Matrix Methods (Notes Only)

1. Matrix Methods(Notes Only) MAE 316 – Strength of Mechanical Components NC State University Department of Mechanical and Aerospace Engineering Matrix Methods

2. Stiffness Matrix Formation • Consider an “element”, which is a section of a beam with a “node” at each end. • If any external forces or moments are applied to the beam, there will be shear forces and moments at each end of the element. • Sign convention – deflection is positive downward, rotation (slope) is positive clockwise. L M1 1 2 M2 x V1 V2 y (+v) Note: For the element, V and M are internal shear and bending moment. Matrix Methods

3. Stiffness Matrix Formation • Integrate the load-deflection differential equation to find expressions for shear force, bending moment, slope, and deflection. Matrix Methods

4. Stiffness Matrix Formation • Express slope and deflection at each node in terms of integration constants c1, c2, c3, and c4. Note: νandθ (deflection and slope) are the same in the element as for the whole beam. Matrix Methods

5. Stiffness Matrix Formation • Written in matrix form Matrix Methods

6. Stiffness Matrix Formation • Solve for integration constants. Matrix Methods

7. Stiffness Matrix Formation • Express shear forces and bending moments in terms of the constants. Matrix Methods

8. Stiffness Matrix Formation • This can also be expressed in matrix form. • Beam w/ one element: matrix equation can be used alone to solve for deflections, slopes and reactions for the beam. • Beam w/ multiple elements: combine matrix equations for each element to solve for deflections, slopes and reactions for the beam (will cover later). Matrix Methods

9. Examples • Cantilever beam with tip load P 1 2 L Matrix Methods

10. Examples • Cantilever beam with tip moment Mo 1 2 L Matrix Methods

11. Examples • Cantilever beam with roller support and tip moment (statically indeterminate) 2 Mo 1 L Matrix Methods

12. Multiple Beam Elements • Matrix methods can also be used for beams with two or more elements. • We will develop a set of equations for the simply supported beam shown below. P 1 2 3 Element 2 Element 1 L1 L2 Matrix Methods

13. Multiple Beam Elements • The internal shear and bending moment equations for each element can be written as follows. Element 1 Element 2 Matrix Methods

14. Multiple Beam Elements • Now, let’s examine node 2 more closely by drawing a free body diagram of an infinitesimal section at node 2. • As Δx→0, the following equilibrium conditions apply. • In other words, the sum of the internal shear forces and bending moments at each node are equal to the external forces and moments at that node. P M12 M21 2 M12 M21 V12 V21 V21 V12 Δx Matrix Methods

15. Multiple Beam Elements • The two equilibrium equations can be written in matrix form in terms of displacements and slopes. Matrix Methods

16. Multiple Beam Elements • Combining the equilibrium equations with the element equations, we get: • Repeat: When the equations are combined for the entire beam, the summed internal shear and moments equal the external forces. Matrix Methods

17. Multiple Beam Elements • Finally, apply boundary conditions and external moments • v1=v3=0 (cancel out rows & columns corresponding to v1 and v3) • M11=M22=0 (set equal to zero in force and moment vector) • End up with the following system of equations. Matrix Methods

18. Multiple Beam Elements • This assembly procedure can be carried out very systematically on a computer. • Define the following (e represents the element number) Matrix Methods

19. Multiple Beam Elements • For the simply supported beam discussed before, we can now formulate the unconstrained system equations. Where: v1, θ1, R1, T1= displacement, slope, force and moment at node 1 v2, θ2, R2, T2= displacement, slope, force and moment at node 2 v3, θ3, R3, T3= displacement, slope, force and moment at node 3 Matrix Methods

20. Multiple Beam Elements • Now apply boundary conditions, external forces, and moments. Matrix Methods

21. Multiple Beam Elements • We are left with the following set of equations, known as the constrained system equations. • The matrix components are exactly the same as in the matrix equations derived previously (slide 17). Matrix Methods

22. Examples • Simply supported beam with mid-span load P 1 2 3 L/2 L/2 Matrix Methods

23. Distributed Loads • Many beam deflection applications involve distributed loads in addition to concentrated forces and moments. • We can expand the previous results to account for uniform distributed loads. M2 M1 V1 V2 w x 1 2 L y (+v) Note: V and M are internal shear and bending moment, w is external load. Matrix Methods

24. Distributed Loads • Integrate the load-deflection differential equation to find expressions for shear force, bending moment, slope, and deflection. Matrix Methods

25. Distributed Loads • Express slope and deflection at each node in terms of integration constants c1, c2, c3, and c4. Note: νandθ (deflection and slope) are the same in the element as for the whole beam. Matrix Methods

26. Distributed Loads • Written in matrix form Matrix Methods

27. Distributed Loads • Solve for integration constants. Matrix Methods

28. Distributed Loads • Express shear forces and bending moments in terms of the constants. Matrix Methods

29. Distributed Loads • This can be expressed in matrix form. • This matrix equation contains an additional term – known as the vector of equivalent nodal loads – that accounts for the distribution load w. Matrix Methods

30. Examples • Propped cantilever beam with uniform load 2 1 w L Matrix Methods

31. Examples • Cantilever beam with uniform load 2 1 w L Matrix Methods

32. Examples • Cantilever beam with moment and partial uniform load w 3 1 2 Mo L1 L2 Matrix Methods

33. Finite Element Analysis of Beams • Everything we have learned so far about matrix methods is foundational for finite element analysis (FEA) of simple beams. • For complex structures, FEA is often performed using computer software programs, such as ANSYS. • FEA is used to calculate and plot deflection, stress, and strain for many different applications. • FEA is covered in more depth in Chapter 19 in the textbook. Matrix Methods

34. Finite Element Analysis of Beams P w 1 3 5 4 2 Nodes: 5 Elements: 4 kunconstrained: 10 x 10 Apply B.C.’s: v1=v5=0 θ5=0 kconstrained: 7 x 7 Matrix Methods

35. Finite Element Analysis of Beams w P 5 3 1 4 2 Nodes: 5 Elements: 4 kunconstrained: 10 x 10 Apply B.C.’s: v1=v3=v5=0 θ1=0 kconstrained: 6 x 6 Matrix Methods