Verification Examples

Transcription

1 Verification Examples 01

2 AxisVM 11 Verification Examples Linear static...3 Supported bar with concentrated loads....4 Thermally loaded bar structure...5 Continously supported beam with constant distributed load...6 External prestessed beam...9 Periodically supported infinite membrane wall with constant distributed load Clamped beam examination with plane stress elements...13 Clamped thin square plate...16 Plate with fixed support and constant distributed load...18 Annular plate All edges simply supported plate with partial distributed load....1 Clamped plate with linear distributed load...3 Hemisphere displacement...5 Nonlinear static...7 3D beam structure...8 Plate with fixed end and bending moment...30 Dynamic...33 Deep simply supported beam...34 Clamped thin rhombic plate...37 Cantilevered thin square plate...39 Cantilevered tapered membrane....4 Flat grillages Stability...49 Simply supported beam...50 Simply supported beam...5 Design...53 N-M interaction curve of cross-section EC, EN : RC beam deflection according to EC, EN : Required steel reinforcement of RC plate according to EC, EN : Interaction check of beam under biaxial bending EC3, EN : Interaction check of beam under normal force, bending and shear force EC3, EN : Buckling resistance of simply supported I beam EC3, EN : Buckling resistance of simply supported T beam EC3, EN : Buckling of a hollow cross-section beam EC3, EN : Lateral torsional buckling of a beam EC3, EN : Interaction check of beam in section class 4. EC3, EN :005, EN : Earth-quake design using response-spectrum method... 80

3 AxisVM 11 Verification Examples 3 Linear static

4 AxisVM 11 Verification Examples 4 Software Release Number: R3 Date: Tested by: InterCAD Page number: File name: beam1.axs Thema Analysis Type Geometry Supported bar with concentrated loads. Linear analysis. Side view Section Area 1,0 m Loads Boundary Conditions Material Properties Element types Mesh Axial direction forces P 1-00 N, P 100 N, P 3-40 N Fix ends, at R 1 and R 5. E 0000 kn / cm ν 0,3 Beam element Target Results R 1, R 5 support forces Theory AxisVM % R 1 [N] -,00 -,00 0,00 R 5 [N] 118,00 118,00 0,00

5 AxisVM 11 Verification Examples 5 Software Release Number: R3 Date: Tested by: InterCAD Page number: File name: beam.axs Thema Analysis Type Geometry Thermally loaded bar structure. Linear analysis. Sections: Steel: A S π x 10-4 m Copper: A C π x 10-4 m 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. Steel: E S 0700 kn / cm, ν 0,3, α S 1, x 10-5 C -1 Copper: E C kn / cm, ν 0,3, α C 1,7 x 10-5 C -1 Beam element S max in the three bars. Theory AxisVM % Steel S max [MPa] ,10 Cooper S max [MPa] ,19

6 AxisVM 11 Verification Examples 6 Software Release Number: R3 Date: Tested by: InterCAD Page number: File name: beam3.axs Thema Analysis Type Geometry Continously supported beam with point loads. Linear analysis. Side view (Section width 1,00 m, height 1 0,30 m, height 0,60 m) Loads Boundary Conditions Material Properties Element types Target Results P kn, P -150 kn, P kn, P kn Elastic supported. From A to D is K z 5000 kn/m/m. From D to F is K z kn/m/m. E 3000 kn/cm ν 0,3 Three node beam element. Shear deformation is taken into account. e z, M y, V z, R z Diagram e z Diagram M y Results

7 AxisVM 11 Verification Examples 7 Diagram V z Diagram R Reference AxisVM e [%] e A [m] 0,006 0,006 0,00 e B [m] 0,009 0,009 0,00 e C [m] 0,014 0,014 0,00 e D [m] 0,015 0,015 0,00 e E [m] 0,015 0,015 0,00 e F [m] 0,013 0,013 0,00 Reference AxisVM e [%] M A [KNm] 0,0 0, 0,00 M B [KNm] 88,5 87,1-1,58 M C [KNm] 636, 630,8-0,85 M D [KNm] 33,8 330,1-0,81 M E [KNm] 164, 163,0-0,73 M F [KNm] 0,0 0,4 0,00

8 AxisVM 11 Verification Examples 8 Results Reference AxisVM e [%] V A [KN] 0,0 0,1 0,00 V B [KN] 11,1 113,1 0,89 V C [KN] 646,8 647, 0,06 V D [KN] 335,0 334,9-0,03 V E [KN] 67,8 67,5-0,11 V F [KN] 0,0-0,1 0,00 Reference AxisVM e [%] R A [KN/m ] 145,7 154,0 5,70 R B [KN/m ] 19,5 19,4-0,05 R C [KN/m ] 343,8 346,0 0,64 R D [KN/m ] 386,9 386,4-0,13 R E [KN/m ] 4,5 4,7 0,09 R F [KN/m ] 01, 00,8-0,0

9 AxisVM 11 Verification Examples 9 Software Release Number: R3 Date: Tested by: InterCAD Page number: File name: beam4.axs Thema Analysis Type Geometry External prestessed 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 I z 0,E-5 m 4 Truss -5, 3-6 A 3,48E-3 m I z 0,E-5 m 4 Beam 1-4 A 1,1516E- m I z,174e-4 m 4 Mesh Element types Three node beam element, 1-5, 5-6, 6-4, 1-4 (shear deformation is taken into account) Truss element -5, 3-6 Target N X at beam 6-7 M y,max at beam -3 e z at node

10 AxisVM 11 Verification Examples 10 Results ,600,000 4,000,000 8,000 Z X Diagram e z ROBOT V6 AxisVM % N x [kn] 584,56 584,80 0,04 M y [knm] 49,6 49,60 0,68 e z [mm] -0,541-0,5469 0,89

11 AxisVM 11 Verification Examples 11 Software Release Number: R3 Date: Tested by: InterCAD Page number: 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 E 880 kn / cm ν 0,16 Parabolic quadrilateral membrane (plane stress) Target S xx at 1-10 nodes (1-5 at middle, 6-10 at support)

12 AxisVM 11 Verification Examples 1 Results Node Analytical [kn/cm ] AxisVM [kn/cm ] % 1 0,1313 0,131-0,08 0,0399 0,0395-1,00 3-0,0093-0,0095,15 4-0,041-0,0413 0,4 5-0,1073-0,1071-0,19 6-0,9317-0,9175-1,5 7 0,0401 0,046 6,3 8 0,0465 0,0469 0,86 9 0,0538 0,0538 0, ,149 0,147-0,16 Reference: Dr. Bölcskey Elemér Dr. Orosz Árpád: Vasbeton szerkezetek Faltartók, Lemezek, Tárolók

13 AxisVM 11 Verification Examples 13 Software Release Number: R3 Date: Tested by: InterCAD Page number: 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. E 880 kn / cm ν 0 Parabolic quadrilateral membrane (plane stress) 0,375 Clamped edge 1 0,500 C 3,000 0,50 Z X Side view

14 AxisVM 11 Verification Examples 14 Target Results τ xy, max at section C Diagram τ xy 5,14 791,56 Z Y 5,8 Diagram τ xy at section C

15 AxisVM 11 Verification Examples 15 V 65,65 kn ( from beam theory) S ' y 0, m 3 b 0,5 m I y 0, m 4 τ xy V S b I ' y y 65,65 0, ,5 0, ,5 kn / m AxisVM result τ xy 791,6 kn / m Difference +0,5 % AxisVM result V nxy 65, 34 kn Difference +0,43 %

16 AxisVM 11 Verification Examples 16 Software Release Number: R3 Date: Tested by: InterCAD Page number: 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 ez ez fix fiy fiz 0 along all edges E 0000 kn / cm ν 0,3 Plate element (Parabolic quadrilateral, heterosis) 4,000 Y 4,000 Target X Displacement of middle of the plate

17 AxisVM 11 Verification Examples 17 Results -0,019-0,01-0,006-0,001-0,043-0,0-0,006-0,043-0,01-0,065-0,084-0,081-0,06-0,04-0,065-0,04-0,15-0,087-0,019-0,06-0,081-0,15-0,087-0,156-0,01-0,081-0,06-0,087-0,006-0,156-0,187-0,168-0,065-0,168-0,043-0,001-0,04-0,087-0,37-0,156-0,0-0,168-0,37-0,15-0,006-0,57-0,084-0,019-0,081-0,043-0,168-0,57-0,57-0,307-0,01-0,37-0,01-0,065-0,187-0,156-0,15-0,065-0,019-0,043-0,57-0,337-0,006-0,15-0,337-0,337-0,37-0,307-0,37-0,156-0,081-0,0-0,084-0,04-0,001-0,187-0,337-0,043-0,383-0,168-0,087-0,006-0,15-0,383-0,57-0,06-0,337-0,01-0,065-0,156-0,37-0,307-0,383-0,168-0,087-0,57-0,06-0,019-0,081-0,383-0,337-0,168-0,57-0,337-0,337-0,04-0,087-0,37-0,156-0,081-0,04-0,06-0,087-0,168-0,57-0,187-0,156-0,37-0,307-0,15-0,065-0,06-0,081-0,15-0,019-0,084-0,04-0,065-0,043-0,043-0,01-0,019-0,0-0,01-0,006-0,006-0,019-0,04-0,001-0,06 Y Z X 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 Displacements 0, Diff1 [%] Diff [%] -5,00-10,00-15,00 Mesh density

18 AxisVM 11 Verification Examples 18 Software Release Number: R3 Date: Tested by: InterCAD Page number: File name: plate_1.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 Material E 990 kn/cm Properties ν 0,16 Element Parabolic triangle plate element types Mesh Target Results Maximal ez (found at Node1) and maximal m x (found at Node) Component Nastran AxisVM % ez,max [mm] -1,613-1,593-1,4 mx,max [knm/m] 3,060 3,059-0,03

19 AxisVM 11 Verification Examples 19 Software Release Number: R3 Date: Tested by: InterCAD Page number: 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 Material Properties Element types E 880 kn / cm ν 0,3 Plate element (parabolic quadrilateral, heterosis)

20 AxisVM 11 Verification Examples 0 Mesh 3,000 1,000 Y 4,000 Target S max, e max X 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.) 77,68 76,10 -,03 b.) 6,76 0,84 -,61 c.) 355,17 35,89-0,64 d.) 3,8 3,4 0,60 e.) 44,6 44,50 0,54 f.) 13,19 13,17-0,0 g.) 11,14 111,94-0,18 h.) 16,83 16,81-0,0 Reference: S. Timoshenko and S. Woinowsky-Krieger: Theory of Plates And Shells

21 AxisVM 11 Verification Examples 1 Software Release Number: R3 Date: Tested by: InterCAD Page number: 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 ϕ 0 perpendicular the edges (hard support) E 880 kn / cm ν 0,3 Plate element (Heterosis) 10,000 Y 5,000 X

22 AxisVM 11 Verification Examples Target Results m x, max, m y, 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

23 AxisVM 11 Verification Examples 3 Software Release Number: R3 Date: Tested by: InterCAD Page number: 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 E 880 kn / cm ν 0,3 Plate element (Heterosis) q ,000 4 Y 10,000 X

24 AxisVM 11 Verification Examples 4 Target m x, m y 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

25 AxisVM 11 Verification Examples 5 Software Release Number: R3 Date: Tested by: InterCAD Page number: 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 C,0 kn,0 kn Z A B X Y

26 AxisVM 11 Verification Examples 6 Boundary Conditions Material Properties Element types Target ex ey ez 0 at A ex ey ez 0 at B E 685 kn / cm ν 0,3 Shell element 1.) guadrilateral parabolic.) triangle parabolic e x at point A Results e x [m] e [%] Theory 0,185 AxisVM quadrilateral 0,185 0,00 AxisVM triangle 0,18-1,6

27 AxisVM 11 Verification Examples 7 Nonlinear static

28 AxisVM 11 Verification Examples 8 Software Release Number: R3 Date: Tested by: InterCAD Page number: 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 ν 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

29 AxisVM 11 Verification Examples 9 Results Comparison with the results obtained using Nastran V4 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

30 AxisVM 11 Verification Examples 30 Software Release Number: R3 Date: Tested by: InterCAD Page number: File name: nonlin.axs Thema Analysis Type Geometry Plate with fixed end and bending moment. Geometrical nonlinear analysis. Edge1 1,0 m Edge 1,0 m Z Y X 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 E 0000 N / mm ν 0 Plate thickness: 150 mm Rib on Edge: circular D 500 mm (for distributing load to the mid-side-node) Parabolic quadrilateral shell (heterosis) Rib on Edge for distributing load to the mid-side-node

31 AxisVM 11 Verification Examples 31 Target Results ϕ Z at Edge 5,550 rad Edge1 1,0 m Edge 1,0 m Z Y X κ I I E z M plate plate plate plate Theoretical results based on the differential equation of the flexible beam: M E ϕ κ l l plate plate 3 a b N m 1 m.6 10 M l ϕ z I platee Nm.6 10 ϕ z plate plate rad Comparison the AxisVM result with the theoretical one: Component Theory AxisVM % fiz [rad] 5,5467 5,550 0,06

32 AxisVM 11 Verification Examples 3 BLANK

33 AxisVM 11 Verification Examples 33 Dynamic

34 AxisVM 11 Verification Examples 34 Software Release Number: R3 Date: Tested by: InterCAD Page number: File name: dynam1.axs Thema Analysis Type Geometry Deep simply supported beam. Dynamic analysis. Beam (Axonometric view) Cross section (square,0 m x,0 m) Loads Boundary Conditions Material Properties Element types Target Self-weight ex ey ez fix 0 at A ey ez 0 at B E 0000 kn / cm ν 0,3 ρ 8000 kg / m 3 Three node beam element (shear deformation is taken into account) First 7 mode shapes

35 AxisVM 11 Verification Examples 35 Results 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

36 AxisVM 11 Verification Examples 36 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

37 AxisVM 11 Verification Examples 37 Software Release Number: R3 Date: Tested by: InterCAD Page number: File name: dynam.axs Thema Analysis Type Geometry Clamped thin rhombic plate. Dynamic 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 (ie: ex, ey, fiz constained at all nodes) ez fix fiy 0 along the 4 edges E 0000 kn / cm ν 0,3 ρ 8000 kg / m 3 Parabolic quadrilateral shell element (heterosis) 10,000 10,000 Y X

38 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 11 Verification Examples 38 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

39 AxisVM 11 Verification Examples 39 Software Release Number: R3 Date: Tested by: InterCAD Page number: File name: dynam3.axs Thema Analysis Type Geometry Cantilevered thin square plate. Dynamic 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 ν 0,3 ρ 8000 kg / m 3 Parabolic quadrilateral shell element (heterosis).

40 AxisVM 11 Verification Examples 40 Target First 5 mode shapes Results Mode 1: f 0,4 Hz Mode 3: f,53 Hz Mode 5: f 3,68 Hz

41 AxisVM 11 Verification Examples 41 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

42 AxisVM 11 Verification Examples 4 Software Release Number: R3 Date: Tested by: InterCAD Page number: File name: dynam4.axs Thema Analysis Type Geometry Cantilevered tapered membrane. Dynamic analysis. Side view (thickness 10,0 cm) Loads Boundary Conditions Material Properties Element types Mesh Self-weight ez 0 at all nodes (ie: ez constained at all nodes) ex ey 0 along y-axis E 0000 kn / cm ν 0,3 ρ 8000 kg / m 3 Parabolic quadrilateral membrane (plane stress) X5,000 Y 10,000 1,000

43 AxisVM 11 Verification Examples 43 Target First 4 mode shapes Results 5,000 1,000 10,000 Y X Mode 1: f 44,33 Hz X5,000 Y 10,000 1,000 Mode : f 18,36 Hz

44 AxisVM 11 Verification Examples 44 X5,000 Y 10,000 Mode 3: f 16,48 Hz 1,000 X5,000 Y 10,000 Mode 4: f 41, Hz 1,000 Results Comparison with NAFEMS example 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, -1,96

45 AxisVM 11 Verification Examples 45 Software Release Number: R3 Date: Tested by: InterCAD Page number: File name: dynam5.axs Thema Analysis Type Geometry Flat grillages. Dynamic 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) E 0000 kn / cm G 7690 kn / cm ν 0,3 ρ 7860 kg / m 3 A 0,004 m Ix,5E-5 m 4 Iy Iz 1,5E-5 m 4 Three node beam element (shear deformation is taken into account) 1,000 0,500 4,500,000 1,000 1,500 1,500 1,500 1,000 0,500 Y X

46 AxisVM 11 Verification Examples 46 Target First 3 mode shapes Results 1,605 1,879 1,679 1,638 1,586 1,114 1,41 1,035 Z Y X Mode 1: f 16,90 Hz -1,813 -,065-1,837 0,856,040,54 1,938 Z Y X Mode : f 0,64 Hz -1,130,040-1,581-1,60 1,71 1,585-1,667-1,99-1,845 Z Y X Mode 3: f 51,76 Hz

47 AxisVM 11 Verification Examples 47 Mode Reference AxisVM (Hz) % 1 16,85 16,90 0,30 0,1 0,64, ,30 51,76 -,89 Reference: C.T.F. ROSS: Finite Element Methods In Engineering Science

48 AxisVM 11 Verification Examples 48 BLANK

49 AxisVM 11 Verification Examples 49 Stability

50 AxisVM 11 Verification Examples 50 Software Release Number: R3 Date: Tested by: InterCAD Page number: File name: buckling1.axs Thema Analysis Type Geometry Simply supported beam. Buckling analysis. Front view ,0 S G z ,0 y 10,0 Cross section (I z 168,3 cm 4, I t 1,18 cm 4, I w cm 6 ) Loads Boundary Conditions Material Properties Element types Mesh Bending moment at both ends of beam M A 1,0 knm, M B -1,0 knm ex ey ez 0 at A ex ey ez 0 at B k z k w 1 E 0600 kn / cm ν 0,3 G 793 kg / m Parabolic quadrilateral shell element (heterosis)

51 AxisVM 11 Verification Examples 51 Target M cr? (for lateral torsional buckling) Results Analytical solution M cr π E I L Z W + L I Z π I G I E I t Z π , ,18 M cr kncm 14, ,3 π ,3 knm AxisVM result M cr 15,3 knm Difference +0,6%

52 AxisVM 11 Verification Examples 5 Software Release Number: R3 Date: Tested by: InterCAD Page number: 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 z 1,0 30,0 y z y Loads P -1,0 kn at point B. Section A 1 Section A Cross-sections Boundary Conditions Material Properties Element types Target Results ex ey ez 0 at A ey ez 0 at B E 0000 kn / cm ν 0,3 Beam element P cr? (for inplane buckling) Theory AxisVM e [%] P cr [kn] 3,340 3,337-0,09

53 AxisVM 11 Verification Examples 53 Design

54 AxisVM 11 Verification Examples 54 Software Release Number: R3 Date: Tested by: InterCAD Page number: File name: RC column1.axs Thema N-M interaction curve of cross-section (EN :004). Analysis Type Geometry Linear static analysis+design. φ0 3φ8 Section: 300x400 mm Covering: 40 mm Loads Boundary Conditions Material Properties Target Results Concrete: f cd 14, N/mm e c1 0,00 e cu 0,0035 (parabola-constans σ-ε diagram) Steel: f sd 348 N/mm e su 0,015 Compare the program results with with hand calculation at keypoints of M-N interaction curve. N N [kn] M [knm] M(N) AxisVM e% ,4 +0, ,7-0, ,5 +0, ,4 +0, , +0, ,7-0,6 Reference: Dr. Kollár L. P., Vasbetonszerkezetek I. Műegyetemi kiadó

55 AxisVM 11 Verification Examples 55 Software Release Number: R3 Date: Tested by: InterCAD Page number: File name: beam1.axs Thema RC beam deflection according to EC, EN :004. Analysis Type Geometry Material nonlinear analysis. q 17 kn/m L 5,60 m Side view φ0 35 cm covering 3 cm β 0,5 4φ0 5 cm Section Loads Boundary Conditions Material Properties Element types Target q 17 kn /m distributed load Simply supported beam. Concrete: C5/30, ϕ,1 Steel: B500B Parabolic quadrilateral plate element (Heterosis) e z, max

56 AxisVM 11 Verification Examples 56 Results -0,00-5,39-10,101-14,4-17,393-19,360-0,09-19,360-17,393-14,4-10,101-5,39-0,00 Z X Diagram e z Aproximate calculation: e ζ e + ( 1 ζ ) e 0,06 _ where, II I mm e I is the deflection which was calculated with the uncracked inertia moment e II is the deflection which was calculated with the cracked inertia moment σ sr ζ 1 β σ s Calculation with integral of κ: e 19,8 mm Calculation with AxisVM: e 19,03 mm (different -4,0%)

57 AxisVM 11 Verification Examples 57 Software Release Number: R3 Date: Tested by: InterCAD Page number: File name: beam.axs Thema Required steel reinforcement of RC plate according to EC, EN :004. Analysis Type Geometry Linear analysis. Szabvány : Eurocode Eset : ST1 50 kn 4,0 Y X Side view Cross-section Loads Boundary Conditions Material Properties Element types Mesh Pz -50 kn point load Clamped cantilever plate. Concrete: C5/30 Steel: B500A Parabolic quadrilateral plate element (heterosis) Szabvány : Eurocode Eset : ST1 4,0 Clamped edge 1,0 Y X Top view

58 AxisVM 11 Verification Examples 58 Target A XT steel reinforcement along x direction at the top of the support Results Lineáris számítás Szabvány : Eurocode Eset : ST1 E (W) : 1,09E-11 E (P) : 1,09E-11 E (ER) : 8,49E-13 Komp. : axf [mm /m] 1,0 4,0 Clamped edge ST1, axf: 093 mm /m Z Y X Diagram A XT Calculation according to EC: 5 f cd 1,5 ξ 500 f yd 435 N / mm 1,15 16,6 N / mm c ε E 0,85 0, , cu S c0 εcu ES + f yd d mm 0,54 xc M sd M Rd b xc fcd d 00 knm x c 439 > h 55 xc 55 ξ 0, 0 0,54 47 < c ξ Steel reinforcement is yielding c d A b xc f f ,6 435 cd S yd 099 mm Calculation with AxisVM: A XT 093 mm / m Different -0,3 %

59 AxisVM 11 Verification Examples 59 Software Release Number: R3 Date: Tested by: InterCAD Page number: 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 t f 10 mm t w 7 mm l 6000 mm A 45,95 cm W y,pl 484,1 cm 3 W z,pl 97 cm 3 IPE70 cross section Loads Boundary Conditions Material Properties q y 1,5 kn/m q z 0,4 kn/m e x e y e z 0 at A e y e z 0 at B S 35 E 1000 kn/cm ν 0,3

60 AxisVM 11 Verification Examples 60 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 Mérnök Kamara Tartószerkezeti tagozata, Budapest, 009. Exercise 3.10., page 8. Analitical solution AxisVM e[%] M y,ed [knm] 91,8 91,8 - M z,ed [knm] 6,75 6,75 - M pl,y,rd [knm] 113,74 113,76 +0,0 M pl,z,rd [knm],78,79 +0,04 α - β capacity ratio [-] 0,948 0,947-0,11

61 AxisVM 11 Verification Examples 61 Software Release Number: R3 Date: Tested by: InterCAD Page number: 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 t f 15 mm t w 9 mm l 1400 mm A 78,1 cm A v 4,83 cm I y 5696 cm 3 W y,pl 643 cm 3 IPE70 cross section Loads Boundary Conditions Material Properties Element types Target F z 300 kn at thirds of beam N 500 kn at B e x e y e z 0 at A e y e z 0 at B S 35 E 1000 kn/cm ν 0,3 Beam element Interaction check of axial force, shear force and bending moment.

62 AxisVM 11 Verification Examples 6 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 Mérnök Kamara Tartószerkezeti tagozata, Budapest, 009. Exercise 3.1., page Analytical solution AxisVM results e[%] N Ed [kn] V z,ed [kn] M y,ed [knm] Pure compression N pl,rd [kn] capacity ratio [-] 0,33 0,33 - Pure shear V pl,z,rd [kn] 394, 394,5 +0,08 capacity ratio [-] 0,761 0,761 - Pure bending M pl,y,rd [knm] 176,8 176,7-0,06 capacity ratio [-] 0,79 0,79 - Interaction check Ρ 0,73 0,71-0,73 M V,Rd [knm] 163,96 163,93-0,0 N 0,33 0,33 - A 0,3 0,3 - M NV,Rd [knm] 14, 14, - capacity ratio [-] 0,985 0,984-0,10

63 AxisVM 11 Verification Examples 63 Software Release Number: R3 Date: Tested by: InterCAD Page number: 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 t f 14 mm t w 8 mm l 4500 mm A 94 cm I y 19065,8cm 4 I z 3647,1 cm 4 i y 14,1 cm i z 6, cm Loads Normal force at point A N A -1,0 kn Boundary e y 0 at A Conditions e x e y e z φ x φ z 0 at B k z k w 1 Material S 35 Properties E 1000 kn / cm ν 0,3 Element Beam element types Target Buckling resistance N b,rd? I cross section, symmetric about y and z axis

64 AxisVM 11 Verification Examples 64 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 Mérnök Kamara Tartószerkezeti tagozata, Budapest, 009. Exercise 3.15., P Analytical solution AxisVM e[%] λ y [-] * 0,673 0,673 - λ z [-] 0,771 0,769-0,6 Χ y [-] * 0,8004 0,7989-0,19 Χ z [-] 0,6810 0, ,07 N b,rd [kn] 1504,3 1505,3 +0,07

65 AxisVM 11 Verification Examples 65 Software Release Number: R3 Date: Tested by: InterCAD Page number: 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 t f 16 mm t w 16 mm l 3000 mm A 68,8 cm I y 394,5cm 4 I z 089,48 cm 4 I cs 58,71 cm 4 I w 1108,0 cm 6 i y 5,90 cm i z 5,51 cm Welded T section, symmetric to z but not y Loads Normal force at point A N A -1,0 kn Boundary e y 0 at A Conditions e x e y e z φ x 0 at B k z k w 1 Material S 35 Properties E 1000 kn/cm ν 0,3 Element Beam element types Target Buckling resistance N b,rd?

66 AxisVM 11 Verification Examples 66 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 Mérnök Kamara Tartószerkezeti tagozata, Budapest, 009. Exercise 3.1., P Analitical solution AxisVM e[%] z s [cm] 49,0 49,0 - z w [cm] 4,10 4,04-1,46 i w [cm] * 9,05 9,03-0, λ y [-] 0,54 0,54 - Χ y [-] 0,804 0,8195-0,11 N b,rd,1 [kn] 136,4 135,0-0,11 λ z [-] * 0,667 0,667 - Χ z [-] * 0,743 0, ,19 N b,rd, [kn] * 101,6 103,9 +0,19 * hidden partial results, Axis does not show them among the steel desing results

67 AxisVM 11 Verification Examples 67 Software Release Number: R3 Date: Tested by: InterCAD Page number: 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 t f 10 mm t w 10 mm L 4,000 m A 43,41 cm I y 109,8 cm 4 I z 635,7 cm 4 i y 5,8 mm i z 38,3 mm W el,y 161,3 cm 3 W el,z 17,1 cm 3 W pl,y 05,6 cm 3 W pl,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 N Ed,C 00 kn M Ed,A M Ed,B 0 knm e x e y e z 0, warping free at A e y e z 0, warping free at B S 75 E 1000 kn / cm ν 0,3 Beam element Buckling length: L y L L z L L w L Check for interaction of compression and bending.

68 AxisVM 11 Verification Examples 68 Results Analytical solution: Section class: 1. Compression flexural buckling π E I y π ,8 N cr,y 1567, kn K y L 400 π E I z π ,7 N cr,z 83,5 kn K z L 400 N pl,rd A f y 43,41 7,5 1193,8 kn λy N pl N cry 1193,8 1567,16 0,878 λz N pl N crz 1193,8 1,040 83,48 imperfection factor based on buckling curve a (hot rolled RHS section): α y α z 0,1 1 + α ( λ - 0.) + λ φ 1 χ : φ + φ - λ χ y 0,7516 χ z 0,575 N b,rd χ y A f y γ 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 56,54 knm > M Ed 10 knm γ 1 1,0 C 1 1,000 k k w 1 z M cr C 1 π E I z (kl) kn 4 π ,7cm cm M cr 1,0 (400 cm) M cr 977,41 knm k z k w I w (kl) G I t + I z π E I z cm 4 635,7 cm kn 4 (400 cm) , cm cm + kn 4 π ,7 cm cm

69 AxisVM 11 Verification Examples 69 λ LT W y f y M cr 3 05,6 cm 7,5 kn/cm 977,41kNm λ LT > 0, torsional buckling may occur 0,405 α LT χ LT : 0, α ( λ - 0.) + λ φ LT LT LT φ + 1 φ - λ LT 0,9684 0,5443 M χ 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.: φ 1,0 C my 0,6 + 0,4φ 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 + ( λ LT k yy 1, ,) (0,87-0,) χ y k yy min (1,149;1,178) 1,149 N Ed N Rk < C my 1 + 0,8 / γ M1 χ y 00 < 1, ,8 0, ,78 /1,0 N Ed N Rk / γ M1 00 0, ,78 /1,0 k zy k zy 0,1 λ 1 z C 0,5 χ mlt z N Ed,x N Rk / γ M1 0,1 1, ,0 0,5 0, ,78 /1,0 k zy max (0,9490; 0,9577) 0,9577 0,1 1 C 0,5 χ mlt z 0,1 1 1,0 0,5 N Ed,x N Rk / γ M1 00 0, ,78 /1,0

70 AxisVM 11 Verification Examples 70 N M Ed y,ed + k yy χ y N Rk / γ M1 χ y M y,rk / γ M ,149 0,646 0, ,78 0, ,54 N Ed + k zy χz N Rk / γ M1 M y,ed M y,rk / γ M ,9577 0,6674 0, ,78 0, ,54 Analytical solution AxisVM e [%] N Rk N pl,rd [kn] 1193,8 1193,9 - λ y [-] 0,873 0,870-0,3 λ z [-] 1,04 1,01-0, Χ y [-] 0,7516 0, Χ z [-] 0,575 0,574 - N b,rd [kn] 69,7 69,7 - M c,rd M pl,rd [knm] 56,54 56,54 - C 1 1,000 1,000 - M cr [knm] 977,41 977,40 - λ LT [-] 0,405 0,405 - Χ LT [-] 0,9684 0, M b,rd [knm] 54,76 54,57-0,3 C my [-] 1,0 1,0 - k yy [-] 1,149 1,150 - k zy [-] 0,9577 0, Interaction capacity ratio 1 [-] 0,643 0,643 - Interaction capacity ratio [-] 0,667 0,667 -

71 AxisVM 11 Verification Examples 71 Software Release Number: R3 Date: Tested by: InterCAD Page number: 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 t f 6 mm t w 9,5 mm L 4,000 m A 45,6 cm I y 510,7 cm 4 I z 94,6 cm 4 i y 74 mm i z 45 mm W el,y 93,7 cm 3 W el,z 10,7 cm 3 W pl,y 34,9 cm 3 W pl,z 156,5 cm 3 HEA180 I w 5893 cm 6 I t 15 cm 4 Loads Boundary Conditions Material Properties Element types Axial force at B: N x -80 kn Point load in y direction at the thirds of the beam: F y 5 kn Distributed load in z direction: q z 4,5 knm e x e y e z 0, warping free at A e y e z 0, warping free at B S 35 E 1000 kn / cm ν 0,3 Beam element

72 AxisVM 11 Verification Examples 7 Steel Design Parameters The elastic critical load factor is: α cr 4,8 As α cr 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: L y L L z L LT buckling length: L w L Target Results Buckling check for interaction of axial force and bi-axial bending. Internal forces from the second order analysis N Ed,x 80 kn M Ed,y 9,84 knm M Ed,z 8,81 knm V Ed,y 6,50 kn V Ed,z 9,61 kn

73 AxisVM 11 Verification Examples 73 Analytical solution: Section class: 1. Normal force π E I y N cr,y K y L π E I N z cr,z K z L N pl,rd π ,7 35,3 kn 400 π ,6 1197,7 kn 400 A f y 45,6 3,5 1063,6 kn λ y N pl N cry 1063,6 35,3 0,5719 λ z N pl N crz 1063,6 0, ,7 based on buckling curve b in y direction and c in z direction: χ y 0,8508 χ z 0,5741 χ y A f y 0, ,6cm 3,5kN/cm N b,rd,1 904,9 kn > N Ed,x 80 kn γ 1 1,0 χ A f z y 0, ,6cm 3,5kN/cm N b,rd, 610,6 kn > N Ed,x 80 kn γ 1 1,0 Bending M pl,rd,y M pl,rd,z W 3 pl,y f y 34,9 cm 3,5 kn/cm γ 1 1,0 W f y 3 pl,z 156,5 cm 3,5 kn/cm γ 1 1,0 76,35 knm > M Ed,y 9,84 knm 36,78 knm > M Ed,z 8,81 knm Calculation of the critical moment: C 1 1,13 (due to the M y moment diagram) π E I k z I w (kl) G I M z t cr C 1 + k I (kl) w z π E I z 4 π 1000 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 π 1000 kn/cm 94,6 cm

74 AxisVM 11 Verification Examples 74 For rolled section, the following procedure may be used to determine the reduction factor (EN ,Paragraph ): λ LT 1 + α ( λ φ LT LT χ LT : W y f y M cr φ + φ - 0.4) ,9 cm λ LT λ LT 3,5 kn/cm 174,10 knm 0,8881 0,7090 0,66 M χ 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 M pl,rd,y M pl,rd,z 1063,6 kn 76,35 knm 36,78 knm Equivalent uniform moment factors according to EN Annex B, Table B.3.: ψ 0, α 0 in both directions C C 0,95 + 0,05α 0,95 (distributed load) my mlt C mz 0,90 + 0,10α 0,90 (concentrated load) k yy C my k yy 0, ( λ y - 0,) N Ed,x χ y N Rk / γ M (0,5719-0,) 0, ,6 /1,0 k yy min (1,0593;1,1851) 1,0593 C my N Ed,x 1 + 0,8 χ y N Rk / γ M1 0, ,8 80 0, ,6 /1,0 k zy k zy 0,1 λ 1 z C 0,5 χ mlt z N Ed,x N Rk / γ M1 0,1 0, ,95 0,5 0, ,6 /1,0 k zy max (0,9383; 0,9345) 0,9383 0,1 1 C 0,5 χ mlt z 0,1 1 0,95 0,5 N Ed,x N Rk / γ M1 80 0, ,6 /1,0

75 AxisVM 11 Verification Examples 75 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 γ χ γ χ λ z 0,9374 0,346 0,136 0, ,78 8,81 1, ,35 0,8881 9,84 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 0,6687 0,056 0,1537 0, ,78 8,81 0,858 76,35 0,8881 9,84 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 γ γ χ γ χ γ γ χ γ χ

76 AxisVM 11 Verification Examples 76 Analytical solution AxisVM e [%] N pl,rd [kn] 1063,6 1063,6 - N cr,y [kn] 35,3 35,4 - N cr,z [kn] 1197,7 1197,7 - λ y, rel [-] 0,5719 0, λ z, rel [-] 0,944 0,944 - Χ y [-] 0,8508 0, Χ z [-] 0,5741 0, M pl,rd,y [knm] 76,35 76,36 - M pl,rd,z [knm] 36,78 36,78 - C 1 [-] 1,13 1,15-0,6* M cr [knm] 174,1 173,0-0,63 λ LT, rel [-] 0,66 0, ,3 Χ LT [-] 0,8881 0, ,1 M b,rd [knm] 67,81 67,73-0,1 C my C mlt [-] 0,95 0,95 - C mz [-] 0,90 0,95 +5,5** k yy 1,0593 1, k zz 1,4303 1, ,5*** k yz 0,858 0, ,5*** k zy 0,9383 0, Interaction capacity ratio 1 0,6687 0, ,7*** Interaction capacity ratio 0,9374 0,9564 +,0*** * AxisVM calculates this factor using a closed form expression, while in the hand calculation C 1 was derived from a table. The effect of this on the final result (efficiency) is 10-4, thus on the safe side. ** See EC3 Annex B, Table B.3: the difference is due to the fact, that AxisVM calculates the equivalent uniform moment factor (C my, C mz, C mlt ) 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 C mz value

77 AxisVM 11 Verification Examples 77 Software Release Number: R3 Date: Tested by: InterCAD Page number: 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 t w 8 mm b 30 mm t f 1 mm L 8,000 m A 164,8 cm I y 36159,4 cm 4 W el,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: q z 16,5 knm The internal forces in the mid-section: M Ed,y 1300 knm, N Ed,x kn e x e y e z 0 at A e y e z 0 at B S 355 E 1000 kn / cm ε0,81 ν 0,3 Beam element Check the strength capacity ratios for axial force, bending and interaction.

78 AxisVM 11 Verification Examples 78 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 Mérnök 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 0,43 0,43-0,831 0,858 +3,1 0,931 0,910 -,3 140,0 14,0 +1, ,957,975 +0,6 0,313 0,311-0,6 340,8 34,4 +0,5 99,98 97,46 -, ,6 0, 0, - 0,43 0,43-0,831 0,858 +3,1 0,931 0,910 -,3 139,95 14,0 +1,4-0,969-0,959 +1,0 3,09,84-1,1 1,31 1,45 +1,1 0,739 0,731-1,1 408,6 410,4 +0, ,1 181,5 1766,5-3,1 0,71 0,74 +4,1 0,91 0,94 +3,3 Small differences occur because AxisVM does not take into account welding when calculating the effective section sizes.

79 AxisVM 11 Verification Examples 79

80 AxisVM 11 Verification Examples 80 Software Release Number: R3 Date: Tested by: InterCAD Page number: 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. Code : Eurocode Case : FR + 5,000 90,0 5,196 90,0 6,000 30,0 8,000 7,000 Y X Top view Code : Eurocode Case : FR + 4,000 3,500 Z X Front view

81 AxisVM 11 Verification Examples 81 Code : Eurocode Case : ST1 All nodal masses are MxMyMz kg All beams 60x40 cm Inertia about vertical axis is multiplied by Node D All columns 60x40 cm Column B Column A Support C Y Z X Perspective view All supports are constrained in all directions. exeyezfixfiyfiz0 Section beams: 60x40 cm Ax400 cm Ay000 cm Az000 cm Ix75100 cm4 Iy70000 cm Iz cm4 Section columns: 60x40 cm Ax400 cm Ay000 cm Az000 cm Ix75100 cm4 Iy70000 cm Iz30000 cm4 Loads Nodal masses on eight nodes. MxMyMz kg Model self-weight is excluded. Spectrum for X and Y direction of seismic action: T[s] S d,156 S d [m/s ] 1 0 1,150 0,000, ,6000, ,3000 0, ,0000 0, ,0000 0, ,150 0,709 0,300,0000 T[s] Boundary Conditions Material Properties Nodes at the columns bottom ends are constrained in all directions. exeyezfixfiyfiz0 C5/30 E3050 kn/cm ν 0, ρ 0

82 AxisVM 11 Verification Examples 8 Element types Target Results 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, ,3601 0, ,314 0, ,054 0,054 0 Modal participating mass ratios in X and Y directions Mode εx SAP000 εx AxisVM Difference % εy SAP000 εy AxisVM Difference % 1 0,5719 0, ,3153 0, ,03 0,3650 0, ,4761 0,4760-0, ,161 0, ,0460 0, ,0131 0, ,0170 0, ,056 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] 315,11 315,15 +0,01 557,6 557,9 +0,005 Vy [kn] 80,34 80,34 0 3,88 3,88 0 Vz [kn] 53,49 53, ,04 41,04 0 Tx [knm] 34,4 34,41-0,03 34,47 34,46-0,03 My [knm] 65,13 65,1-0, , ,70-0,004 Mz [knm] 61,31 61, ,41 553,41 0 Support forces of Support C Support C SAP000 Support C AxisVM Difference % Rx [kn] 80,34 80,34 0 Ry [kn] 53,49 53,49 0 Rz [kn] 315,11 315,15 +0,01 Rxx [knm] 65,13 65,1-0,00 Ryy [knm] 61,31 61,31 0 Rzz [knm] 34,4 34,41-0,03 Displacements of Node D Node D SAP000 Node D AxisVM Difference % ex [mm] 33,51 33,51 0 ey [mm] 19,944 19,945 +0,005 ez [mm] 0,9 0,9 0 ϕx [rad] 0, , ϕy [rad] 0, , ϕz [rad] 0,0057 0,0057 0

83 AxisVM 11 Verification Examples 83 Normal forces:

84 AxisVM 11 Verification Examples 84 Bending moments:

85 AxisVM 11 Verification Examples 85

86 AxisVM 11 Verification Examples 86 Displacements: