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

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 1 2 M1 V1 M2 V2 x y (+v) Note: For the element, V and M are internal shear and bending moment. Matrix Methods

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

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

Stiffness Matrix Formation Written in matrix form Matrix Methods

Stiffness Matrix Formation Solve for integration constants. Matrix Methods

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

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

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

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

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

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. L1 1 3 2 L2 P Element 1 Element 2 Matrix Methods

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

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. Δx 2 P M12 V12 M21 V21 Matrix Methods

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

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

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

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

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

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

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

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

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. 1 2 L M2 V2 M1 V1 w x y (+v) Note: V and M are internal shear and bending moment, w is external load. Matrix Methods

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

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

Distributed Loads Written in matrix form Matrix Methods

Distributed Loads Solve for integration constants. Matrix Methods

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

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

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

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

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

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

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

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