Transcript v 1 - Department of Mechanical and Aerospace Engineering

Matrix Methods
(Notes Only)
MAE 316 – Strength of Mechanical Components
NC State University Department of Mechanical and Aerospace Engineering
1
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
M2
2
x
V1
V2
y (+v)
Note: For the element, V and M are internal shear and bending moment.
2
Matrix Methods
Stiffness Matrix Formation
Integrate the load-deflection differential equation to find
expressions for shear force, bending moment, slope, and
deflection.

d 4v
EI 4  0
dx
d 3v
EI 3  c1  V
dx
d 2v
EI 2  c1 x  c2   M
dx
dv
x2
EI
 c1  c2 x  c3  EI
dx
2
x3
x2
EIv  c1  c2
 c3 x  c4
6
2
3
Matrix Methods
Stiffness Matrix Formation
Express slope and deflection at each node in terms of
integration constants c1, c2, c3, and c4.

v1  v (0) 
1 
c4
EI
c
dv
(0)  3
dx
EI

1  c1 L3 c2 L2
v2  v( L) 


c
L

c
3
4

EI  6
2


dv
1  c1 L2
 2  ( L) 

c
L

c
2
3

dx
EI  2

Note: ν and θ (deflection and slope) are the same in the element as for the
whole beam.
4
Matrix Methods
Stiffness Matrix Formation
Written in matrix form


 0

 0
 3
 L
 6 EI
 L2

 2 EI
5
0
0
L2
2 EI
L
EI
0
1
EI
L
EI
1
EI
1 
EI 
 c1   v1 
0    
 c2    1 
1  c3   v2 
   

EI  c   
 4   2 
0

Matrix Methods
Stiffness Matrix Formation
Solve for integration constants.

 12EI
 c1   3
   L
 c2    6 EI
 c    L2
 3  0
c 
 4 
 EI
6
6 EI
L2
 4 EI
L
EI
 12EI
L3
6 EI
L2
0
0
0
6 EI 
 v1 
2

L
 

 2 EI  1 

L   v2 
0  
 2 

0 
Matrix Methods
Stiffness Matrix Formation
Express shear forces and bending moments in terms of the
constants.

V (0)  V1  c1
12 EI
6 EI
12 EI
6 EI
v



v

2
1
1
2
L3
L2
L3
L2
M (0)  M1  c2
V1 
6 EI
4 EI
6 EI
2 EI
v



v

2
1
1
2
2
2
L
L
L
L
V ( L)  V2  c1
M1 
 12 EI
6 EI
12 EI
6 EI
v



v

2
1
1
2
3
2
3
2
L
L
L
L
M ( L)  M 2  c1L  c2
V2 
M2 
7
6 EI
2 EI
6 EI
4 EI
v



v

2
1
1
2
2
2
L
L
L
L
Matrix Methods
Stiffness Matrix Formation
This can also be expressed in matrix form.

6 L  12 6 L  v1   V1 
 12
 6 L 4 L2  6 L 2 L2     M 
EI 
 1    1 
L3  12  6 L 12  6 L  v2   V2 
  


2
2 
 M 

6
L
2
L

6
L
4
L

 2   2 
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).


8
Matrix Methods
Examples
Cantilever beam with tip load

P
1
2
L
9
Matrix Methods
Examples

Cantilever beam with tip moment
1
2
L
10
Matrix Methods
Mo
Examples

Cantilever beam with roller support and tip moment
(statically indeterminate)
2
1
L
11
Matrix Methods
Mo
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
Element 1
1
2
L1
12
Element 2
3
L2
Matrix Methods
Multiple Beam Elements

The internal shear and bending moment equations for each
element can be written as follows.
6 L1
 12
2
 6L
4
L
EI  1
1
3
L1  12  6 L1

2
6
L
2
L
1
 1
Element 2
13
 12
 6 L1
12
 6 L1
6 L1  v11   V11 
 1  1
2 
2 L1 1   M 1 
 1 

1
 6 L1  v2
V2 



2 
4 L1  21   M 21 
6 L2
 12
2
 6L
EI  2 4 L2
3
L2  12  6 L2

2
 6 L2 2 L2
 12
 6 L2
12
 6 L2
Element 1
6 L2  v12   V12 
 2  2
2 
2 L2 1   M 1 
 2 

2 
 6 L2  v2
V2 



2  2 
4 L2  2   M 22 
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.
P
M1 2
V1
M1 2
2
M2 1
2
V12
V21
M21 V2
1
Δx

As Δx→0, the following equilibrium conditions apply.
V21  V12  P
M 21  M 12  0

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.
14
Matrix Methods
Multiple Beam Elements

The two equilibrium equations can be written in matrix
form in terms of displacements and slopes.
 12EI

 V21  V12   L13
 1

 M  M 2    6 EI
1 
 2

2
L
 1
15
6 EI
2
L1
2 EI
L1

12EI 12EI

3
3
L1
L2
6 EI 6 EI
 2  2
L1
L2
6 EI 6 EI
 2
2
L1
L2
4 EI 4 EI

L1
L2

Matrix Methods
12EI
3
L2
6 EI
 2
L2

 v1 
 
6 EI  1 

2 
L2  v2   P 
 
2 EI  2   0 
L2  v3 
 
 
 3
Multiple Beam Elements

Combining the equilibrium equations with the element equations, we get:
 12EI

3
 L1
 6 EI

2
L
1

  12EI
 L3
1

 6 EI
 L12

 0


 0


6 EI
2
L1
4 EI
L1
6 EI
 2
L1
2 EI
L1
0
0
12EI
3
L1
6 EI
 2
L1
12EI 12EI

3
3
L1
L2
6 EI 6 EI
 2  2
L1
L2
12EI

3
L2
6 EI
2
L2

6 EI
2
L1
2 EI
L1
6 EI 6 EI
 2  2
L1
L2
4 EI 4 EI

L1
L2
6 EI
 2
L2
2 EI
L2
0
0
12EI
3
L2
6 EI
 2
L2
12EI
3
L2
6 EI
 2
L2





0  v   V 1 
1
1
   1 
6 EI  1   M 1 

  
2
L2  v2   P 


2 EI  2   0 
L2  v3   V 2 
   2 
6 EI    2 
 2   3   M 2 
L2 
4 EI 
L2 
0
Repeat: When the equations are combined for the entire beam, the
summed internal shear and moments equal the external forces.
16
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.
 4 EI
 L
 1
 6 EI
 L 2
1

 2 EI

 L1

 0

17

6 EI
L12
12 EI 12 EI

3
L1
L23

6 EI 6 EI
 2
2
L1
L2
6 EI
L2 2
2 EI
L1

6 EI 6 EI
 2
2
L1
L2
4 EI 4 EI

L1
L2
2 EI
L2

0 

6 EI  1   0 
L2 2   v2   P 



0

2 EI  2
   
L2   3   0 
4 EI 

L2 
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)
 V1 


 M1 
e
f 
V2 


M 
 2
18
 v1 
 
 1 
e
d  
v
 2
 
 2
6 L  12 6 L 
 12


2
2
EI  6 L 4 L  6 L 2 L 
e
k  3
L  12  6 L 12  6 L 


 6 L 2 L2  6 L 4 L2 


Matrix Methods
Multiple Beam Elements

For the simply supported beam discussed before, we can now
formulate the unconstrained system equations.
1
 k11
 1
 k 21
 k1
 31
1
 k 41

 0
 0

1
k12
1
k13
1
k14
0
1
k 22
1
k32
1
k 23
1
k33
 k112
1
k 24
1
k34
 k122
0
k132
1
k 42
1
2
k 43
 k 21
1
2
k 44
 k 22
2
k 23
0
0
k312
2
k 41
k322
2
k 42
k332
2
k 43
0  v1   R1 
   
0  1   T1 
k142  v2   R2 
  
2
k 24  2   T2 
 R 
2 
v
k34  3   3 
2 
 T 

k 44
 3   3 
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
19
Matrix Methods
Multiple Beam Elements

Now apply boundary conditions, external forces, and
moments.
v1  v3  0
T1  T2  T3  0
V2  P
1
 k11
 1
 k 21
 k1
 31
1
 k 41

 0
 0

20
1
k12
1
k13
1
k14
0
1
k 22
1
k32
1
k 23
1
k33
 k112
1
k 24
1
k34
 k122
0
k132
1
k 42
1
2
k 43
 k 21
1
2
k 44
 k 22
2
k 23
0
0
k312
2
k 41
k322
2
k 42
k332
2
k 43
0  0   R1 
   
0  1   0 
k142  v2   P 
  
2
k 24  2   0 
   
k342  0   R3 
2 
  
k 44
  3   0 
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).
 k 221
k231
k 241
0  1   0 
 1

   
1
2
1
2
2
 k32 k33  k11 k34  k12 k14  v2   P 
 k 1 k 1  k 2 k 1  k 2 k 2     0 
 42 43 21 44 22 24  2   


2
2
2 

0

0
k 41
k 42
k44  3   

21
Matrix Methods
Examples

Simply supported beam with mid-span load
P
1
2
L/2
22
3
L/2
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.
w
1
x
2
M1 V
1
V2
M2
L
y (+v)
Note: V and M are internal shear and bending moment, w is external load.
23
Matrix Methods
Distributed Loads

Integrate the load-deflection differential equation to find
expressions for shear force, bending moment, slope, and
deflection.
d 4v
EI 4  w
dx
d 3v
EI 3  wx  c1  V
dx
d 2 v wx 2
EI 2 
 c1 x  c2   M
dx
2
dv wx 3
x2
EI

 c1  c2 x  c3  EI
dx
6
2
wx 4
x3
x2
EIv 
 c1  c2
 c3 x  c4
24
6
2
24
Matrix Methods
Distributed Loads

Express slope and deflection at each node in terms of
integration constants c1, c2, c3, and c4.
v1  v (0) 
1 
c4
EI
c
dv
(0)  3
dx
EI

1  wL4 c1L3 c2 L2
v2  v( L) 


 c3 L  c4 

EI  24
6
2


dv
1  wL3 c1 L2
 2  ( L) 

 c2 L  c3 

dx
EI  6
2

Note: ν and θ (deflection and slope) are the same in the element as for the
whole beam.
25
Matrix Methods
Distributed Loads

Written in matrix form

 0

 0
 3
 L
 6 EI
 L2

 2 EI
26
0
0
L2
2 EI
L
EI
0
1
EI
L
EI
1
EI
1 
v1


EI 

 c1  
1

0   
wL4 
  c2   
1  c3   v2  24EI 

  
3

EI  c  
wL 
 4    2 

6 EI 

0 

Matrix Methods
Distributed Loads

Solve for integration constants.
 12EI
 c1   3
   L
 c2    6 EI
 c    L2
 3  0
c 
 4 
 EI
27
6 EI
L2
 4 EI
L
EI
0
 12EI
L3
6 EI
L2
0
0
v1

6 EI 


2

1
L



4 
 2 EI 
wL
 v 
L  2 24EI 
0 
3 
wL 

0   2  6 EI 
Matrix Methods
Distributed Loads

Express shear forces and bending moments in terms of the
constants.
V (0)  V  c
1
1
wL 12 EI
6 EI
12 EI
6 EI
 3 v1  2 1  3 v2  2  2
2
L
L
L
L
M (0)  M1  c2
V1 
wL2 6 EI
4 EI
6 EI
2 EI
M1 
 2 v1 
1  2 v2 
2
12
L
L
L
L
V ( L)  V2  wL  c1
wL  12 EI
6 EI
12 EI
6 EI

v



v

2
1
1
2
2
L3
L2
L3
L2
wL2
M ( L)   M 2  
 c1 L  c2
12
V2 
wL2 6 EI
2 EI
6 EI
4 EI
M2 
 2 v1 
1  2 v2 
2
12
L
L
L
L
28
Matrix Methods
Distributed Loads

This can be expressed in matrix form.
6 L  12 6 L  v1   V1 
 12
 6 


 6 L 4 L2  6 L 2 L2     M 
EI 
 1    1   wL  L 
L3  12  6 L 12  6 L  v2   V2  12  6 
  




2
2 






M
6
L
2
L

6
L
4
L

L

 2   2 



This matrix equation contains an additional term – known as
the vector of equivalent nodal loads – that accounts for the
distribution load w.
29
Matrix Methods
Examples

Propped cantilever beam with uniform load
w
2
1
L
30
Matrix Methods
Examples

Cantilever beam with uniform load
w
2
1
L
31
Matrix Methods
Examples

Cantilever beam with moment and partial uniform
load
w
1
Mo
L1
32
3
2
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.
33
Matrix Methods
Finite Element Analysis of Beams
P
w
1
34
2
3
4
Nodes:
Elements:
kunconstrained:
5
4
10 x 10
Apply B.C.’s:
v1=v5=0
θ5=0
kconstrained:
7x7
Matrix Methods
5
Finite Element Analysis of Beams
w
1
35
P
2
3
4
Nodes:
Elements:
kunconstrained:
5
4
10 x 10
Apply B.C.’s:
v1=v3=v5=0
θ1=0
kconstrained:
6x6
Matrix Methods
5