals to C4 timber. Properties Section MM 7s/40 XLAM section with x oriented top layer grain direction and Service Class, Properties producing an overall thickness of 40 mm. Element Plate element (Parabolic quadrilateral, heterosis type) types Mesh Target Average element length is 0. m. Displacement and stresses at node 397 (x = 8.0 m, y = m), material stiffness matrix, shear correction factors. Displacements and stresses with Axis VM, ANSYS and from Navier solution 1

20 AxisVM X4 Verification Examples 0 x y Description AxisVM ANSYS Diff [%] Navier solution Diff [%] 8.0 m m e z [mm] σ x maximum 8.0 m m [N/mm ] 8.0 m m σ y maximum [N/mm ] 8.0 m m τ xz maximum [N/mm ] 8.0 m m τ yz maximum [N/mm ] Material stiffness matrix and shear correction factors from ANSYS Material stiffness matrix and shear correction factos from Axis VM 3 Membrane terms

21 AxisVM X4 Verification Examples 1 Bending terms Shear terms 4 Shear correction factors 1 : with 50 harmonic terms included : units in [N] and [mm] 3 : units in [kn] and [m] 4 : corrected shear stiffnesses

22 AxisVM X4 Verification Examples Software Release Number: X4r1 Date: Tested by: InterCAD File name: plate.axs Thema Analysis Type Geometry Plate with fixed support and constant distributed load. Linear analysis. Top view (thickness = 15,0 cm) Loads P = -5 kn / m Boundary ex = ey = ez = fix = fiy = fiz = 0 along all edges Conditions Nodal DOF: Plate in X-Y plane Material E = 990 kn/cm Properties = 0,16 Element Parabolic triangle plate element types Mesh Target Results Maximal ez (found at Node1) and maximal mx (found at Node) Component Nastran AxisVM % ez,max [mm] mx,max [knm/m]

23 AxisVM X4 Verification Examples 3 Software Release Number: X4r1 Date: Tested by: InterCAD File name: plate3.axs Thema Analysis Type Geometry Annular plate. Linear analysis. Top view (thickness =,0 cm) Loads Boundary Conditions Edge load: Q = 100 kn / m Distributed load: q = 100 kn / m Nodal DOF: Plate in X-Y plane Material Properties Element types E = 880 kn / cm = 0,3 Plate element (parabolic quadrilateral, heterosis type)

24 AxisVM X4 Verification Examples 4 Mesh Target Smax, emax Results Theory AxisVM Model S max S max % [kn/cm] [kn/cm] a.),8,78-1,4 b.) 6,88 6,76-1,74 c.) 14, 14,10-0,84 d.) 1,33 1,33 0,00 e.),35,5-4,6 f.) 9,88 9,88 0,00 g.) 4,79 4,76-0,63 h.) 7,86 7,86 0,00 Theory AxisVM Model e max e max % [mm] [mm] a.) b.) c.) d.) e.) f.) g.) h.) Reference: S. Timoshenko and S. Woinowsky-Krieger: Theory of Plates And Shells

25 AxisVM X4 Verification Examples 5 Software Release Number: X4r1 Date: Tested by: InterCAD File name: plate4.axs Thema Analysis Type Geometry All edges simply supported plate with partial distributed load. Linear analysis. Top view (thickness =,0 cm) Loads Boundary Conditions Material Properties Element types Mesh Distributed load: q = -10 kn / m (middle of the plate at,0 x,0 m area) a.) ex = ey = ez = 0 along all edges (soft support) b.) ex = ey = ez = 0 along all edges = 0 perpendicular the edges (hard support) Nodal DOF: Plate in X-Y plane E = 880 kn / cm = 0,3 Plate element (Heterosis type)

26 AxisVM X4 Verification Examples 6 Target Results mx, max, my, max a.) Moment Theory AxisVM % m x, max [knm/m] 7,4 7,34 1,38 m y, max [knm/m] 5,3 5,39 1,3 b.) Moment Theory AxisVM % m x, max [knm/m] 7,4 7,8 0,55 m y, max [knm/m] 5,3 5,35 0,56 Reference: S. Timoshenko and S. Woinowsky-Krieger: Theory of Plates And Shells

27 AxisVM X4 Verification Examples 7 Software Release Number: X4r1 Date: Tested by: InterCAD File name: plate5.axs Thema Analysis Type Geometry Clamped plate with linear distributed load. Linear analysis. Top view (thickness =,0 cm) Loads Distributed load: q = -10 kn / m Boundary Conditions Material Properties Element types Mesh ex = ey = ez = fix = fiy= fiz = 0 along all edges Nodal DOF: Plate in X-Y plane E = 880 kn / cm = 0,3 Plate element (Heterosis type)

28 AxisVM X4 Verification Examples 8 Target mx, my Results Reference: Results Theory AxisVM % m x,1 [knm/m] 11,50 11,48-0,17 m y,1 [knm/m] 11,50 11,48-0,17 m x, [knm/m] 33,40 33,3-0,51 m x,3 [knm/m] 17,90 17,83-0,39 m y,4 [knm/m] 5,70 5,53-0,66 S. Timoshenko and S. Woinowsky-Krieger: Theory of Plates And Shells

29 AxisVM X4 Verification Examples 9 Software Release Number: X4r1 Date: Tested by: InterCAD File name: hemisphere.axs Thema Analysis Type Geometry Hemisphere displacement. Linear analysis. Hemisphere (Axonometric view) t = 0,04 m Loads Point load P =,0 kn

30 AxisVM X4 Verification Examples 30 Boundary Conditions Material Properties Element types Target ex = ey = ez = fix = fiy = fiz= 0 at C Symmetry in X-Z plane on A-C edge Symmetry in Y-Z plane on B-C edge E = 685 kn / cm = 0,3 Shell element 1.) guadrilateral parabolic.) triangle parabolic ex at point A Results e x [m] e [%] Theory 0,185 AxisVM quadrilateral 0,185 0,00 AxisVM triangle 0,18-1,6

31 AxisVM X4 Verification Examples 31 Nonlinear static Software Release Number: X4r1

32 AxisVM X4 Verification Examples 3 Date: Tested by: InterCAD File name: nonlin1.axs Thema Analysis Type Geometry 3D beam structure. Geometrical nonlinear analysis. 1,73 m F y =-300,00 kn F z =-600,00 kn Node1 F y =-300,00 kn F z =-600,00 kn 3,000 m 1,73 m Beam1 D Y X 1,73 m 3,000 m 1,73 m A F z =-600,00 kn C B 4,000 m Z Z Y X X Loads Boundary Conditions Material Properties Cross- Section Properties Element types Target Py = -300 kn Pz = -600 kn ex = ey = ez = 0 at A, B, C and D S 75 E = 1000 kn / cm = 0,3 HEA 300 Ax = cm ; Ix = 85.3 cm 4 ; Iy = cm 4 ; Iz = cm 4 Beam ex, ey, ez, at Node1 Nx, Vy, Vz, Tx, My, Mz of Beam1 at Node1 Results Comparison with the results obtained using Nastran V4

33 AxisVM X4 Verification Examples 33 Component Nastran AxisVM % ex [mm] 17,898 17,881-0,09 ey [mm] -75,70-75,663-0,05 ez [mm] -4,63-4,597-0,06 Nx [kn] -83,15-83,5 0,04 Vy [kn] -8,09-8,10 0,04 Vx [kn] -106,57-106,48-0,08 Tx [knm] -4,57-4,57 0,00 My [knm] -519,00-518,74-0,05 Mz [knm] 148,94 148,91-0,0

34 AxisVM X4 Verification Examples 34 Software Release Number: X4r1 Date: Tested by: InterCAD File name: nonlin.axs Thema Analysis Type Geometry Plate with fixed end and bending moment. Geometrical nonlinear analysis. Loads Boundary Conditions Material Properties Cross Section Properties Element types Mz = 600 knm (x1300 Nm) acting on Edge ex = ey = ez = fix = fiy = fiz = 0 along Edge1 (Use Constrained nodes instead of line support; Nodal DOF on Edge 1: Fixed node) E = 0000 N / mm = 0 Plate thickness: 150 mm Rib on Edge: circular D = 500 mm (for distributing load to the mid-side-node) Parabolic quadrilateral shell (heterosis type) Rib on Edge for distributing load to the mid-side-node

35 AxisVM X4 Verification Examples 35 Target Results Z at Edge I I E z plate plate plate plate M.610 Theoretical results based on the differential equation of the flexible beam: M E plate plate 3 ab N m 1 m M z I platee Nm.610 z plate plate rad Comparison the AxisVM result with the theoretical one: Component Theory AxisVM % fiz [rad] 5,5467 5,550 0,06

36 AxisVM X4 Verification Examples 36 Software Release Number: X4r1 Date: Tested by: InterCAD File name: Plastic_1.axs Thema Analysis Type Geometry Plastic material Nonlinear static analysis Cross-section: D = 30mm Loads Boundary Conditions Material Properties Element types Target Axial force at A: N Solution control: Displacement at A ex = ey = ez = 0 at B, C and D S 35 E = 1000 kn / cm = 0,3 Linear elastic perfectly plastic material model Truss element Check the load vertical displacement (A) curve Results Analytical results: [u;f(u)] AxisVM: [Axisi,1; Axisi,0]

37 AxisVM X4 Verification Examples 37 Software Release Number: X4r1 Date: Tested by: InterCAD File name: Plastic_.axs Thema Analysis Type Geometry Clamped beam with plastic material under cyclic loading Nonlinear static analysis Fz A L = 100 cm B Nx Cross-section: Z = 30mm Loads Boundary Conditions Material Properties Element types Target Results Nx = 63,333 kn; Fz =,666 kn Solution control: Displacement at B ez = -70 mm Increment function: 1, ,5 0-0, ,5 ex = ey = ez = fix = fiy = fiz = 0 at A Steel E = kn/cm ; ET = 1000 kn/cm ; y = 10 kn/cm = 0,3 Linear elastic plastic material model Hardening rule: Isotropic hardening Beam element Check the load displacements and beam strains curves AxisVM: Beam element Rib element (shear deformation is taken into account)

38 Fz [kn] Fz [kn] Fx [kn] AxisVM X4 Verification Examples 38 ANSYS 14.0 Beam element (unrestrained warping) Axis beam ANSYS beam Axis Rib -400 ex [mm] Axis beam ANSYS beam Axis Rib -0 ez [mm] Axis beam ANSYS beam Axis Rib -0 ey [mm]

39 Fz [kn] AxisVM X4 Verification Examples ,004-0,003-0,00-0, ,001 0,00 0,003 0,004 0,005-5 Axis beam ANSYS beam Axis Rib exx []

40 AxisVM X4 Verification Examples 40 Software Release Number: X4r3 Date: Tested by: InterCAD File name: matnl_01_xx (xx element type) Thema Analysis Type Geometry Clamped beam with symmetric nonlinear material model Nonlinear static analysis Loads Boundary Conditions Material Properties Fz = 00 kn Solution control: Force Increment function: Equal increments ex = ey = ez = fix = fiy = fiz = 0 at A Steel Strain energy based (NLE) Steel Von Mises (VM) Material model function: σ = 400 ε Discrete function assignment per ε = [] = 0,3 Element types / File name Target Beam/Rib element Plate element (heterosis type) matnl_01_beam-rib_nle.axs, matnl_01_beam-rib_vm.axs matnl_01_plate_nle.axs, matnl_01_plate_vm.axs Membrane element matnl_01_membrane_nle.axs, matnl_01_membrane_vm.axs Check vertical displacements (B) and stresses (A)

41 AxisVM X4 Verification Examples 41 Results Analytical background: Appendix A; Yield criterion Strain energy based Von Mises Type of element eb [mm] [%] eb [mm] [%] Analytical 156,4 156,4 Beam 157,98 1,01 158,6 1,41 Rib 158,48 1,33 159,14 1,75 Plate TRIA 158,98 1,65 158,68 1,46 Plate QUAD 159,04 1,69 158,81 1,54 Membrane TRIA 158,34 1,4 159,14 1,75 Membrane QUAD 158,79 1,53 159,6,05 Yield criterion Strain energy based Von Mises Type of element A [kn/cm ] [%] A [kn/cm ] [%] Analytical Beam 49,9-0,0 48,88 -,4 Rib 49,9-0,0 48,88 -,4 Plate TRIA 49,84-0,3 49,6-0,80 Plate QUAD 50,5 1,00 50,7 1,40 Membrane TRIA 49,75-0,50 47,44-5,1 Membrane QUAD 49,66-0,68 47,04-5,9

42 AxisVM X4 Verification Examples 4 Software Release Number: X4r3 Date: Tested by: InterCAD File name: matnl_0_xx (xx element type) Thema Analysis Type Geometry Clamped beam with asymmetric nonlinear material model Nonlinear static analysis Loads Fz = 100 N; Solution control: Force Increment function: Equal increments Boundary ex = ey = ez = fix = fiy = fiz = 0 at A Conditions Material Properties Concrete Bresler-Pister (BP) Other Strain energy based (NLE) E = 8600 N/mm ; ET = 0 N/mm ; yt = 1,6 N/mm ;yc = 16 N/mm ; CyB = 1, (Bresler-Pister); = 0; Element types / File name Target Beam/Rib element matnl_0_beam-rib_nle.axs, matnl_0_beam-rib_bp.axs Shell element matnl_0_shell_nle.axs, matnl_0_shell_bp.axs (heterosis type) Check vertical displacements (B) and stresses (C), length of plastic zone (x)

43 AxisVM X4 Verification Examples 43 Results Analytical background: Appendix A; Yield criterion Strain energy based Bresler Pister Type of element eb [mm] [%] eb [mm] [%] Analytical,833,833 Beam,798-1,4,796-1,31 Rib,841 0,8,837 0,14 Shell,899,33,79-1,45 Yield criterion Strain energy based Bresler Pister Type of element C,min [kn/cm ] [%] C,min [kn/cm ] Analytical 4,98 4,98 [%] Beam 5,03 1,00 5,0 0,8 Rib 4,90-1,61 4,90-1,61 Shell 5,01 0,60 4,90-1,61 Axis shell NLE model top and bottom Sxx [N/mm ]

44 AxisVM X4 Verification Examples 44 Effective plastic strain of Beam and shell (top) with Bresler-Pister material Analytical result for x = 1,111 m (plastic zone)

45 AxisVM X4 Verification Examples 45 Software Release Number: X4r3 Date: Tested by: InterCAD File name: matnl_03_xx (xx element type) Thema Analysis Type Geometry Clamped beam with only compression material model Nonlinear static analysis Loads Boundary Conditions Material Properties Fz =00 N; N = 5000 N Solution control: Force Increment function: Equal increments ex = ey = ez = fix = fiy = fiz = 0 at A Concrete Bresler-Pister (BP) Other Strain energy based (NLE) E = 8600 N/mm ; ET = 0 N/mm ; yt = 0,016 N/mm ;yc = 16 N/mm ; CyB = 1, (Bresler-Pister); = 0; Element types / File name Target Beam/Rib element matnl_03_beam-rib_nle.axs, matnl_03_beam-rib_bp.axs Shell element matnl_03_shell_nle.axs, matnl_03_shell_bp.axs (heterosis type) Check vertical displacements (B) and stresses (C), length of plastic zone (x)

46 AxisVM X4 Verification Examples 46 Results Analytical background: Appendix A; Yield criterion Strain energy based Bresler Pister Type of element eb [mm] [%] eb [mm] [%] Analytical 0,475 0,475 Beam 0,467-1,76 0,466-1,97 Rib 0,475-0,07 0,473-0,50 Shell 0,483 1,61 0,471-0,9 Yield criterion Strain energy based Bresler Pister Type of element C,min [N/mm ] [%] C,min [N/mm ] Analytical 1,097 1,097 [%] Beam 1,088-0,8 1,086-1,0 Rib 1,067 -,73 1,066 -,8 Shell 1,085-1,09 1,076-1,97 Axis shell NLE model top and bottom Sxx [N/mm ]

47 AxisVM X4 Verification Examples 47 Effective plastic strain of Beam and shell (top) with Bresler-Pister material Analytical result for x = 1,113 m (plastic zone)

48 AxisVM X4 Verification Examples 48 BLANK

49 AxisVM X4 Verification Examples 49 Dynamic Software Release Number: X4r1

50 AxisVM X4 Verification Examples 50 Date: Tested by: InterCAD File name: dynam1.axs Thema Analysis Type Geometry Deep simply supported beam. Vibration analysis. Beam (Axonometric view) Cross section (square,0 m x,0 m) Loads Boundary Conditions Material Properties Element types Target Self-weight (Other option: Apply Masses only option on Vibration analysis window) ex = ey = ez = fix = 0 at A ey = ez = 0 at B E = 0000 kn / cm = 0,3 = 8000 kg / m 3 Rib elemen: Three node beam element (shear deformation is taken into account) First 7 mode shapes Results

51 AxisVM X4 Verification Examples 51 Mode 1: f = 43,16 Hz Mode : f = 43,16 Hz Mode 3: f = 14,01 Hz Mode 4: f = 15,50 Hz Mode 5: f = 15,50 Hz Mode 6: f = 93,55 Hz Mode 7: f = 93,55 Hz

52 AxisVM X4 Verification Examples 5 Results Comparison with NAFEMS example Mode NAFEMS (Hz) AxisVM (Hz) % 1 4,65 43,16-1,0 4,65 43,16-1,0 3 15,00 14,01 0, ,31 15,50 -, ,31 15,50 -, ,55 93,55-3, ,55 93,55-3,16

53 AxisVM X4 Verification Examples 53 Software Release Number: X4r1 Date: Tested by: InterCAD File name: dynam.axs Thema Analysis Type Geometry Clamped thin rhombic plate. Vibration analysis. Top view of plane (thickness = 5,0 cm) Loads Boundary Conditions Material Properties Element types Mesh Self-weight ex = ey = fiz = 0 at all nodes (i.e.: ex, ey, fiz constrained at all nodes; Nodal DOF: Plate in X-Y plane) ez = fix = fiy = 0 along the 4 edges (Line support) E = 0000 kn / cm = 0,3 = 8000 kg / m 3 Parabolic quadrilateral shell element (heterosis type)

54 er 0,506 0,470 0,433 0,397 0,361 0,35 0,89 0,53 0,17 0,181 0,144 0,108 0,07 0,036 0 er 0,486 0,451 0,416 0,38 0,347 0,31 0,78 0,43 0,08 0,174 0,139 0,104 0,069 0,035 0 er 0,498 0,46 0,47 0,391 0,356 0,30 0,84 0,49 0,13 0,178 0,14 0,107 0,071 0,036 0 er 0,463 0,49 0,396 0,363 0,330 0,97 0,64 0,31 0,198 0,165 0,13 0,099 0,066 0,033 0 er er 0,449 0,417 0,385 0,353 0,31 0,89 0,57 0,5 0,19 0,160 0,18 0,096 0,064 0,03 0 0,50 0,483 0,446 0,409 0,37 0,335 0,97 0,60 0,3 0,186 0,149 0,11 0,074 0,037 0 AxisVM X4 Verification Examples 54 Target First 6 mode shapes Results Mode 1: f = 8,0 Hz Mode : f = 13,0 Hz Mode 3: f = 18,41 Hz Mode 4: f = 19,33 Hz Mode 5: f = 4,6 Hz Mode 6: f = 8,4 Hz Results Comparison with NAFEMS example Mode NAFEMS (Hz) AxisVM (Hz) % 1 7,94 8,0 1,01 1,84 13,0 1, ,94 18,41,6 4 19,13 19,33 1,05 5 4,01 4,6,54 6 7,9 8,4 1,15

55 AxisVM X4 Verification Examples 55 Software Release Number: X4r1 Date: Tested by: InterCAD File name: dynam3.axs Thema Analysis Type Geometry Cantilevered thin square plate. Vibration analysis. Top view (thickness = 5,0 cm) Loads Boundary Conditions Material Properties Element types Mesh Self-weight ex = ey = ez = fix = fiy = fiz = 0 along y-axis E = 0000 kn / cm = 0,3 = 8000 kg / m 3 Parabolic quadrilateral shell element (heterosis type).

56 AxisVM X4 Verification Examples 56 Target First 5 mode shapes Results Mode 1: f = 0,4 Hz Mode 3: f =,53 Hz Mode 5: f = 3,68 Hz

57 AxisVM X4 Verification Examples 57 Mode : f = 1,0 Hz Mode 4: f = 3, Hz Comparison with NAFEMS example Mode NAFEMS (Hz) AxisVM (Hz) % 1 0,41 0,40-0,4 1,09 1,00-0,87 3,580,530-1,94 4 3,310 3,0 -,7 5 3,750 3,680-1,87

58 AxisVM X4 Verification Examples 58 Software Release Number: X4r1 Date: Tested by: InterCAD File name: dynam4.axs Thema Analysis Type Geometry Cantilevered tapered membrane. Vibration analysis. Side view (thickness = 10,0 cm) Loads Boundary Conditions Material Properties Element types Mesh Self-weight Edge 1: Nodal DOF: Fixed node Other nodes: DOF: (f f C C C C) (f: free; C: constrained) E = 0000 kn / cm = 0,3 = 8000 kg / m 3 Parabolic quadrilateral membrane (plane stress)

59 AxisVM X4 Verification Examples 59 Target First 4 mode shapes Results 5,000 1,000 10,000 X Mode 1: f = 44,50 Hz Mode : f = 18,60 Hz

60 AxisVM X4 Verification Examples 60 Mode 3: f = 16,48 Hz Mode 4: f = 41,46 Hz Results Comparison with NAFEMS example Software Release Number: R3 Software Release Number: X4r1 Mode NAFEMS (Hz) AxisVM (Hz) % 1 44,6 44,33-0,65 130,03 18,36-1,8 3 16,70 16,48-0, ,05 41,46-1,87

61 AxisVM X4 Verification Examples 61 Date: Tested by: InterCAD File name: dynam5.axs Thema Analysis Type Geometry Flat grillages. Vibration analysis. Top view Loads Boundary Conditions Material Properties Cross Section Element types Mesh Self-weight ex = ey = ez = 0 at the ends (simple supported beams) Nodal DOF: Grillage in X-Y plane E = 0000 kn / cm G = 7690 kn / cm = 0,3 = 7860 kg / m 3 A = 0,004 m Ix =,5E-5 m 4 Iy = Iz = 1,5E-5 m 4 Rib element: Three node beam element (shear deformation is taken into account)

62 AxisVM X4 Verification Examples 6 Target First 3 mode shapes Results Mode 1: f = 16,90 Hz Mode : f = 0,64 Hz Mode 3: f = 51,76 Hz

63 AxisVM X4 Verification Examples 63 Reference: Mode Reference AxisVM (Hz) % 1 16,85 16,90 0,30 0,1 0,64, ,30 51,76 -,89 C.T.F. ROSS: Finite Element Methods In Engineering Science

64 AxisVM X4 Verification Examples 64 BLANK

65 AxisVM X4 Verification Examples 65 Stability Software Release Number: X4r1

66 AxisVM X4 Verification Examples 66 Date: Tested by: InterCAD File name: buckling1.axs Thema Analysis Type Geometry Simply supported beam. Buckling analysis. Front view Cross section(iz =168,3 cm 4, It =1,18 cm 4, Iw =16667 cm 6 ) Loads Boundary Conditions Material Properties Element types Mesh Bending moment at both ends of beam MA = 1,0 knm, MB = -1,0 knm (Moments are applied as surface edge loads) ex = ey = ez = 0 at A ex = ey = ez = 0 at B kz = kw = 1 E = 0600 kn / cm = 0,3 Parabolic quadrilateral shell element (heterosis type)

67 AxisVM X4 Verification Examples 67 Target Mcr =? (for lateral torsional buckling) Results Analytical solution M cr E I Z W L IZ I L G It E I Z , ,18 M cr 1451 kncm 14, , ,3 knm AxisVM result Mcr = 15,3 knm Difference +0,6%

68 AxisVM X4 Verification Examples 68 Software Release Number: X4r1 Date: Tested by: InterCAD File name: buckling.axs Thema Analysis Type Geometry Simply supported beam. Buckling analysis. Front view (L = 1,0 m) S G 1 10,0 S G 1 10, z 1,0 30,0 y z y Loads P = -1,0 kn at point B. Section A1 Cross-sections Section A Boundary Conditions Material Properties Element types Target Results ex = ey = ez = 0 at A ey = ez = 0 at B E = 0000 kn / cm = 0,3 Beam element Pcr =? (for inplane buckling) Theory AxisVM e [%] P cr [kn] 3,340 3,337-0,09

69 AxisVM X4 Verification Examples 69 Design Software Release Number: X4r1

70 AxisVM X4 Verification Examples 70 Date: Tested by: InterCAD File name: RC column1.axs Thema N-M interaction curve of cross-section (EN :004). Analysis Type Geometry Linear static analysis+design Section: 300x400 mm Covering: 40 mm Loads Boundary Conditions Material Properties Target Results Arbitrary. Arbitrary. Concrete: fcd=14, N/mm ec1=0,00 ecu=0,0035 (parabola-constans - diagram) Steel: fsd=348 N/mm esu=0,015 Compare the program results with with hand calculation at keypoints of M-N interaction curve. N N [kn] M [knm] N AxisVM M(N) AxisVM e % ,5 +61,4 +0, ,7-0, ,5 +0, ,5-61,4 +0, , +0, ,7-0,6 Reference: Dr. Kollár L. P., Vasbetonszerkezetek I. Műegyetemi kiadó

71 AxisVM X4 Verification Examples 71 Software Release Number: X4r Date: Tested by: InterCAD File name: RCbeam.axs Thema RC beam deflection according to EC, EN :010. Analysis Type Geometry Materially nonlinear analysis. q = 17 kn/m L = 5.60 m Side view 0 35 cm covering = 3 cm = cm Section Loads Boundary Conditions Material Properties Element types Target q = 17 kn /m distributed load Simply supported beam. Concrete: C5/30, =.1 Steel: B500B ε=0.4 shrinkage strain Simple 1DOF beam elements (Euler-Bernoulli beam) ez, max

72 AxisVM X4 Verification Examples 7 Results without shrinkage Diagram ez Hand calculation: e eii ( 1 ) ei mm where, M cr 1 M ei is the deflection which was calculated based on uncracked section eii is the deflection which was calculated based on cracked section Calculation with AxisVM: e = 19.9 mm (difference < 1%) Results with shrinkage Hand calculation: e eii ( 1 ) ei 3. 54mm where, M cr 1 M ei is the deflection which was calculated based on uncracked section eii is the deflection which was calculated based on cracked section Calculation with AxisVM: e =.1 mm (difference ~ -6%)

73 AxisVM X4 Verification Examples 73 Software Release Number: X4r Date: Tested by: InterCAD File name: RC_Slab_1.axs Thema RC one-way slab deflection according to EC, EN :010. Analysis Type Geometry Materially nonlinear analysis. q = 17 kn/m L = 5.60 m Side view Section: 5 cm 1 cm As = 16/00 = m Loads Boundary Conditions Material Properties Element types Target q = 17 kn /m distributed load Simply supported one-way slab. Concrete: C5/30, =.1, ν = 0.0 Steel: B500B ε=0.4 shrinkage strain triangle shell elements ez, max

74 AxisVM X4 Verification Examples 74 Results - without shrinkage NL behaviour in ε-n + κ-m Diagram ez NL behaviour in κ-m only Hand calculation: e eii ( 1 ) ei mm where, M cr 1 M ei is the deflection which was calculated based on uncracked section eii is the deflection which was calculated based on cracked section Calculation with AxisVM: e = mm (NL ε-n + κ-m) (difference < -1%) e = mm (NL κ-m) (difference ~ +3%) Results - with shrinkage Hand calculation: e eii ( 1 ) ei mm where, M cr 1 M ei is the deflection which was calculated based on uncracked section eii is the deflection which was calculated based on cracked section Calculation with AxisVM: e = 45.0 mm (NL ε-n + κ-m) (difference ~ -5%) e = 49.4 mm (NL κ-m) (difference ~ +4%)

75 AxisVM X4 Verification Examples 75 Software Release Number: X4r Date: Tested by: InterCAD File name: RC_Slab_.axs Thema RC two-way slab deflection according to EC, EN :010. Analysis Type Geometry Materially nonlinear analysis. x y h=50mm; q=17kn/m 5.6m axb=φ10/100; d=0mm ayb=φ10/100; d=10mm 8m Loads Boundary Conditions Material Properties Element types Target = 0.5 q = 17 kn /m distributed load Simply supported two-way slab. Concrete: C5/30, =.1, ν = 0.0 (due to the comparison with hand calculation based on Marcus method) Steel: B500B ε=0.4 shrinkage strain triangle shell elements ez, max

76 AxisVM X4 Verification Examples 76 Results - without shrinkage Hand calculation: e eii ( 1 ) ei mm where, NL behaviour in ε-n + κ-m Diagram ez M cr 1 M ei is the deflection which was calculated based on uncracked section eii is the deflection which was calculated based on cracked section Calculation with AxisVM: e = 9 mm (difference ~ -14%) (note: hand calculation gives conservative result because the effect of Mxy is not considered) Results - with shrinkage Hand calculation: e eii ( 1 ) ei mm where, M cr 1 M ei is the deflection which was calculated based on uncracked section eii is the deflection which was calculated based on cracked section Calculation with AxisVM: e = ~35 mm (difference ~ -13%) (note: hand calculation gives conservative result because the effect of Mxy is not considered)

77 AxisVM X4 Verification Examples 77 Software Release Number: X4r1 Date: Tested by: InterCAD File name: RCcolumn.axs. RCLcolumn.axs Thema Nonlinear analysis of RC columns according to EC, EN :010. Analysis Type Geometry Materially and geometrically nonlinear analysis. Loads Boundary Conditions Material Properties Element types Target Concentrated force on the top Cantilever Concrete: C5/30, =,0 Steel: B500B Simple 1DOF beam elements (Euler-Bernoulli beam) ez, max

78 AxisVM X4 Verification Examples 78 Results

79 AxisVM X4 Verification Examples 79 Software Release Number: X4r1 Date: Tested by: InterCAD File name: RCcolumn.axs Thema Axially loaded RC column check according to EC, EN :010. Analysis Type Geometry Materially and geometrically linear analysis. Loads Material Properties Concentrated force NEd = 70 kn Bending moments: at the top MyEd = 48 knm at the bottom MyEd = 66 knm Concrete: C30/37, =,0 Steel: B500B Element Simple 1DOF beam elements (Euler-Bernoulli beam) types Target Calculate eccentricities according to EN 199-1

80 AxisVM X4 Verification Examples 80 Results [cm] Hand calculation AxisVM e0 ei e etot e0 ei e etot e % x-z Bottom plane Middle ** 0** * ** 0** * 0 Top x-y plane Bottom * * 0 Middle * * 0 Top * * 0 * due to minimal eccentricity requirement ** due to the buckling

81 AxisVM X4 Verification Examples 81 Software Release Number: X4r1 Date: Tested by: InterCAD File name: Rccolumn3.axs Thema Axially loaded RC column check according to EC, EN :010. Analysis Type Geometry Materially and geometrically linear analysis. Loads Material Properties Concentrated force NEd = 70 kn Bending moments: at the top MzEd = 40 knm at the bottom MzEd = 40 knm Concrete: C30/37, =,0 Steel: B500B Element Simple 1DOF beam elements (Euler-Bernoulli beam) types Target Calculate eccentricities according to MSZ EN 199-1

82 AxisVM X4 Verification Examples 8 Results [cm] Hand calculation AxisVM e0 ei e etot e0 ei e etot e % x-z Bottom ** * ** * 0 plane Middle * * 0 Top ** * ** * 0 x-y plane Bottom Middle ~0 ** ** ** ** Top ** 0** * ** 0** * 0 * due to minimal eccentricity requirement ** due to the buckling

83 AxisVM X4 Verification Examples 83 Software Release Number: X4r1 Date: Tested by: InterCAD File name: beam.axs Thema Required steel reinforcement of RC plate according to EC, EN :004. Analysis Type Geometry Linear analysis. Side view Cross-section Loads Boundary Conditions Material Properties Element types Mesh Pz = -50 kn point load Clamped cantilever plate. Fix line support on clamped edge. Nodal DOF: Plate in X-Y plane Concrete: C5/30 Steel: B500A Parabolic quadrilateral plate element (heterosis type) Top view

84 AxisVM X4 Verification Examples 84 Target AXT steel reinforcement along x direction at the top of the support Results Diagram AXT Calculation according to EC: f cd 5 1, ,15 16,6 N / mm f yd 435 N / mm c E 0,850, , cu S c0 cu ES f yd d = = 47 mm 0,54 xc M sd M Rd b xc fcd d 00 knm x c 439 h 55 xc 55 0, 0 0,54 47 c Steel reinforcement is yielding c d A S b xc f f yd cd , mm Calculation with AxisVM: AXT = 093 mm / m Different = -0,3 %

85 AxisVM X4 Verification Examples 85 Software Release Number: X4r1 Date: Tested by: InterCAD File name: 3_10 Plastic biaxial bending interaction.axs Thema Interaction check of simply supported beam under biaxial bending (EN ). Analysis Type Geometry Steel Design h = 70 mm b = 135 mm tf = 10 mm tw = 7 mm l = 6000 mm A = 45,95 cm Wy,pl = 484,1 cm 3 Wz,pl = 97 cm 3 IPE70 cross section Loads Boundary Conditions Material Properties qy = 1,5 kn/m qz = 0,4 kn/m ex = ey = ez = 0 at A ey = ez = 0 at B S 35 E = 1000 kn/cm = 0,3

86 AxisVM X4 Verification Examples 86 Element types Target Results Beam element Interaction check taking into account plastic resistances Analytical solution in the following book: Dunai, L., Horváth, L., Kovács, N., Verőci, B., Vigh, L. G.: Acélszerkezetek méretezése az Eurocode 3 alapján, Gyakorlati útmutató (Design of steel structures according to Eurocode 3, ) Magyar Mérnöki Kamara Tartószerkezeti tagozata, Budapest, 009. Exercise 3.10., page 8. Analitical solution AxisVM e[%] My,Ed [knm] 91,8 91,8 - Mz,Ed [knm] 6,75 6,75 - Mpl,y,Rd [knm] 113,74 114,57 +0,07 Mpl,z,Rd [knm],78,78 +0,00 α - β capacity ratio [-] 0,948 0,

87 AxisVM X4 Verification Examples 87 Software Release Number: X4r1 Date: Tested by: InterCAD File name: 3_1 _MNV_Interaction.axs Thema Analysis Type Geometry Interaction check of simply supported beam under normal force, bending and shear force. (EN , EN ) Steel Design h = 00 mm b = 00 mm tf = 15 mm tw = 9 mm l = 1400 mm A = 78,1 cm Av = 4,83 cm Iy = 5696 cm 3 Wy,pl = 643 cm 3 IPE70 cross section Loads Boundary Conditions Material Properties Element types Target Fz = 300 kn at thirds of beam N = 500 kn at B ex = ey = ez = fix = 0 at A ey = ez = fix =0 at B S 35 E = 1000 kn/cm = 0,3 Beam element Interaction check of axial force, shear force and bending moment.

88 AxisVM X4 Verification Examples 88 Results Analytical solution in the following book: Dunai, L., Horváth, L., Kovács, N., Verőci, B., Vigh, L. G.: Acélszerkezetek méretezése az Eurocode 3 alapján, Gyakorlati útmutató (Design of steel structures according to Eurocode 3, ) Magyar Mérnöki Kamara Tartószerkezeti tagozata, Budapest, 009. Exercise 3.1., page Analytical solution AxisVM results e[%] NEd [kn] Vz,Ed [kn] My,Ed [knm] Pure compression Npl,Rd [kn] ,6-0,0- capacity ratio [-] 0,33 0,33 - Pure shear Vpl,z,Rd [kn] 394, 394,5 +0,08 capacity ratio [-] 0,761 0,761 - Pure bending Mpl,y,Rd [knm] 176,8 178,4 +0,9 capacity ratio [-] 0,79 0,79 - Interaction check 0,73 0,71-0,73 MV,Rd [knm] 163,96 165,57 +0,98 n 0,33 0,33 - a 0,3 0,3 - MNV,Rd [knm] 14, 143,67 +1,03 capacity ratio [-] 0,985 0,974-1,1

89 AxisVM X4 Verification Examples 89 Software Release Number: X4r1 Date: Tested by: InterCAD File name: 3_15 Központosan nyomott rúd - I szelvény.axs Thema Buckling resistance of simply supported beam (EN ). Analysis Type Geometry Steel Design h = 300 mm b = 50 mm tf = 14 mm tw = 8 mm l = 4500 mm A = 94 cm Iy = 19065,8cm 4 Iz = 3647,1 cm 4 iy = 14,1 cm iz = 6, cm I cross section, symmetric about y and z axis Loads Normal force at point A NA= -1,0 kn Boundary ey = 0 at A Conditions ex = ey = ez = fix = fiz= 0 at B kz = kw = 1 Material S 35 Properties E = 1000 kn / cm = 0,3 Element Beam element types Target Buckling resistance Nb,Rd =? Results Analytical solution in the following book:

90 AxisVM X4 Verification Examples 90 Dunai, L., Horváth, L., Kovács, N., Verőci, B., Vigh, L. G.: Acélszerkezetek méretezése az Eurocode 3 alapján, Gyakorlati útmutató (Design of steel structures according to Eurocode 3, ) Magyar Mérnöki Kamara Tartószerkezeti tagozata, Budapest, 009. Exercise 3.15., P Analytical solution AxisVM e[%] y [-] 0,673 0,673 - z [-] 0,771 0,769-0,6 Χy [-] 0,8004 0,7989-0,19 Χz [-] 0,6810 0, ,07 Nb,Rd [kn] 1504,3 1505,3 +0,07

91 AxisVM X4 Verification Examples 91 Software Release Number: X4r1 Date: Tested by: InterCAD File name: 3_1 Központosan nyomott rúd - T szelvény.axs Thema Buckling resistance of simply supported beam (EN ). Analysis Type Geometry Steel Design h = 180 mm b = 50 mm tf = 16 mm tw = 16 mm l = 3000 mm A = 68,8 cm Iy = 394,5cm 4 Iz = 089,48 cm 4 Ics= 58,71 cm 4 Iw = 1108,0 cm 6 iy = 5,90 cm iz = 5,51 cm Welded T section, symmetric to z but not y Loads Normal force at point A NA= -1,0 kn Boundary ez = ey = 0 at A Conditions ex = ey = ez = fix = 0 at B kz = kw = 1 Material S 35 Properties E = 1000 kn/cm = 0,3 Element Beam element types Target Buckling resistance Nb,Rd =? Results Analytical solution in the following book:

92 AxisVM X4 Verification Examples 9 Dunai, L., Horváth, L., Kovács, N., Verőci, B., Vigh, L. G.: Acélszerkezetek méretezése az Eurocode 3 alapján, Gyakorlati útmutató (Design of steel structures according to Eurocode 3, ) Magyar Mérnöki Kamara Tartószerkezeti tagozata, Budapest, 009. Exercise 3.1., P Analitical solution AxisVM e[%] zs [cm] 49,0 49,0 - zw [cm] 4,10 4,04-1,46 iw [cm] * 9,05 9,03-0, y [-] 0,54 0,54 - Χy [-] 0,804 0,8195-0,11 Nb,Rd,1 [kn] 136,4 135,0-0,11 TF [-] * 0,667 0,667 - ΧTF [-] 0,743 0, ,19 Nb,Rd, [kn] 101,6 103,9 +0,19 * hidden partial results, Axis does not show them among the steel design results

93 AxisVM X4 Verification Examples 93 Software Release Number: X4r1 Date: Tested by: InterCAD File name: Külpontosan nyomott rúd - RHS szelvény.axs Topic Buckling of a hollow cross-section beam (EN ). Analysis Type Geometry Steel Design h = 150 mm b = 100 mm tf = 10 mm tw = 10 mm L = 4,000 m A = 43,41 cm Iy = 109,8 cm 4 Iz = 635,7 cm 4 iy = 5,8 mm iz = 38,3 mm Wel,y = 161,3 cm 3 Wel,z = 17,1 cm 3 Wpl,y = 05,6 cm 3 Wpl,z = 154,6 cm 3 RHS 150x100x10,0 cross section (hot rolled) Loads Boundary Conditions Material Properties Element types Steel Design Parameters Target Bending moment at both ends of beam and axial force NEd,C = 00 kn MEd,A = MEd,B = 0 knm ex = ey = ez = 0, warping free at A ey = ez = 0, warping free at B S 75 E = 1000 kn / cm = 0,3 Beam element Buckling length: Ly = L Lz = L Lw = L Check for interaction of compression and bending.

94 AxisVM X4 Verification Examples 94 Results Analytical solution: Section class: 1. Compression flexural buckling E I y ,8 N cr,y 1567, kn K y L 400 E I z ,7 N cr,z 83,5 kn K z L 400 N pl,rd A f y 43,41 7,5 1193,8 kn y N pl N cry 1193,8 1567,16 0,878 z N pl N crz 1193,8 1,040 83,48 imperfection factor based on buckling curve a (hot rolled RHS section): y z 0,1 1 ( - 0.) 1 : - y 0,7516 z 0,575 N b,rd y A f y 1 0,575 43,41cm 7,5 kn/cm 1,0 69,7 kn N Ed,x 00 kn Bending lateral torsional buckling M pl,rd,y W 3 pl,y f y 05,6 cm 7,5 kn/cm 1 1,0 C 1 1,000 k k w 1 z 56,54 knm M Ed 10 knm M cr M cr M cr C 1 E I z (kl) kn ,7cm cm 1,0 (400 cm) 977,41kNm k z k w I w (kl) G I t I z E I z cm 4 635,7 cm kn 4 (400 cm) , cm cm kn ,7 cm cm

95 AxisVM X4 Verification Examples 95 LT W y f y M cr 3 05,6 cm 7,5 kn/cm 0, ,41kNm LT 0, torsional buckling may occur LT LT : 0,76 1 ( - 0.) LT LT LT 1 0, LT 0,5443 M M knm knm b Rd LT 0, ,54 54,, pl, Rd, y 76 Interaction of bending and buckling N Rk A f y 43,41cm 7,5 kn/cm M y,rk M pl,rd,y 56,54kNm 1193,8 kn Equivalent uniform moment factors according to EN Annex B, Table B.3.: 1,0 C my 0,6 0,4 1,0 0,4 For members susceptible to torsional deformations the interaction factors may be calculated according to EN Annex B, Table B..: k yy C my 1 ( LT k yy 1,0 1-0,) (0,87-0,) y k yy min (1,149;1,178) 1,149 N Ed N Rk C my 1 0,8 / M1 y 00 1,0 1 0,8 0, ,78 /1,0 N Ed N Rk / M1 00 0, ,78 /1,0 k zy k zy 0,1 1 z C 0,5 mlt z N Ed,x N Rk / M1 0,11, ,0 0,5 0, ,78 /1,0 k zy max (0,9490; 0,9577) 0,9577 0,1 1 C 0,5 mlt z 0,1 1 1,0 0,5 N Ed,x N Rk / M1 00 0, ,78 /1,0

96 AxisVM X4 Verification Examples 96 N M Ed y,ed k yy y N Rk / M1 y M y,rk / M ,149 0,646 0, ,78 0, ,54 N Ed k zy z N Rk / M1 M y,ed M y,rk / M ,9577 0,6674 0, ,78 0, ,54 Analytical solution AxisVM e [%] NRk = Npl,Rd [kn] 1193,8 1193,9 - y [-] 0,873 0,870-0,3 z [-] 1,04 1,01-0, Χy [-] 0,7516 0, Χz [-] 0,575 0,574 - Nb,Rd [kn] 69,7 69,7 - Mc,Rd = Mpl,Rd [knm] 56,54 56,54 - C1 1,000 1,000 - Mcr [knm] 977,41 977,40 - LT [-] 0,405 0,405 - ΧLT [-] 0,9684 0, Mb,Rd [knm] 54,76 54,75 - Cmy [-] 1,0 1,0 - kyy [-] 1,149 1,150 - kzy [-] 0,9577 0, Interaction capacity ratio 1 [-] 0,643 0,643 - Interaction capacity ratio [-] 0,667 0,667 -

97 AxisVM X4 Verification Examples 97 Software Release Number: X4r1 Date: Tested by: InterCAD File name: 3_6 Külpontosan nyomott rúd - I szelvény.axs Thema Lateral torsional buckling of a beam (EN ). Analysis Type Geometry Steel Design h = 171 mm b = 180 mm tf = 6 mm tw = 9,5 mm L = 4,000 m A = 45,6 cm Iy = 510,7 cm 4 Iz = 94,6 cm 4 iy = 74 mm iz = 45 mm Wel,y = 93,7 cm 3 Wel,z = 10,7 cm 3 Wpl,y = 34,9 cm 3 Wpl,z = 156,5 cm 3 HEA180 Iw = 5893 cm 6 It = 15 cm 4 Loads Boundary Conditions Material Properties Element types Axial force at B: Nx = -80 kn Point load in y direction at the thirds of the beam: Fy = 5 kn Distributed load in z direction: qz = 4,5 knm ex = ey = ez = 0, warping free at A ey = ez = 0, warping free at B S 35 E = 1000 kn / cm = 0,3 Beam element

98 AxisVM X4 Verification Examples 98 Steel Design Parameters The elastic critical load factor is: αcr = 4,8 As αcr = 4,8 < 15 II. order analysis is required. For this, the beam element needs to be meshed. Divison of the beam element into 4. Buckling length: Ly = L Lz = L LT buckling length: Lw = L Target Results Buckling check for interaction of axial force and bi-axial bending. Internal forces from the second order analysis NEd,x = 80 kn MEd,y = 9,81 knm MEd,z = 8,88 knm VEd,y = 6,5 kn VEd,z = 9,61 kn

99 AxisVM X4 Verification Examples 99 Analytical solution: Section class: 1. Normal force E I y N cr,y K y L E I N z cr,z K z L N pl,rd ,7 35,3 kn ,6 1197,7 kn 400 A f y 45,6 3,5 1063,6 kn y N pl N cry 1063,6 35,3 0,5719 z N pl N crz 1063,6 0, ,7 based on buckling curve b in y direction and c in z direction: y 0,8508 z 0,5741 N b,rd,1 N b,rd, y A f y 0, ,6cm 3,5kN/cm 904,9 kn N Ed,x 80 kn 1 1,0 A f z y 0, ,6cm 3,5kN/cm 610,6 kn N Ed,x 80 kn 1 1,0 Bending M pl,rd,y M pl,rd,z W pl,y 1 W pl,z f y 1 f y 3 34,9 cm 3,5 kn/cm 1, ,5 cm 3,5 kn/cm 1, 0 76,35 knm M Ed,y 9,81kNm 36,78 knm M Ed,z 8,88 knm Calculation of the critical moment: C 1 1,13 (due to the My moment diagram) E I k z I w (kl) G I M z t cr C 1 k I (kl) w z E I z kn/cm 94,6 cm M cr 1,13 (400 cm) M cr 174,1 knm cm (400 cm) 8077 kn/cm 15 cm ,6 cm 1000 kn/cm 94,6 cm

100 AxisVM X4 Verification Examples 100 For rolled section, the following procedure may be used to determine the reduction factor (EN ,Paragraph ): LT 1 ( LT LT LT : W y f y M cr 3 34,9 cm 3,5 kn/cm 0,66 174,10 knm - 0.4) 0.75 LT 1 0, LT 0,7090 M M knm knm b Rd LT 0, ,35 67,, pl, Rd, y 81 Interaction of axial force and bi-axial bending N Rk M y,rk M z,rk N pl,rd 1063,6 kn M pl,rd,y 76,35 knm M pl,rd,z 36,78 knm Equivalent uniform moment factors according to EN Annex B, Table B.3.: 0, 0 C my Cmz in both directions C 0,95 0,05 0,95 (distributed load) mlt 0,90 0,10 0,90 (concentrated load) k yy C my 1 ( y k yy 0,95-0,) y N Ed,x N Rk 80 1 (0,5719-0,) 0,95 1 0,8 0, ,6 /1,0 k yy min (1,0593;1,1851) 1,0593 C my 1 0,8 / M1 y N Ed,x N Rk / M1 80 0, ,6 /1,0 0,1 k zy 1 z C 0,5 mlt z k zy N Ed,x N Rk 0,1 0, , ,95 0,5 0, ,6 /1,0 0,95 0,5 k zy max (0,9383; 0,9345) 0,9383 0,1 1 / C 0,5 M1 mlt z N Ed,x N Rk / M1 80 0, ,6 /1,0

101 AxisVM X4 Verification Examples 101 0,858 zz 0,6 k k yz 1,4303 min (1,4303;1,478) k zz 1063,6 /1,0 0, ,4 1 0, ,6 /1,0 0, ,6) 0,944 - ( 1 0,90 k zz M1 / Rk N Ed,x N 1,4 1 mz C M1 / Rk N Ed,x N 0,6) - ( 1 mz C k zz z z z ,78 8,88 1, ,35 0,8881 9,81 0, ,6 0, M1 / z,rk M z,ed M k zz M1 / y,rk M LT y,ed M k zy M1 / Rk N z Ed,x N 0, ,78 8,88 0,858 76,35 0,8881 9,81 1, ,6 0, M1 / z,rk M z,ed M k yz M1 / y,rk M LT y,ed M k yy M1 / Rk N y Ed,x N,

102 AxisVM X4 Verification Examples 10 Analytical solution AxisVM e [%] Npl,Rd [kn] 1063,6 1063,6 - Ncr,y [kn] 35,3 35,4 - Ncr,z [kn] 1197,7 1197,7 - λy, rel [-] 0,5719 0, λz, rel [-] 0,944 0,944 - Χy [-] 0,8508 0, Χz [-] 0,5741 0, Mpl,Rd,y [knm] 76,35 77,17 +1 Mpl,Rd,z [knm] 36,78 36,78 - C1 [-] 1,13 1,13 Mcr [knm] 174,1 173,3-0,5 λlt, rel [-] 0,66 0,667 +0,7 ΧLT [-] 0,8881 0, ,3 Mb,Rd [knm] 67,81 68,353 +0,8 Cmy = CmLt [-] 0,95 0,95 - Cmz [-] 0,90 0,95 +5,5** kyy 1,0593 1, kzz 1,4303 1, ,5*** kyz 0,858 0, ,5*** kzy 0,9383 0, Interaction capacity ratio 1 0,6687 0, ,7*** Interaction capacity ratio 0,9374 0,9564 +,0*** ** See EC3 Annex B, Table B.3: the difference is due to the fact, that AxisVM calculates the equivalent uniform moment factor (Cmy, Cmz, CmLT) for both uniform load and concentrated load, and then takes the higher value. The effect on the final result (efficiency) is +1~%. *** the difference is due to the different Cmz value

103 AxisVM X4 Verification Examples 103 Software Release Number: X4r1 Date: Tested by: InterCAD File name: Double-symmetric I - Class 4.axs Thema Interaction check of beam in section class 4 (EN , EN ) Analysis Type Geometry Steel Design h = 114 mm tw = 8 mm b = 30 mm tf = 1 mm L = 8,000 m A = 164,8 cm Iy = 36159,4 cm 4 Wel,y = 5803,6 cm 3 Double-symmetric welded I shape Loads Boundary Conditions Material Properties Element types Target Axial force at B: N Ed,C = 700 kn Distributed load in z direction: qz = 16,5 knm The internal forces in the mid-section: MEd,y = 1300 knm, NEd,x = kn ex = ey = ez = fix = 0 at A ey = ez = fix = 0 at B S 355 E = 1000 kn / cm ε=0,81 = 0,3 Beam element Check the strength capacity ratios for axial force, bending and interaction.

104 AxisVM X4 Verification Examples 104 Results Analytical solution in the following book: Dunai, L., Horváth, L., Kovács, N., Verőci, B., Vigh, L. G.: Acélszerkezetek méretezése az Eurocode 3 alapján, Gyakorlati útmutató (Design of steel structures according to Eurocode 3, ) Magyar Mérnöki Kamara Tartószerkezeti tagozata, Budapest, 009. Exercise 3.4., P Exercise 3.6., P Exercise 3.13., P. 34. Analytical solution AxisVM e [%] Uniform compression Uniform bending k σ,flange [ ] 0,43 0,43 - λ p,flange [ ] 0,831 0,858 +3,1 ρ flange [ ] 0,931 0,910 -,3 b eff,f [cm] 140,0 14,0 +1,4 k σ,web [ ] λ p,web [ ],957,975 +0,6 ρ web [ ] 0,313 0,311-0,6 b eff,web [cm] 340,8 34,4 +0,5 A eff [cm ] 99,98 97,46 -,6 N eff [kn/cm ] [kn] ,6 capacity ratio: N 0, 0, - k σ,flange [ ] 0,43 0,43 - λ p,flange [ ] 0,831 0,858 +3,1 ρ flange [ ] 0,931 0,910 -,3 b eff,f [cm] 139,95 14,0 +1,4 Ψ [ ] -0,969-0,959 +1,0 k σ,web [ ] 3,09,84-1,1 λ p,web [ ] 1,31 1,45 +1,1 ρ web [ ] 0,739 0,731-1,1 b eff,web [cm] 408,6 410,4 +0,4 W eff,y,min [cm 3 ] ,1 M y,eff,rd [knm] 181,5 1766,5-3,1 capacity ratio: M 0,71 0,74 +4,1 capacity ratio: N M interaction 0,91 0,94 +3,3 Small differences occur because AxisVM does not take into account welding when calculating the effective section sizes.

105 AxisVM X4 Verification Examples 105 Software Release Number: X4r1 Date: Tested by: InterCAD Reference: Jean-Marc Franssen, Paulo Villa Real: Fire Design of Steel Structures (Example 5.3) File name: steel_fire.axs Thema Fire design of steel elements Unprotected column under axial compression (EN ) Analysis Steel Design Type Geometry Length: L = 3.5m Section: HE180B Loads Boundary Conditions Material Properties Element types Axial force at A: N fi,ed = 495 kn R30 required fire resistance ex = ey = ez = fix = 0 at B ey = ez = fix = 0 at A S75 E = 1000 kn / cm = 0,3 Beam element Results Analytical solution AxisVM e [%] d [ C] cr [ C] k sh [-] A / V [1/m] z, fi [-] N b, fi,rd [kn]

106 AxisVM X4 Verification Examples 106 Software Release Number: X4r1 Date: Tested by: InterCAD Reference: Jean-Marc Franssen, Paulo Villa Real: Fire Design of Steel Structures (Example 5.6) File name: steel_fire.axs Thema Fire design of steel elements Unrestrained beam (EN ) Analysis Type Geometry Steel Design qfi,ed = 1.48 kn/m L = 5,0 m Section: IPE 300 Side view Loads Boundary Conditions Material Properties Element types Target Distributed load: qfi,ed = 1.48 kn/m ex = ey = ez = fix = 0 at A ey = ez = fix = 0 at B S 35 E = 1000 kn / cm = 0,3 Beam element Evaluate the critical temperature. Results Analytical solution AxisVM e [%] cr [ C] LT, [1/m] LT, fi [-] M b, fi,rd [knm]

107 AxisVM X4 Verification Examples 107 Software Release Number: X4r1 Date: Tested by: InterCAD Reference: Jean-Marc Franssen, Paulo Villa Real: Fire Design of Steel Structures (Example 5.7) File name: steel_fire.axs Thema Fire design of steel elements Unrestrained beam-column (EN ) Analysis Type Geometry Steel Design Length: L= 6.0m Section IPE 450 Loads Boundary Conditions Material Properties Element types Target Axial force at B: N fi,ed = kn Distributed load: qfi,ed = kn/m ex = ey = ez = fix = 0 at A ey = ez = fix = 0 at B S 35 E = 1000 kn / cm = 0,3 Beam element Evaluate the critical temperature.

108 AxisVM X4 Verification Examples 108 Results Analytical solution AxisVM e [%] cr [ C] (no LTB) cr [ C] z, fi [-] (51 C) LT, fi [-] (51 C) LT [-] (51 C) k LT [-] (51 C)

109 AxisVM X4 Verification Examples 109 Software Release Number: X4r1 Date: Tested by: InterCAD Reference: Jean-Marc Franssen, Paulo Villa Real: Fire Design of Steel Structures (Example 5.8) File name: steel_fire.axs Thema Analysis Type Geometry Fire design of steel elements Beam-column with restrained lateral displacements (EN ) Steel Design (Jean-Marc Franssen, Paulo Villa Real: Fire Design of Steel Structures) Length: L= 3.0m Section HE 00B Loads Boundary Conditions Material Properties Element types Axial force: N fi,ed = 800 kn Bending moment: M y,fi,ed = +/-50 knm ex = ey = ez = fix = 0 at A ey = ez = fix = 0 at B S 35 E = 1000 kn / cm = 0,3 Beam element Results Analytical solution AxisVM e [%] cr [ C] (with buckling) k y [-] y, fi [-] V pl, fi,rd [kn] N pl, fi,rd [kn] ~0.0 M N, fi,rd [knm] cr [ C] (without buckling; M+N)

110 AxisVM X4 Verification Examples 110 Software Release Number: X4r1 Date: Tested by: InterCAD File name: Earthquake-01-EC.axs Thema Analysis Type Geometry Earth-quake design using response-spectrum method. Linear frequency analysis with 5 modes. Linear static analysis. Top view Front view

111 AxisVM X4 Verification Examples 111 Perspective view Section beams: 60x40 cm Ax=400 cm Ay=000 cm Az=000 cm Ix=75100 cm4 Iy=70000 cm Iz= cm4 Section columns: 60x40 cm Ax=400 cm Ay=000 cm Az=000 cm Ix=75100 cm4 Iy=70000 cm Iz=30000 cm4 Loads Nodal masses on eight nodes. Mx=My=Mz= kg Model self-weight is excluded. qd = 1 Spectrum for X and Y direction of seismic action: T[s] S d 1 0 1,150 0,000, ,6000, ,3000 0, ,0000 0, ,0000 0, ,156 S d [m/s ] 1,150 0,709 0,300,0000 T[s] Boundary Conditions Material Properties Nodes at the columns bottom ends are constrained in all directions. ex=ey=ez=fix=fiy=fiz=0 C5/30 E=3050 kn/cm =0, = 0

112 AxisVM X4 Verification Examples 11 Element types Target Results Rib element: Three node straight prismatic beam element. Shear deformation is taken into account. Compare the model results with SAP000 v6.13 results. The results are combined for all modes and all direction of spectral acceleration. CQC combination are used for modes in each direction of acceleration. SRSS combination are used for combination of directions. Period times of first 5 modes Mode T[s] SAP000 T[s] AxisVM Difference [%] 1 0,7450 0, ,7099 0, ,01 3 0,3601 0, ,314 0, ,054 0,053 +0,05 Modal participating mass ratios in X and Y directions Mode X SAP000 X AxisVM Difference % Y SAP000 Y AxisVM Difference % 1 0,5719 0,573 +0,07 0,3153 0,3151-0,06 0,3650 0,3647-0,08 0,4761 0, , ,161 0, ,0460 0, , 0,0131 0, ,0170 0, ,056 0, Summ 1,0000 1, ,9868 0, Internal forces at the bottom end of Column A and Column B Column A SAP000 Column A AxisVM Difference % Column B SAP000 Column B AxisVM Difference % Nx [kn] 315,11 311,63 +1,11 557,6 551,68 +1,011 Vy [kn] 80,34 80,35-0,003 3,88 3,89-0,004 Vz [kn] 53,49 53,5-0,011 41,04 41,11-0,017 Tx [knm] 34,4 34,4 0 34,47 34,47 0 My [knm] 65,13 65,19-0, , ,88-0,013 Mz [knm] 61,31 61,33-0, ,41 553,43-0,004 Support forces of Support C Support C SAP000 Support C AxisVM Difference % Rx [kn] 80,34 80,35-0,004 Ry [kn] 53,49 53,5-0,011 Rz [kn] 315,11 311,63 +1,116 Rxx [knm] 65,13 65,19-0,01 Ryy [knm] 61,31 61,33-0,003 Rzz [knm] 34,4 34,4 0 Displacements of Node D Node D SAP000 Node D AxisVM Difference % ex [mm] 33,51 33,5-0,003 ey [mm] 19,944 19,945-0,005 ez [mm] 0,9 0,6 +1,37 X [rad] 0, , Y [rad] 0, , Z [rad] 0,0057 0,0057 0

113 AxisVM X4 Verification Examples 113 Normal forces:

114 AxisVM X4 Verification Examples 114 Bending moments:

115 AxisVM X4 Verification Examples 115

116 AxisVM X4 Verification Examples 116 Displacements:

117 AxisVM X4 Verification Examples 117 Software Release Number: X4r1 Date: Tested by: InterCAD File name: XLAM_Example_.axs Thema Analysis Type Geometry Design of an XLAM shell (EC5). Linear analysis. Top view (x-y plane) Loads P = -100 kn concentrated force acting at point (x = 9.0 m, y =.5 m) Pz = kn/m uniform load PX = kn/m line load on the right edge Boundary Conditions Material Properties Section Properties Element types Mesh ez =0 along all edges ex = 0 along the left edge ey = 0 along the top edge The remaining DOFs are left free. Material quality equals to C4 timber. MM 7s/40 XLAM section with x oriented top layer grain direction and Service Class, producing an overall thickness of 40 mm. Shell element (Parabolic quadrilateral, heterosis type) Target Average element length is 0. m. Efficiency at node 540 (x = m, y = 3.15 m). Stresses calculated with Axis VM

118 AxisVM X4 Verification Examples 118 Normal stresses in x direction from bending and normal forces [N/mm ] Normal stresses in y direction from bending [N/mm ] Shear stresses in x and y direction [N/mm ] Rolling shear stresses in x and y direction [N/mm ] Design value of strength [N/mm ]