Download Structural dynamics and earthquake engineering

Structural dynamics and earthquake engineering
1. Define the concept of dynamic degree of freedom. Give some examples of single degree
of freedom systems and multi degree of freedom systems.
Answer: Dynamic degrees of freedom are a set of independent displacements/rotations that
completely define the displaced position of the mass with respect to its initial position.
Examples:
• A vertical cantilever with the mass concentrated at its tip can be
idealised as a single degree of freedom (SDOF) system. The
degree of freedom is represented by the lateral displacement u
of the mass.
•
A multi-storey frame with the masses concentrated the storey
levels can be idealised as a multi degree of freedom (MDOF)
system. The degrees of freedom are the lateral displacements of
the storey masses. The system in the next figure has 4 dynamic
degrees of freedom – displacements u1 – u4.
4
u3
u2
u1
2. Write the equation of motion of a single degree of freedom systems subjected to a
dynamic force and explain its terms. Show using a sketch an example of a dynamic
system like this.
Answer: The equation of motion of a single degree of freedom (SDOF) system subjected to
an external force p(t) is the following:
mu&&( t ) + cu& ( t ) + ku ( t ) = p ( t )
•
The term mu&&( t ) represents the inertia force due to acceleration u&&( t ) of the mass m.
•
The term cu& ( t ) represents the viscous damping force equal to the product between the
damping coefficient c and velocity u& ( t ) .
•
The term ku ( t ) represents the reactions produced by the system with stiffness k due to the
displacement u ( t ) .
In the next figure is shown a SDOF system with mass m,
damping coefficient c and stiffness k, subjected to the
external force p(t).
3. Explain the lateral force method used to assess the seismic response of multi-storey
structures. State the limitations of the method.
Answer: The lateral force method (LFM) is a simplified structural analysis method which can
be used in the case of multi-storey structures whose response is governed by the fundamental
mode of vibration. According to P100-1/2006, the LFM can be used only for structures
which:
• are regular in elevation,
• have a period of vibration less than 1.5 sec. and
• are less than 30 m high.
The LFM is essentially a modal response spectrum analysis, which considers only the
fundamental mode of vibration. The method consists in a static analysis of the structure
subjected to the lateral forces Fi. These forces are determined by distributing along the height
of the structure the base shear force Fb, which is determined according to the following
formula:
Fb = Sd (T1 ) mλ
where: Sd (T1 ) is the design spectral acceleration corresponding
to the fundamental period of vibration T1; m is the total mass of
the building; λ is a correction factor.
Fi
mi
zi
In the simplified variant, lateral forces Fi are determined
according to the following expression:
Fi = Fb
mi zi
N
∑m z
i i
i =1
where mi is the mass of storey i, and zi is the height of storey i
with respect to the base of the structure.
Fb
4. Discuss the measures for conceptual seismic design of buildings from the point of view of
torsional resistance and stiffness.
Answer: Lateral force resisting system should be as symmetrical as possible (see the figure
below), in order to obtain as small difference as possible between the centre of mass (CM)
and centre of rigidity (CR) of a structure. If an eccentricity between the CM and CR is
present, the floor will be subjected to a torsional motion, in addition to the translational one.
This results in larger displacements at the flexible side of the building in the direction of the
seismic action. Additionally, displacements perpendicular to the direction of seismic action
will be present.
2x
2x
D2y
Fx
CR=CM
Fx
CM
CR
e0y
D1y
Y
D1x
X
D1x
Structures which are torsionally flexible have larger displacements and forces in the
components located on the perimeter of the building, and a non uniform distribution of
deformations and stresses in structural members. Lateral force resisting systems should be
located on the perimeter of the building (see figure below) in order to obtain increased
torsional strength and stiffness.
lateral-force
resisting system
lateral-force
resisting system
gravity-force
resisting system
gravity-force
resisting system
Structure with low torsional stiffness
Structure with large torsional stiffness
5. Which are the essential differences between the design concepts of dissipative and lowdissipative structural behaviour from the following points of view:
- calculation of the design seismic action;
- design of structural components.
Answer: Design of structures based on the concept of dissipative structural behaviour is based
on values of the behaviour factor q substantially larger than 1, which results in low values of
the design seismic action. This leads to structures with low strength, which should be
compensated through adequate ductility. Dissipative structural components are designed for
forces in the seismic design situation and should fulfil a set of additional rules intended to
provide them with a ductile response. Plastic deformations should be prevented in the nondissipative components through an overstrength with respect to the dissipative ones. The
design forces in the non-dissipative components are determined based on capacity design,
corresponding to yielded and strain hardened dissipative components.
In the case of low-dissipative structural behaviour, the design seismic action is determined
based on values of the behaviour factor q between 1 and 1.5 (in P100-1/2006 only q=1 is
allowed). In this case the structure relies on strength to resist the seismic action. Structural
components are designed for the forces in the seismic design situation, similarly to the design
in the persistent (fundamental) design situation. Special design rules intended to provide for a
ductile response of the structure are not needed in this case.