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8-Beam Element Dr. Ahmet Zafer Şenalp Mechanical Engineering Department Gebze Institute of Technology.

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1 8-Beam Element Dr. Ahmet Zafer Şenalp Mechanical Engineering Department Gebze Institute of Technology ME 520 Fundamentals of Finite Element Analysis

2 L: Length I: Moment of inertia of the cross-sectional area E: Elastic modulus v=v(x): Deflection (lateral displacement) of the neutral axis F=F(x): Shear force M=M(x): Moment about z-axis Simple Plane Beam Element ME 520 Dr. Ahmet Zafer Şenalp 2Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element : Rotation about the z-axis

3 Elementary Beam Theory: Direct Method Using the results from elementary beam theory to compute each column of the stiffness matrix. Element stiffness equation (local node: i, j or 1, 2): Simple Plane Beam Element ME 520 Dr. Ahmet Zafer Şenalp 3Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element

4 Example 1 ME 520 Dr. Ahmet Zafer Şenalp 4Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element Find; (a)The deflection and rotation at the center node (b) the reaction forces and moments at the two ends Solution: Connectivity table: E#N1N

5 Example 1 ME 520 Dr. Ahmet Zafer Şenalp 5Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element Boundary conditions:  Displacement boundary conditions:  Force boundary conditions: a)Element Stiffness Matrices:

6 Example 1 ME 520 Dr. Ahmet Zafer Şenalp 6Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element Global FE equation is: Applying BC’s: Reaction Forces:

7 Example 1 ME 520 Dr. Ahmet Zafer Şenalp 7Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element Stresses in the beam at the two ends can be calculated using the formula, Note that the FE solution is exact according to the simple beam theory, since no distributed load is present between the nodes.

8 Equivalent Nodal Loads of Distributed Transverse Load ME 520 Dr. Ahmet Zafer Şenalp 8Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element 2 element case:

9 Example 2 ME 520 Dr. Ahmet Zafer Şenalp 9Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element Find ; (a)The deflection and rotation at the right end (b)The reaction force and moment at the left end. Solution: Connectivity table: Equivalent system: E#N1N2 112

10 Example 2 ME 520 Dr. Ahmet Zafer Şenalp 10Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element Boundary conditions:  Displacement boundary conditions:  Force boundary conditions: The structure FE equation:

11 Example 2 ME 520 Dr. Ahmet Zafer Şenalp 11Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element Reaction forces: This force vector gives the total effective nodal forces which include the equivalent nodal forces for the distributed lateral load p given by : The correct reaction forces can be obtained as follows,

12 Example 3 ME 520 Dr. Ahmet Zafer Şenalp 12Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element Given; Find ; (a)Deflections, rotations (b)reaction forces Solution: Connectivity table: E#N1N

13 Example 3 ME 520 Dr. Ahmet Zafer Şenalp 13Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element Boundary conditions:  Displacement boundary conditions:  Force boundary conditions : The spring stiffness matrix : Adding this stiffness matrix to the global FE equation:

14 Example 3 ME 520 Dr. Ahmet Zafer Şenalp 14Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element Aplying BC’s:

15 Example 3 ME 520 Dr. Ahmet Zafer Şenalp 15Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element Reaction Forces: Checking the results: Draw free body diagram of the beam:

16 ME 520 Dr. Ahmet Zafer Şenalp 16Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element It is clear that the beam element has 4 degrees of freedom (2 at each node) The sign convension used is that the displacement is positive if it points upwards and the rotation is positive if it is counterclockwise. For a structure with n nodes, the global stiffness matrix K will be of size 2nx2n. 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

17 ME 520 Dr. Ahmet Zafer Şenalp 17Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element Finally once the unkown displacements and and reactions are found, the force is obtained for each element as follows: where f is the 4x1 nodal force vector in the element and u is the 4x1 element displacement vector. The first and second elements in each vector are the transverse displacement and rotation, respectively, at the first node, while the third and fourth elements in each vector are the transverse displacement and rotation, respectively, at the second node. Solution procedure with matlab Solution procedure with matlab

18 Matlab functions used ME 520 Dr. Ahmet Zafer Şenalp 18Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element The 5 Matlab functions used for the beam element are: BeamElementStiffness(E,I,L) This function returns the element stiffness matrix for a beam element with modulus of elasticity E, moment of inertia I, and length L. The size of the element stiffness matrix is 4 x 4. Function contents: function y = BeamElementStiffness(E,I,L) %BeamElementStiffness This function returns the element % stiffness matrix for a beam % element with modulus of elasticity E, % moment of inertia I, and length L. % The size of the element stiffness % matrix is 4 x 4. y = E*I/(L*L*L)*[12 6*L -12 6*L ; 6*L 4*L*L -6*L 2*L*L ; *L 12 -6*L ; 6*L 2*L*L -6*L 4*L*L];

19 Matlab functions used ME 520 Dr. Ahmet Zafer Şenalp 19Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element BeamAssemble(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 2nx2n global stiffness matrix K after the element stiffness matrix k is assembled. Function contents: function y = BeamAssemble(K,k,i,j) %BeamAssemble 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.

20 Matlab functions used ME 520 Dr. Ahmet Zafer Şenalp 20Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element K(2*i-1,2*i-1) = K(2*i-1,2*i-1) + k(1,1); K(2*i-1,2*i) = K(2*i-1,2*i) + k(1,2); K(2*i-1,2*j-1) = K(2*i-1,2*j-1) + k(1,3); K(2*i-1,2*j) = K(2*i-1,2*j) + k(1,4); K(2*i,2*i-1) = K(2*i,2*i-1) + k(2,1); K(2*i,2*i) = K(2*i,2*i) + k(2,2); K(2*i,2*j-1) = K(2*i,2*j-1) + k(2,3); K(2*i,2*j) = K(2*i,2*j) + k(2,4); K(2*j-1,2*i-1) = K(2*j-1,2*i-1) + k(3,1); K(2*j-1,2*i) = K(2*j-1,2*i) + k(3,2); K(2*j-1,2*j-1) = K(2*j-1,2*j-1) + k(3,3); K(2*j-1,2*j) = K(2*j-1,2*j) + k(3,4); K(2*j,2*i-1) = K(2*j,2*i-1) + k(4,1); K(2*j,2*i) = K(2*j,2*i) + k(4,2); K(2*j,2*j-1) = K(2*j,2*j-1) + k(4,3); K(2*j,2*j) = K(2*j,2*j) + k(4,4); y = K;

21 Matlab functions used ME 520 Dr. Ahmet Zafer Şenalp 21Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element 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 4x1 element force vecor 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;

22 Matlab functions used ME 520 Dr. Ahmet Zafer Şenalp 22Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element 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')

23 Matlab functions used ME 520 Dr. Ahmet Zafer Şenalp 23Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element 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')

24 ME 520 Dr. Ahmet Zafer Şenalp 24Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element Consider the beam as shown Given E=210 GPa I=60x10 -6 m 4 P=20 kN L=2 m Determine: a)the global stiffness matrix for the structure b)vertical displacement 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 4 with Matlab

25 ME 520 Dr. Ahmet Zafer Şenalp 25Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element Solution of Example 4 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#N1N

26 ME 520 Dr. Ahmet Zafer Şenalp 26Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element Solution of Example 4 with Matlab Step 2-Copying relevant files and starting Matlab Create a directory Copy BeamElementStiffness.m BeamAssemble.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

27 ME 520 Dr. Ahmet Zafer Şenalp 27Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element Solution of Example 4 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. Each matrix has size 4x4. Enter the data >>E=210e6 >>I=60e-6 >>L=2 >>k1=BeamElementStiffness(E,I,L) k1 =

28 ME 520 Dr. Ahmet Zafer Şenalp 28Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element Solution of Example 4 with Matlab >>k2=BeamElementStiffness(E,I,L) k2 = Step 4-Assembling the global stiffness matrix: Since the structure has 3 nodes, the size of the global stiffness matrix is 6x6. >>K=zeros(6,6) >>K=BeamAssemble(K,k1,1,2) >>K=BeamAssemble(K,k2,2,3) K =

29 ME 520 Dr. Ahmet Zafer Şenalp 29Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element Solution of Example 4 with Matlab Step 5-Applying the boundary conditions: Finite element equation for the problem is; The boundary conditions for the problem are;

30 ME 520 Dr. Ahmet Zafer Şenalp 30Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element Solution of Example 4 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 3 to 4 and column 6, row 6 and columns 3 to 4, and row 6 and column 6. Therefore we obtain:

31 ME 520 Dr. Ahmet Zafer Şenalp 31Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element Solution of Example 4 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(3:4,3:4) K(3:4,6) ; K(6,3:4) K(6,6)] k = >>f=[-20 ; 0 ; 0] f = -20 0

32 ME 520 Dr. Ahmet Zafer Şenalp 32Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element Solution of Example 4 with Matlab >>u=k\f u = 1.0e-03 * It is now clear that the vertical displacement at node 2= m (downward) rotation at node 2 = rad (clockwise) rotation at node 3 = rad (counterclockwise) 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.

33 ME 520 Dr. Ahmet Zafer Şenalp 33Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element Solution of Example 4 with Matlab >>U=[0 ; 0 ; u(1) ; u(2) ; 0 ; u(3)] U = 1.0e-03 * >>F=K*U F =

34 ME 520 Dr. Ahmet Zafer Şenalp 34Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element Solution of Example 4 with Matlab thus the recations are; Force at node 1=13.75 kN Moment at node 1=15 kNm (countereclockwise) Force at node 3=6.25 kN 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)] u1 = 1.0e-03 *

35 ME 520 Dr. Ahmet Zafer Şenalp 35Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element Solution of Example 4 with Matlab >> u2=[U(3) ; U(4) ; U(5) ; U(6)] u2 = 1.0e-03 * >>f1 =BeamElementForces(k1,u1) f1 =

36 ME 520 Dr. Ahmet Zafer Şenalp 36Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element Solution of Example 4 with Matlab >>f2 =BeamElementForces(k2,u2) f2 = Shear force at centilever region=13.75 kN Bending moment at centilever region=15 kNm Shear force at pin joint=6.25 kN Finally we call the Matlab functions BeamElementShearDiagram and BeamElementMomentDiagram, respectively for each element.

37 ME 520 Dr. Ahmet Zafer Şenalp 37Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element Solution of Example 4 with Matlab >>BeamElementShearDiagram(f1,L)

38 ME 520 Dr. Ahmet Zafer Şenalp 38Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element Solution of Example 4 with Matlab >>BeamElementShearDiagram(f2,L)

39 ME 520 Dr. Ahmet Zafer Şenalp 39Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element Solution of Example 4 with Matlab >>BeamElementMomentDiagram(f1, L)

40 ME 520 Dr. Ahmet Zafer Şenalp 40Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element Solution of Example 4 with Matlab >>BeamElementMomentDiagram(f2, L)

41 ME 520 Dr. Ahmet Zafer Şenalp 41Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element Consider the beam as shown Given E=210 GPa I=5x10 -6 m 4 w=7 kN/m Determine: a)the global stiffness matrix for the structure b)rotations at nodes 1, 2 and 3 c)the reactions at nodes 1, 2, 3 and 4 d)the forces (shears and moments) in each element e)the shear force diagram for each element f)the bending moment diagram for each element Solution of Example 5 with Matlab

42 ME 520 Dr. Ahmet Zafer Şenalp 42Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element Solution of Example 5 with Matlab Solution: Step 1-Discretizing the domain: 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 beam with eqivalent nodal load is shown below:

43 ME 520 Dr. Ahmet Zafer Şenalp 43Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element Solution of Example 5 with Matlab The units used in Matlab calculations are kN and meter. The element connectivity is: E#N1N

44 ME 520 Dr. Ahmet Zafer Şenalp 44Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element Solution of Example 5 with Matlab Step 2-Copying relevant files and starting Matlab Create a directory Copy BeamElementStiffness BeamAssemble BeamElementForces BeamElementShearDiagram BeamElementMomentDiagram 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

45 ME 520 Dr. Ahmet Zafer Şenalp 45Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element Solution of Example 5 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. Each matrix has size 4x4. Enter the data >>E=210e6 >>I=5e-6 >>L1=3 >>L2=4 >>L3=2 >>k1=BeamElementStiffness(E,I,L1) k1 = 1.0e+03 *

46 ME 520 Dr. Ahmet Zafer Şenalp 46Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element Solution of Example 5 with Matlab >>k2=BeamElementStiffness(E,I,L2) k2 = 1.0e+03 * >>k3=BeamElementStiffness(E,I,L3) k3 =

47 ME 520 Dr. Ahmet Zafer Şenalp 47Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element Solution of Example 5 with Matlab Step 4-Assembling the global stiffness matrix: Since the structure has 4 nodes, the size of the global stiffness matrix is 8x8. >>K=zeros(8,8) >>K=BeamAssemble(K,k1,1,2) >>K=BeamAssemble(K,k2,2,3) >>K=BeamAssemble(K,k3,3,4) K = 1.0e+03 *

48 ME 520 Dr. Ahmet Zafer Şenalp 48Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element Solution of Example 5 with Matlab Step 5-Applying the boundary conditions: Finite element equation for the problem is; The boundary conditions for the problem are;

49 ME 520 Dr. Ahmet Zafer Şenalp 49Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element Solution of Example 5 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 2, 4 and 6 and columns2, 4 and 6. Therefore we obtain: The solution of the above system is obtained using Matlab as follows. Note that the ‘\’ operator is used for Gaussian elimination. >>k=[K(2,2) K(2,4) K(2,6) ; K(4,2) K(4,4) K(4,6) ; K(6,2) K(6,4) K(6,6)] k =

50 ME 520 Dr. Ahmet Zafer Şenalp 50Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element Solution of Example 5 with Matlab >>f=[0 ; ; 9.333] f = >>u=k\f u = It is now clear that rotation at node 1 = rad (counterclockwise) rotation at node 2 = rad (clockwise) rotation at node 3 = rad (counterclockwise)

51 ME 520 Dr. Ahmet Zafer Şenalp 51Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element Solution of Example 5 with Matlab Step 7-Post-processing: In this step we obtain the reactions at nodes 1, 2, 3 and 4 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. >>U=[0 ;u(1) ;0 ; u(2) ; 0 ; u(3); 0 ; 0] U =

52 ME 520 Dr. Ahmet Zafer Şenalp 52Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element Solution of Example 5 with Matlab >>F=K*U F = thus the recations are; Force at node 1= kN Force at node 2= kN Force at node 3= kN Force at node 4= kN Moment at node 4 (at fixed support)= kNm (counterclockwise)

53 ME 520 Dr. Ahmet Zafer Şenalp 53Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element Solution of Example 5 with Matlab 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 BeamElementForces. >> u1=[U(1) ; U(2) ; U(3) ; U(4)] u1 = >> u2=[U(3) ; U(4) ; U(5) ; U(6)] u2 = >> u3=[U(5) ; U(6) ; U(7) ; U(8)] u3 =

54 ME 520 Dr. Ahmet Zafer Şenalp 54Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element Solution of Example 5 with Matlab >>f1 =BeamElementForces(k1,u1) f1 = >>f2 =BeamElementForces(k2,u2) f2 = >>f3 =BeamElementForces(k3,u3) f3 =

55 ME 520 Dr. Ahmet Zafer Şenalp 55Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element Solution of Example 5 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 equivalent nodal loads given in equation (**). This is performed using Matlab as follows: >>f2=f2-[-14 ; ; -14 ; 9.333] f2 = Element 1 has a shear force of kN and a bending moment of 0 kNm at its left end while it has a shear force of kN and a bending moment of kNm at its right end. Element 2 has a shear force of kN and a bending moment of kNm at its left end while it has a shear force of kN and a bending moment of kNm at its right end.

56 ME 520 Dr. Ahmet Zafer Şenalp 56Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element Solution of Example 5 with Matlab Element 3 has a shear force of kN and a bending moment of kNm at its left end while it has a shear force of kN and a bending moment of kNm at its right end. Obviously the roller at the left end has zero moment. Finally we call the Matlab functions BeamElementShearDiagram and BeamElementMomentDiagram, respectively for each element.

57 ME 520 Dr. Ahmet Zafer Şenalp 57Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element Solution of Example 5 with Matlab >>BeamElementShearDiagram(f1,L1)

58 ME 520 Dr. Ahmet Zafer Şenalp 58Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element Solution of Example 5 with Matlab >>BeamElementShearDiagram(f2,L2)

59 ME 520 Dr. Ahmet Zafer Şenalp 59Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element Solution of Example 5 with Matlab >>BeamElementShearDiagram(f3,L3)

60 ME 520 Dr. Ahmet Zafer Şenalp 60Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element Solution of Example 5 with Matlab >>BeamElementMomentDiagram(f1, L1)

61 ME 520 Dr. Ahmet Zafer Şenalp 61Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element Solution of Example 5 with Matlab >>BeamElementMomentDiagram(f2, L2)

62 ME 520 Dr. Ahmet Zafer Şenalp 62Mechanical Engineering Department, GYTE 8-Beam Element 8-Beam Element Solution of Example 5 with Matlab >>BeamElementMomentDiagram(f3, L3)


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