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Response Spectrum Analysis Theory, Benefits and Limitations
Emrah Erduran, PhD NORSAR / International Center of Geohazards (ICG), Kjeller, Norway
Table of contents
1. Introduction
2. Design spectrum for elastic analysis
3. Free Vibration
3.1. SDOF
4.1. MDOF
4. Modal Analysis under Earthquake Forces
5. Response Spectrum Analysis (RSA)
6. RSA in EC-8
7. Tutorial
8. Concluding Remarks
Structural Analysis under Seismic Action
The aim of structural analysis under seismic action is to compute the design actions (forces and displacements) on the building components and the entire system
G + ΨEi ⋅ Q General types of analysis methods specified in EC8:
linear-elastic methods:
(1) lateral force method of analysis
(2) response spectrum analysis
non-linear (inelastic) methods
(3) non-linear static ('pushover') analysis
(4) non-linear time history analysis
i j
plastic hinge region
lpl lpl
lpl = 0.5∙d to d
with: d - depth of section
Moment
Curvature
Non-linear Models
Curvature
Mom
ent
E, I, A
to avoid explicit inelastic structural analysis in design, the capacity of the structure to dissipate energy is accounted for by using a reduced response spectrum by q
q is an approximation of the ratio of seismic forces that the structure would experience if its response would be completely elastic to the seismic forces used for the design
for 0 ≤ T ≤ TB :
for TB ≤ T ≤ TC :
for TC ≤ T ≤ TD :
for TD ≤ T ≤ 4 s :
−⋅+⋅⋅= )
.()(
3252
32
qTTSaTS
Bgd
qSaTS gd
52.)( ⋅⋅=
⋅⋅⋅=
TT
qSa C
g52.
⋅⋅⋅⋅= 2
52T
TTq
Sa DCg
.
Design spectrum for elastic analysis
Period T [sec]
Spec
tral
acc
eler
atio
n S d
q = 1
q = 2
q = 4
TB TC TD ga⋅≥ β
)(TSd
)(TSd
ga⋅≥ β
with: β = 0.20 (lower bound factor)
(EN 1998-1:2004 3.2.2.2)
Free Vibration - SDOF
For an SDOF system, the equation of motion:
For an undamped system, the equation of motion reduces to:
When an undamped SDOF system is disturbed with an initial displacement and released, the system osciallates with an harmonic motion.
Disp
lace
men
t
Time
Free Vibration - MDOF
Equation of motion for MDOF:
with:
Assumption: [C] = zero matrix ! – Undamped system
[ ] { } [ ] { } [ ] { } 0=⋅+⋅+⋅ uKuCuM
[ ]
=
imm
mm
M
000000000000
3
2
1
[ ]
=
nnn
n
ccc
ccc
C
................
....
1
33
22
111
[ ]
=
nnn
n
kkk
kkk
K
................
....
1
33
22
111
m2
m3
m1
Free Vibration - MDOF
Contrary to SDOF systems, for a MDOF system, the motion of each mass (or each floor) is NOT a simple harmonic motion, when an arbitrary initial deflection is applied to the system.
An associated period (or frequency) cannot be defined.
An undamped MDOF will undergo simple harmonic motion with an associated frequency only if the initial deformations are arranged in an approprioate distribution
m2
m3
m1
φn,1
φj+1,1
φj,1
φn,2
φj+1,2
φj,2
φn,3
φj+1,3
φj,3
NATURAL MODES OF VIBRATION
or
MODE SHAPES
Free Vibration - MDOF
At any given instant:
wn: natural circular frequency of vibration
Tn: natural period of vibration; time required for one cycle of simple harmonic motion
m2
m3
m1
φn,1
φj+1,1
φj,1
φn,2
φj+1,2
φj,2
φn,3
φj+1,3
φj,3
Determination of Mode Shapes
Procedure:
modal segmentation: =>
derive circular frequencies ωi / periods Ti and mode shapes φi
or obtain them from finite element software (e.g. ROBOT)
[ ] [ ] 02 =⋅− MK ω mk=ω
Modal Analysis of MDOF systems under earthquake forces
=
m2
m3
m1
φn,1
φj+1,1
φj,1
φn,2
φj+1,2
φj,2
φn,3
φj+1,3
φj,3
m2
m3
m1 T1
T2 T3
Modal Analysis of MDOF systems under earthquake forces
=
φn,1
φj+1,1
φj,1
φn,2
φj+1,2
φj,2
φn,3
φj+1,3
φj,3
m2
m3
m1
T1
T2
T3
qi(t) can be computed fairly easily!
Modal Analysis of MDOF systems under earthquake forces
Modal analysis of MDOF systems allows us to conduct a simplified analysis instead of solving a system of nxn differential equations.
However, we still need to solve n differential equations!!!
Modal analysis provides us with the response parameters (e.g. member forces, displacements, etc...) at every time step of an earthquake record.
Typically, this time step is between 0.005 seconds and 0.02 seconds.
Do we really need this information?
ENGINEERS NEED TO KNOW ONLY THE MAXIMUM PROBABLE ACTION FOR DESIGN!
RESPONSE SPECTRUM!
Excursion
Response spectrum:
used in earthquake engineering (exclusively)
describes the maximum response of a SDOF system to a particular input motion (i.e. the respective accelerogram)
dependent on damping ratio ξ (1–10 %)
response spectra reflect the maximum response to simple structures (SDOF)
T1 T2 T3 T4 T5 T6 T7
Sa,1
Sa,2 Sa,3
Sa,4
Sa,5
Sa,6
Sa,7
Spec
tral a
ccel
erat
ion
S a
Period T
maximum value
response action
earthquake impact
Response spectrum:
Response Spectrum Analysis
Procedure:
Given: - circular frequencies ωi / periods Ti
- mode shapes φi
modal participation factors γi :
design spectral accelerations Sa(Ti ) for each mode i :
{ }
= +
1
11
1
1
,
,
,
n
j
j
φφφ
φ
∑
∑
=
=
⋅
⋅= n
jijj
n
jijj
i
m
m
1
2
1
,
,
φ
φγ
φn,1
φj+1,1
φj,1
T1
Period T [sec]
Spec
tral
acc
eler
atio
n S a
T2 T3 T1
Sa,d (T3)
Sa,d (T2) Sa.d (T1)
Response Spectrum Analysis
Procedure:
lateral story loads Fj ,i :
)(,,, idaiijjij TSmF ⋅⋅⋅= γφ
Mode shape i: 1 2 3
Fn,1
Fj+1,1
Fj,1
Fn,2
Fj+1,2
Fj,2
Fn,3
Fj+1,3
Fj,3
φn,1
φj+1,1
φj,1
φn,2
φj+1,2
φj,2
φn,3
φj+1,3
φj,3
resulting shear forces Fb,m, that will be used in the design: ????
Modal Combination Rules
Square root of sum of squares (SRSS):
Simple
Fairly ‘accurate’ for buildings with well-seperated frequencies.
Should be avoided if the modal frequencies are close to each other.
Absolute Sum(ABSSUM):
Very Simple (Primitive (?))
Conservative... REALLY Conservative!!
Modal Combination Rules
Complete Quadratic Combination:
Mathematically complex
Leads to the most reliable solution for buildings with closely spaced natural frequencies .
... as well as well-seperated frequencies
ρn,m is the cross-modal coefficient
Response spectrum analysis in EC-8
Criteria:
shall be applied if the criteria for analysis method (1) are not fulfilled, this means if:
4 ⋅ TC T1 > 2.0 sec
Fb
1st mode
response of all modes shall be considered that contribute significantly to the global building response (i.e., important for buildings of a certain height)
those modes shall be considered for which:
(1) the sum of the modal masses is at least 90% of the total building mass or
(2) the modal mass is larger than 5% of the total building mass
∑mi ≥ 0.9 ⋅ mtot
mi ≥ 0.05 ⋅ mtot
(EN 1998-1:2004 4.3.3.3)
Response spectrum analysis in EC-8
Criteria (cont'd):
if the '90%' and the '5%' criteria is not fulfilled (e.g. for buildings prone to torsional effects), those modes shall be considered for which:
with: k - number of modes taken into account n - story number (from above foundation to top) Tk - period of vibration of mode k
n = 4
k ≥ 3 ⋅ √n and Tk ≤ 0.20 s
Mode shape: 1 2 3 4
Period Tk : 0.85 s 0.29 s 0.16 s 0.07 s
Example: 16-story building
k ≥ 3 ⋅ √16 = 12 and T12 = 0.002 s ≤ 0.20 s → twelve modes shall be
considered !!
Methods of analysis
General types of analysis methods specified in EC8:
Regularity Allowed simplification Behavior factor (for linear analysis) Plan Elevation Model Linear-elastic analysis
● ● planar
lateral force reference value
● ○ modal decreased value (⋅0.8)
○ ● spatial
lateral force reference value
○ ○ modal decreased value (⋅0.8)
(EN 1998-1:2004 4.2.3)
3-story RC frame building (residential use) behavior factor q = 4 ground motion: agR = 0.3 g residential use: γI = 1.0 structural parameters: E = 2.1 ⋅ 108 kN/m2 I = 2.679 ⋅ 10-5 m4 h = 3.0 m k = 12 ⋅ EI/h3
m = 50 tons = 50 kNs2/m
Tutorial – VIII – 'Modal RS method'
1. Setting up the differential equation of motion:
cf. Tutorial 4.2
m3 = m
m2 = 1.5m
m1 = 2m
h h h
k3 = k
k2 = 2k
k1 = 3k
[ ] { } [ ] { } [ ] { } 0=⋅+⋅+⋅ uKuCuM if [C] = 0 : [ ] { } [ ] { } 0=⋅+⋅ uKuM
[ ]
⋅=
=
1000510002
000000
3
2
1
.mm
mm
M [ ]
−−−
−⋅=
−−+−
−+=
110132
025
0
0
33
3322
221
kkkkkkk
kkkK
Tutorial – VIII – 'Modal RS method'
2. Modal segmentation:
cf. Tutorial 4.2
⇒ [ ] [ ] 02 =⋅− MK ω0
05132
0225
2
2
2
=−−−−−
−−
ωω
ω
mkkkmkk
kmk.
3. Modal circular frequencies ωi and periods Ti :
ω1 = 4.19 s-1 → T1 = 1.50 sec ω2 = 8.97 s-1 → T2 = 0.70 sec ω3 = 13.3 s-1 → T3 = 0.47 sec
4. Eigenmodes:
{ }
=
0016440300
1
...
φ { }
−−
=00160106760
2
...
φ { }
−=
001572
472
3
..
.φ
Tutorial – VIII – 'Modal RS method'
5. Modal participation factors γi :
cf. Tutorial 4.2
α1 = 100 ⋅ 0.3 + 75 ⋅ 0.644 + 50 ⋅ 1.0 = 128.3 kNs2/m α2 = –100 ⋅ 0.676 – 75 ⋅ 0.601 + 50 ⋅ 1.0 = -62.7 kNs2/m α3 = 100 ⋅ 2.47 – 75 ⋅ 2.57 + 50 ⋅ 1.0 = 104.3 kNs2/m M1
* = 100 ⋅ 0.32 + 75 ⋅ 0.6442 + 50 ⋅ 1.02 = 90.0 kNs2/m M2
* = 100 ⋅ 0.6762 + 75 ⋅ 0.6012 + 50 ⋅ 1.02 = 122.8 kNs2/m M3
* = 100 ⋅ 2.472 + 75 ⋅ 2.572 + 50 ⋅ 1.02 = 1155.0 kNs2/m → γ1 = 128.3 / 90.0 = 1.426 → γ2 = -62.7 / 122.8 = –0.511 → γ3 = 104.3 / 1155.0 = 0.090
*
,
,
i
in
jijj
n
jijj
i Mm
mα
φ
φγ =
⋅
⋅=
∑
∑
=
=
1
2
1
Tutorial – VIII – 'Modal RS method'
6. Design spectral accelerations Sa(Ti ) for each mode i : T1 = 1.50 sec : Check: Sa,d (T) = 0.846 m/s2 ≥ β ∙ ag = 0.20 ∙ 2.943 = 0.5886 m/s2 T2 = 0.70 sec : Check: Sa,d (T) = 1.813 m/s2 ≥ β ∙ ag = 0.20 ∙ 2.943 = 0.5886 m/s2 T3 = 0.47 sec :
cf. Tutorial 4.2
2
1
846050160
045215101943252 sm
TT
qSaTS C
gda /...
.
..)..(
.)(, =
⋅⋅⋅⋅=
⋅⋅⋅=
2
1
81317060
045215101943252 sm
TT
qSaTS C
gda /...
.
..)..(
.)(, =
⋅⋅⋅⋅=
⋅⋅⋅=
21152045215101943252 sm
qSaTS gda /.
.
..)..(
.)(, =⋅⋅⋅=⋅⋅=
Tutorial – VIII – 'Modal RS method'
5. Lateral story loads Fj,i : F1,1
= 100 ⋅ 0.30 ⋅ 1.426 ⋅ 0.846 = 36.2 kN F2,1
= 75 ⋅ 0.644 ⋅ 1.426 ⋅ 0.846 = 58.3 kN F3,1
= 50 ⋅ 1.00 ⋅ 1.426 ⋅ 0.846 = 60.3 kN F1,2
= 100 ⋅ (–0.676) ⋅ (–0.511) ⋅ 1.813 = 62.6 kN F2,2
= 75 ⋅ (–0.601) ⋅ (–0.511) ⋅ 1.813 = 41.8 kN F3,2
= 50 ⋅ 1.00 ⋅ (–0.511) ⋅ 1.813 = –46.3 kN F1,3
= 100 ⋅ 2.47 ⋅ 0.090 ⋅ 2.115 = 47.0 kN F2,3
= 75 ⋅ (–2.57) ⋅ 0.090 ⋅ 2.115 = –36.7 kN F3,3
= 50 ⋅ 1.00 ⋅ 0.090 ⋅ 2.115 = 9.5 kN
cf. Tutorial 4.2
)(,,, idaiijjij TSmF ⋅⋅⋅= γφF3,1= 60.3
F2,1 = 58.3
F1,1 = 36.2
F3,2 = –46.3
F2,2 = 41.8
F1,2 = 62.6
F3,3 = 9.5
F2,3 = –36.7
F1,3 = 47.0
Tutorial – VIII – 'Modal RS method'
5. Maximum shear forces Fb :
cf. Tutorial 4.2
60.3
118.6
154.8
-46.3
-4.5
58.1
9.5
-27.2
19.8
76.6
121.7
166.5
⇒
2
1imb
n
imb FF ,,, ∑
=
=
Conclusion
• Response Spectrum Analysis (RSA) is an elastic method of analysis and lies in between equivalent force method of analysis and nonlinear analysis methods in terms of complexity.
• RSA is based on the structural dynamics theory and can be derived from the basic principles (e.g. Equation of motion).
• RSA, unlike equivalent force method, considers the influence of several modes on the seismic behaviour of the building.
• Damping of the structures is inherently taken into account by using a design (or response) spectrum with a predefined damping level.
• The maximum response of each mode is an exact solution.
• The sole approximation used in RSA is the combination of modal responses.