Module 4: Buckling of 2D Simply Supported Beam

Transcription

1 Module 4: Buckling of D Simply Supported Beam Table of Contents Page Number Problem Description Theory Geometry 4 Preprocessor 7 Element Type 7 Real Constants and Material Properties 8 Meshing 9 Solution 11 Static Solution 11 Eigenvalue 14 Mode Shape 1 General Postprocessor 1 Results 18 Validation 18 UCONN ANSYS Module Page 1

2 Problem Description: y x Nomenclature: L =00mm b =10mm h =1 mm P=1N E=00,000 Length of beam Cross Section Base Cross Section Height Applied Force =0. Poisson s Ratio of Steel Young s Modulus of Steel at Room Temperature Moment of Inertia This module is a simply supported beam subject to two opposite edge compressions until the material buckles. Buckling is inherently non-linear, but we linearize the problem through the Eigenvalue method. This solution is an overestimate of the theoretical value since it does not consider imperfections and nonlinearities in the structure such as warping and manufacturing defects. We model the beam with D elements. Theory Buckling load Hooke s Law equates stress as shown: (4.1) Deriving both sides of equation.1 it shows (4.) By solving for equilibrium: (4.) Equation. is a nonlinear equation, however this equation can be linearized using eigenvalues. UCONN ANSYS Module Page

3 Since: (4.4) Then: Plugging in Equation.1 for stress we find: (4.) (4.) Plugging Equation. into Equation., Equation. becomes (4.7) Which is simplifies to: (4.8) By integrating two times Equation.8 becomes (4.9) At the fixed end (x=0), v=0,, thus 0 At the supported end (x=l), v=0,, thus 0 Equation.9 becomes (4.10) Equation. represents the Differential Equation for a Sin Wave (4.11) A and B are arbitrary constants which are calculated based on Boundary Conditions. At the fixed end (x=0), v=0 proving B=0. Equation.11 becomes But A cannot equal zero or this problem is trivial. At the supported end (x=l), v=0 Equation 1 becomes (4.1) (4.1) Since A cannot equal zero, ( ) must equal zero: Sin(nπ)=0 for n=(0, 1,,, 4..) UCONN ANSYS Module Page

4 So: n 0 or it is trivial (4.14) We are interested in finding P which is the Critical Buckling Load. Since n can be any integer greater than zero and a continuous beam has theoretically infinite degrees of freedom there are infinite amount of eigenvalues ( ). Where the lowest Buckling Load is at This is an over estimate so there are certain correction factors (C) to account for this. (C) is dependent on the beam constraints. Where C=1 for a fixed-simply supported beam. So the Critical Buckling Load is (4.1) (4.1) (4.17) = N (4.18) Geometry Opening ANSYS Mechanical APDL 1. On your Windows 7 Desktop click the Start button. Under Search Programs and Files type ANSYS. Click on Mechanical APDL (ANSYS) to start ANSYS. This step may take time. 1 UCONN ANSYS Module Page 4

5 Preferences 1. Go to Main Menu -> Preferences. Check the box that says Structural. Click OK 1 Title: To add a title 1. Utility Menu -> ANSYS Toolbar -> type /prep7 -> enter. Utility Menu -> ANSYS Toolbar -> type /Title, Title Name -> enter UCONN ANSYS Module Page

6 Key points Since we will be using D Elements, our goal is to model the length and width of the beam. 1. Go to Main Menu -> Preprocessor -> Modeling -> Create -> Keypoints -> On Working Plane. Click Global Cartesian. In the box underneath, write: 0,0,0 This will create a keypoint at the Origin. 4. Click Apply. Repeat Steps and 4 for the following points in order: 0,0,10 00,0,10 00,0,0. Click Ok 7. The Triad in the top left corner is blocking keypoint 1. To get rid of the triad, type /triad,off in Utility Menu -> Command Prompt 7 8. Go to Utility Menu -> Plot -> Replot Areas 1. Go to Main Menu -> Preprocessor -> Modeling -> Create -> Areas -> Arbitrary -> Through KPs. Select Pick. Select Min, Max, Inc 4. In the space below, type 1,4,1. If you are familiar with programming, the Min,Max, Inc acts as a FOR loop, connecting nodes 1 through 4 incrementing (Inc) by 1.. Click OK. Go to Plot -> Areas 7. Click the Top View tool 4 UCONN ANSYS Module Page

7 Your beam should look as below: Saving Geometry We will be using the geometry we have just created for modules. Thus it would be convenient to save the geometry so that it does not have to be made again from scratch. 1. Go to File -> Save As. Under Save Database to pick a name for the Geometry. For this tutorial, we will name the file Buckling simply supported. Under Directories: pick the Folder you would like to save the.db file to. 4. Click OK 4 Preprocessor Element Type 1. Go to Main Menu -> Preprocessor -> Element Type -> Add/Edit/Delete. Click Add. Click Shell -> 4node Click OK 4 UCONN ANSYS Module Page 7

8 SHELL181 is suitable for analyzing thin to moderately-think shell structures. It is a 4-node element with six degrees of freedom at each node: translations in the x, y, and z directions, and rotations about the x, y, and z-axes. (If the membrane option is used, the element has translational degrees of freedom only). The degenerate triangular option should only be used as filler elements in mesh generation. This element is well-suited for linear, large rotation, and/or large strain nonlinear applications. Change in shell thickness is accounted for in nonlinear analyses. In the element domain, both full and reduced integration schemes are supported. SHELL181 accounts for follower (load stiffness) effects of distributed pressures Real Constants and Material Properties Now we will add the thickness to our beam. 1. Go to Main Menu -> Preprocessor -> Real Constants -> Add/Edit/Delete. Click Add. Click OK 4. Under Real Constants for SHELL18 -> Shell thickness at node I TK(I) enter 1 for the thickness. Click OK. Click Close 4 UCONN ANSYS Module Page 8

9 Now we must specify Young s Modulus and Poisson s Ratio 1. Go to Main Menu -> Preprocessor -> Material Props -> Material Models. Go to Material Model Number 1 -> Structural -> Linear -> Elastic -> Isotropic. Input E for the Young s Modulus (Steel in mm) in EX. 4. Input 0. for Poisson s Ratio in PRXY. Click OK. of Define Material Model Behavior window 4 Meshing 1. Go to Main Menu -> Preprocessor -> Meshing -> Mesh Tool. Go to Size Controls: -> Global -> Set. Under NDIV No. of element divisions put. This will create a mesh of elements across the smallest thickness, 4 in total. 4. Click OK. Click Mesh. Click Pick All 4 UCONN ANSYS Module Page 9

10 7. Go to Utility Menu -> Plot -> Nodes 8. Go to Utility Menu -> Plot Controls -> Numbering 9. Check NODE Node Numbers to ON 10. Click OK 9 10 The resulting graphic should be as shown using the Top View : This is one of the main advantages of ANSYS Mechanical APDL vs ANSYS Workbench in that we can visually extract the node numbering scheme. As shown, ANSYS numbers nodes at the left corner, the right corner, followed by filling in the remaining nodes from left to right. UCONN ANSYS Module Page 10

11 Solution There are two types of solution menus that ANSYS APDL provides; the Abridged solution menu and the Unabridged solution menu. Before specifying the loads on the beam, it is crucial to be in the correct menu. Go to Main Menu -> Solution -> Unabridged menu This is shown as the last tab in the Solution menu. If this reads Abridged menu you are already in the Unabridged solution menu. Static Solution Analysis Type 1. Go to Main Menu -> Solution -> Analysis Type -> New Analysis. Choose Static. Click OK 4. Go to Main Menu -> Solution -> Analysis Type ->Analysis Options. Under [SSTIF][PSTRES] Stress stiffness or prestress select Prestress ON. Click OK Prestress is the only change necessary in this window and it is a crucial step in obtaining a final result for eigenvalue buckling. UCONN ANSYS Module Page 11

12 Displacement 1. Go to Main Menu -> Solution ->Define Loads ->Apply -> Structural ->Displacement -> On Nodes. Select Min, Max, Inc. In the space below, type 1,,1. If you are familiar with programming, the Min,Max, Inc acts as a FOR loop, connecting nodes 1 through incrementing by 1. This selects the nodes on the far left of the beam 9 4. Click OK. Under Lab DOFs to be constrained select UX, UY and ROTX. Under VALUE Displacement value enter Click OK 8. Go to Main Menu -> Solution -> Define Loads ->Apply ->Structural -> Displacement -> On Nodes 9. Select Pick -> Box 10. Box nodes: 4, 4 and 44 (far right nodes) 11. Click OK 1. Under Lab DOFs to be constrained highlight UY and ROTX 1. Under VALUE Displacement value enter Click OK 1. Go to Main Menu -> Solution -> Define Loads ->Apply ->Structural -> Displacement -> On Nodes 1. Select Pick -> Single 17. Select nodes and 4 (one node on the left of the beam and one node on the right) 18. Under Lab DOFs to be constrained highlight UZ 19. Under VALUE Displacement value enter 0 UCONN ANSYS Module Page 1

13 0. Click OK This creates the fixed end on the left and roller support on the right, while constraining movement in the Z-direction WARNING: UX, UY and ROTX might already be highlighted, if so, leave UY and ROTX highlighted and click UX to remove it from the selection. Failure to only constrain UY and ROTX will result in incorrect results. Loads 1. Go to Main Menu -> Solution -> Define Loads -> Apply -> Structural ->Force/Moment -> On Nodes. Select Pick -> Single -> List of Items. In the space provided, type 4. Press OK. 4. Under Direction of force/mom select FX. Under VALUE Force/moment value enter Click OK USEFUL TIP: The force value is only a magnitude of 1 because eigenvalues are calculated by a factor of the load applied, so having a force of 1 will make the eigenvalue answer equal to the critical load. UCONN ANSYS Module Page 1

14 Solve 1. Go to Main Menu -> Solution -> Solve -> Current LS. Go to Main Menu -> Finish Eigenvalue 1. Go to Main Menu -> Solution -> Analysis Type -> New Analysis. Choose Eigen Buckling. Click OKGo to Main Menu -> Solution -> Analysis Type ->Analysis Options 4. Under NMODE No. of modes to extract input 4. Click OK. Go to Main Menu -> Solution -> Solve -> Current LS 7. Go to Main Menu -> Finish UCONN ANSYS Module Page 14

15 Mode Shape 1. Go to Main Menu -> Solution -> Analysis Type -> ExpansionPass. Click [EXPASS] Expansion pass to ensure this is turned on. Click OK 4. Go to Main Menu -> Solution -> Load Step Opts -> ExpansionPass -> Single Expand -> Expand Modes. Under NMODE No. of modes to expand input 4. Click OK 7. Go to Main Menu -> Solution -> Solve -> Current LS 8. Go to Main Menu -> Finish UCONN ANSYS Module Page 1

16 General Postprocessor Buckling Load Now that ANSYS has solved these three analysis lets extract the lowest eigenvalue. This represents the lowest force to cause buckling. Go to Main Menu -> General Postproc -> List Results -> Detailed Summary Results for Buckling Load: P= N Mode Shape To view the deformed shape of the buckled beam vs. original beam: 1. Go to Main Menu -> General Postproc -> Read Results -> By Pick. Select the lowest Eigenvalue: Set 1 -> Click Read UCONN ANSYS Module Page 1

17 . Click Close 4. Go to Main Menu -> General Postproc -> Plot Results -> Deformed Shape. Under KUND Items to be plotted select Def + undeformed. Click OK The graphics area should look as below using the Oblique view: UCONN ANSYS Module Page 17

18 Results The percent error (%E) in our model can be defined as: ( ) = 0.1% This shows that there is no error baseline using one dimensional elements. Validation Critical Buckling Load Theoretical 0 Elements 80 Elements 4 Elements N Percent Error 0%.048% 0.1%.4% This table provides the critical buckling loads and corresponding error from the Theory (Euler), and two different ANSYS results; one with 80 elements and one with 4 elements. This is to prove mesh independence, showing with increasing mesh size, the answer approaches the theoretical value. The results here show that using a coarse mesh of 4 elements creates an unacceptable error. As mesh is refined it converges to a more accurate answer. The eigenvalue buckling method over-estimates the real life buckling load. This is due to the assumption of a perfect structure, disregarding flaws and nonlinearities in the material. There is no such thing as a perfect structure so the structure will never actually reach the eigenvalue load that is calculated. Thus, it is important to consider conservative factors of safety into your design for safe measure. UCONN ANSYS Module Page 18