Response Spectrum Analysis.pdf

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

Linear Models

lpl

lpl i E, I, A

j

Non-linear Models

lpl

Moment

i Moment

lpl E, I, A plastic hinge region

lpl = 0.5∙d to d Curvature

Curvature

with: d - depth of section

j

Design spectrum for elastic analysis

(EN 1998-1:2004 3.2.2.2)

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 :

 2 T 2.5 2  Sd (T ) = a g ⋅ S ⋅  + ⋅ ( − )  3 TB q 3  Sd ( T ) = a g ⋅ S ⋅

2.5 q

2.5  TC  = a ⋅ S ⋅ ⋅  g  q T  Sd (T )  ≥ β ⋅ag 2.5  TC ⋅ TD  a S = ⋅ ⋅ ⋅ g  2 q  T  Sd (T )  ≥ β ⋅ag

Spectral acceleration Sd



q=1 q=2 q=4

TB

TC

TD

Period T [sec]

with: β = 0.20 (lower bound factor)

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.

Displacement



Time

Free Vibration - MDOF m3 m2 m1



Equation of motion for MDOF:



with:  m1  0 [M ] =  0  0 

0 m2 0 0

0 0 m3 0

0  0 0  m i 

[M ]⋅ {u} + [C ]⋅ {u} + [K ]⋅ {u} = 0  c 11 ..  .. c 22 [C ] =  .. ..  c  n1 ..

.. .. c 33 ..

c 1n   ..  ..   c nn 

 k11 ..  .. k22 [K ] =  .. ..  k  n1 ..

Assumption: [C] = zero matrix ! – Undamped system

.. .. k33 ..

k1n   ..  ..   knn 

Free Vibration - MDOF φn,1

m3 m2 m1

φj+1,1 φj,1

φn,2 φj+1,2 φj,2

φn,3 φj+1,3 φj,3



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

NATURAL MODES OF VIBRATION or

MODE SHAPES

Free Vibration - MDOF φn,1

m3 m2 m1

φj+1,1 φj,1



At any given instant:



wn: natural circular frequency of vibration

 Tn: natural period

φn,2 φj+1,2 φj,2

φn,3 φj+1,3 φj,3

of vibration; time required for one cycle of simple harmonic motion

Determination of Mode Shapes Procedure:

[K ] − ω 2 ⋅ [M ] = 0



modal segmentation:

=>



derive circular frequencies ωi / periods Ti and mode shapes φi



or obtain them from finite element software (e.g. ROBOT)

ω = km

Modal Analysis of MDOF systems under earthquake forces m3 m2 m1

φn,1 φj+1,1 φj,1

m3 m2 m1

φn,2

φn,3 φj+1,3

φj+1,2

φj,3

φj,2

= T1

T2

T3

Modal Analysis of MDOF systems under earthquake forces φn,1 φj+1,1 φj,1

m3 m2

φn,2

φn,3 φj+1,3

φj+1,2

φj,3

φj,2

=

m1

T1

qi(t) can be computed fairly easily!

T2

T3

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)

maximum value earthquake impact

Sa,2 Spectral acceleration Sa



response action

Sa,3

Sa,1

Response spectrum:

Sa,4 Sa,5

Sa,6 Sa,7 T1

T2

T3

T4 Period T

T5

T6

T7

Response Spectrum Analysis Procedure:

 φ j ,1    {φ1} = φ j +1,1  φ   n ,1 

- mode shapes φi n

modal participation factors γi :

γi =

∑m

j

⋅ φ j ,i

∑m

j

⋅ φ 2j ,i

j =1 n j =1



design spectral accelerations Sa(Ti ) for each mode i :

φj+1,1 φj,1

T1

Spectral acceleration Sa

Given: - circular frequencies ωi / periods Ti



φn,1

Sa,d (T3) Sa,d (T2) Sa.d (T1)

T3

T2

T1

Period T [sec]

Response Spectrum Analysis Procedure: 

F j ,i = m j ⋅ φ j ,i ⋅ γ i ⋅ S a ,d ( Ti )

lateral story loads Fj ,i : Mode shape i:

φn,1

3

φj+1,1 φj,1

Fj+1,1 Fj,1

2

φn,2

φn,3 φj+1,3

φj+1,2

φj,3

φj,2 Fn,2

Fn,1



1

Fn,3 Fj+1,2 Fj,2

Fj+1,3 Fj,3

resulting shear forces Fb,m, that will be used in the design: ????

Modal Combination Rules 



Absolute Sum(ABSSUM):



Very Simple (Primitive (?))



Conservative... REALLY Conservative!!

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.

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

(EN 1998-1:2004 4.3.3.3)

Criteria: 

shall be applied if the criteria for analysis method (1) are not fulfilled, this means if: T1 >

Fb

4 ⋅ TC 2.0 sec

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

∑mi ≥ 0.9 ⋅ mtot

(2) the modal mass is larger than 5% of the total building mass

mi ≥ 0.05 ⋅ mtot

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: k ≥ 3 ⋅ √n and Tk ≤ 0.20 s with:

n=4

k - number of modes taken into account n - story number (from above foundation to top) Tk - period of vibration of mode k

Mode shape: 1

2

3

4 Example: 16-story building k ≥ 3 ⋅ √16 = 12 and T12 = 0.002 s ≤ 0.20 s → twelve modes shall be considered !!



Period Tk :

0.85 s

0.29 s

0.16 s

0.07 s

Methods of analysis

(EN 1998-1:2004 4.2.3)

General types of analysis methods specified in EC8:

Regularity

Allowed simplification

Plan

Elevation

●

●

●

○

○

●

○

○

Model planar

spatial

Linear-elastic analysis

Behavior factor (for linear analysis)

lateral force

reference value

modal

decreased value (⋅0.8)

lateral force

reference value

modal

decreased value (⋅0.8)

Tutorial – VIII – 'Modal RS method' cf. Tutorial 4.2

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

m3 = m

h

m2 = 1.5m

h

m1 = 2m

h

k3 = k k2 = 2k k1 = 3k

1. Setting up the differential equation of motion:

[M ]⋅ {u} + [C ]⋅ {u} + [K ]⋅ {u} = 0  m1  [M ] =  0 0 

0 m2 0

0  2 0 0    0  = m ⋅  0 1.5 0  0 0 1 m 3   

if [C] = 0 :  k1 + k2  [K ] =  − k2  0 

[M ]⋅ {u} + [K ]⋅ {u} = 0 − k2 k 2 + k3 − k3

0   5 −2 0     − k3  = k ⋅  − 2 3 − 1  0 −1 1  k3   

Tutorial – VIII – 'Modal RS method' cf. Tutorial 4.2

2. Modal segmentation:

[K ] − ω 2 ⋅ [M ] = 0

⇒

5k − 2mω 2 − 2k 0

− 2k 3k − 1.5mω 2 −k

3. Modal circular frequencies ωi and periods Ti :

ω1 = 4.19 s-1 ω2 = 8.97 s-1 ω3 = 13.3 s-1

→ → →

T1 = 1.50 sec T2 = 0.70 sec T3 = 0.47 sec

4. Eigenmodes:  0.30    {φ1} =  0.644   1.00   

 − 0.676    {φ2 } =  − 0.601   1.00   

 2.47    {φ3} =  − 2.57   1.00   

0 −k = 0 k − mω 2

Tutorial – VIII – 'Modal RS method' 5. Modal participation factors γi : n

γi =

∑m j =1 n

∑m j =1

j

⋅ φ j ,i =

j

⋅φ

2 j ,i

αi

Mi *

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

cf. Tutorial 4.2

Tutorial – VIII – 'Modal RS method' 6. Design spectral accelerations Sa(Ti ) for each mode i :

T1 = 1.50 sec :

S a ,d ( T ) = a g ⋅ S ⋅

cf. Tutorial 4.2

2.5  TC  2.5  0.6  2 ⋅ ⋅   = ( 2.943 ⋅1.0 ) ⋅1.15 ⋅ = . / 0 846 m s q  T1  4.0 1.50 

Check: Sa,d (T) = 0.846 m/s2 ≥ β ∙ ag = 0.20 ∙ 2.943 = 0.5886 m/s2

T2 = 0.70 sec :

S a ,d ( T ) = a g ⋅ S ⋅

2.5  TC  2.5  0.6  ⋅   = 1.813 m / s 2 ⋅   = ( 2.943 ⋅1.0 ) ⋅1.15 ⋅ q  T1  4.0  0.7 

Check: Sa,d (T) = 1.813 m/s2 ≥ β ∙ ag = 0.20 ∙ 2.943 = 0.5886 m/s2

T3 = 0.47 sec :

S a ,d ( T ) = a g ⋅ S ⋅

2.5 2.5 = ( 2.943 ⋅1.0 ) ⋅1.15 ⋅ = 2.115 m / s 2 q 4.0

Tutorial – VIII – 'Modal RS method' cf. Tutorial 4.2

5. Lateral story loads Fj,i :

F j ,i = m j ⋅ φ j ,i ⋅ γ i ⋅ S a ,d ( Ti ) 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

F3,1= 60.3 F2,1 = 58.3 F1,1 = 36.2

F3,2 = –46.3

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

F2,2 = 41.8 F1,2 = 62.6

F3,3 = 9.5

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

F2,3 = –36.7 F1,3 = 47.0

Tutorial – VIII – 'Modal RS method' cf. Tutorial 4.2

5. Maximum shear forces Fb :

9.5

-46.3

60.3

118.6

154.8

-27.2

-4.5

58.1

76.6

121.7

⇒ 19.8

166.5

Fb , m =

n

∑ i =1

Fb2, m ,i

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.

Conclusion (cont.d) • ABSSUM is the most conservative modal combination rul. Too conservative? • SRSS is the most popular modal combination rule due to its simplicity and ‘accuracy’ for buildings with well-seperated frequencies. • CQC is regarded as the most viable option for any structure.