Buckling AH-ALBERTA Web:

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1 Buckling Intrductin This tutrial was created using ANSYS 7.0 t slve a simple buckling prblem. It is recmmended that yu cmplete the NnLinear Tutrial prir t beginning this tutrial Buckling lads are critical lads where certain types f structures becme unstable. Each lad has an assciated buckled mde shape; this is the shape that the structure assumes in a buckled cnditin. There are tw primary means t perfrm a buckling analysis: 1. Eigenvalue Eigenvalue buckling analysis predicts the theretical buckling strength f an ideal elastic structure. It cmputes the structural eigenvalues fr the given system lading and cnstraints. This is knwn as classical Euler buckling analysis. Buckling lads fr several cnfiguratins are readily available frm tabulated slutins. Hwever, in real-life, structural imperfectins and nnlinearities prevent mst real-wrld structures frm reaching their eigenvalue predicted buckling strength; ie. it ver-predicts the expected buckling lads. This methd is nt recmmended fr accurate, real-wrld buckling predictin analysis. 2. Nnlinear Nnlinear buckling analysis is mre accurate than eigenvalue analysis because it emplys nnlinear, large-deflectin, static analysis t predict buckling lads. Its mde f peratin is very simple: it gradually increases the applied lad until a lad level is fund whereby the structure becmes unstable (ie. suddenly a very small increase in the lad will cause very large deflectins). The true nn-linear nature f this analysis thus permits the mdeling f gemetric imperfectins, lad perterbatins, material nnlinearities and gaps. Fr this type f analysis, nte that small ffaxis lads are necessary t initiate the desired buckling mde. This tutrial will use a steel beam with a 10 mm X 10 mm crss sectin, rigidly cnstrained at the bttm. The required lad t cause buckling, applied at the tp-center f the beam, will be calculated. 1 Eigenvalue Buckling Analysis Preprcessing: Defining the Prblem 1. Open preprcessr menu /PREP7 2. Give example a Title Utility Menu > File > Change Title... /title,eigen-value Buckling Analysis 3. Define Keypints Preprcessr > Mdeling > Create > Keypints > In Active CS... K,#,X,Y

2 We are ging t define 2 Keypints fr this beam as given in the fllwing table: Keypints Crdinates (x,y) 1 (0,0) 2 (0,100) 4. Create Lines Preprcessr > Mdeling > Create > Lines > Lines > In Active Crd L,1,2 Create a line jining Keypints 1 and 2 5. Define the Type f Element Preprcessr > Element Type > Add/Edit/Delete... Fr this prblem we will use the BEAM3 (Beam 2D elastic) element. This element has 3 degrees f freedm (translatin alng the X and Y axes, and rtatin abut the Z axis). 6. Define Real Cnstants Preprcessr > Real Cnstants... > Add... In the 'Real Cnstants fr BEAM3' windw, enter the fllwing gemetric prperties: i. Crss-sectinal area AREA: 100 ii. Area mment f inertia IZZ: iii. Ttal Beam Height HEIGHT: 10 This defines a beam with a height f 10 mm and a width f 10 mm. 7. Define Element Material Prperties Preprcessr > Material Prps > Material Mdels > Structural > Linear > Elastic > Istrpic In the windw that appears, enter the fllwing gemetric prperties fr steel: i. Yung's mdulus EX: ii. Pissn's Rati PRXY: Define Mesh Size Preprcessr > Meshing > Size Cntrls > ManualSize > Lines > All Lines... Fr this example we will specify an element edge length f 10 mm (10 element divisins alng the line). 9. Mesh the frame Preprcessr > Meshing > Mesh > Lines > click 'Pick All' LMESH,ALL Slutin Phase: Assigning Lads and Slving 1. Define Analysis Type Slutin > Analysis Type > New Analysis > Static ANTYPE,0 2. Activate prestress effects T perfrm an eigenvalue buckling analysis, prestress effects must be activated. Yu must first ensure that yu are lking at the unabridged slutin menu s that yu can select Analysis Optins in the Analysis Type submenu. The last ptin in the slutin menu will either be 'Unabridged menu' (which means yu are currently lking at the abridged versin) r 'Abriged Menu' (which means yu are lking at the unabridged menu). If yu are lking at the abridged menu, select the unabridged versin. Select Slutin > Analysis Type > Analysis Optins In the fllwing windw, change the [SSTIF][PSTRES] item t 'Prestress ON', which ensures the stress stiffness matrix is calculated. This is required in eigenvalue buckling analysis. 2

3 3. Apply Cnstraints Slutin > Define Lads > Apply > Structural > Displacement > On Keypints Fix Keypint 1 (ie all DOF cnstrained). 4. Apply Lads Slutin > Define Lads > Apply > Structural > Frce/Mment > On Keypints The eignenvalue slver uses a unit frce t determine the necessary buckling lad. Applying a lad ther than 1 will scale the answer by a factr f the lad. Apply a vertical (FY) pint lad f -1 N t the tp f the beam (keypint 2). The applied lads and cnstraints shuld nw appear as shwn in the figure belw Slve the System

4 Slutin > Slve > Current LS 6. Exit the Slutin prcessr Clse the slutin menu and click at the bttm f the Main Menu. Nrmally at this pint yu enter the pstprcessing phase. Hwever, with a buckling analysis yu must re-enter the slutin phase and specify the buckling analysis. Be sure t clse the slutin menu and re-enter it r the buckling analysis may nt functin prperly. 7. Define Analysis Type Slutin > Analysis Type > New Analysis > Eigen Buckling ANTYPE,1 8. Specify Buckling Analysis Optins Select Slutin > Analysis Type > Analysis Optins Cmplete the windw which appears, as shwn belw. Select 'Blck Lanczs' as an extractin methd and extract 1 mde. The 'Blck Lanczs' methd is used fr large symmetric eigenvalue prblems and uses the sparse matrix slver. The 'Subspace' methd culd als be used, hwever it tends t cnverge slwer as it is a mre rbust slver. In mre cmplex analyses the Blck Lanczs methd may nt be adequate and the Subspace methd wuld have t be used. 9. Slve the System Slutin > Slve > Current LS 10. Exit the Slutin prcessr Clse the slutin menu and click at the bttm f the Main Menu. Again it is necessary t exit and re-enter the slutin phase. This time, hwever, is fr an expansin pass. An expansin pass is necessary if yu want t review the buckled mde shape(s). 11. Expand the slutin Select Slutin > Analysis Type > Expansin Pass... and ensure that it is n. Yu may have t select the 'Unabridged Menu' again t make this ptin visible. Select Slutin > Lad Step Opts > ExpansinPass > Single Expand > Expand Mdes... Cmplete the fllwing windw as shwn t expand the first mde 4

5 12. Slve the System Slutin > Slve > Current LS Pstprcessing: Viewing the Results 1. View the Buckling Lad T display the minimum lad required t buckle the beam select General Pstprc > List Results > Detailed Summary. The value listed under 'TIME/FREQ' is the lad (41,123), which is in Newtns fr this example. If mre than ne mde was selected in the steps abve, the crrespnding lads wuld be listed here as well. /POST1 SET,LIST 2. Display the Mde Shape Select General Pstprc > Read Results > Last Set t bring up the data fr the last mde calculated. Select General Pstprc > Plt Results > Defrmed Shape 5 Nn-Linear Buckling Analysis Ensure that yu have cmpleted the NnLinear Tutrial prir t beginning this prtin f the tutrial Preprcessing: Defining the Prblem 1. Open preprcessr menu /PREP7 2. Give example a Title Utility Menu > File > Change Title...

6 /TITLE, Nnlinear Buckling Analysis 3. Create Keypints Preprcessr > Mdeling > Create > Keypints > In Active CS K,#,X,Y We are ging t define 2 keypints (the beam vertices) fr this structure t create a beam with a length f 100 millimeters: Keypint Crdinates (x,y) 1 (0,0) 2 (0,100) 4. Define Lines Preprcessr > Mdeling > Create > Lines > Lines > Straight Line Create a line between Keypint 1 and Keypint 2. L,1,2 5. Define Element Types Preprcessr > Element Type > Add/Edit/Delete... Fr this prblem we will use the BEAM3 (Beam 2D elastic) element. This element has 3 degrees f freedm (translatin alng the X and Y axis's, and rtatin abut the Z axis). With nly 3 degrees f freedm, the BEAM3 element can nly be used in 2D analysis. 6. Define Real Cnstants Preprcessr > Real Cnstants... > Add... In the 'Real Cnstants fr BEAM3' windw, enter the fllwing gemetric prperties: i. Crss-sectinal area AREA: 100 ii. Area Mment f Inertia IZZ: iii. Ttal beam height HEIGHT: 10 This defines an element with a slid rectangular crss sectin 10 x 10 millimeters. 7. Define Element Material Prperties Preprcessr > Material Prps > Material Mdels > Structural > Linear > Elastic > Istrpic In the windw that appears, enter the fllwing gemetric prperties fr steel: i. Yung's mdulus EX: 200e3 ii. Pissn's Rati PRXY: Define Mesh Size Preprcessr > Meshing > Size Cntrls > Lines > All Lines... Fr this example we will specify an element edge length f 1 mm (100 element divisins alng the line). ESIZE,1 9. Mesh the frame Preprcessr > Meshing > Mesh > Lines > click 'Pick All' LMESH,ALL Slutin: Assigning Lads and Slving 1. Define Analysis Type Slutin > New Analysis > Static ANTYPE,0 2. Set Slutin Cntrls Select Slutin > Analysis Type > Sl'n Cntrl... The fllwing image will appear: 6

7 Ensure the fllwing selectins are made under the 'Basic' tab (as shwn abve) A. Ensure Large Static Displacements are permitted (this will include the effects f large deflectin in the results) B. Ensure Autmatic time stepping is n. Autmatic time stepping allws ANSYS t determine apprpriate sizes t break the lad steps int. Decreasing the step size usually ensures better accuracy, hwever, this takes time. The Autmatic Time Step feature will determine an apprpriate balance. This feature als activates the ANSYS bisectin feature which will allw recvery if cnvergence fails. C. Enter 20 as the number f substeps. This will set the initial substep t 1/20 th f the ttal lad. D. Enter a maximum number f substeps f This stps the prgram if the slutin des nt cnverge after 1000 steps. E. Enter a minimum number f substeps f 1. F. Ensure all slutin items are writen t a results file. Ensure the fllwing selectin is made under the 'Nnlinear' tab (as shwn belw) G. Ensure Line Search is 'On'. This ptin is used t help the Newtn-Raphsn slver cnverge. H. Ensure Maximum Number f Iteratins is set t

8 NOTE There are several ptins which have nt been changed frm their default values. Fr mre infrmatin abut these cmmands, type help fllwed by the cmmand int the cmmand line. 3. Apply Cnstraints Slutin > Define Lads > Apply > Structural > Displacement > On Keypints Fix Keypint 1 (ie all DOFs cnstrained). 4. Apply Lads Slutin > Define Lads > Apply > Structural > Frce/Mment > On Keypints Place a -50,000 N lad in the FY directin n the tp f the beam (Keypint 2). Als apply a -250 N lad in the FX directin n Keypint 2. This hrizntal lad will persuade the beam t buckle at the minimum buckling lad. The mdel shuld nw lk like the windw shwn belw Slve the System

9 Slutin > Slve > Current LS The fllwing will appear n yur screen fr NnLinear Analyses This shws the cnvergence f the slutin. General Pstprcessing: Viewing the Results 1. View the defrmed shape T view the element in 2D rather than a line: Utility Menu > PltCtrls > Style > Size and Shape and turn 'Display f element' ON (as shwn belw). 9

10 General Pstprc > Plt Results > Defrmed Shape... > Def + undefrmed PLDISP,1 10 View the deflectin cntur plt General Pstprc > Plt Results > Cntur Plt > Ndal Slu... > DOF slutin, UY PLNSOL,U,Y,0,1

11 Other results can be btained as shwn in previus linear static analyses. Time Histry Pstprcessing: Viewing the Results As shwn, yu can btain the results (such as deflectin, stress and bending mment diagrams) the same way yu did in previus examples using the General Pstprcessr. Hwever, yu may wish t view time histry results such as the deflectin f the bject ver time. 1. Define Variables Select: Main Menu > TimeHist Pstpr. The fllwing windw shuld pen autmatically. If it des nt pen autmatically, select Main Menu > TimeHist Pstpr > Variable Viewer 11 Click the add buttn in the upper left crner f the windw t add a variable.

12 Duble-click Ndal Slutin > DOF Slutin > Y-Cmpnent f displacement (as shwn belw) and click OK. Pick the uppermst nde n the beam and click OK in the 'Nde fr Data' windw. T add anther variable, click the add buttn again. This time select Reactin Frces > Structural Frces > Y-Cmpnent f Frce. Pick the lwermst nde n the beam and click OK. On the Time Histry Variable windw, click the circle in the 'X-Axis' clumn fr FY_3. This will make the reactin frce the x-variable. The Time Histry Variables windw shuld nw lk like this: 2. Graph Results ver Time Click n UY_2 in the Time Histry Variables windw. Click the graphing buttn in the Time Histry Variables windw. The labels n the plt are nt updated by ANSYS, s yu must change them manually. Select Utility Menu > Plt Ctrls > Style > Graphs > Mdify Axes and re-label the X and Y-axis apprpriately. 12

13 The plt shws hw the beam became unstable and buckled with a lad f apprximately 40,000 N, the pint where a large deflectin ccured due t a small increase in frce. This is slightly less than the eigen-value slutin f 41,123 N, which was expected due t nn-linear gemetry issues discussed abve. Cmmand File Mde f Slutin The abve example was slved using a mixture f the Graphical User Interface (r GUI) and the cmmand language interface f ANSYS. This prblem has als been slved using the ANSYS cmmand language interface that yu may want t brwse. Cpy and paste fllwing cde int Ntepad r a similar text editr and save it t yur cmputer. Nw g t 'File > Read input frm...' and select the file. /CLEAR! These tw cmmands clear current data /TITLE,Eigenvalue Buckling Analysis 13 /PREP7 ET,1,BEAM3 R,1,100, ,10 MP,EX,1, MP,PRXY,1,0.3 K,1,0,0 K,2,0,100 L,1,2 ESIZE,10 LMESH,ALL,ALL /SOLU ANTYPE,STATIC! Enter the preprcessr! Define the element f the beam t be buckled! Real Cnsts: type 1, area (mm^2), I (mm^4), height (mm)! Yung's mdulus (in MPa)! Pissn's rati! Define the gemetry f beam (100 mm high)! Draw the line! Set element size t 1 mm! Mesh the line! Enter the slutin mde! Befre yu can d a buckling analysis, ANSYS! needs the inf frm a static analysis

14 PSTRES,ON DK,1,ALL FK,2,FY,-1 /SOLU ANTYPE,BUCKLE BUCOPT,LANB,1 /SOLU EXPASS,ON MXPAND,1 /POST1 SET,LIST SET,LAST PLDISP NnLinear Buckling /CLEAR! Prestress can be accunted fr - required! during buckling analysis! Cnstrain the bttm f beam! Lad the tp vertically with a unit lad.! This is dne s the eigenvalue calculated! will be the actual buckling lad, since! all lads are scaled during the analysis.! Enter the slutin mde again t slve buckling! Buckling analysis! Buckling ptins - subspace, ne mde! Re-enter slutin mde t expand inf - necessary! An expantin pass will be perfrmed! Specifies the number f mdes t expand! Enter pst-prcessr! List eigenvalue slutin - Time/Freq listing is the! frce required fr buckling (in N fr this case).! Read in data fr the desired mde! Plts the deflected shape! These tw cmmands clear current data /TITLE, Nnlinear Buckling Analysis /PREP7! Enter the preprcessr ET,1,BEAM3! Define element as beam3 MP,EX,1, MP,PRXY,1,0.3 R,1,100, ,10 K,1,0,0,0 K,2,0,100,0 L,1,2 ESIZE,1 LMESH,ALL! Yung's mdulus (in Pa)! Pissn's rati! area, I, height! Lwer nde! Upper nde (100 mm high)! Draws line! Sets element size t 1 mm! Mesh line 14 /SOLU ANTYPE,STATIC NLGEOM,ON OUTRES,ALL,ALL NSUBST,20 NEQIT,1000 AUTOTS,ON LNSRCH,ON /ESHAPE,1! Static analysis (nt buckling)! Nn-linear gemetry slutin supprted! Stres bunches f utput! Lad brken int 5 lad steps! Use 20 lad steps t find slutin! Aut time stepping! Plts the beam as a vlume rather than line

15 DK,1,ALL,0 FK,2,FY, FK,2,FX,-250! Cnstrain bttm! Apply lad slightly greater than predicted! required buckling lad t upper nde! Add a hrizntal lad (0.5% FY) t initiate! buckling /POST26! Time histry pst prcessr RFORCE,2,1,F,Y! Reads frce data in variable 2 NSOL,3,2,U,Y! Reads y-deflectin data int var 3 XVAR,2! Make variable 2 the x-axis PLVAR,3! Plts variable 3 n y-axis /AXLAB,Y,DEFLECTION /AXLAB,X,LOAD /REPLOT! Changes y label! Changes X label Reference 15