1. 9-Beam Element with Axial Force e-mail: Dr. Ahmet Zafer Şenalp e-mail: azsenalp@gmail.comazsenalp@gmail.com Mechanical Engineering Department Gebze Technical University ME 520 Fundamentals of Finite Element Analysis

2. Apply the formula, To derive this, we introduce the shape functions: Then, we can represent the deflection as: which is a cubic function. Notice that, Formal Approach ME 520 Dr. Ahmet Zafer Şenalp 2Mechanical Engineering Department, GTU 9-Beam Element with Axial Force 9-Beam Element with Axial Force

3. which implies that the rigid body motion is represented by the assumed deformed shape of the beam. Curvature of the beam is, where the strain-displacement matrix B is given by, Formal Approach ME 520 Dr. Ahmet Zafer Şenalp 3Mechanical Engineering Department, GTU 9-Beam Element with Axial Force 9-Beam Element with Axial Force

4. Strain energy stored in the beam element is: We conclude that the stiffness matrix for the simple beam element is Formal Approach ME 520 Dr. Ahmet Zafer Şenalp 4Mechanical Engineering Department, GTU 9-Beam Element with Axial Force 9-Beam Element with Axial Force

5. Combining the axial stiffness (bar element), we obtain the stiffness matrix of a general 2-D beam element: Formal Approach ME 520 Dr. Ahmet Zafer Şenalp 5Mechanical Engineering Department, GTU 9-Beam Element with Axial Force 9-Beam Element with Axial Force

6. ME 520 Dr. Ahmet Zafer Şenalp 6Mechanical Engineering Department, GTU It is clear that the beam element has 6 degrees of freedom (3 at each node) For a structure with n nodes, the global stiffness matrix K will be of size 3nx3n. The global stiffness matrix K is obtained by making calls to the Matlab function BeamAssemble which is written for this purpose. Once the global stiffness matrix; K is obtained we have the following structure equation; where U is the global nodal displacement vector and F is the global nodal force vector. At this step boundary conditions are applied manually to the vectors U and F. Then the matrix equation is solved by partitioning and Gaussion elimination. Solution procedure with matlab Solution procedure with matlab 9-Beam Element with Axial Force 9-Beam Element with Axial Force

7. ME 520 Dr. Ahmet Zafer Şenalp 7Mechanical Engineering Department, GTU Finally once the unknown displacements and and reactions are found, the force is obtained for each element as follows: where f is the 6x1 nodal force vector in the element and u is the 6x1 element displacement vector. The first, second and third elements in each vector are the axial displacement, transverse displacement and rotation, respectively, at the first node, while the fourth fifth and sixth elements in each vector are the axial displacement, transverse displacement and rotation, respectively, at the second node. Solution procedure with matlab Solution procedure with matlab 9-Beam Element with Axial Force 9-Beam Element with Axial Force

8. Matlab functions used ME 520 Dr. Ahmet Zafer Şenalp 8Mechanical Engineering Department, GTU The 5 Matlab functions used for the beam element are: BeamElementStiffness_axial_az( E,A,I,L) This function returns the element stiffness matrix for a beam element with axial force with modulus of elasticity E, area A, moment of inertia I, and length L. The size of the element stiffness matrix is 6 x 6. Function contents: function y = BeamElementStiffness_axial_az(E,A,I,L) %BeamElementStiffness This function returns the element % stiffness matrix for a beam % element with axial force % with modulus of elasticity E, % with area A % moment of inertia I, and length L. % The size of the element stiffness % matrix is 6 x 6. c=E*A/L; c12=2*E*I/(L); c14=4*E*I/(L); c2=6*E*I/(L*L); c3=12*E*I/(L*L*L); y = [c 0 0 -c 0 0; 0 c3 c2 0 -c3 c2; 0 c2 c14 0 -c2 c12; -c 0 0 c 0 0; 0 -c3 -c2 0 c3 -c2; 0 c2 c12 0 -c2 c14]; 9-Beam Element with Axial Force 9-Beam Element with Axial Force

9. Matlab functions used ME 520 Dr. Ahmet Zafer Şenalp 9Mechanical Engineering Department, GTU BeamAssemble_axial(K,k,i,j) This function assembles the element stiffness matrix k of the beam element with nodes i and j into the global stiffness matrix K. This function returns the 3nx3n global stiffness matrix K after the element stiffness matrix k is assembled. Function contents: function y = BeamAssemble_axial(K,k,i,j) %BeamAssemble_axial This function assembles the element stiffness % matrix k of the beam element with nodes % i and j into the global stiffness matrix K. % This function returns the global stiffness % matrix K after the element stiffness matrix % k is assembled. K(3*i-2,3*i-2) = K(3*i-2,3*i-2) + k(1,1); K(3*i-2,3*i-1) = K(3*i-2,3*i-1) + k(1,2); K(3*i-2,3*i) = K(3*i-2,3*i) + k(1,3); K(3*i-2,3*j-2) = K(3*i-2,3*j-2) + k(1,4); K(3*i-2,3*j-1) = K(3*i-2,3*j-1) + k(1,5); K(3*i-2,3*j) = K(3*i-2,3*j) + k(1,6); K(3*i-1,3*i-2) = K(3*i-1,3*i-2) + k(2,1); K(3*i-1,3*i-1) = K(3*i-1,3*i-1) + k(2,2); K(3*i-1,3*i) = K(3*i-1,3*i) + k(2,3); K(3*i-1,3*j-2) = K(3*i-1,3*j-2) + k(2,4); K(3*i-1,3*j-1) = K(3*i-1,3*j-1) + k(2,5); K(3*i-1,3*j) = K(3*i-1,3*j) + k(2,6); 9-Beam Element with Axial Force 9-Beam Element with Axial Force

10. Matlab functions used ME 520 Dr. Ahmet Zafer Şenalp 10Mechanical Engineering Department, GTU K(3*i,3*i-2) = K(3*i,3*i-2) + k(3,1); K(3*i,3*i-1) = K(3*i,3*i-1) + k(3,2); K(3*i,3*i) = K(3*i,3*i) + k(3,3); K(3*i,3*j-2) = K(3*i,3*j-2) + k(3,4); K(3*i,3*j-1) = K(3*i,3*j-1) + k(3,5); K(3*i,3*j) = K(3*i,3*j) + k(3,6); K(3*j-2,3*i-2) = K(3*j-2,3*i-2) + k(4,1); K(3*j-2,3*i-1) = K(3*j-2,3*i-1) + k(4,2); K(3*j-2,3*i) = K(3*j-2,3*i) + k(4,3); K(3*j-2,3*j-2) = K(3*j-2,3*j-2) + k(4,4); K(3*j-2,3*j-1) = K(3*j-2,3*j-1) + k(4,5); K(3*j-2,3*j) = K(3*j-2,3*j) + k(4,6); K(3*j-1,3*i-2) = K(3*j-1,3*i-2) + k(5,1); K(3*j-1,3*i-1) = K(3*j-1,3*i-1) + k(5,2); K(3*j-1,3*i) = K(3*j-1,3*i) + k(5,3); K(3*j-1,3*j-2) = K(3*j-1,3*j-2) + k(5,4); K(3*j-1,3*j-1) = K(3*j-1,3*j-1) + k(5,5); K(3*j-1,3*j) = K(3*j-1,3*j) + k(5,6); K(3*j,3*i-2) = K(3*j,3*i-2) + k(6,1); K(3*j,3*i-1) = K(3*j,3*i-1) + k(6,2); K(3*j,3*i) = K(3*j,3*i) + k(6,3); K(3*j,3*j-2) = K(3*j,3*j-2) + k(6,4); K(3*j,3*j-1) = K(3*j,3*j-1) + k(6,5); K(3*j,3*j) = K(3*j,3*j) + k(6,6); y = K; 9-Beam Element with Axial Force 9-Beam Element with Axial Force

11. Matlab functions used ME 520 Dr. Ahmet Zafer Şenalp 11Mechanical Engineering Department, GTU BeamElementForces(k,u) This function calculates the element element force vector using the element stiffness matrix k and the element displacement vector u. It returns the 6x1 element force vector f Function contents: function y = BeamElementForces(k,u) %BeamElementForces This function returns the element nodal force % vector given the element stiffness matrix k % and the element nodal displacement vector u. y = k * u; 9-Beam Element with Axial Force 9-Beam Element with Axial Force

12. Matlab functions used ME 520 Dr. Ahmet Zafer Şenalp 12Mechanical Engineering Department, GTU BeamElementShearDiagram(f, L) This function plots the shear force diagram for the beam element with nodal force vector f and length L. Function contents: function y = BeamElementShearDiagram(f, L) %BeamElementShearDiagram This function plots the shear force % diagram for the beam element with nodal % force vector f and length L. x = [0 ; L]; z = [f(1) ; -f(3)]; hold on; title('Shear Force Diagram'); plot(x,z); y1 = [0 ; 0]; plot(x,y1,'k') 9-Beam Element with Axial Force 9-Beam Element with Axial Force

13. Matlab functions used ME 520 Dr. Ahmet Zafer Şenalp 13Mechanical Engineering Department, GTU BeamElementMomentDiagram(f, L) This function plots the bending moment diagram for the beam element with nodal force vector f and length L. Function contents: function y = BeamElementMomentDiagram(f, L) %BeamElementMomentDiagram This function plots the bending moment % diagram for the beam element with nodal % force vector f and length L. x = [0 ; L]; z = [-f(2) ; f(4)]; hold on; title('Bending Moment Diagram'); plot(x,z); y1 = [0 ; 0]; plot(x,y1,'k') 9-Beam Element with Axial Force 9-Beam Element with Axial Force

14. ME 520 Dr. Ahmet Zafer Şenalp 14Mechanical Engineering Department, GTU Consider the beam as shown Given E=210 GPa A=0.025 m 2 I=60x10 -6 m 4 P1=20 kN P2=30 kN L=2 m Determine: a)the global stiffness matrix for the structure b)horizontal and vertical displacements at node 2 c)rotations at nodes 2 and 3 d)the reactions at nodes 1 and 3 e)the forces (shears and moments) in each element f)the shear force diagram for each element g)the bending moment diagram for each element Solution of Example 1 with Matlab 9-Beam Element with Axial Force 9-Beam Element with Axial Force

15. ME 520 Dr. Ahmet Zafer Şenalp 15Mechanical Engineering Department, GTU Solution of Example 1 with Matlab Solution: Use the 7 steps to solve the problem using beam element. Step 1-Discretizing the domain: We will put a node (node2) at the location of the concentrated force so that we may determine the required quantities (displacements, rotation, shear, moment) at that point. The domain is subdivided into two elements and three nodes. The units used in Matlab calculations are kN and meter. The element connectivity is: E#N1N2 112 223 9-Beam Element with Axial Force 9-Beam Element with Axial Force

16. ME 520 Dr. Ahmet Zafer Şenalp 16Mechanical Engineering Department, GTU Solution of Example 1 with Matlab Step 2-Copying relevant files and starting Matlab Create a directory Copy BeamElementStiffness_axial_az.m BeamAssemble_axial.m BeamElementForces.m BeamElementShearDiagram.m BeamElementMomentDiagram.m files under the created directory Open Matlab; Open ‘Set Path’ command and by using ‘Add Folder’ command add the current directory. Start solving the problem in Command Window: >>clearvars >>clc 9-Beam Element with Axial Force 9-Beam Element with Axial Force

17. ME 520 Dr. Ahmet Zafer Şenalp 17Mechanical Engineering Department, GTU Solution of Example 1 with Matlab Step 3-Writing the element stiffness matrices: The two element stiffness matrices k 1 and k 2 are obtained by making calls to the Matlab function BeamElementStiffness_axial_az. Each matrix has size 6x6. Enter the data >>E=210e6 >>A=0.025 >>I=60e-6 >>L=2 >>k1=BeamElementStiffness_axial_az(E,A,I,L) k1 = 2625000 0 0 -2625000 0 0 0 18900 18900 0 -18900 18900 0 18900 25200 0 -18900 12600 -2625000 0 0 2625000 0 0 0 -18900 -18900 0 18900 -18900 0 18900 12600 0 -18900 25200 9-Beam Element with Axial Force 9-Beam Element with Axial Force

18. ME 520 Dr. Ahmet Zafer Şenalp 18Mechanical Engineering Department, GTU Solution of Example 1 with Matlab >>k2=BeamElementStiffness_axial_az(E,A,I,L) k2 = 2625000 0 0 -2625000 0 0 0 18900 18900 0 -18900 18900 0 18900 25200 0 -18900 12600 -2625000 0 0 2625000 0 0 0 -18900 -18900 0 18900 -18900 0 18900 12600 0 -18900 25200 Step 4-Assembling the global stiffness matrix: Since the structure has 3 nodes, the size of the global stiffness matrix is 9x9. >>K=zeros(9,9) >>K=BeamAssemble_axial(K,k1,1,2) >>K=BeamAssemble_axial(K,k2,2,3) K = 2625000 0 0 -2625000 0 0 0 0 0 0 18900 18900 0 -18900 18900 0 0 0 0 18900 25200 0 -18900 12600 0 0 0 -2625000 0 0 5250000 0 0 -2625000 0 0 0 -18900 -18900 0 37800 0 0 -18900 18900 0 18900 12600 0 0 50400 0 -18900 12600 0 0 0 -2625000 0 0 2625000 0 0 0 0 0 0 -18900 -18900 0 18900 -18900 0 0 0 0 18900 12600 0 -18900 25200 9-Beam Element with Axial Force 9-Beam Element with Axial Force

19. ME 520 Dr. Ahmet Zafer Şenalp 19Mechanical Engineering Department, GTU Solution of Example 1 with Matlab Step 5-Applying the boundary conditions: Finite element equation for the problem is; The boundary conditions for the problem are; 9-Beam Element with Axial Force 9-Beam Element with Axial Force

20. ME 520 Dr. Ahmet Zafer Şenalp 20Mechanical Engineering Department, GTU 9-Beam Element with Axial Force 9-Beam Element with Axial Force Solution of Example 1 with Matlab Inserting the above conditions into finite element equation Step 6-Solving the equations: Solving the above system of equations will be performed by partitioning (manually) and Gaussian elimination (with Matlab) First we partition the above equation by extracting the submatrices in rows 4 to 6 and columns 4 to 6 and row 9 column 9. Therefore we obtain:

21. ME 520 Dr. Ahmet Zafer Şenalp 21Mechanical Engineering Department, GTU 9-Beam Element with Axial Force 9-Beam Element with Axial Force Solution of Example 1 with Matlab The solution of the above system is obtained using Matlab as follows. Note that the ‘\’ operator is used for Gaussian elimination. >>k=[K(4:6,4:6) K(4:6,9) ; K(9,4:6) K(9,9)] k = 5250000 0 0 0 0 37800 0 18900 0 0 50400 12600 0 18900 12600 25200 >>f=[30; -20 ; 0 ; 0] f = -20 0

22. ME 520 Dr. Ahmet Zafer Şenalp 22Mechanical Engineering Department, GTU 9-Beam Element with Axial Force 9-Beam Element with Axial Force Solution of Example 1 with Matlab >>u=k\f u = 1.0e-03 * 0.0057 -0.9259 -0.1984 0.7937 Step 7-Post-processing: In this step we obtain the reactions at nodes 1 and 3 and the forces (shears and moments) in each beam element using Matlab as follows. First we set up the global nodal displacement vector U, then we calculate the nodal force vector F.

23. ME 520 Dr. Ahmet Zafer Şenalp 23Mechanical Engineering Department, GTU 9-Beam Element with Axial Force 9-Beam Element with Axial Force Solution of Example 1 with Matlab >>U=[0 ; 0 ; 0; u(1) ; u(2) ; u(3); 0; 0; u(4)] U = 1.0e-03 * 0 0.0057 -0.9259 -0.1984 0 0.7937 >>F=K*U F = -15.0000 13.7500 15.0000 30.0000 -20.0000 0 -15.0000 6.2500 -0.0000

24. ME 520 Dr. Ahmet Zafer Şenalp 24Mechanical Engineering Department, GTU 9-Beam Element with Axial Force 9-Beam Element with Axial Force Solution of Example 1 with Matlab Next we set up the element nodal displacement vectors u 1 and u 2 then we calculate the element force vectors f 1 and f 2 by making calls to the Matlab function BeamElementForces. >> u1=[U(1) ; U(2) ; U(3) ; U(4) ; U(5) ; U(6)] u1 = 1.0e-03 * 0 0.0057 -0.9259 -0.1984

25. ME 520 Dr. Ahmet Zafer Şenalp 25Mechanical Engineering Department, GTU 9-Beam Element with Axial Force 9-Beam Element with Axial Force Solution of Example 1 with Matlab >> u2=[U(4) ; U(5) ; U(6) ; U(7) ; U(8) ; U(9)] u2 = 1.0e-03 * 0.0057 -0.9259 -0.1984 0 0.7937 >>f1 =BeamElementForces(k1,u1) f1 = -15.0000 13.7500 15.0000 -13.7500 12.5000

26. ME 520 Dr. Ahmet Zafer Şenalp 26Mechanical Engineering Department, GTU 9-Beam Element with Axial Force 9-Beam Element with Axial Force Solution of Example 1 with Matlab >>f2 =BeamElementForces(k2,u2) f2 = 15.0000 -6.2500 -12.5000 -15.0000 6.2500 -0.0000 Finally we call the Matlab functions BeamElementShearDiagram and BeamElementMomentDiagram, respectively for each element.

27. ME 520 Dr. Ahmet Zafer Şenalp 27Mechanical Engineering Department, GTU 9-Beam Element with Axial Force 9-Beam Element with Axial Force Solution of Example 1 with Matlab >>BeamElementShearDiagram(f1,L)

28. ME 520 Dr. Ahmet Zafer Şenalp 28Mechanical Engineering Department, GTU 9-Beam Element with Axial Force 9-Beam Element with Axial Force Solution of Example 1 with Matlab >>BeamElementShearDiagram(f2,L)

29. ME 520 Dr. Ahmet Zafer Şenalp 29Mechanical Engineering Department, GTU 9-Beam Element with Axial Force 9-Beam Element with Axial Force Solution of Example 1 with Matlab >>BeamElementMomentDiagram(f1, L)

30. ME 520 Dr. Ahmet Zafer Şenalp 30Mechanical Engineering Department, GTU 9-Beam Element with Axial Force 9-Beam Element with Axial Force Solution of Example 1 with Matlab >>BeamElementMomentDiagram(f2, L)