Module 3: Buckling of 1D Simply Supported Beam

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1 Module : Buckling of 1D 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 15 General Postprocessor 16 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 In this module, we model a simply supported steel beam with compressive loads. This module stresses the importance of buckling in designs involving linear compressive loading and serves as a starting point for later modules which deal with more complex buckling considerations. 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. Theory Buckling load Hooke s Law equates stress as shown: (.1) Deriving both sides of equation.1 it shows (.) By solving for equilibrium: (.) Equation. is a nonlinear equation, however this equation can be linearized using eigenvalues. UCONN ANSYS Module Page

3 Since: (.4) Then: Plugging in Equation.1 for stress we find: (.5) (.6) Plugging Equation.6 into Equation., Equation.6 becomes (.7) Which is simplifies to: (.8) By integrating two times Equation.8 becomes (.9) At the fixed end (x=0), v=0,, thus 0 At the supported end (x=l), v=0,, thus 0 Equation.9 becomes (.10) Equation.6 represents the Differential Equation for a Sin Wave (.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 (.1) (.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 (.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 (.15) (.16) (.17) = N (.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 5

Key points Since we will be using 1D Elements, our goal is to model the length of the beam. 1. Go to Main Menu -> Preprocessor -> Modeling -> Create -> Keypoints -> On Working Plane.

Repeat Steps and 4 for 00,0,0 6. Click Ok 7. The Triad in the top left corner is blocking keypoint 1.

6 Key points Since we will be using 1D Elements, our goal is to model the length 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 key point at the origin. 4. Click Apply 5. Repeat Steps and 4 for 00,0,0 6. 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 6 7 Line 8. Go to Utility Menu -> Plot -> Replot 1. Go to Main Menu -> Preprocessor -> Modeling -> Create -> Lines -> Lines -> Straight Line. Select Pick. Select List of Items 4. Type 1, for points previously generated. 5. Click Ok 5 4 UCONN ANSYS Module Page 6

Under Save Database to pick a name for the Geometry. For this tutorial, we will name the file Buckling simply supported.

7 The resulting graphic should be as shown: 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 Beam -> D Elastic 4. Click OK 4 UCONN ANSYS Module Page 7

Beam is a uniaxial element with tension, compression, and bending capabilities.

rotation about the nodal z-axis Real Constants and Material Properties Now we will dimension our beam. 1.

Click OK 5. Under Cross-sectional area AREA enter 10 6. Under Area moment of inertia IZZ Enter 10/1 7.

8 Beam is a uniaxial element with tension, compression, and bending capabilities. The element has three degrees of freedom at each node: translations in the nodal x and y directions and rotation about the nodal z-axis Real Constants and Material Properties Now we will dimension our beam. 1. Go to Main Menu -> Preprocessor -> Real Constants -> Add/Edit/Delete. Click Add. Choose Type 1 Beam 4. Click OK 5. Under Cross-sectional area AREA enter Under Area moment of inertia IZZ Enter 10/1 7. Under Total beam height HEIGHT enter Click OK 9. Close out of the Real Constants window UCONN ANSYS Module Page 8

Now we must specify Young s Modulus and Poisson s Ratio 1.

for Poisson s Ratio in PRXY 5. Click OK 6.

Go to Main Menu -> Preprocessor -> Meshing -> Mesh Tool.

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 E5 for the Young s Modulus (Steel) in EX. 4. Input 0. for Poisson s Ratio in PRXY 5. Click OK 6. of Define Material Model Behavior window Meshing 1. Go to Main Menu -> Preprocessor -> Meshing -> Mesh Tool. Go to Size Controls: -> Global -> Set. Under NDIV No. of element divisions put 10. This will create a mesh of a total 10 elements 4. Click OK 5. Click Mesh 6. Click Pick All 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: ANSYS numbers nodes from the left extreme to the right extreme and then numbers from left to right. UCONN ANSYS Module Page 10

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.

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 5. Under [SSTIF][PSTRES] Stress stiffness or prestress select Prestress ON 6. 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. 6 5 UCONN ANSYS Module Page 11

Go to Main Menu -> Solution -> Define Loads ->Apply ->Structural -> Displacement -> On Nodes 8. Select Pick -> Single -> and click node 9. Click OK 10.

12 Displacement 1. Go to Main Menu -> Solution -> Define Loads ->Apply ->Structural -> Displacement -> On Nodes. Select Pick -> Single -> and click node 1. Click OK 4. Under Lab DOFs to be constrained select UX and UY 5. Under VALUE Displacement value enter 0 6. Click OK Go to Main Menu -> Solution -> Define Loads ->Apply ->Structural -> Displacement -> On Nodes 8. Select Pick -> Single -> and click node 9. Click OK 10. Under Lab DOFs to be constrained select only UY 11. Under VALUE Displacement value enter 0 1. Click OK WARNING: UX and UY might already be highlighted, if so, leave UY highlighted and click UX to remove it from the selection. Failure to only constrain UY will result in incorrect results. UCONN ANSYS Module Page 1

Click OK 4. Under Direction of force/mom select FX 5. Under VALUE Force/moment value enter -1 6.

13 The graphics area should look as below: Loads 1. Go to Main Menu -> Solution -> Define Loads ->Apply ->Structural -> Force/Moment -> On Nodes. Select Pick -> Single -> and click node. Click OK 4. Under Direction of force/mom select FX 5. Under VALUE Force/moment value enter Click OK The graphics area should look as below: 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 not skew the eigenvalue answer. UCONN ANSYS Module Page 1

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 OK 4.

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 OK 4. Go to Main Menu -> Solution -> Analysis Type ->Analysis Options 5. Under NMODE No. of modes to extract input 1 6. Click OK Go to Main Menu -> Solution -> Solve -> Current LS 8. Go to Main Menu -> Finish UCONN ANSYS Module Page 14

Go to Main Menu -> Solution -> Load Step Opts -> ExpansionPass -> Single Expand -> Expand Modes 5.

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 5. Under NMODE No. of modes to expand input 1 6. Click OK Go to Main Menu -> Solution -> Solve -> Current LS 8. Go to Main Menu -> Finish UCONN ANSYS Module Page 15

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.

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 -> First Set. Go to Main Menu -> General Postproc -> Plot Results -> Deformed Shape. Under KUND Items to be plotted select Def + undeformed 4. Click OK 4 UCONN ANSYS Module Page 16

17 The graphics area should look as below: UCONN ANSYS Module Page 17

18 Results The percent error (%E) in our model can be defined as: ( ) = 0% As one can see, eigenvalue problems are very accurate for one dimensional elements and are solved quickly by the solver. Validation Theoretical 10 Elements Elements Critical Buckling Load N Percent Error 0% 0% 0.751% This table provides the critical buckling loads and corresponding error from the Theory (Euler), and two different ANSYS results; one with elements and one with 10 elements. This is to prove mesh independence, showing with increasing mesh size, the answer approaches the theoretical value. The results here show that even using a coarse mesh of elements the error baseline is minimal. 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 beam so the structure will never actually reach the eigenvalue load that is calculated. UCONN ANSYS Module Page 18

Module 1.2: Moment of a 1D Cantilever Beam

Module 1.2: Moment of a 1D Cantilever Beam Module 1.: Moment of a 1D Cantilever Beam Table of Contents Page Number Problem Description Theory Geometry Preprocessor 6 Element Type 6 Real Constants and Material Properties 7 Meshing 9 Loads 10 Solution