Matrix Methods (Notes Only)

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Matrix Methods (Notes Only). MAE 316 – Strength of Mechanical Components NC State University Department of Mechanical and Aerospace Engineering. Stiffness Matrix Formation. Consider an “element”, which is a section of a beam with a “node” at each end.

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### 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

M1

1

2

M2

x

V1

V2

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: νandθ (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

P

1

2

L

Matrix Methods

Examples
• Cantilever beam with tip moment

Mo

1

2

L

Matrix Methods

Examples
• Cantilever beam with roller support and tip moment (statically indeterminate)

2

Mo

1

L

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.

P

1

2

3

Element 2

Element 1

L1

L2

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.

P

M12

M21

2

M12

M21

V12

V21

V21

V12

Δx

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

P

1

2

3

L/2

L/2

Matrix Methods

• 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

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

Matrix Methods

• 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

• Written in matrix form

Matrix Methods

• Solve for integration constants.

Matrix Methods

• Express shear forces and bending moments in terms of the constants.

Matrix Methods

• 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

2

1

w

L

Matrix Methods

Examples
• Cantilever beam with uniform load

2

1

w

L

Matrix Methods

Examples
• Cantilever beam with moment and partial uniform load

w

3

1

2

Mo

L1

L2

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

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

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