EQE 6108
Dynamics of Structures
Lecture 1 - Fundamentals of structural dynamics
Dr. Shafayat Bin Ali
Assistant Professor (Research)
Institute of Earthquake Engineering Research (IEER)
Chittagong University of Engineering & Technology (CUET)
Chittagong-4349, Bangladesh
Dynamics:
The term dynamic may be defined simply as time varying; thus a dynamic load is any load of
which its magnitude, direction, and/or position varies with time. Similarly, the structural
response to a dynamic load, i.e., the resulting stresses and deflections, is also time varying, or
dynamic.
Dynamic loads on structures originate from a variety of sources such as
- human activities (e.g., walking, running, dancing or skipping
- working machines, inside a building or in the surroundings
- construction work (e.g., piling, mechanical excavations or drilling and blasting activities)
- moving loads on bridges
- car, train or airplane accidents
- impact loads (e.g., falling debris)
- collapse of a structural element
- wind loads, wind gusts
- air blast pressure
- loss of support because of ground failure
- earthquake
Types of Dynamic loading:
1. Prescribed dynamic loading: If the time variation of loading is fully known, even though it may
be highly oscillatory or irregular in character, it is referred as a prescribed dynamic loading.
(a) The simplest periodic
loading has the
sinusoidal variation
which is termed simple
harmonic.
(b) Other forms of periodic
loading, e.g., those
caused by hydrodynamic
pressures generated by a
propeller at the stern of a
ship
(c) Nonperiodic loadings
may be either short
duration impulsive
loadings or long duration
general forms of loads.
2. Random dynamic loading:
If the time variation is not completely known but can be defined in a statistical sense,
the loading is termed a random dynamic loading.
• The ground motion time history for a hypothetical future earthquake is random.
• No one knows exactly what it will look like
• Modeling can be done:
* The magnitude and duration of a future earthquake
* The frequency content of the earthquake
Approaches for evaluating structural response to dynamic loads:
1. Deterministic: The analysis of the response of any specified structural system to
a prescribed dynamic loading is defined as a deterministic analysis. A
deterministic analysis leads directly to displacement time-histories
corresponding to the prescribed loading history.
2. Nondeterministic: The analysis of the response of any specified structural system
to a random dynamic loading is defined as a nondeterministic analysis. A
nondeterministic analysis provides only statistical information about the
displacements resulting from the statistically defined loading.
* The choice of method to be used in any given case depends upon how the loading
is defined.
* In general, structural response to any dynamic loading is expressed basically in
terms of the displacements of the structure.
Difference between Dynamic and Static problem:
Dynamic
Static
1. Both loading and response vary with time, it 1. Static problem has single solution
is evident that a dynamic problem does not
have a single solution (the analyst must
establish
a
succession
of
solutions
corresponding to all times of interest in the
response history)
2. The resulting response of structure depend 2. The resulting response of structure
not only upon the load but also upon inertial depends on the load
forces which oppose/resist the accelerations.
3. A dynamic analysis is more complex and
time consuming.
3. Static analysis is relatively simple.
Basic
Inertial Force:
An inertial force is a force that resists a change in velocity of an object. It is equal to—and in the
opposite direction of—an applied force, as well as a resistive force.
The concept is based on Newton's Laws of Motion, including the Law of Inertia and the ActionReaction Law.
•
Every object in a state of uniform motion tends to remain in that state of motion unless an
external force is applied to it (Law of Inertia).
•
The relationship between an object's mass m, its acceleration a, and the applied force F is:
F = ma.
•
If a force is applied to an object, there is an equal and opposite reaction (Action-Reaction
Law).
That equal and opposite reaction is called the inertial force. It is equal to −F = ma.
Simple Structures:
• If the mass of structure is distributed continuously along its length, the displacements
and accelerations must be defined for each point along the axis the analysis must be
formulated in terms of partial differential equations because position along the span as
well as time must be taken as independent variables.
• If one assumes the mass of the beam to be concentrated at discrete points the
analytical problem becomes greatly simplified because inertial forces develop only at
these mass points.
• Thus a structure is termed as simple when it can be idealized as a concentrated or
lumped mass ‘m’ supported by a massless structure with stiffness ‘k’ in the lateral
direction.
Degree-of-freedom (DOF):
The number of independent displacements required to define the displaced positions
of all the masses relative to their original position is called the number of degrees of
freedom (DOFs) for dynamic analysis.
• The three masses in the system of Fig are fully concentrated and are constrained so that the
corresponding mass points translate only in a vertical direction, this would be called a three
degree of freedom (3 DOF) system.
• If the masses are not fully concentrated so that they possess finite rotational inertia, the
rotational displacements of the three points will also have to be considered, in which case
the system has 6 DOF.
• If axial distortions of the beam are significant, translation displacements parallel with the
beam axis will also result giving the system 9 DOF.
• If the structure can deform in three-dimensional space, each mass will have 6 DOF; then
the system will have 18 DOF.
Single-Degree-of-freedom (DOF):
Consider the one-story frame of Fig, constrained to move only in the direction of the
excitation.
• The static analysis problem has to be formulated with three DOFs—lateral
displacement and two joint rotations—to determine the lateral stiffness
• In contrast, the structure has only one DOF—lateral displacement—for dynamic
analysis if it is idealized with mass concentrated at one location, typically the roof
level. Thus we call this a single-degree-of-freedom (SDF) system.
Force–displacement Relation:
• In figure the system subjected to an externally applied static force fs along the DOF
u as shown.
• The internal force resisting the displacement u is equal and opposite to the external
force fs.
• The force displacement relation would be linear at small deformations but would be
nonlinear at large deformation.
Linearly Elastic Systems:
• For a linear system the relationship between the lateral force fs and resulting
deformation u is linear.
• This linear relationship implies that fs is a single-valued function of u (i.e., the
loading and unloading curves are identical).
• A system is said to be elastic; hence the term linearly elastic system to emphasize
both properties.
fs = ku
where k is the lateral stiffness of the system; its units are force/length
Inelastic Systems:
• Determined by experiments
• The initial loading curve is nonlinear at the larger amplitudes of deformation, and
the unloading and reloading curves differ from the initial loading branch; such a
system is said to be inelastic.
• This implies that the force–deformation relation is path dependent, i.e., it depends
on whether the deformation is increasing or decreasing.
• Thus the resisting force is an implicit function of deformation
fs = fs(u)
Determination of Inelastic system:
The force–deformation relation for the idealized
one-story frame deforming into the inelastic range
can be determined in one of two ways:
(a) One approach is to use methods of nonlinear
static structural analysis.
• For example, in analyzing a steel structure with
an assumed stress–strain law, the analysis keeps
track of the initiation and spreading of yielding at
critical locations and formation of plastic hinges
to obtain the initial loading curve (o–a).
• The unloading (a–c) and reloading (c–a) curves
can be computed similarly or can be defined from
the initial loading curve using existing
hypotheses.
(b) Another approach is to define the inelastic forcedeformation relation as an idealized version of the
experimental data.
Damping:
• The process by which vibration steadily diminishes in amplitude is called damping.
• The kinetic energy and strain energy of the vibrating system are dissipated by
various damping mechanisms.
• Most of the energy dissipation presumably arises from the thermal effect of repeated
elastic straining of the material and from the internal friction when a solid is
deformed.
• In a vibrating building these include friction at steel connections, opening and
closing of microcracks in concrete, and friction between the structure itself and
nonstructural elements such as partition walls
• It seems impossible to identify or describe mathematically the damping in actual
structures is usually represented in a highly idealized manner.
• The damping coefficient is selected so that the vibrational energy it dissipates is
equivalent to the energy dissipated in all the damping mechanisms, combined,
present in the actual structure.
• This idealization is therefore called equivalent viscous damping
Damping Force:
• Figure shows a linear viscous damper subjected to a force fD along the DOF u. The
internal force in the damper is equal and opposite to the external force fD.
• As shown the damping force fD is related to the velocity across the linear viscous
damper by
• where the constant c is the viscous damping coefficient; it has units of force ×
time/length
• Unlike the stiffness of a structure, the damping coefficient cannot be calculated from
the dimensions of the structure and the sizes of the structural elements.
• Vibration experiments on actual structures provide the data for evaluating the
damping coefficient
• Nonlinearity of the damping property is usually not considered explicitly in dynamic
analyses.
• It may be handled indirectly by selecting a value for the damping coefficient that is
appropriate for the expected deformation amplitude.
• Damping energy dissipated during one deformation cycle between deformation
limits is given by the area within the hysteresis loop.
Formulation of Equations of Motion:
• The primary objective of a deterministic structural-dynamic analysis is the
evaluation of the displacement time histories of a given structure subjected to a
given time-varying loading.
• In most cases, an approximate analysis involving only a limited number of
degrees of freedom will provide sufficient accuracy; thus, the problem can be
reduced to the determination of the time histories of these selected displacement
components.
• The mathematical expressions defining the dynamic displacements are called the
equations of motion of the structure, and the solution of these equations of motion
provides the required displacement time histories.
• The formulation of the equations of motion of a dynamic system is possibly the
most important, and sometimes the most difficult, phase of the entire analysis
procedure.
Equation of Motion: External Force
Using Newton’s Second Law of Motion
Dynamic Equilibrium
0
Stiffness, Damping and Mass Components
• The system as the combination of three pure components:
1. The stiffness component is related to the displacement: the frame without
damping or mass
2. The damping component is related to the velocity: the frame with its damping
property but no stiffness or mass
3. The mass component is related to the acceleration: the roof mass without the
stiffness or damping of the frame
• The external force p(t) distributed among the three components of the structure and
fS + fD + fI must equal the applied force p(t).
MASS–SPRING–DAMPER System
• The classic SDF system is the mass–spring–damper system
• If we consider the spring and damper to be massless, the mass to be rigid, and all
motion to be in the direction of the x-axis, we have an SDF system.
• Figure shows the forces acting on the mass; these include the elastic resisting force,
fS = ku exerted by a linear spring of stiffness k, and the damping resisting force,
fD = cu due to a linear viscous damper.
• It is clear that the equation of motion derived earlier for the idealized one-story
frame of is also valid for the mass–spring–damper system.
Equation of Motion: Earthquake Excitation
Rotational Component of Earthquake Excitation
• Although the rotational components of ground motion are not measured during
earthquakes, they can be estimated from the measured translational components.
• For this purpose, consider the cantilever tower, which may be considered as an
idealization of the water tank, subjected to base rotation θg. The total displacement
u(t) of the mass is made up of two parts: u associated with structural deformation
and a rigid-body component hθg, where h is the height of the mass above the base.
At each instant of time these displacements are related by
• Putting all these equations together leads to
• The effective earthquake force associated with ground rotation is
Basic
Methods of Solution of Differential Equation
Mark Distribution (300 marks)
1. Attendance : 10%
2. Quiz : 20%
3. Assignment : 20%
4. Final exam: 50%
Assignment 1:
Date of submission: 7-9-2023
Book: Dynamics of Structures – Anil K. Chopra (4th edition)
Examples: 1.1, 1.2, 1.3, 1.4, 1.5, 1.6
Problems: 1.1, 1.2, 1.3, 1.4, 1.13, 1.14, 1.15, 1.16
Total: 14 math