Dynamic Response Analysis of

1
Chapter 2 Dynamic Response Analysis of Bridge
Structures
2008 Course
Kazuhiko Kawashima
Tokyo Institute of Technology
2
Uniqueness of Bridges for Structural Analysis
Bridges are

• long in the longitudinal direction, and
  structural
• properties and soil condition are not the same
  along the
• axial axis
• consists of many structural types (deck bridges,
  arch
• bridges, cable stayed bridges, suspension
  bridges, .)
• have various shapes (straight, skewed, curved,
• separation into several segments, and
  combination of
• those types)
• have various heights (short bridges and high
  bridges)
• are made of various materials (RC, PC, steel,
• composites)

3
Types of Analysis
4
Idealization of a Bridge

• Idealization of a superstructure substructures
• Idealization of foundations
• Idealization of soil response
• Multiple excitation effect (out of coherent GM)

5
Analytical Model of Superstructures
6
Analytical Model of Substructures
7
Idealization of Substructures
Soil springs
8
3D FEM Idealization
9
Plastic Deformation of Columns
10
Idealization of Bridges
Stiffness Idealization
Total stiffness matrix
Element stiffness matrix
Time dependent stiffness
Mass Idealization
Total mass matrix Element mass matrix
11
Viscous damping is not the only one mechanism of
producing damping force of bridges, but there are
various sources energy dissipation mechanism
which contribute to damping
12
Why is viscous damping force assumed in most
analyses?

• Closed form solution can be easily available by
  assuming viscous damping force, however it cannot
  be obtained in other expressions.

13
Why is viscous damping force assumed in most
analyses? (continued)

• Damping ratio can be a common value which can be
  used in a wide range of application.
• It is known based on the past studies that
  viscous damping force can represent the
  structural response with reasonable accuracy if
  other type of energy dissipation mechanism is
  appropriately idealized in terms of viscous
  damping force.

14
How other types of energy dissipation are
idealized in terms of viscous damping force?
Equivalent damping force
15
Idealization of Damping Force

• There are various sources which contribute to
  energy
• dissipation.
• It is common to idealize the energy dissipation
• in terms of the viscous damping.

• Since the valuation of element damping matrix is
  generally difficult, the system damping ratio is
  often assumed by the Rayleigh damping as

(2.5)
16
Orthogonarity condition
17
(No Transcript)
18
This assumption sometimes results in a
problem, because

• Solution becomes unstable sometimes
• Does not capture the fact that inelastic
  response of
• structural members dissipate energy which results
  in an
• increase of damping ratio

19
How can we determine the modal damping ratios by
assigning damping ratios of each structural
components?

• Theoretically, damping ratio is defined for a
  SDOF system. If we can assume the oscillation of
  each structural component as a SDOF system, it
  may be possible to assign a damping ratio for
  each structural component.

20
How can we determine the modal damping ratios by
assigning damping ratios of each structural
components? (continued)

• There is not a single method which is exact and
  easy for implementation for design purpose.
• Following empirical methods are frequently used
• Strain energy proportional method
• Kinematic energy proportional method

21
Strain Energy Proportional Method
22
Because
Strain energy of m-th element for k-th mode is
Therefore, the total energy dissipation of the
system is
23
(No Transcript)
24
Kinematic Energy Proportional Damping Ratio
25
Which is better for determining modal damping
ratios between the strain energy proportional
method and kinematic energy proportional method?

• Damping ratios of the structural components where
  large strain energy is developed are emphasized
  in the strain energy proportional method.

Plastic deformation of columns
Plastic deformation of foundations soils

• Strain energy proportional method is better in a
• system in which hysteretic energy dissipation is
  predominant

26
Which is better for determining modal damping
ratios between the strain energy proportional
method and kinematic energy proportional method?

• Damping ratios of the structural components with
  larger kinematic energy are emphasized in the
  kinematic energy proportional method.

• Kinematic energy proportional method is better in
  a
• system in which hysteretic energy dissipation is
  less significant

27
Equations of Motion
Single-degree-of-freedom oscillator
28
Equations of Motion (continued)
Multi-degree-of-freedom system
29
Equations of Motion (continued)
Multiple Excitation
free nodal points
non-zero support displacements
30
quasi-static displacements
(2.11)
dynamic displacements
From definition,
31
The equation of motions can be written by
enlarging the mass, damping and stiffness
matrices as well as the dynamic load vector to
account for the nb support displacements
(2.12)
The equations of motion associated with n free
nodal point displacement become
(2.13)
32
Substitution of Eq. (2.11) into Eq. (2.13) yields
By definition of the quasi-static displacement
(2.15)
0
small compared to inertia force
(2.14)
33
From
(2.16)
(2.17)
nb x 1
n x n
n x nb
34
Multiple Support Excitation
(2.17)
(2.18)
(2.19)
35
Rigid Support Excitation
n x n
n x 3
3 x 1
36
Linear Analysis
a) Natural Mode Shapes and Natural Frequencies
37
Linear Analysis (continued)
where,
38
where,
Only assumption
39
Solving equation of motion for a SDOF system
Time History Analysis
Direct integration method
Determine
Determine force, stress and strain
40
Response Spectral Method
Equation of motion for the i-th generalized
coordinate is
Instead of solving the equation, we can have the
peak qi using displacement response spectrum as
(2.31)
41
Then the maximum displacement of the ith mode can
be written as
(2.32)
Therefore, the peak displacement u can be
obtained by taking root of square summation of
each mode
(2.33)
Forces of the i-th mode
42
Nonlinear Dynamic Response Analysis
Equations of motion in the incremental form
(2.47)
43
Nonlinear Dynamic Response Analysis (continued)
Newmarks generalized acceleration method
44
Newmarks generalized acceleration method
(continued)
Constant Acceleration Method

45
Newmarks generalized acceleration method
(continued)
Linear Acceleration Method
46
Newmarks generalized acceleration method
(continued)
(2.50)
constant acceleration method
linear acceleration method
47
Newmarks generalized acceleration method
(continued)
(2.51)
linear acceleration method
constant acceleration method
48
Newmarks generalized acceleration method
(continued)
(2.47)
(2.51)
(2.52)
where,
(2.53)
(2.54)
49
Accuracy of Computed Responses
Overshooting
50
Accuracy of Computed Responses (continued)
Computed restoring force
Unbalance force
51
Accuracy of Computed Responses (continued)
. . .
If error gt tolerance,
52
Accuracy of Computed Responses (continued)
If error gt tolerance,

• re-compute using a smaller time increment of
  numerical integration-This is always effective to
  improve the stability and accuracy of solution.
• add unbalanced forces to the incremental forces
  at the next time step
• use a numerical iteration

53
Add unbalanced forces to the incremental forces
at the next time step
Unbalance Force at time t
Incremental Equations of Motion
Add the unbalance force to the incremental
external force
54
Add unbalanced forces to the incremental forces
at the next time step (continued)
This method is effective when unloading and
reloading are important
55
Numerical Iteration for the Equilibrium of
Equations of Motion
56
Computer Soft-wares for Dynamic Response Analysis
General Purpose Soft-wares

• ASKA
• DYNA
• ABAQUA
• SAP2000

• Multi-purposes
• Well maintenance
• Some kind of consensus for the results
• Not so easy to modify
• User routines may be included depending on
  programs

57
Computer Soft-wares for Dynamic Response
Analysis (continued)
Hand-made

• Easy to include problem-oriented routines
• Difficult for maintenance for the use of other
  groups and long-term maintenance
• Few consensus for the results

• Open-base forum for source code
• Prepare well documented manual and example
  problems

58
Structures of Computer Programs for Dynamic
Response Analysis
Input structural shape (coordinate) properties
Input ground motions
Form time invariant structural properties such as
mass matrix, stiffness of elastic elements
At time t
Form time dependent properties such as stiffness
of nonlinear element
Solve
(2.52)
59
Check the accuracy by Eq. (2.62) or similar forms
If the accuracy is not enough, use small smaller
time increment, or add unbalance force to the
next incremental force, or iteration
Store the responses on a file
Repeat until the end of ground motion