numerical methods in applied structural mechanics
DESCRIPTION
Examples of Geometric NonlinearityTRANSCRIPT
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Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon
Lecture notes:
Prof. Maurício V. Donadon
NUMERICAL METHODS IN APPLIED STRUCTURAL
MECHANICS
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Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon
Examples of Geometric Nonlinearity
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Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon
Shallow truss problem
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Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon
Shallow truss problem
Initial configurationz
w
W
L
sK
E, A
θ
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Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon
Shallow truss problem formulation
( )sin s sN z wW N K w K wL
1/ 2 1/ 22 2 2 2 2
1/ 2 22 2
( ) 12
z w L z L zw wl lz L
Strain-displacement relationship for the bars
Equilibrium equation
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Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon
Shallow truss problem formulation
2
2
12
zw wN EA EAl l
2 2 33
3 12 2 s
EAW z w zw w K wL
Internal force in the bar
Resultant equilibrium equation
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Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon
Shallow truss problem simulation
Truss dimensions and properties:
• EA = 50 MN
• z = 25 mm
• L = 2500 mm
• Ks = 1.35 N/mm
• Ks = 1.0 N/mm
• Ks = 0.0 N/mm
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Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon
Shallow truss problem simulation
Solution methods to be tested under load control:
• Incremental solutions (EULER)
• Iterative solutions (N-R)
• Combined incremental/iterative solutions (EULER + N-R)
• Quasi-static solutions (DYNAMIC RELAXATION)
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Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon
Shallow truss problem simulationANALYTICAL SOLUTION
0 0.5 1 1.5 2-0.2
0
0.2
0.4
0.6
0.8
-w/z
-(WL3 )/(
EAz3 )
Ks=0.0Ks=1.00Ks=1.35
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Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon
Shallow truss problemEULER METHOD
0 0.5 1 1.50
0.2
0.4
0.6
0.8
1
-w/z
(-WL3 )/(
EAz3 )
Ks=1.35
Exact solutionLoad increment = 10 NLoad increment = 5.0 NLoad increment = 1.0 NLoad increment = 0.1 N
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Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon
Shallow truss problemEULER METHOD
0 10 20 30 40 50-10
-8
-6
-4
-2
0
2
Fe [N]
g(w
) [N
]
Ks=1.35
Load increment = 10 NLoad increment = 5 NLoad increment = 1 NLoad increment = 0.1 N
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Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon
0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
1
-w/z
(-WL3 )/(
EAz3 )
Ks=1.0
Exact solutionLoad increment = 10 NLoad increment = 5 NLoad increment = 1 NLoad increment = 0.1 N
Shallow truss problemEULER METHOD
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Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon
Shallow truss problemEULER METHOD
0 10 20 30 40 50-20
-15
-10
-5
0
5
10
Fe [N]
g(w
) [N
]
Ks=1.0
Load increment = 10 NLoad increment = 5 NLoad increment = 1 NLoad increment = 0.1 N
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Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon
Shallow truss problemEULER METHOD
0 0.5 1 1.5 2 2.5-0.2
0
0.2
0.4
0.6
0.8
-w/z
-(WL3 )/(
EAz3 )
Ks=0.0
Exact solutionLoad increment = 0.1 NLoad increment = 0.01 N
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Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon
Shallow truss problemN-R METHOD
0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
1
-w/z
(-WL3 )/(
EAz3 )
Ks=1.35Tol=1.0e-3
Exact solutionLoad increment = 10 NLoad increment = 1 N
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Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon
Shallow truss problemN-R METHOD
0 10 20 30 40 50-6
-4
-2
0
2
4
6
8
10 x 10-8
Fe [N]
g(w
) [N
]
Load increment = 10 NLoad increment = 1 N
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Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon
Shallow truss problemN-R METHOD
0 10 20 30 40 502
3
4
5
6
7
8
Fe [N]
Itera
tions
Ks=1.35
Load increment = 10 NLoad increment = 1 N
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Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon
Shallow truss problemN-R METHOD
0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
1
-w/z
(-WL3 )/(
EAz3 )
Ks=1.0Tol=1.0e-3
Exact solutionLoad increment = 10 NLoad increment = 1 N
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Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon
Shallow truss problemN-R METHOD
0 10 20 30 40 50-0.5
0
0.5
1
1.5
2
2.5
3 x 10-4
Fe [N]
g(w
) [N
]
Load increment = 10 NLoad increment = 1 N
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Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon
Shallow truss problemN-R METHOD
0 10 20 30 40 500
5
10
15
20
Fe [N]
Itera
tions
Ks=1.0
Load increment = 10 NLoad increment = 1 N
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Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon
Shallow truss problemN-R METHOD
0 0.5 1 1.5 2 2.5-0.2
0
0.2
0.4
0.6
0.8
-w/z
(-WL3 )/(
EAz3 )
Ks=0.0Tol=1.0e-3
Exact solutionLoad increment = 10 NLoad increment = 1 NLoad increment = 0.1 N
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Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon
0 0.2 0.4 0.6 0.8 1-50
-40
-30
-20
-10
0
Normalized time (t/tmax)
Forc
e (N
)
Shallow truss problemDYNAMIC RELAXATION – Ks=1.35
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Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon
Shallow truss problemDYNAMIC RELAXATION
0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
1
-w/z
(-WL3 )/(
EAz3 )
Ks=1.35
Exact solution500 N/s50 N/s5 N/s
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Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon
Shallow truss problemDYNAMIC RELAXATION – 500 N/s – Ks=1.35
0 0.02 0.04 0.06 0.08 0.10
100
200
300
400
500
600
Time (s)
U, K
Internal energyKinetic energy
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Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon
Shallow truss problemDYNAMIC RELAXATION – 50 N/s – Ks=1.35
0 0.2 0.4 0.6 0.8 10
200
400
600
800
Time (s)
U, K
Internal energyKinetic energy
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Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon
Shallow truss problemDYNAMIC RELAXATION – 5 N/s – Ks=1.35
0 2 4 6 8 100
200
400
600
800
Time (s)
U, K
Internal energyKinetic energy
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Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon
Shallow truss problemDYNAMIC RELAXATION – Ks=1.0 – Ks=1.35
0 0.2 0.4 0.6 0.8 1-50
-40
-30
-20
-10
0
Normalized time (t/tmax)
Forc
e (N
)
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Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon
Shallow truss problemDYNAMIC RELAXATION – Ks=1.0
0 0.5 1 1.5 2 2.50
0.2
0.4
0.6
0.8
1
-w/z
(-WL3 )/(
EAz3 )
Ks=1.0
Exact solution50 N/s5 N/s0.5 N/s
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Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon
Shallow truss problemDYNAMIC RELAXATION – Ks=1.0
0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
1
-w/z
(-WL3 )/(
EAz3 )
Ks=1.0
Exact solution0.5 N/s (Alpha=36.9, Beta=0.0247)
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Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon
Shallow truss problemDYNAMIC RELAXATION – Ks=1.0
0 20 40 60 80 1000
200
400
600
800
Time (s)
U, K
, Ud
Internal energyKinetic energyDissipated energy
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Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon
Shallow truss problemDYNAMIC RELAXATION – Ks=0.0
0 0.5 1 1.5 2 2.5 3-0.2
0
0.2
0.4
0.6
0.8
-w/z
(-WL3 )/(
EAz3 )
Exact solutionDynamic relaxation - 0.5 N/s
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Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon
Shallow truss problemDYNAMIC RELAXATION – Ks=0.0 – Disp. control
0 20 40 60 80 100-70
-60
-50
-40
-30
-20
-10
0
Time (s)
w (m
m)
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Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon
Shallow truss problemDYNAMIC RELAXATION – Ks=0.0 – Disp. control
0 0.5 1 1.5 2 2.5 3-0.2
0
0.2
0.4
0.6
0.8
1
1.2
-w/z
(-WL3 )/(
EAz3 )
Dynamic relaxation (Fe = 0.0)Exact solution
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Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon
Shallow truss problemARC-LENGTH METHOD – Ks=1.35
0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
1
-w/z
(-WL3 )/(
EAz3 )
Exact solutionArc-length method (Arc-length=1)
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Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon
Shallow truss problemARC-LENGTH METHOD – Ks=1.0
0 0.5 1 1.5 2 2.50
0.2
0.4
0.6
0.8
1
-w/z
(-WL3 )/(
EAz3 )
Exact solutionArc-length method (Arc-length=1)
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Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon
Shallow truss problemARC-LENGTH METHOD – Ks=0.0
0 0.5 1 1.5 2 2.5-0.5
0
0.5
1
-w/z
(-WL3 )/(
EAz3 )
Exact solutionArc-length method (Arc-length=1)