MENG IN STRUCTURAL
ENGINEERING
CEE 6241 P7
CALCULATION OF STRUCTURAL RESPONSE
By
Dr M N Mulenga
FEIZ, MASCE, RENG
SOME ACRONYMS
• AIJ Architectural Institute of Japan
• JDD Japanese Damping Database
2
Introduction
• Earthquake loading poses the structural
analyst with one of the most challenging
problems in engineering. A violent and
essentially unpredictable dynamic ground
motion imposes extreme cyclic loads on
engineering materials whose response under
such conditions is complex and incompletely
understood
3
Response
4
Resonance
5
Rigid Systems
• Very rigid systems with low periods track the
ground motion closely. The normalised
response therefore tends to unity as the
system period tends to zero, or (equivalently)
as the ground motion period becomes very
long in comparison to the period of the
structure.
6
Flexible Systems
• Very flexible springs, on the other hand, act to
isolate their masses from the input motion and so
response tends to zero where the period of the
structure is very long compared to that of the
ground motion. In other words, response
becomes small for very long-period structures or
for very short-period motions.
• This is the principle behind, for example, isolation
mounts for rotating machinery and also seismic
isolation systems for earthquake-resistant
buildings.
7
Response
• Figure 3.1 describes the steady-state response to
constant-amplitude single period motions. By
contrast, earthquakes are transient phenomena
and the associated ground motions contain a
range of periods. Nevertheless, certain periods
tend to predominate, depending chiefly on the
magnitude of the earthquake and the soil
conditions at the site. The match between these
predominant periods and the periods of a
particular structure is crucial in determining its
response.
8
Response Spectrum for a typical
Earthquake
9
Damping in Buildings
Reduction of intensity with time or spatial
propagation
 Cease of vibration with time
 Reduction of wind-induced/earthquake
induced vibration
 Increase of onset wind speed of aerodynamic
instability
10
Damping in Buildings
Estimation of damping
 No theoretical method
 Based on full-scale data
Significant scatter
11
Factors
• Structural Materials
• Soil & Foundations
• Architectural Finishing
• Joints
• Non-structural Members
• Vibration Amplitude
• Non-stationarity of Excitations
• Vibration Measuring Methods
• Damping Evaluation Techniques
12
Physical Causes of Damping in
Buildings
13
Internal Friction
14
Plasticity Damping
15
Force-Deformation Characteristics
16
Radiation Damping
17
External Viscous Damping
18
Fluid Dynamic Damping (Aerodynamic
Damping)
19
Damping and Building Vibration
20
Damping Ratio of Buildings
21
Current Design Damping Values
22
Damping Values-Steel Buildings
23
Damping Values-RC Buildings
24
DIN 1055 (TEIL 4)
25
26
27
28
29
30
31
References of Damping Data
32
33
34
35
36
37
38
39
40
41
42
43
44
45
Structural Periods
The period of an undamped mass supported on a
spring is equal to:
T = 2π/(√k/m)
as shown in Fig. 3.1.
Doubling the mass therefore increases the period
by about 40%, and the same is true if the stiffness is
halved. More complex structures can have their
natural periods determined from their mass and
stiffness.
46
Approximation for T
Approximation for T for buildings with a regular
distribution of mass and stiffness is:
T ≈ 2/(√δ)
(Eq. 1)
where δ is the lateral deflection in metres of the
top of the building when subjected to its gravity
loads acting horizontally (see for example
Eurocode 8 Part 1 (CEN 2004) equation (4.9) )
47
Seismic Analysis
Two basic methods are widely used for
dynamic seismic analysis, namely,
• Response Spectrum, and
• Time History methods
48
Response Spectrum Method
• Response Spectrum
methods allows
determination of
maximum modal
response of a singly
supported structural
system or a multiple
supported system
where all supports
receive the same
excitation.
49
Time History Method
• Time History method of
analysis permits the
simultaneous
application of different
excitations at each
support point of
uncoupled model of
the system of interest .
50
Earthquake Response Spectra
• Calculating the earthquake response, even of a simple structure
idealised as a linear spring/mass/dashpot system (Fig. 3.2), is
complex. Response spectrum analysis provides a much simpler
method for calculating just the maximum response of the system
during the earthquake, without having to calculate behaviour at
other times.
• Since the maximum response is usually the quantity of greatest
engineering interest, this is both useful and convenient.
• The method relies on the prior calculation of the maximum
response of a series of simple systems with a range of periods,
from short to long and with various levels of damping. The
maxima (called spectral values) are then plotted against the
natural period of the system to produce the response spectrum
shown in Fig. 3.2.
51
Response Spectrum
52
Earthquake Response Spectra (Cont’d)
• Spectra can be plotted for spectral acceleration,
velocity or displacement.
• It can be shown that the spectral (i.e. peak) response of
all idealised linear systems with the same period and
percentage of critical damping is the same for a given
earthquake motion. Thus, a 10-tonne mass with 5%
damping and 1-second period deflects and accelerates
just as much as a 10 kg mass with the same damping
and period when subjected to, say, the motions
recorded during the El Centro earthquake of 1940.
• The response spectrum therefore becomes a powerful
and versatile design tool.
53
Knowing the mass, damping and period of a
structure (providing it can be idealised as a simple
linear spring/mass/dashpot system) and given an
acceleration response spectrum, the following
quantities of interest to the designer, can be derived:
F = m Sa
(Eq 2)
where F is the peak spring force, m is the mass
and Sa is the spectral acceleration.
Sd = F/k
(Eq 3)
where Sd is peak deflection, F is peak spring
force and k is the spring constant.
54
Combining equations ( 1), ( 2) and ( 3) gives
Sd = mSa/(4π2m/T2)
Sd =SaT2/4π2
(Eq 4)
Together, equations (2) and (4) show that with
an earthquake response spectrum, two of the
quantities of most use to earthquake engineers,
namely peak force and peak deflection in a
given earthquake – can be derived for a simple
structure, provided its mass, natural period and
damping are known.
55
Example 8.1
For the Spectral response given in the
figure 3.5, determine the peak
acceleration and peak displacement.
56
57
Example 8.2
The water tank shown in the figure is subjected to the El
Centro earthquake excitation, as shown in the response
spectra. The mass of tank is 7500 kg and the supporting
column is a hollow circular reinforced concrete column,
with outside diameter 800mm and wall thickness
100mm. If the Young’s Modulus of Concrete is 25,000
MPa, and degree of damping,  is 5%, determine:
(a)
(b)
(c)
The maximum relative displacement, xmax
The maximum shear force in the column
The maximum bending stress in the column of
height is 7.5m.
58
Water Tank
59
60
Solution
Mass=7500 kg
Ec= 25,000 MPa
Stiffness of column
Do= 0.8m, Di=800-2x100=600mm=0.6m
I =π(Do4- Di4)/64= 13.744x10-3 m4
Column Stiffness, k
k =3EI/l3
= 3x13.744x10-3x25000x106/7.53
= 2.443x106 N/m
=√k/m=18.05 rads/sec, T=2𝛑/= 0.35 sec
61
Max relative displacement
 = 0.05, T=0.35 sec,
Sd=0.8 in =20.32mm =0.02032m
Max shear force in the column
V = k x Sd
= 2.443x106xx0.02032
= 49641.76 N = 49.642 kN
Maximum Bending Stress in the column
σ = My/I= (Vh)x(Do/2)/I
= (49.642x103x7.5)x0.4/(13.744x10-3)
= 10.836 x106 N/m2 = 10.836 MPa
62
References
• Clough and Penzien
• Chopra
63
END
64