matrix methods (notes only) mae 316 – strength of mechanical components nc state university...
TRANSCRIPT
Matrix Methods(Notes Only)
MAE 316 – Strength of Mechanical ComponentsNC State University Department of Mechanical and Aerospace Engineering
Matrix Methods1
Stiffness Matrix Formation
Matrix Methods2
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 2M1
V1
M2
V2
x
y (+v)
Note: For the element, V and M are internal shear and bending moment.
Stiffness Matrix Formation
Matrix Methods3
Integrate the load-deflection differential equation to find expressions for shear force, bending moment, slope, and deflection.
04
4
dx
vdEI
Vcdx
vdEI 13
3
Mcxcdx
vdEI 212
2
EIcxcx
cdx
dvEI 32
2
1 2
43
2
2
3
1 26cxc
xc
xcEIv
Stiffness Matrix Formation
Matrix Methods4
Express slope and deflection at each node in terms of integration constants c1, c2, c3, and c4.
EI
cvv 4
1 )0(
EI
c
dx
dv 31 )0(
43
22
31
2 26
1)( cLc
LcLc
EILvv
32
21
2 2
1)( cLc
Lc
EIL
dx
dv
Note: ν and θ (deflection and slope) are the same in the element as for the whole beam.
Stiffness Matrix Formation
Matrix Methods5
Written in matrix form
2
2
1
1
4
3
2
1
2
23
01
2
1
26
01
00
1000
v
v
c
c
c
c
EIEI
L
EI
LEIEI
L
EI
L
EI
LEI
EI
Stiffness Matrix Formation
Matrix Methods6
Solve for integration constants.
2
2
1
1
22
2323
4
3
2
1
000
000
2646
612612
v
v
EI
EIL
EI
L
EI
L
EI
L
EIL
EI
L
EI
L
EI
L
EI
c
c
c
c
Stiffness Matrix Formation
Matrix Methods7
Express shear forces and bending moments in terms of the constants.
11)0( cVV
222312131
612612 L
EIv
L
EI
L
EIv
L
EIV
21)0( cMM
2221121
2646 L
EIv
L
EI
L
EIv
L
EIM
12)( cVLV
222312132
612612 L
EIv
L
EI
L
EIv
L
EIV
212)( cLcMLM
2221122
4626 L
EIv
L
EI
L
EIv
L
EIM
Stiffness Matrix Formation
Matrix Methods8
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).
2
2
1
1
2
2
1
1
22
22
3
4626
612612
2646
612612
M
V
M
V
v
v
LLLL
LL
LLLL
LL
L
EI
Cantilever beam with tip load
Examples
Matrix Methods9
L
1 2
P
Cantilever beam with tip moment
Examples
Matrix Methods10
L
1 2Mo
Cantilever beam with roller support and tip moment (statically indeterminate)
Examples
Matrix Methods11
L
1
2Mo
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.
Multiple Beam Elements
Matrix Methods12
L1
1 32
L2
PElement 1 Element 2
The internal shear and bending moment equations for each element can be written as follows.
Multiple Beam Elements
Matrix Methods13
12
12
11
11
12
12
11
11
211
211
11
211
211
11
31
4626
612612
2646
612612
M
V
M
V
v
v
LLLL
LL
LLLL
LL
L
EI
22
22
21
21
22
22
21
21
222
222
22
222
222
22
32
4626
612612
2646
612612
M
V
M
V
v
v
LLLL
LL
LLLL
LL
L
EI
Element 1
Element 2
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.
Multiple Beam Elements
Matrix Methods14
Δx
2
P
M12
V12
M21
V21
M12
V12
M2
1
V21
021
12
21
12
MM
PVV
The two equilibrium equations can be written in matrix form in terms of displacements and slopes.
Multiple Beam Elements
Matrix Methods15
026446626
612661212612
3
3
2
2
1
1
22221
22
211
21
22
32
22
21
32
31
21
31
21
12
21
12 P
v
v
v
L
EI
L
EI
L
EI
L
EI
L
EI
L
EI
L
EI
L
EIL
EI
L
EI
L
EI
L
EI
L
EI
L
EI
L
EI
L
EI
MM
VV
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.
Multiple Beam Elements
Matrix Methods16
22
22
11
11
3
3
2
2
1
1
2222
22
22
32
22
32
22221
22
211
21
22
32
22
21
32
31
21
31
1211
21
21
31
21
31
0
462600
61261200
26446626
612661212612
002646
00612612
M
V
P
M
V
v
v
v
L
EI
L
EI
L
EI
L
EIL
EI
L
EI
L
EI
L
EIL
EI
L
EI
L
EI
L
EI
L
EI
L
EI
L
EI
L
EIL
EI
L
EI
L
EI
L
EI
L
EI
L
EI
L
EI
L
EIL
EI
L
EI
L
EI
L
EIL
EI
L
EI
L
EI
L
EI
Finally, apply boundary conditions and external moments v1=v3=0 (cancel out rows & columns corresponding to v1
and v3)
M11=M2
2=0 (set equal to zero in force and moment vector)
End up with the following system of equations.
Multiple Beam Elements
Matrix Methods17
21 1 1
12 3 3 2 2 21 1 2 1 2 2 2
22 2
1 1 2 1 2 2 3
22 2 2
4 6 20
06 12 12 6 6 6
2 6 6 4 4 2 0
0
6 2 40
EI EI EI
L L L
EI EI EI EI EI EI
L L L L L L v P
EI EI EI EI EI EI
L L L L L L
EI EI EI
L L L
This assembly procedure can be carried out very systematically on a computer.
Define the following (e represents the element number)
Multiple Beam Elements
Matrix Methods18
2
2
1
1
M
V
M
V
f e
2
2
1
1
v
v
d e
22
22
3
4626
612612
2646
612612
LLLL
LL
LLLL
LL
L
EIk e
For the simply supported beam discussed before, we can now formulate the unconstrained system equations.
Multiple Beam Elements
Matrix Methods19
3
3
2
2
1
1
3
3
2
2
1
1
244
243
242
241
234
233
232
231
224
223
222
144
221
143
142
141
214
213
212
134
211
133
132
131
124
123
122
121
114
113
112
111
00
00
00
00
T
R
T
R
T
R
v
v
v
kkkk
kkkk
kkkkkkkk
kkkkkkkk
kkkk
kkkk
Where: v1, θ1, R1, T1 = displacement, slope, force and moment at node 1v2, θ2, R2, T2 = displacement, slope, force and moment at node 2v3, θ3, R3, T3 = displacement, slope, force and moment at node 3
Now apply boundary conditions, external forces, and moments.
Multiple Beam Elements
Matrix Methods20
0
0
0
0
0
00
00
00
00
3
1
3
2
2
1
244
243
242
241
234
233
232
231
224
223
222
144
221
143
142
141
214
213
212
134
211
133
132
131
124
123
122
121
114
113
112
111
R
P
R
v
kkkk
kkkk
kkkkkkkk
kkkkkkkk
kkkk
kkkk
PV
TTT
vv
2
321
31
0
0
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).
Multiple Beam Elements
Matrix Methods21
0
0
0
0
0
3
2
2
1
244
242
241
224
222
144
221
143
142
214
212
134
211
133
132
124
123
122
Pv
kkk
kkkkkk
kkkkkk
kkk
Simply supported beam with mid-span load
Examples
Matrix Methods22
L/2
1 32
L/2
P
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.
Distributed Loads
Matrix Methods23
1 2
L
M2V2
M1 V1
w
x
y (+v)
Note: V and M are internal shear and bending moment, w is external load.
Distributed Loads
Matrix Methods24
Integrate the load-deflection differential equation to find expressions for shear force, bending moment, slope, and deflection.
wdx
vdEI
4
4
Vcwxdx
vdEI 13
3
Mcxcwx
dx
vdEI 21
2
2
2
2
EIcxcx
cwx
dx
dvEI 32
2
1
3
26
43
2
2
3
1
4
2624cxc
xc
xc
wxEIv
Distributed Loads
Matrix Methods25
Express slope and deflection at each node in terms of integration constants c1, c2, c3, and c4.
EI
cvv 4
1 )0(
EI
c
dx
dv 31 )0(
43
22
31
4
2 2624
1)( cLc
LcLcwL
EILvv
32
21
3
2 26
1)( cLc
LcwL
EIL
dx
dv
Note: ν and θ (deflection and slope) are the same in the element as for the whole beam.
Distributed Loads
Matrix Methods26
Written in matrix form
EI
wLEI
wLv
v
c
c
c
c
EIEI
L
EI
LEIEI
L
EI
L
EI
LEI
EI
6
24
01
2
1
26
01
00
1000
3
2
4
2
1
1
4
3
2
1
2
23
Distributed Loads
Matrix Methods27
Solve for integration constants.
EI
wLEI
wLv
v
EI
EIL
EI
L
EI
L
EI
L
EIL
EI
L
EI
L
EI
L
EI
c
c
c
c
6
24
000
000
2646
612612
3
2
4
2
1
1
22
2323
4
3
2
1
Distributed Loads
Matrix Methods28
Express shear forces and bending moments in terms of the constants.
11)0( cVV
222312131
612612
2
L
EIv
L
EI
L
EIv
L
EIwLV
21)0( cMM
222112
2
1
2646
12
L
EIv
L
EI
L
EIv
L
EIwLM
12)( cwLVLV
222312132
612612
2
L
EIv
L
EI
L
EIv
L
EIwLV
21
2
2 12)( cLc
wLMLM
222112
2
2
4626
12
L
EIv
L
EI
L
EIv
L
EIwLM
Distributed Loads
Matrix Methods29
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.
L
LwL
M
V
M
V
v
v
LLLL
LL
LLLL
LL
L
EI
6
6
12
4626
612612
2646
612612
2
2
1
1
2
2
1
1
22
22
3
Propped cantilever beam with uniform load
Examples
Matrix Methods30
12
L
w
Cantilever beam with uniform load
Examples
Matrix Methods31
12
L
w
Cantilever beam with moment and partial uniform load
Examples
Matrix Methods32
13
L1
w
2
L2
Mo
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.
Finite Element Analysis of Beams
Matrix Methods33
Finite Element Analysis of Beams
Matrix Methods34
Nodes: 5Elements: 4kunconstrained: 10 x 10
Apply B.C.’s: v1=v5=0θ5=0
kconstrained: 7 x 7
15
w
2 3 4
P
Finite Element Analysis of Beams
Matrix Methods35
15
w
2 3 4
P
Nodes: 5Elements: 4kunconstrained: 10 x 10
Apply B.C.’s: v1=v3=v5=0θ1=0
kconstrained: 6 x 6