1. 10-Beam Elements in 2-D Space (Plane Frame Element) 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. General assumptions: Members in a frame are considered to be rigidly connected. Both forces and moments can be transmitted through their joints. We need the general beam element (combinations of bar and simple beam elements) to model frames FE Analysis of Frame Structures ME 520 Dr. Ahmet Zafer Şenalp 2Mechanical Engineering Department, GTU 10-Beam Elements in 2-D Space (Plane Frame Element) (Plane Frame Element)

3. The plane frame element is a 2 dimensional finite element with both local and global coordinates. The plane frame element has modulus of elasticity; E moment of inertia; I cross-sectional area; A Length; L Each plane frame element has 2 nodes and is inclined with an angle  measured counterclockwise from the positive global X axis as shown in figure below. Let In this case the element stiffness matrix is given by the following matrix including axial deformation. FE Analysis of Frame Structures ME 520 Dr. Ahmet Zafer Şenalp 3Mechanical Engineering Department, GTU 10-Beam Elements in 2-D Space (Plane Frame Element) (Plane Frame Element)

4. FE Analysis of Frame Structures ME 520 Dr. Ahmet Zafer Şenalp 4Mechanical Engineering Department, GTU 10-Beam Elements in 2-D Space (Plane Frame Element) (Plane Frame Element)

5. It is clear that the plane frame element has 6 degrees of freedom (3 at each node) (2 displacements and 1 rotation). For a structure with n nodes, the global stiffness matrix K will be of size 3nx3n. The global stiffness matrix K is assembled by making calls to the Matlab function PlaneFrameAssemble which is written specifically for this purpose. Once the global stiffness matrix K is obtained we have the following structure equation: where U: global nodal displacement vector F: global nodal force vector At this step the boundary conditions are applied manually to the vectors U and F. Then the above matrix equation is solved by partitioning and Gaussian elimination. Solution procedure with matlab ME 520 Dr. Ahmet Zafer Şenalp 5Mechanical Engineering Department, GTU 10-Beam Elements in 2-D Space (Plane Frame Element) (Plane Frame Element)

6. Finally once the unknown displacements and reactions are found, the nodal force vector is obtained for each element as follows: Solution procedure with matlab ME 520 Dr. Ahmet Zafer Şenalp 6Mechanical Engineering Department, GTU 10-Beam Elements in 2-D Space (Plane Frame Element) (Plane Frame Element) is the 6x1 nodal force vector is the 6x1 element displacement vector

7. The first and second elements in each vector are the two displacements while the third element is the rotation, respectively, at the first node, while the fourth and the fifth elements in each vector are the two displacements while the sixth element is the rotation, respectively, at the second node. If there is an inclined support at one of the nodes of the plane frame then the global stiffness matrix needs to be modified using the following equation: where : 3n x 3n transformation matrix that is obtained by making a call to the Matlab function PlaneFrameInclinedSupport. The inclined support in a plane frame is handled in the same way it is handled in a plane truss except that the size of the transformation matrix is now 3n x 3n instead of 2n x 2n. The inclined support is assumed to be at node i with an angle of inclination alpha as shown below. Solution procedure with matlab ME 520 Dr. Ahmet Zafer Şenalp 7Mechanical Engineering Department, GTU 10-Beam Elements in 2-D Space (Plane Frame Element) (Plane Frame Element)

8. Matlab functions used ME 520 Dr. Ahmet Zafer Şenalp 8Mechanical Engineering Department, GTU The 8 Matlab functions used for the plane frame element are: PlaneFrameElementLength(x1,y1,x2,y2) This function returns the length of the plane frame element whose first node has coordinates (x1,y1) and second node has coordinates (x2,y2). Function contents: function y = PlaneFrameElementLength(x1,y1,x2,y2) %PlaneFrameElementLength This function returns the length of the % plane frame element whose first node has % coordinates (x1,y1) and second node has % coordinates (x2,y2). y = sqrt((x2-x1)*(x2-x1) + (y2-y1)*(y2-y1)); 10-Beam Elements in 2-D Space (Plane Frame Element) (Plane Frame Element)

9. Matlab functions used ME 520 Dr. Ahmet Zafer Şenalp 9Mechanical Engineering Department, GTU PlaneFrameElementStiffness(E,A,I,L,theta) This function calculates the element stiffness matrix for each plane frame element with modulus of elasticity E, cross-sectional area A, moment of inertia I, length L, and angle theta (in degrees). The size of the element stiffness matrix is 6 x 6. Function contents: function y = PlaneFrameElementStiffness(E,A,I,L,theta) %PlaneFrameElementStiffness This function returns the element % stiffness matrix for a plane frame % element with modulus of elasticity E, % cross-sectional area A, moment of % inertia I, length L, and angle % theta (in degrees). % The size of the element stiffness % matrix is 6 x 6. x = theta*pi/180; C = cos(x); S = sin(x); w1 = A*C*C + 12*I*S*S/(L*L); w2 = A*S*S + 12*I*C*C/(L*L); w3 = (A-12*I/(L*L))*C*S; w4 = 6*I*S/L; w5 = 6*I*C/L; y = E/L*[w1 w3 -w4 -w1 -w3 -w4 ; w3 w2 w5 -w3 -w2 w5 ; -w4 w5 4*I w4 -w5 2*I ; -w1 -w3 w4 w1 w3 w4 ; -w3 -w2 -w5 w3 w2 -w5 ; -w4 w5 2*I w4 -w5 4*I]; 10-Beam Elements in 2-D Space (Plane Frame Element) (Plane Frame Element)

10. Matlab functions used ME 520 Dr. Ahmet Zafer Şenalp 10Mechanical Engineering Department, GTU PlaneFrameAssemble(K,k,i,j) This function assembles the element stiffness matrix k of the plane frame element joining nodes i and j into the global stiffness matrix K. It returns the 3n x 3n global stiffness matrix K every time an element stiffness matrix k is assembled. Function contents: function y = PlaneFrameAssemble(K,k,i,j) %PlaneFrameAssemble This function assembles the element stiffness % matrix k of the plane frame 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); 10-Beam Elements in 2-D Space (Plane Frame Element) (Plane Frame Element)

11. Matlab functions used ME 520 Dr. Ahmet Zafer Şenalp 11Mechanical 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; 10-Beam Elements in 2-D Space (Plane Frame Element) (Plane Frame Element)

12. Matlab functions used ME 520 Dr. Ahmet Zafer Şenalp 12Mechanical Engineering Department, GTU PlaneFrameElementForces(E,A,I,L,theta,u) This function calculates the element force vector using the modulus of elasticity E, the cross-sectional area A, the moment of inertia I, the length L, the angle theta (in degrees), and the element nodal displacement vector u. It returns 6x1 element force vector f. Function contents: function y = PlaneFrameElementForces(E,A,I,L,theta,u) %PlaneFrameElementForces This function returns the element force % vector given the modulus of elasticity E, % the cross-sectional area A, the moment of % inertia I, the length L, the angle theta % (in degrees), and the element nodal % displacement vector u. x = theta * pi/180; C = cos(x); S = sin(x); w1 = E*A/L; w2 = 12*E*I/(L*L*L); w3 = 6*E*I/(L*L); w4 = 4*E*I/L; w5 = 2*E*I/L; kprime = [w1 0 0 -w1 0 0 ; 0 w2 w3 0 -w2 w3 ; 0 w3 w4 0 -w3 w5 ; -w1 0 0 w1 0 0 ; 0 -w2 -w3 0 w2 -w3 ; 0 w3 w5 0 -w3 w4]; T = [C S 0 0 0 0 ; -S C 0 0 0 0 ; 0 0 1 0 0 0 ; 0 0 0 C S 0 ; 0 0 0 -S C 0 ; 0 0 0 0 0 1]; y = kprime*T* u; 10-Beam Elements in 2-D Space (Plane Frame Element) (Plane Frame Element)

13. Matlab functions used ME 520 Dr. Ahmet Zafer Şenalp 13Mechanical Engineering Department, GTU PlaneFrameElementAxialDiagram(f, L) This function plots the axial force diagram for the plane frame element with nodal force vector f and length L. Function contents: function y = PlaneFrameElementAxialDiagram(f, L) %PlaneFrameElementAxialDiagram This function plots the axial force % diagram for the plane frame element % with nodal force vector f and length % L. x = [0 ; L]; z = [-f(1) ; f(4)]; hold on; title('Axial Force Diagram'); plot(x,z); y1 = [0 ; 0]; plot(x,y1,'k') 10-Beam Elements in 2-D Space (Plane Frame Element) (Plane Frame Element)

14. Matlab functions used ME 520 Dr. Ahmet Zafer Şenalp 14Mechanical Engineering Department, GTU PlaneFrameElementShearDiagram(f, L) This function plots the shear force diagram for the plane frame element with nodal force vector f and length L. Function contents: function y = PlaneFrameElementShearDiagram(f, L) %PlaneFrameElementShearDiagram This function plots the shear force % diagram for the plane frame element % with nodal force vector f and length % L. x = [0 ; L]; z = [f(2) ; -f(5)]; hold on; title('Shear Force Diagram'); plot(x,z); y1 = [0 ; 0]; plot(x,y1,'k') 10-Beam Elements in 2-D Space (Plane Frame Element) (Plane Frame Element)

15. Matlab functions used ME 520 Dr. Ahmet Zafer Şenalp 15Mechanical Engineering Department, GTU PlaneFrameElementMomentDiagram(f, L) This function plots the bending moment diagram for the plane frame element with nodal force vector f and length L. Function contents: function y = PlaneFrameElementMomentDiagram(f, L) %PlaneFrameElementMomentDiagram This function plots the bending % moment diagram for the plane frame % element with nodal force vector f % and length L. x = [0 ; L]; z = [-f(3) ; f(6)]; hold on; title('Bending Moment Diagram'); plot(x,z); y1 = [0 ; 0]; plot(x,y1,'k') 10-Beam Elements in 2-D Space (Plane Frame Element) (Plane Frame Element)

16. Matlab functions used ME 520 Dr. Ahmet Zafer Şenalp 16Mechanical Engineering Department, GTU PlaneFrameInclinedSupport(T,i,alpha) This function calculates the transformation matrix T of the inclined support at node i with angle of inclination alpha (in degrees). Function contents: function y = PlaneFrameInclinedSupport(T,i,alpha) %PlaneFrameInclinedSupport This function calculates the % transformation matrix T of the inclined % support at node i with angle of % inclination alpha (in degrees). x = alpha*pi/180; T(3*i-2,3*i-2) = cos(x); T(3*i-2,3*i-1) = sin(x); T(3*i-2,3*i) = 0; T(3*i-1,3*i-2) = -sin(x); T(3*i-1,3*i-1) = cos(x); T(3*i-1,3*i) = 0; T(3*i,3*i-2) = 0; T(3*i,3*i-1) = 0; T(3*i,3*i) = 1; y = T; 10-Beam Elements in 2-D Space (Plane Frame Element) (Plane Frame Element)

17. Solution of Example 1 with Matlab ME 520 Dr. Ahmet Zafer Şenalp 17Mechanical Engineering Department, GTU 10-Beam Elements in 2-D Space (Plane Frame Element) (Plane Frame Element) Given; E=210 GPa, A=2x10 -2 m 2, I=5x10 -5 m 4 Find; (a)the global stiffness matrix for the structure (b)the displacements and rotations at nodes 2 and 3 (c)the reactions at nodes 1 and 4 (d)the axial force, shear force and bending moment in each element (e)the axial force diagram for each element (f)the shear force diagram for each element (g)the bending moment diagram for each element

18. ME 520 Dr. Ahmet Zafer Şenalp 18Mechanical Engineering Department, GTU Solution of Example 1 with Matlab Solution: Use the 7 steps to solve the problem using plane frame element. Step 1-Discretizing the domain: This problem is already discretized. The domain is subdivided into 3 elements and 4 nodes. E#N1N2 112 223 334 10-Beam Elements in 2-D Space (Plane Frame Element)

19. ME 520 Dr. Ahmet Zafer Şenalp 19Mechanical Engineering Department, GTU Solution of Example 1 with Matlab Step 2-Copying relevant files and starting Matlab Create a directory Copy PlaneFrameAssemble.m PlaneFrameElementAxialDiagram.m PlaneFrameElementForces.m PlaneFrameElementLength.m PlaneFrameElementMomentDiagram.m PlaneFrameElementShearDiagram.m PlaneFrameElementStiffness.m PlaneFrameInclinedSupport.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 10-Beam Elements in 2-D Space (Plane Frame Element)

20. ME 520 Dr. Ahmet Zafer Şenalp 20Mechanical Engineering Department, GTU Solution of Example 1 with Matlab Step 3-Writing the element stiffness matrices: The three element stiffness matrices k 1, k 2 and k 3 are obtained by making calls to the Matlab function PlaneFrameElementStiffness. Each matrix has size 6x6. Enter the data >>E=210e6 >>A=2e-2 >>I=5e-5 >>L1=3 >>L2=4 >>L3=3 >>k1= PlaneFrameElementStiffness(E,A,I,L1,90) k1 = 1.0e+06 * 0.0047 0.0000 -0.0070 -0.0047 -0.0000 -0.0070 0.0000 1.4000 0.0000 -0.0000 -1.4000 0.0000 -0.0070 0.0000 0.0140 0.0070 -0.0000 0.0070 -0.0047 -0.0000 0.0070 0.0047 0.0000 0.0070 -0.0000 -1.4000 -0.0000 0.0000 1.4000 -0.0000 -0.0070 0.0000 0.0070 0.0070 -0.0000 0.0140 10-Beam Elements in 2-D Space (Plane Frame Element)

21. ME 520 Dr. Ahmet Zafer Şenalp 21Mechanical Engineering Department, GTU Solution of Example 1 with Matlab >>k2= PlaneFrameElementStiffness(E,A,I,L2,0) k2 = 1.0e+06 * 1.0500 0 0 -1.0500 0 0 0 0.0020 0.0039 0 -0.0020 0.0039 0 0.0039 0.0105 0 -0.0039 0.0053 -1.0500 0 0 1.0500 0 0 0 -0.0020 -0.0039 0 0.0020 -0.0039 0 0.0039 0.0053 0 -0.0039 0.0105 >>k3= PlaneFrameElementStiffness(E,A,I,L3,270) k3 = 1.0e+06 * 0.0047 0.0000 0.0070 -0.0047 -0.0000 0.0070 0.0000 1.4000 -0.0000 -0.0000 -1.4000 -0.0000 0.0070 -0.0000 0.0140 -0.0070 0.0000 0.0070 -0.0047 -0.0000 -0.0070 0.0047 0.0000 -0.0070 -0.0000 -1.4000 0.0000 0.0000 1.4000 0.0000 0.0070 -0.0000 0.0070 -0.0070 0.0000 0.0140 10-Beam Elements in 2-D Space (Plane Frame Element)

22. ME 520 Dr. Ahmet Zafer Şenalp 22Mechanical Engineering Department, GTU Solution of Example 1 with Matlab Step 4-Assembling the global stiffness matrix: Since the structure has 4 nodes, the size of the global stiffness matrix is 12x12. >>K=zeros(12,12) >>K=PlaneFrameAssemble(K,k1,1,2) >>K=PlaneFrameAssemble(K,k2,2,3) >>K=PlaneFrameAssemble(K,k3,3,4) K = 1.0e+06 * 0.0047 0.0000 -0.0070 -0.0047 -0.0000 -0.0070 0 0 0 0 0 0 0.0000 1.4000 0.0000 -0.0000 -1.4000 0.0000 0 0 0 0 0 0 -0.0070 0.0000 0.0140 0.0070 -0.0000 0.0070 0 0 0 0 0 0 -0.0047 -0.0000 0.0070 1.0547 0.0000 0.0070 -1.0500 0 0 0 0 0 -0.0000 -1.4000 -0.0000 0.0000 1.4020 0.0039 0 -0.0020 0.0039 0 0 0 -0.0070 0.0000 0.0070 0.0070 0.0039 0.0245 0 -0.0039 0.0053 0 0 0 0 0 0 -1.0500 0 0 1.0547 0.0000 0.0070 -0.0047 -0.0000 0.0070 0 0 0 0 -0.0020 -0.0039 0.0000 1.4020 -0.0039 -0.0000 -1.4000 -0.0000 0 0 0 0 0.0039 0.0053 0.0070 -0.0039 0.0245 -0.0070 0.0000 0.0070 0 0 0 0 0 0 -0.0047 -0.0000 -0.0070 0.0047 0.0000 -0.0070 0 0 0 0 0 0 -0.0000 -1.4000 0.0000 0.0000 1.4000 0.0000 0 0 0 0 0 0 0.0070 -0.0000 0.0070 -0.0070 0.0000 0.0140 10-Beam Elements in 2-D Space (Plane Frame Element)

23. ME 520 Dr. Ahmet Zafer Şenalp 23Mechanical Engineering Department, GTU Solution of Example 1 with Matlab Step 5-Applying the boundary conditions: Finite element equation for the problem is; 10-Beam Elements in 2-D Space (Plane Frame Element)

24. ME 520 Dr. Ahmet Zafer Şenalp 24Mechanical Engineering Department, GTU Solution of Example 1 with Matlab The boundary conditions for the problem are; 10-Beam Elements in 2-D Space (Plane Frame Element)

25. ME 520 Dr. Ahmet Zafer Şenalp 25Mechanical Engineering Department, GTU Solution of Example 1 with Matlab 10-Beam Elements in 2-D Space (Plane Frame Element)

26. ME 520 Dr. Ahmet Zafer Şenalp 26Mechanical Engineering Department, GTU Solution of Example 1 with Matlab 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 9 and columns 4 to 9 Therefore we obtain: >>k=K(4:9,4:9) k = 1.0e+06 * 1.0547 0.0000 0.0070 -1.0500 0 0 0.0000 1.4020 0.0039 0 -0.0020 0.0039 0.0070 0.0039 0.0245 0 -0.0039 0.0053 -1.0500 0 0 1.0547 0.0000 0.0070 0 -0.0020 -0.0039 0.0000 1.4020 -0.0039 0 0.0039 0.0053 0.0070 -0.0039 0.0245 10-Beam Elements in 2-D Space (Plane Frame Element)

27. ME 520 Dr. Ahmet Zafer Şenalp 27Mechanical Engineering Department, GTU Solution of Example 1 with Matlab >>f=[-20; 0 ; 0 ; 0; 0; 12] The solution of the system is obtained using Matlab as follows. Note that the ‘\’ operator is used for Gaussian elimination. >>u=k\f u = -0.0038 -0.0000 0.0008 -0.0038 0.0000 0.0014 10-Beam Elements in 2-D Space (Plane Frame Element)

28. ME 520 Dr. Ahmet Zafer Şenalp 28Mechanical Engineering Department, GTU Solution of Example 1 with Matlab Step 7-Post-processing: In this step we obtain the reactions at nodes 1 and 4 and the forces (axial forces, shears and moments) in each plane frame element using Matlab as follows. First we set up the global nodal displacement vector U, then we calculate the nodal force vector F. >>U=[0 ; 0 ; 0; u ; 0; 0; 0] U = 0 -0.0038 -0.0000 0.0008 -0.0038 0.0000 0.0014 0 10-Beam Elements in 2-D Space (Plane Frame Element)

29. ME 520 Dr. Ahmet Zafer Şenalp 29Mechanical Engineering Department, GTU Solution of Example 1 with Matlab >>F=K*U F = 12.1897 8.5865 -21.0253 -20.0000 -0.0000 0.0000 12.0000 7.8103 -8.5865 -16.6286 10-Beam Elements in 2-D Space (Plane Frame Element)

30. ME 520 Dr. Ahmet Zafer Şenalp 30Mechanical Engineering Department, GTU Solution of Example 1 with Matlab Thus the horizontal reactions at nodes 1 and 4 are forces of 12.1897 kN (to the right) and 7.8103 kN (to the right), respectively. The vertical reactions at nodes 1 and 4 are forces of 8.5865 kN(upward) and 8.5865 kN (downward), respectively. The moments at nodes 1 and 4 are moments of 21.0253 kNm (clockwise) and 16.6286 kNm (clockwise), respectively. It is clear that force equilibrium is satisfied. Next we set up the element nodal displacement vectors u 1 u 2 and u 3 then we calculate the element force vectors f 1 f 2 and f 3 by making calls to the Matlab function PlaneFrameElementForces. 10-Beam Elements in 2-D Space (Plane Frame Element)

31. ME 520 Dr. Ahmet Zafer Şenalp 31Mechanical Engineering Department, GTU Solution of Example 1 with Matlab >> u1=[U(1) ; U(2) ; U(3) ; U(4) ; U(5) ; U(6)] u1 = 0 -0.0038 -0.0000 0.0008 >> u2=[U(4) ; U(5) ; U(6) ; U(7) ; U(8) ; U(9)] u2 = -0.0038 -0.0000 0.0008 -0.0038 0.0000 0.0014 10-Beam Elements in 2-D Space (Plane Frame Element)

32. ME 520 Dr. Ahmet Zafer Şenalp 32Mechanical Engineering Department, GTU Solution of Example 1 with Matlab >> u3=[U(7) ; U(8) ; U(9) ; U(10) ; U(11) ; U(12)] u3 = -0.0038 0.0000 0.0014 0 >>f1=PlaneFrameElementForces(E,A,I,L1,90,u1) f1 = 8.5865 -12.1897 -21.0253 -8.5865 12.1897 -15.5438 10-Beam Elements in 2-D Space (Plane Frame Element)

33. ME 520 Dr. Ahmet Zafer Şenalp 33Mechanical Engineering Department, GTU Solution of Example 1 with Matlab >>f2=PlaneFrameElementForces(E,A,I,L2,0,u2) f2 = -7.8103 8.5865 15.5438 7.8103 -8.5865 18.8023 >>f3=PlaneFrameElementForces(E,A,I,L3,270,u3) f3 = -8.5865 -7.8103 -6.8023 8.5865 7.8103 -16.6286 10-Beam Elements in 2-D Space (Plane Frame Element)

34. ME 520 Dr. Ahmet Zafer Şenalp 34Mechanical Engineering Department, GTU Solution of Example 1 with Matlab The forces for each element are given above. Element 1 has an axial force of 8.5865 kN, shear force of -12.1897 kN bending moment of -21.0253 kNm at its left end while it has an axial force of -8.5865 kN, shear force of 12.1897 kN bending moment of -15.5438 kNm at its right end. Element 2 has an axial force of -7.8103 kN, shear force of 8.5865 kN bending moment of 15.5438 kNm at its left end while it has an axial force of 7.8103 kN, shear force of -8.5865 kN bending moment of 18.8023 kNm at its right end. 10-Beam Elements in 2-D Space (Plane Frame Element)

35. ME 520 Dr. Ahmet Zafer Şenalp 35Mechanical Engineering Department, GTU Solution of Example 1 with Matlab Element 3 has an axial force of -8.5865 kN, shear force of -7.8103 kN bending moment of -6.8023 kNm at its left end while it has an axial force of 8.5865 kN, shear force of 7.8103 kN bending moment of -16.6286 kNm at its right end. Finally we call the Matlab functions PlaneFrameElementAxialDiagram, PlaneFrameElementShearDiagram, PlaneFrameElementMomentDiagram to draw the axial force diagram, shear force diagram and bending moment diagram, respectively, for each element. This process is illustrated below. 10-Beam Elements in 2-D Space (Plane Frame Element)

36. ME 520 Dr. Ahmet Zafer Şenalp 36Mechanical Engineering Department, GTU Solution of Example 1 with Matlab Element 3 has an axial force of -8.5865 kN, shear force of -7.8103 kN bending moment of -6.8023 kNm at its left end while it has an axial force of 8.5865 kN, shear force of 7.8103 kN bending moment of -16.6286 kNm at its right end. Finally we call the Matlab functions PlaneFrameElementAxialDiagram, PlaneFrameElementShearDiagram, PlaneFrameElementMomentDiagram to draw the axial force diagram, shear force diagram and bending moment diagram, respectively, for each element. This process is illustrated below. 10-Beam Elements in 2-D Space (Plane Frame Element)

37. ME 520 Dr. Ahmet Zafer Şenalp 37Mechanical Engineering Department, GTU Solution of Example 1 with Matlab >>PlaneFrameElementAxialDiagram(f1,L1) 10-Beam Elements in 2-D Space (Plane Frame Element)

38. ME 520 Dr. Ahmet Zafer Şenalp 38Mechanical Engineering Department, GTU Solution of Example 1 with Matlab >>PlaneFrameElementAxialDiagram(f2,L2) 10-Beam Elements in 2-D Space (Plane Frame Element)

39. ME 520 Dr. Ahmet Zafer Şenalp 39Mechanical Engineering Department, GTU Solution of Example 1 with Matlab >>PlaneFrameElementAxialDiagram(f3,L3) 10-Beam Elements in 2-D Space (Plane Frame Element)

40. ME 520 Dr. Ahmet Zafer Şenalp 40Mechanical Engineering Department, GTU Solution of Example 1 with Matlab >>PlaneFrameElementShearDiagram(f1,L1) 10-Beam Elements in 2-D Space (Plane Frame Element)

41. ME 520 Dr. Ahmet Zafer Şenalp 41Mechanical Engineering Department, GTU Solution of Example 1 with Matlab >>PlaneFrameElementShearDiagram(f2,L2) 10-Beam Elements in 2-D Space (Plane Frame Element)

42. ME 520 Dr. Ahmet Zafer Şenalp 42Mechanical Engineering Department, GTU Solution of Example 1 with Matlab >>PlaneFrameElementShearDiagram(f3,L3) 10-Beam Elements in 2-D Space (Plane Frame Element)

43. ME 520 Dr. Ahmet Zafer Şenalp 43Mechanical Engineering Department, GTU Solution of Example 1 with Matlab >>PlaneFrameElementMomentDiagram(f1,L1) 10-Beam Elements in 2-D Space (Plane Frame Element)

44. ME 520 Dr. Ahmet Zafer Şenalp 44Mechanical Engineering Department, GTU Solution of Example 1 with Matlab >>PlaneFrameElementMomentDiagram(f2,L2) 10-Beam Elements in 2-D Space (Plane Frame Element)

45. ME 520 Dr. Ahmet Zafer Şenalp 45Mechanical Engineering Department, GTU Solution of Example 1 with Matlab >>PlaneFrameElementMomentDiagram(f3,L3) 10-Beam Elements in 2-D Space (Plane Frame Element)

46. Solution of Example 2 with Matlab ME 520 Dr. Ahmet Zafer Şenalp 46Mechanical Engineering Department, GTU 10-Beam Elements in 2-D Space (Plane Frame Element) (Plane Frame Element) Given; E=200 GPa, A=4x10 -2 m 2, I=1x10 -6 m 4 Find; (a)the global stiffness matrix for the structure (b)the displacements and rotations at node 2 (c)the reactions at nodes 1 and 3 (d)the axial force, shear force and bending moment in each element (e)the axial force diagram for each element (f)the shear force diagram for each element (g)the bending moment diagram for each element

47. ME 520 Dr. Ahmet Zafer Şenalp 47Mechanical Engineering Department, GTU Solution of Example 2 with Matlab Solution: Use the 7 steps to solve the problem using plane frame element. We need first to replace the distributed loading on element 2 by equivalent nodal loads. This is performed as follows for element 2 with a uniformly distributed load. The resulting plane frame with equivalent nodal loads is shown as; 10-Beam Elements in 2-D Space (Plane Frame Element)

48. ME 520 Dr. Ahmet Zafer Şenalp 48Mechanical Engineering Department, GTU Solution of Example 2 with Matlab Step 1-Discretizing the domain: This problem is already subdivided into 2 elements and 3 nodes. E#N1N2 112 223 10-Beam Elements in 2-D Space (Plane Frame Element)

49. ME 520 Dr. Ahmet Zafer Şenalp 49Mechanical Engineering Department, GTU Solution of Example 2 with Matlab Step 2-Copying relevant files and starting Matlab Create a directory Copy PlaneFrameAssemble.m PlaneFrameElementAxialDiagram.m PlaneFrameElementForces.m PlaneFrameElementLength.m PlaneFrameElementMomentDiagram.m PlaneFrameElementShearDiagram.m PlaneFrameElementStiffness.m PlaneFrameInclinedSupport.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 10-Beam Elements in 2-D Space (Plane Frame Element)

50. ME 520 Dr. Ahmet Zafer Şenalp 50Mechanical Engineering Department, GTU Solution of Example 2 with Matlab Step 3-Writing the element stiffness matrices: The twoelement stiffness matrices k 1, and k 2 are obtained by making calls to the Matlab function PlaneFrameElementStiffness. Each matrix has size 6x6. Enter the data >>E=200e6 >>A=4e-2 >>I=1e-6 >>L1=PlaneFrameElementLength(0,3,2,0) L1 = 3.6056 >>L2=4 >>theta1=360-atan(3/2)*180/pi theta1 = 303.6901 >>theta2=0 10-Beam Elements in 2-D Space (Plane Frame Element)

51. ME 520 Dr. Ahmet Zafer Şenalp 51Mechanical Engineering Department, GTU Solution of Example 2 with Matlab >>k1= PlaneFrameElementStiffness(E,A,I,L1,theta1) k1 = 1.0e+06 * 0.6827 -1.0240 0.0001 -0.6827 1.0240 0.0001 -1.0240 1.5361 0.0001 1.0240 -1.5361 0.0001 0.0001 0.0001 0.0002 -0.0001 -0.0001 0.0001 -0.6827 1.0240 -0.0001 0.6827 -1.0240 -0.0001 1.0240 -1.5361 -0.0001 -1.0240 1.5361 -0.0001 0.0001 0.0001 0.0001 -0.0001 -0.0001 0.0002 >>k2= PlaneFrameElementStiffness(E,A,I,L2,theta2) k2 = 1.0e+06 * 2.0000 0 0 -2.0000 0 0 0 0.0000 0.0001 0 -0.0000 0.0001 0 0.0001 0.0002 0 -0.0001 0.0001 -2.0000 0 0 2.0000 0 0 0 -0.0000 -0.0001 0 0.0000 -0.0001 0 0.0001 0.0001 0 -0.0001 0.0002 10-Beam Elements in 2-D Space (Plane Frame Element)

52. ME 520 Dr. Ahmet Zafer Şenalp 52Mechanical Engineering Department, GTU Solution of Example 2 with Matlab 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=PlaneFrameAssemble(K,k1,1,2) >>K=PlaneFrameAssemble(K,k2,2,3) K = 1.0e+06 * 0.6827 -1.0240 0.0001 -0.6827 1.0240 0.0001 0 0 0 -1.0240 1.5361 0.0001 1.0240 -1.5361 0.0001 0 0 0 0.0001 0.0001 0.0002 -0.0001 -0.0001 0.0001 0 0 0 -0.6827 1.0240 -0.0001 2.6827 -1.0240 -0.0001 -2.0000 0 0 1.0240 -1.5361 -0.0001 -1.0240 1.5361 0.0000 0 -0.0000 0.0001 0.0001 0.0001 0.0001 -0.0001 0.0000 0.0004 0 -0.0001 0.0001 0 0 0 -2.0000 0 0 2.0000 0 0 0 0 0 0 -0.0000 -0.0001 0 0.0000 -0.0001 0 0 0 0 0.0001 0.0001 0 -0.0001 0.0002 10-Beam Elements in 2-D Space (Plane Frame Element)

53. ME 520 Dr. Ahmet Zafer Şenalp 53Mechanical Engineering Department, GTU Solution of Example 2 with Matlab Step 5-Applying the boundary conditions: Finite element equation for the problem is; 10-Beam Elements in 2-D Space (Plane Frame Element)

54. ME 520 Dr. Ahmet Zafer Şenalp 54Mechanical Engineering Department, GTU Solution of Example 2 with Matlab The boundary conditions for the problem are; 10-Beam Elements in 2-D Space (Plane Frame Element)

55. ME 520 Dr. Ahmet Zafer Şenalp 55Mechanical Engineering Department, GTU Solution of Example 2 with Matlab 10-Beam Elements in 2-D Space (Plane Frame Element)

56. ME 520 Dr. Ahmet Zafer Şenalp 56Mechanical Engineering Department, GTU Solution of Example 2 with Matlab 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 Therefore we obtain: >>k=K(4:6,4:6) k = 1.0e+06 * 2.6827 -1.0240 -0.0001 -1.0240 1.5361 0.0000 -0.0001 0.0000 0.0004 >>f=[0 ; -16 ; -10.667] 10-Beam Elements in 2-D Space (Plane Frame Element)

57. ME 520 Dr. Ahmet Zafer Şenalp 57Mechanical Engineering Department, GTU Solution of Example 2 with Matlab The solution of the system is obtained using Matlab as follows. Note that the ‘\’ operator is used for Gaussian elimination. >>u=k\f u = -0.0000 -0.0253 It is now clear that the horizontal and vertical displacements at node 2 are -0.0000 and -0.0000 m respectively (both almost zero), while the rotation at node 2 is 0.0253 rad (clockwise). 10-Beam Elements in 2-D Space (Plane Frame Element)

58. ME 520 Dr. Ahmet Zafer Şenalp 58Mechanical Engineering Department, GTU Solution of Example 2 with Matlab Step 7-Post-processing: In this step we obtain the reactions at nodes 1 and 3 and the forces (axial forces, shears and moments) in each plane frame element using Matlab as follows. First we set up the global nodal displacement vector U, then we calculate the nodal force vector F. >>U=[0 ; 0 ; 0; u ; 0; 0; 0] U = 0 -0.0000 -0.0253 0 10-Beam Elements in 2-D Space (Plane Frame Element)

59. ME 520 Dr. Ahmet Zafer Şenalp 59Mechanical Engineering Department, GTU Solution of Example 2 with Matlab >>F=K*U F = -12.2064 14.1031 -2.8039 0.0000 -16.0000 -10.6670 12.2064 1.8969 -2.5295 10-Beam Elements in 2-D Space (Plane Frame Element)

60. ME 520 Dr. Ahmet Zafer Şenalp 60Mechanical Engineering Department, GTU Solution of Example 2 with Matlab Thus the horizontal and vertical reactions at node 1 are forces of 12.2064 kN (to the left) and 14.1031 kN (upward), respectively. Thus the horizontal and vertical reactions at node 3 are forces of 12.2064 kN (to the right) and 1.8969 kN (upward), respectively. The moments at nodes 1 and 3 are 2.8039 kNm (clockwise) and 2.5295 kNm (clockwise), respectively. It is clear that force equilibrium is satisfied. 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 PlaneFrameElementForces. 10-Beam Elements in 2-D Space (Plane Frame Element)

61. ME 520 Dr. Ahmet Zafer Şenalp 61Mechanical Engineering Department, GTU Solution of Example 2 with Matlab >> u1=[U(1) ; U(2) ; U(3) ; U(4) ; U(5) ; U(6)] u1 = 0 -0.0000 -0.0253 >> u2=[U(4) ; U(5) ; U(6) ; U(7) ; U(8) ; U(9)] u2 = -0.0000 -0.0253 0 10-Beam Elements in 2-D Space (Plane Frame Element)

62. ME 520 Dr. Ahmet Zafer Şenalp 62Mechanical Engineering Department, GTU Solution of Example 2 with Matlab >>f1=PlaneFrameElementForces(E,A,I,L1,theta1,u1) f1 = -18.5054 -2.3333 -2.8039 18.5054 2.3333 -5.6090 >>f2=PlaneFrameElementForces(E,A,I,L2,theta2,u2) f2 = -12.2064 -1.8969 -5.0580 12.2064 1.8969 -2.5295 10-Beam Elements in 2-D Space (Plane Frame Element)

63. ME 520 Dr. Ahmet Zafer Şenalp 63Mechanical Engineering Department, GTU Solution of Example 2 with Matlab Note that the forces for element 2 need to be modified because of the distributed load. In order to obtain the correct forces for element 2 we need to subtract from f2 the vector of eqivalent nodal loads given in equation (**). This is performed using Matlab as follows: >>f2=f2-[0; -16; -10.667; 0; -16; 10.667] f2 = -12.2064 14.1031 5.6090 12.2064 17.8969 -13.1965 10-Beam Elements in 2-D Space (Plane Frame Element)

64. ME 520 Dr. Ahmet Zafer Şenalp 64Mechanical Engineering Department, GTU Solution of Example 2 with Matlab The forces for each element are given above. Element 1 has an axial force of -18.5865 kN, shear force of -2.3333 kN bending moment of -2.8039 kNm at its left end while it has an axial force of 18.5054 kN, shear force of 2.3333 kN bending moment of -5.6090 kNm at its right end. Element 2 has an axial force of -12.2064 kN, shear force of 14.1031 kN bending moment of 5.6090 kNm at its left end while it has an axial force of 12.2064 kN, shear force of 17.8969 kN bending moment of -13.1965 kNm at its right end. 10-Beam Elements in 2-D Space (Plane Frame Element)

65. ME 520 Dr. Ahmet Zafer Şenalp 65Mechanical Engineering Department, GTU Solution of Example 2 with Matlab Finally we call the Matlab functions PlaneFrameElementAxialDiagram, PlaneFrameElementShearDiagram, PlaneFrameElementMomentDiagram to draw the axial force diagram, shear force diagram and bending moment diagram, respectively, for each element. This process is illustrated below. 10-Beam Elements in 2-D Space (Plane Frame Element)

66. ME 520 Dr. Ahmet Zafer Şenalp 66Mechanical Engineering Department, GTU Solution of Example 2 with Matlab >>PlaneFrameElementAxialDiagram(f1,L1) 10-Beam Elements in 2-D Space (Plane Frame Element)

67. ME 520 Dr. Ahmet Zafer Şenalp 67Mechanical Engineering Department, GTU Solution of Example 2 with Matlab >>PlaneFrameElementAxialDiagram(f2,L2) 10-Beam Elements in 2-D Space (Plane Frame Element)

68. ME 520 Dr. Ahmet Zafer Şenalp 68Mechanical Engineering Department, GTU Solution of Example 2 with Matlab >>PlaneFrameElementShearDiagram(f1,L1) 10-Beam Elements in 2-D Space (Plane Frame Element)

69. ME 520 Dr. Ahmet Zafer Şenalp 69Mechanical Engineering Department, GTU Solution of Example 2 with Matlab >>PlaneFrameElementShearDiagram(f2,L2) 10-Beam Elements in 2-D Space (Plane Frame Element)

70. ME 520 Dr. Ahmet Zafer Şenalp 70Mechanical Engineering Department, GTU Solution of Example 2 with Matlab >>PlaneFrameElementMomentDiagram(f1,L1) 10-Beam Elements in 2-D Space (Plane Frame Element)

71. ME 520 Dr. Ahmet Zafer Şenalp 71Mechanical Engineering Department, GTU Solution of Example 2 with Matlab >>PlaneFrameElementMomentDiagram(f2,L2) 10-Beam Elements in 2-D Space (Plane Frame Element)