Verification Examples

Verification Examples 2008

AxisVM 9 Verification Examples 2 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....11 Clamped beam examination with plane stress elements...13 Clamped thin square plate...16 Plate with fixed support and constant distributed load...18 Annular plate....19 All edges simply supported plate with partial distributed load....21 Clamped plate with linear distributed load...23 Hemisphere displacement...25 Nonlinear static...27 3D beam structure...28 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....42 Flat grillages....45 Stability...49 Simply supported beam...50 Simply supported beam...52 Design...53 N-M interaction curve of cross-section (EN 1992-1-1:2004)...54 RC beam deflection according to EC2, EN 1992-1-1:2004....55 Required steel reinforcement of RC plate according to EC2, EN 1992-1-1:2004...57 Earth-quake design using response-spectrum method....59

AxisVM 9 Verification Examples 3 Linear static

AxisVM 9 Verification Examples 4 Software Release Number: R2 Date: 12. 03. 2008. 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 2 Loads Boundary Conditions Material Properties Element types Mesh Axial direction forces P 1 = -200 N, P 2 = 100 N, P 3 = -40 N Fix ends, at R 1 and R 5. E = 20000 kn / cm 2 ν = 0,3 Beam element Target Results R 1, R 5 support forces Theory AxisVM % R 1 [N] 118,00 118,00 0,00 R 5 [N] 22,00 22,00 0,00

AxisVM 9 Verification Examples 5 Software Release Number: R2 Date: 12. 03. 2008. Tested by: InterCAD Page number: File name: beam2.axs Thema Analysis Type Geometry Thermally loaded bar structure. Linear analysis. Sections: Steel: A S = π x 10-4 m 2 Copper: A C = π x 10-4 m 2 Side view Loads Boundary Conditions Material Properties Element types Target Results P = -12 kn (Point load) Temperature rise of 10 C in the structure after assembly. The upper end of bars are fixed. Steel: E S = 20700 kn / cm 2, ν = 0,3, α S = 1,2 x 10-5 C -1 Copper: E C = 11040 kn / cm 2, 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] 23824000 23848000 0,10 Cooper S max [MPa] 7185300 7198900 0,19

AxisVM 9 Verification Examples 6 Software Release Number: R2 Date: 12. 03. 2008. 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 2 = 0,60 m) Loads Boundary Conditions Material Properties Element types Target Results P 1 = -300 kn, P 2 = -1250 kn, P 3 = -800 kn, P 4 = -450 kn Elastic supported. From A to D is K z = 25000 kn/m/m. From D to F is K z = 15000 kn/m/m. E = 3000 kn/cm 2 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

AxisVM 9 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 0,00 M B [KNm] 88,5 87,5-1,13 M C [KNm] 636,2 632,8-0,53 M D [KNm] 332,8 329,7-0,93 M E [KNm] 164,2 163,3-0,55 M F [KNm] 0,0 0,0 0,00

AxisVM 9 Verification Examples 8 Results Reference AxisVM e [%] V A [KN] 0,0 0,0 0,00 V B [KN] 112,1 113,1 0,89 V C [KN] 646,8 647,2 0,06 V D [KN] 335,0 334,9-0,03 V E [KN] 267,8 267,5-0,11 V F [KN] 0,0 0,0 0,00 Reference AxisVM e [%] R A [KN/m 2 ] 145,7 154,0 5,70 R B [KN/m 2 ] 219,5 219,4-0,05 R C [KN/m 2 ] 343,8 346,0 0,64 R D [KN/m 2 ] 386,9 386,4-0,13 R E [KN/m 2 ] 224,5 224,7 0,09 R F [KN/m 2 ] 201,2 200,8-0,20

AxisVM 9 Verification Examples 9 Software Release Number: R2 Date: 12. 03. 2008. 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 = -6,52E-3 at beam 5-6 ey = ez = = 0 at node 1 ex = ey = ez = 0 at node 4 E = 2,1E11 N / m 2 Beam 1-5, 5-6, 6-4 A = 4,5E-3 m 2 I z = 0,2E-5 m 4 Truss 2-5, 3-6 A = 3,48E-3 m 2 I z = 0,2E-5 m 4 Beam 1-4 A = 1,1516E-2 m 2 I z = 2,174E-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 2-5, 3-6 Target N X at beam 6-7 M y,max at beam 2-3 e z at node 2

AxisVM 9 Verification Examples 10 Results 1 2 3 4 5 6 0,600 2,000 4,000 2,000 8,000 Z X Diagram e z ROBOT V6 AxisVM % N x [kn] 584,56 585,70 0,19 M y [knm] 49,26 49,60 0,68 e z [mm] -0,5421-0,5469 0,89

AxisVM 9 Verification Examples 11 Software Release Number: R2 Date: 12. 03. 2008. 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 = 200 kn / m Side view (thickness = 20,0 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 2 ν = 0,16 Parabolic quadrilateral membrane (plane stress) Target S xx at 1-10 nodes (1-5 at middle, 6-10 at support)

AxisVM 9 Verification Examples 12 Results Node Analytical [kn/cm 2 ] AxisVM [kn/cm 2 ] % 1 0,1313 0,1310-0,23 2 0,0399 0,0395-1,00 3-0,0093-0,0095 2,15 4-0,0412-0,0410-0,49 5-0,1073-0,1070-0,28 6-0,9317-0,9270-0,50 7 0,0401 0,0426 6,23 8 0,0465 0,0469 0,86 9 0,0538 0,0540 0,37 10 0,1249 0,1240-0,72 Reference: Dr. Bölcskey Elemér Dr. Orosz Árpád: Vasbeton szerkezetek Faltartók, Lemezek, Tárolók

AxisVM 9 Verification Examples 13 Software Release Number: R2 Date: 12. 03. 2008. Tested by: InterCAD Page number: File name: plane2.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 = -25 kn/m Both ends built-in. E = 880 kn / cm 2 ν = 0 Parabolic quadrilateral membrane (plane stress) 0,375 Clamped edge 1 0,500 C 3,000 0,250 Z X Side view

AxisVM 9 Verification Examples 14 Target Results τ xy, max at section C Diagram τ xy 5,14 791,56 Z Y 5,28 Diagram τ xy at section C

AxisVM 9 Verification Examples 15 V = 65,625 kn ( from beam theory) S ' y = 0,0078125 m 3 b = 0,25 m I y = 0,00260416 m 4 τ xy V S = b I ' y y 65,625 0,0078125 = 0,25 0,00260416 = 787,5 kn / m 2 AxisVM result τ xy = 791,6 kn / m 2 Difference = +0,52 % AxisVM result = V nxy = 65, 34 kn Difference = +0,43 %

AxisVM 9 Verification Examples 16 Software Release Number: R2 Date: 12. 03. 2008. 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 = 20000 kn / cm 2 ν = 0,3 Plate element (Parabolic quadrilateral, heterosis) 4,000 Y 4,000 Target X Displacement of middle of the plate

AxisVM 9 Verification Examples 17 Results -0,019-0,012-0,006-0,001-0,043-0,022-0,006-0,043-0,012-0,065-0,084-0,081-0,026-0,024-0,065-0,024-0,125-0,087-0,019-0,026-0,081-0,125-0,087-0,156-0,012-0,081-0,026-0,087-0,006-0,156-0,187-0,168-0,065-0,168-0,043-0,001-0,024-0,087-0,237-0,156-0,022-0,168-0,237-0,125-0,006-0,257-0,084-0,019-0,081-0,043-0,168-0,257-0,257-0,307-0,012-0,237-0,012-0,065-0,187-0,156-0,125-0,065-0,019-0,043-0,257-0,337-0,006-0,125-0,337-0,337-0,237-0,307-0,237-0,156-0,081-0,022-0,084-0,024-0,001-0,187-0,337-0,043-0,383-0,168-0,087-0,006-0,125-0,383-0,257-0,026-0,337-0,012-0,065-0,156-0,237-0,307-0,383-0,168-0,087-0,257-0,026-0,019-0,081-0,383-0,337-0,168-0,257-0,337-0,337-0,024-0,087-0,237-0,156-0,081-0,024-0,026-0,087-0,168-0,257-0,187-0,156-0,237-0,307-0,125-0,065-0,026-0,081-0,125-0,019-0,084-0,024-0,065-0,043-0,043-0,012-0,019-0,022-0,012-0,006-0,006-0,019-0,024-0,001-0,026 Y Z X Displacements Mode Mesh Book 1 Timoshenko 2 AxisVM Diff 1 [%] Diff 2 [%] 1 2x2 0,402 0,420 4,48 10,53 2 4x4 0,416 0,369-11,30-2,89 3 8x8 0,394 0,38 0,381-3,30 0,26 4 12x12 0,387 0,383-1,03 0,79 5 16x16 0,385 0,383-0,52 0,79 References: 1.) The Finite Element Method (Fourth Edition) Volume 2. /O.C. Zienkiewicz and R.L. Taylor/ McGraw-Hill Book Company 1991 London 2.) Result of analytical solution of Timoshenko Convergency 15,00 10,00 5,00 Displacements 0,00 1 2 3 4 5 Diff1 [%] Diff2 [%] -5,00-10,00-15,00 Mesh density

AxisVM 9 Verification Examples 18 Software Release Number: R2 Date: 12. 03. 2008. Tested by: InterCAD Page number: File name: plate2_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 2 Boundary ex = ey = ez = fix = fiy = fiz = 0 along all edges Conditions Material E = 990 kn/cm 2 Properties ν = 0,16 Element Parabolic triangle plate element types Mesh Target Results Maximal ex (found at Node1) and maximal m x (found at Node2) Component Nastran AxisVM % ez,max [mm] -1,613-1,593-1,24 mx,max [knm/m] 3,060 3,059-0,03

AxisVM 9 Verification Examples 19 Software Release Number: R2 Date: 12. 03. 2008. Tested by: InterCAD Page number: File name: plate3.axs Thema Analysis Type Geometry Annular plate. Linear analysis. Top view (thickness = 22,0 cm) Loads Boundary Conditions Edge load: Q = 100 kn / m Distributed load: q = 100 kn / m 2 Material Properties Element types E = 880 kn / cm 2 ν = 0,3 Plate element (parabolic quadrilateral, heterosis)

AxisVM 9 Verification Examples 20 Mesh 3,000 1,000 Y 4,000 Target S max, e max X Results Theory AxisVM Model S max S max % [kn/cm2] [kn/cm2] a.) 2,82 2,78-1,42 b.) 6,88 6,76-1,74 c.) 14,22 14,10-0,84 d.) 1,33 1,33 0,00 e.) 2,35 2,25-4,26 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 76,10-2,03 b.) 226,76 220,84-2,61 c.) 355,17 352,89-0,64 d.) 23,28 23,42 0,60 e.) 44,26 44,50 0,54 f.) 123,19 123,17-0,02 g.) 112,14 111,94-0,18 h.) 126,83 126,81-0,02 Reference: S. Timoshenko and S. Woinowsky-Krieger: Theory of Plates And Shells

AxisVM 9 Verification Examples 21 Software Release Number: R2 Date: 12. 03. 2008. 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 = 22,0 cm) Loads Boundary Conditions Material Properties Element types Mesh Distributed load: q = -10 kn / m 2 (middle of the plate at 2,0 x 2,0 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 2 ν = 0,3 Plate element (Heterosis) 10,000 Y 5,000 X

AxisVM 9 Verification Examples 22 Target Results m x, max, m y, max a.) Moment Theory AxisVM % m x, max [knm/m] 7,24 7,34 1,38 m y, max [knm/m] 5,32 5,39 1,32 b.) Moment Theory AxisVM % m x, max [knm/m] 7,24 7,28 0,55 m y, max [knm/m] 5,32 5,35 0,56 Reference: S. Timoshenko and S. Woinowsky-Krieger: Theory of Plates And Shells

AxisVM 9 Verification Examples 23 Software Release Number: R2 Date: 12. 03. 2008. Tested by: InterCAD Page number: File name: plate5.axs Thema Analysis Type Geometry Clamped plate with linear distributed load. Linear analysis. Top view (thickness = 22,0 cm) Loads Distributed load: q = -10 kn / m 2 Boundary Conditions Material Properties Element types Mesh ex = ey = ez = fix = fiy= fiz = 0 along all edges E = 880 kn / cm 2 ν = 0,3 Plate element (Heterosis) q 3 1 2 10,000 4 Y 10,000 X

AxisVM 9 Verification Examples 24 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,2 [knm/m] 33,40 33,23-0,51 m x,3 [knm/m] 17,90 17,83-0,39 m y,4 [knm/m] 25,70 25,53-0,66 S. Timoshenko and S. Woinowsky-Krieger: Theory of Plates And Shells

AxisVM 9 Verification Examples 25 Software Release Number: R2 Date: 12. 03. 2008. 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 = 2,0 kn C 2,0 kn 2,0 kn Z A B X Y

AxisVM 9 Verification Examples 26 Boundary Conditions Material Properties Element types Target ex = ey = ez = 0 at A ex = ey = ez = 0 at B E = 6825 kn / cm 2 ν = 0,3 Shell element 1.) guadrilateral parabolic 2.) triangle parabolic e x at point A Results e x [m] e [%] Theory 0,185 AxisVM quadrilateral 0,186 0,54 AxisVM triangle 0,185 0,00

AxisVM 9 Verification Examples 27 Nonlinear static

AxisVM 9 Verification Examples 28 Software Release Number: R2 Date: 12. 03. 2008. Tested by: InterCAD Page number: File name: nonlin1.axs Thema Analysis Type Geometry 3D beam structure. Geometrical nonlinear analysis. 1,732 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 1,732 m Beam1 D Y X 1,732 m 3,000 m 1,732 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 275 E = 21000 kn / cm 2 ν = 0,3 HEA 300 Ax = 112.56 cm 2 ; Ix = 85.3 cm 4 ; Iy = 18268.0 cm 4 ; Iz = 6309.6 cm 4 Beam ex, ey, ez, at Node1 Nx, Vy, Vz, Tx, My, Mz of Beam1 at Node1

AxisVM 9 Verification Examples 29 Results Comparison with the results obtained using Nastran V4 Component Nastran AxisVM % ex [mm] 17,898 17,881-0,09 ey [mm] -75,702-75,663-0,05 ez [mm] -42,623-42,597-0,06 Nx [kn] -283,15-283,25 0,04 Vy [kn] -28,09-28,10 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] 148,94 148,91-0,02

AxisVM 9 Verification Examples 30 Software Release Number: R2 Date: 12. 03. 2008. Tested by: InterCAD Page number: File name: nonlin2.axs Thema Analysis Type Geometry Plate with fixed end and bending moment. Geometrical nonlinear analysis. Edge1 1,0 m Edge2 12,0 m Z Y X Loads Boundary Conditions Material Properties Cross Section Properties Element types Mz = 2600 knm (2x1300 Nm) acting on Edge2 ex = ey = ez = fix = fiy = fiz = 0 along Edge1 E = 20000 N / mm 2 ν = 0 Plate thickness: 150 mm Rib on Edge2: circular D = 500 mm (for distributing load to the mid-side-node) Parabolic quadrilateral shell (heterosis) Rib on Edge2 for distributing load to the mid-side-node

AxisVM 9 Verification Examples 31 Target Results ϕ Z at Edge2 5,5502 rad Edge1 1,0 m Edge2 12,0 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 1 0.15 = = 12 12 10 2 = 2 10 N m = 12 m = 2.6 10 M l ϕ z = I platee Nm 2.6 10 ϕ z = 2.8125 10 6 6 4 3 12 2 10 plate plate = 2.8125 10 10 4 = 5.5467 rad Comparison the AxisVM result with the theoretical one: Component Theory AxisVM % fiz [rad] 5,5467 5,5502 0,06

AxisVM 9 Verification Examples 32 BLANK

AxisVM 9 Verification Examples 33 Dynamic

AxisVM 9 Verification Examples 34 Software Release Number: R2 Date: 12. 03. 2008. 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 2,0 m x 2,0 m) Loads Boundary Conditions Material Properties Element types Target Self-weight ex = ey = ez = fix = 0 at A ey = ez = 0 at B E = 20000 kn / cm 2 ν = 0,3 ρ = 8000 kg / m 3 Three node beam element (shear deformation is taken into account) First 7 mode shapes

AxisVM 9 Verification Examples 35 Results Mode 1: f = 43,16 Hz Mode 2: f = 43,16 Hz Mode 3: f = 124,01 Hz Mode 4: f = 152,50 Hz Mode 5: f = 152,50 Hz Mode 6: f = 293,55 Hz Mode 7: f = 293,55 Hz

AxisVM 9 Verification Examples 36 Results Comparison with NAFEMS example Mode NAFEMS (Hz) AxisVM (Hz) % 1 42,65 43,16-1,20 2 42,65 43,16-1,20 3 125,00 124,01 0,79 4 148,31 152,50-2,83 5 148,31 152,50-2,83 6 284,55 293,55-3,16 7 284,55 293,55-3,16

AxisVM 9 Verification Examples 37 Software Release Number: R2 Date: 12. 03. 2008. Tested by: InterCAD Page number: File name: dynam2.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 = 20000 kn / cm 2 ν = 0,3 ρ = 8000 kg / m 3 Parabolic quadrilateral shell element (heterosis) 10,000 10,000 Y X

er 0,506 0,470 0,433 0,397 0,361 0,325 0,289 0,253 0,217 0,181 0,144 0,108 0,072 0,036 0 er 0,486 0,451 0,416 0,382 0,347 0,312 0,278 0,243 0,208 0,174 0,139 0,104 0,069 0,035 0 er 0,498 0,462 0,427 0,391 0,356 0,320 0,284 0,249 0,213 0,178 0,142 0,107 0,071 0,036 0 er 0,463 0,429 0,396 0,363 0,330 0,297 0,264 0,231 0,198 0,165 0,132 0,099 0,066 0,033 0 er er 0,449 0,417 0,385 0,353 0,321 0,289 0,257 0,225 0,192 0,160 0,128 0,096 0,064 0,032 0 0,520 0,483 0,446 0,409 0,372 0,335 0,297 0,260 0,223 0,186 0,149 0,112 0,074 0,037 0 AxisVM 9 Verification Examples 38 Target First 6 mode shapes Results Mode 1: f = 8,02 Hz Mode 2: f = 13,02 Hz Mode 3: f = 18,41 Hz Mode 4: f = 19,33 Hz Mode 5: f = 24,62 Hz Mode 6: f = 28,24 Hz Results Comparison with NAFEMS example Mode NAFEMS (Hz) AxisVM (Hz) % 1 7,94 8,02 1,01 2 12,84 13,02 1,40 3 17,94 18,41 2,62 4 19,13 19,33 1,05 5 24,01 24,62 2,54 6 27,92 28,24 1,15

AxisVM 9 Verification Examples 39 Software Release Number: R2 Date: 12. 03. 2008. 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 = 20000 kn / cm 2 ν = 0,3 ρ = 8000 kg / m 3 Parabolic quadrilateral shell element (heterosis).

AxisVM 9 Verification Examples 40 Target First 5 mode shapes Results Mode 1: f = 0,42 Hz Mode 3: f = 2,53 Hz Mode 5: f = 3,68 Hz

AxisVM 9 Verification Examples 41 Mode 2: f = 1,02 Hz Mode 4: f = 3,22 Hz Comparison with NAFEMS example Mode NAFEMS (Hz) AxisVM (Hz) % 1 0,421 0,420-0,24 2 1,029 1,020-0,87 3 2,580 2,530-1,94 4 3,310 3,220-2,72 5 3,750 3,680-1,87

AxisVM 9 Verification Examples 42 Software Release Number: R2 Date: 12. 03. 2008. 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 = 20000 kn / cm 2 ν = 0,3 ρ = 8000 kg / m 3 Parabolic quadrilateral membrane (plane stress) X5,000 Y 10,000 1,000

AxisVM 9 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 2: f = 128,36 Hz

AxisVM 9 Verification Examples 44 X5,000 Y 10,000 Mode 3: f = 162,48 Hz 1,000 X5,000 Y 10,000 Mode 4: f = 241,22 Hz 1,000 Results Comparison with NAFEMS example Mode NAFEMS (Hz) AxisVM (Hz) % 1 44,62 44,33-0,65 2 130,03 128,36-1,28 3 162,70 162,48-0,14 4 246,05 241,22-1,96

AxisVM 9 Verification Examples 45 Software Release Number: R2 Date: 12. 03. 2008. 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 = 20000 kn / cm 2 G = 7690 kn / cm 2 ν = 0,3 ρ = 7860 kg / m 3 A = 0,004 m 2 Ix = 2,5E-5 m 4 Iy = Iz = 1,25E-5 m 4 Three node beam element (shear deformation is taken into account) 1,000 0,500 4,500 2,000 1,000 1,500 1,500 1,500 1,000 0,500 Y X

AxisVM 9 Verification Examples 46 Target First 3 mode shapes Results 1,605 1,879 1,679 1,638 1,586 1,114 1,241 1,035 Z Y X Mode 1: f = 16,90 Hz -1,813-2,065-1,837 0,856 2,040 2,254 1,938 Z Y X Mode 2: f = 20,64 Hz -1,130 2,040-1,581-1,620 1,721 1,585-1,667-1,992-1,845 Z Y X Mode 3: f = 51,76 Hz

AxisVM 9 Verification Examples 47 Mode Reference AxisVM (Hz) % 1 16,85 16,90 0,30 2 20,21 20,64 2,13 3 53,30 51,76-2,89 Reference: C.T.F. ROSS: Finite Element Methods In Engineering Science

AxisVM 9 Verification Examples 48 BLANK

AxisVM 9 Verification Examples 49 Stability

AxisVM 9 Verification Examples 50 Software Release Number: R2 Date: 12. 03. 2008. Tested by: InterCAD Page number: File name: buckling1.axs Thema Analysis Type Geometry Simply supported beam. Buckling analysis. Front view 2 2 7 3 6 20,0 S G 1 1 8 z 5 9 4 1,0 y 10,0 Cross section (I z =168,3 cm 4, I t =12,18 cm 4, I w =16667 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 = 20600 kn / cm 2 ν = 0,3 G = 7923 kg / m 2 Parabolic quadrilateral shell element (heterosis)

AxisVM 9 Verification Examples 51 Target M cr =? (for lateral torsional buckling) Results Analytical solution M cr π E I L 2 2 Z W = + 2 2 L I Z π I G I E I t Z 2 2 π 20600 168,3 16667 200 7923 12,18 M cr = + = 12451 kncm = 124, 51 2 2 200 168,3 π 20600 168,3 knm AxisVM result M cr = 125,3 knm Difference +0,6%

AxisVM 9 Verification Examples 52 Software Release Number: R2 Date: 12. 03. 2008. Tested by: InterCAD Page number: File name: buckling2.axs Thema Analysis Type Geometry Simply supported beam. Buckling analysis. Front view (L = 1,0 m) 1 2 5 2 5 2 S G 1 10,0 2 S G 1 3 4 3 4 z 12,0 30,0 y z y Loads P = -1,0 N at point B. Section A 1 Section A 2 Cross-sections Boundary Conditions Material Properties Element types Target Results ex = ey = ez = 0 at A ey = ez = 0 at B E = 20000 kn / cm 2 ν = 0,3 Beam element P cr =? (for inplane buckling) Theory AxisVM e [%] P cr [N] 3340,0 3337,0-0,09

AxisVM 9 Verification Examples 53 Design

AxisVM 9 Verification Examples 54 Software Release Number: R2 Date: 12. 03. 2008. Tested by: InterCAD Page number: File name: RC column1.axs Thema N-M interaction curve of cross-section (EN 1992-1-1:2004). Analysis Type Geometry Linear static analysis+design. 2φ20 3φ28 Section: 300x400 mm Covering: 40 mm Loads Boundary Conditions Material Properties Target Results Concrete: f cd =14,2 N/mm 2 e c1 =0,0007 e cu =0,0035 (rectangular σ-ε diagram) Steel: f sd =348 N/mm 2 e su =0,015 Compare the program results with with hand calculation at keypoints of M-N interaction curve. N 1 2 6 5 3 4 N [kn] M [knm] M(N) AxisVM e% 1-2561 +61 +61,4 +0,7 2-1221 +211 +212,3 +0,6 3 0 +70 +70,5 +0,7 4 +861-61 -61,4 +0,7 5 0-190 -191,2 +0,6 6-362 +211 +212,3 +0,6 Reference: Dr. Kollár L. P., Vasbetonszerkezetek I. Műegyetemi kiadó

AxisVM 9 Verification Examples 55 Software Release Number: R2 Date: 12. 03. 2008. Tested by: InterCAD Page number: File name: beam1.axs Thema RC beam deflection according to EC2, EN 1992-1-1:2004. Analysis Type Geometry Material nonlinear analysis. q = 17 kn/m L = 5,60 m Side view 2φ20 35 cm covering = 3 cm β = 0,5 4φ20 25 cm Section Loads Boundary Conditions Material Properties Element types Target q = 17 kn /m distributed load Simply supported beam. Concrete: C25/30, ϕ = 2,1 Steel: B500B Parabolic quadrilateral plate element (Heterosis) e z, max

AxisVM 9 Verification Examples 56 Results -0,002-5,239-10,101-14,242-17,393-19,360-20,029-19,360-17,393-14,242-10,101-5,239-0,002 Z X Diagram e z Aproximate calculation: e = ζ e + ( 1 ζ ) e = 20,06 _ 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 2 σ sr ζ = 1 β σ s Calculation with integral of κ: e = 19,82 mm Calculation with AxisVM: e = 20,03 mm (different +1,1%)

AxisVM 9 Verification Examples 57 Software Release Number: R2 Date: 12. 03. 2008. Tested by: InterCAD Page number: File name: beam2.axs Thema Required steel reinforcement of RC plate according to EC2, EN 1992-1-1:2004. 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: C25/30 Steel: B500A Parabolic quadrilateral plate element (heterosis) Szabvány : Eurocode Eset : ST1 4,0 Clamped edge 1,0 Y X Top view

AxisVM 9 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 2 /m] 1,0 4,0 Clamped edge ST1, axf: 2093 mm 2 /m Z Y X Diagram A XT Calculation according to EC2: 25 f cd = 1,5 ξ 500 f yd = = 435 N / mm 1,15 2 2 = 16,6 N / mm c ε E 0,85 0,0035 20000 = 0,0035 20000 + 435 cu S c0 = = εcu ES + f yd d = 300 53 = 247 mm 0,54 xc M sd = M Rd = b xc fcd d 200 2 = knm x c 439 > h = 55 = xc 55 ξ = 0,22 0 0,54 247 = < = c ξ Steel reinforcement is yielding c d A b xc f = f 55 1000 16,6 = 435 cd S = yd 2099 mm 2 Calculation with AxisVM: A XT = 2 2093 mm / m Different = -0,3 %

AxisVM 9 Verification Examples 59 Software Release Number: R2 Date: 12. 03. 2008. 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

AxisVM 9 Verification Examples 60 Code : Eurocode Case : ST1 All nodal masses are Mx=My=Mz=100000 kg All beams 60x40 cm Inertia about vertical axis is multiplied by 1000. Node D All columns 60x40 cm Column B Column A Support C Y Z X Perspective view All supports are constrained in all directions. ex=ey=ez=fix=fiy=fiz=0 Section beams: 60x40 cm Ax=2400 cm2 Ay=2000 cm2 Az=2000 cm2 Ix=751200 cm4 Iy=720000 cm2 Iz=320000000 cm4 Section columns: 60x40 cm Ax=2400 cm2 Ay=2000 cm2 Az=2000 cm2 Ix=751200 cm4 Iy=720000 cm2 Iz=320000 cm4 Loads Nodal masses on eight nodes. Mx=My=Mz=100000 kg Model self-weight is excluded. Spectrum for X and Y direction of seismic action: T[s] S d 2,156 S d [m/s 2 ] 1 0 1,150 2 0,2000 2,156 3 0,6000 2,156 4 1,3000 0,995 5 3,0000 0,300 6 4,0000 0,300...... 1,150 0,709 0,300 2,0000 T[s] Boundary Conditions Material Properties Nodes at the columns bottom ends are constrained in all directions. ex=ey=ez=fix=fiy=fiz=0 C25/30 E=3050 kn/cm2 ν =0,2 ρ = 0

AxisVM 9 Verification Examples 61 Element types Target Results Three node straight prismatic beam element. Shear deformation is taken into account. Compare the model results with SAP2000 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] SAP2000 T[s] AxisVM Difference [%] 1 0,7450 0,7450 0 2 0,7099 0,7099 0 3 0,3601 0,3601 0 4 0,2314 0,2314 0 5 0,2054 0,2054 0 Modal participating mass ratios in X and Y directions Mode εx SAP2000 εx AxisVM Difference % εy SAP2000 εy AxisVM Difference % 1 0,5719 0,5719 0 0,3153 0,3154 +0,03 2 0,3650 0,3650 0 0,4761 0,4760-0,02 3 0 0 0 0,1261 0,1261 0 4 0,0460 0,0460 0 0,0131 0,0131 0 5 0,0170 0,0170 0 0,0562 0,0562 0 Summ 1,0000 1,0000 0 0,9868 0,9868 0 Internal forces at the bottom end of Column A and Column B Column A SAP2000 Column A AxisVM Difference % Column B SAP2000 Column B AxisVM Difference % Nx [kn] 315,11 315,15 +0,01 557,26 557,29 +0,005 Vy [kn] 280,34 280,34 0 232,88 232,88 0 Vz [kn] 253,49 253,49 0 412,04 412,04 0 Tx [knm] 34,42 34,41-0,03 34,47 34,46-0,03 My [knm] 625,13 625,12-0,002 1038,74 1038,70-0,004 Mz [knm] 612,31 612,31 0 553,41 553,41 0 Support forces of Support C Support C SAP2000 Support C AxisVM Difference % Rx [kn] 280,34 280,34 0 Ry [kn] 253,49 253,49 0 Rz [kn] 315,11 315,15 +0,01 Rxx [knm] 625,13 625,12-0,002 Ryy [knm] 612,31 612,31 0 Rzz [knm] 34,42 34,41-0,03 Displacements of Node D Node D SAP2000 Node D AxisVM Difference % ex [mm] 33,521 33,521 0 ey [mm] 19,944 19,945 +0,005 ez [mm] 0,229 0,229 0 ϕx [rad] 0,00133 0,00133 0 ϕy [rad] 0,00106 0,00106 0 ϕz [rad] 0,00257 0,00257 0

AxisVM 9 Verification Examples 62 Normal forces:

AxisVM 9 Verification Examples 63 Bending moments:

AxisVM 9 Verification Examples 64

AxisVM 9 Verification Examples 65 Displacements: