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Prof. Dr.-Ing. Ekkehard Fehling Chair of

Structural Concrete Institute for Structural

Engineering

Basics of Design and Structural Modeling

according to EC 8

Measures of Earthquake -Strength

- Magnitude Richter-Scale logarithmic, for

total Energy of an Earthquake - Intensity measures local Effect,
- oriented towards description by observing

persons

8 Short form of the EMS-98 The short form of the

European Macroseismic Scale, abstracted from the

Core Part, is intended to give a very simplified

and generalized view of the EM Scale. It can,

e.g., be used for educational purposes. This

short form is not suitable for intensity

assignments.

Definition Description of typical observed effects (abstracted)

I Not felt Not felt.

II Scarcely felt Felt only by very few individual people at rest in houses.

III Weak Felt indoors by a few people. People at rest feel a swaying or light trembling.

IV Largely observed Felt indoors by many people, outdoors by very few. A few people are awakened. Windows, doors and dishes rattle.

V Strong Felt indoors by most, outdoors by few. Many sleeping people awake. A few are frightened. Buildings tremble throughout. Hanging objects swing considerably. Small objects are shifted. Doors and windows swing open or shut.

VI Slightly damaging Many people are frightened and run outdoors. Some objects fall. Many houses suffer slight non-structural damage like hair-line cracks and fall of small pieces of plaster.

VII Damaging Most people are frightened and run outdoors. Furniture is shifted and objects fall from shelves in large numbers. Many well built ordinary buildings suffer moderate damage small cracks in walls, fall of plaster, parts of chimneys fall down older buildings may show large cracks in walls and failure of fill-in walls.

VIII Heavily damaging Many people find it difficult to stand. Many houses have large cracks in walls. A few well built ordinary buildings show serious failure of walls, while weak older structures may collapse.

IX Destructive General panic. Many weak constructions collapse. Even well built ordinary buildings show very heavy damage serious failure of walls and partial structural failure.

X Very destructive Many ordinary well built buildings collapse.

XI Devastating Most ordinary well built buildings collapse, even some with good earthquake resistant design are destroyed.

XII Completely devastating Almost all buildings are destroyed.

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Seismologist scientist, describes earthquake as

natural phenomenon Magnitude logarithmic

measure of

total released

energy no information about the

effect on site of building Intensity

information related to site, but purely

phenomenological, no hard

numbers Structural Engineer technician,

wants to construct, make Designs and check them

by numbers, needs real quantitative (physical)

description !

Forces not known initially Deformations depe

nd on seismic input and building ground-

tine historey, peak values

(absolute), accelerations effective

value

inportant before starting any calculation

- planning and architectural layout suited for

earthquake loading - structural layout suited for earthquake action

Regularity of building in plan and elevation

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Typical Mistakes

Insufficient confinement

Single Soft-storey

Short Columns (X-Crack, brittle)

Effect of irregular layout in plan

S

uy

uy due to rotation

ux

Total deformation at distant building corner

Example L-shape in plan

Shape of Building in plan

Layout of floor diaphragms

a) less favourable

d) more favourable

c) less favourable

b) more favourable

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Irregularity in Elevation

Regularity of buildings

- Regularity in Plan
- horizontal stiffness and mass distribution in

two orthogonal - directions
- shape in plan shall be compact
- stiffness of slabs as diaphragms is big

- Regularity in Elevation
- members of lateral stiffening system shall be

continuous from foundation to top of building - Horizontal stiffness and mass distribution are

constant across height

Criteria for Regularity according to EC 8

Criteria for the Regularity according to EC 8

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Analysis Methods

From this V(x), M(x)

1. 2. 3.

Fb

Multimodal Analysis / Response Spectrum-method tak

ing into account multiple mode shapes

Simplified Response Spectrum method equivalent

static force, Also called lateral force analysis

in EC8

(nonlinear) time step analysis for multiple

degree of freedom systems (MDOF)

Reality Engineering model

d displacement

mass m to stiffness k kN/m

ag

ground acceleration ag horizontal,

vertical, (rotations)

g ground

a ?

u

Response

F kN

m

F kN

m

k

a

Dynamics

Statics

ag

known

..

F k u u F / k

F m a m u a u F / m

..

Units m kg oder to a m/s² F kN 1 N

1 kg 1 m/s² 1 kN 1 to 1 m/s²

Newton

du

velocity acceleration

v ------

dt

d2u

dv

a ------ ------

dt

dt2

Free vibration

for c 0 undamped system, free

(undamped) vibration

k

u

u(t)

u(t) u sin (? t f )

m

c

u

Dynamic equilibrium SF 0

t

.

---

m ü c u k u 0

circular frequency

? v---

k m

or in dimensionless notation

? 2 p f

.

ü 2? ? u ?² u 0

modal frequency

f ? / (2 p)

c 2 m ?

with ? ----

(modal) period

T 1/ f

Harmonic Excitation (sinusoidal excitation)

F(t)

u

F(t) F sin (? t)

F(t)

F

t

m k

V

Dynamic equilibrium

max V 1/ (2?)

.

m ü c u k u F(t) 0

1,0

F k

max u ---- V

fExcitation

fmodal

Dynamic Magnification Factor

Earthquake Excitation Character of ground

acceleration filtered white noise, non

stationary, transient

Time Historey of acceleration

- Calculation of
- vibration response
- Duhamel-Integral
- Time Step-Methods, e.g. Central

Difference, Newmark, Wilson-?

Response Spectrum

vibration Response

Earthquake Excitation Response Spectrum

maximum value (absolute)

Base acceleration

Response Spectrum

Multiple Degree of Freedom Oscillator (MDOF)

1

1

1

k13

k23

k33

m3

3

2

m2

u1a

k22

k12

k32

u1

1

m1

k21

k31

k11

Stiffness of springs (Reactions for displacement

1 at one single node, unit-displacement-states)

üg(t)

f23

f33

f13

m3

3

2

f12

f22

f32

m2

u1a

u1

1

f11

f21

f31

m1

Mode Shapes Modal Frequencies f f1

f f2 f f3 f1 lt f2 lt

f3

ug(t)

u1a u1 ug

ü1a ü1 üg

Inertia Force F m a m ü

m1 ü1a m1 (ü1üg)

ground- acceleration

absolute acceleration

relative acceleration

Equilibrium of forces at node 1

m1 ü1 m1 üg (t) k11 u1 k12 u2 k13

u3 0

m1 ü1a

Equilibrium of forces at all nodes

m1 ü1 k11 u1 k12 u2 k13 u3

-m1 üg (t)

m2 ü2 k21 u1 k22 u2 k23 u3

-m2 üg (t)

m3 ü3 k31 u1 k32 u2 k33 u3

-m3 üg (t)

Equliibrium equations for 3 degrees of freedom

m1 ü1 k11 u1 k12 u2 k13 u3

-m1 üg (t)

m2 ü2 k21 u1 k22 u2 k23 u3

-m2 üg (t)

m3 ü3 k31 u1 k32 u2 k33 u3

-m3 üg (t)

using Matrix-Notation

m1 ü1 k11 k12

k13 u1 m1

m2 ü2 k21 k22 k23

u 2 - m2 üg (t)

m3 ü3 k31 k32 k33

u3 m3

M ü K u

- M e üg

M ü K u

- M e üg

scalar

stiff- ness- matrix

mass- matrix

unit- vector, or directional vector of rigid

body displacment due to ug 1

u1

1

u u2

e 1

u3

1

ü1

ü ü2

m1

ü3

M e m2

m3

.

M ü C u K u 0

dampingsmatrix

without external excitation

Linear differential equation - system of 2nd

order

Type of Solution u f e i?t u if ?

e i?t ü - f ?² e i?t

with i v-1

.

Eigenvalues ?k k 1,2, 3

without damping, i.e. for C 0

( - ?² M K) f e i?t 0

det (K - ?² M ) 0

?1 bis ?3

Eigen (modal) frequencies and Mode Shapes

?1 bis ?3

det (K - ?² M ) 0

with fk ?k / (2p) It follows

Eigenfrequencies modal frequencies f1 lt f2

lt f3 Modal Periods T1 gt T2 gt T3

f23

f33

f13

f12

f22

f32

Mode shapes can be obtained mathematically as

Eigenvectors fk fk2

f11

f21

f31

fk1

k1, .. n with n number of DOFs

fk3

- 2. 3. Mode Shape
- (mode shapes)

(from solution of system of equns. after

putting in ?k²)

k-th Mode Shape as vector fk fk2

fk1

k1, .. n with n number of DOFs

fk3

all mode shape vectors in one matrix (Modal

Matrix)

f13 f12 f13

F

f12 f22 f23

F

f1 f2 f3

or

f13 f32 f33

n

after transformation of variables u S

fkyk F y

k1

..

.

FT M F y FT C F y FT K F y -

FT M ex üg (t)

M

C

K

r

Diagonal Matrices with real coefficients if

C a M ß K or if C 0

Vector of Participation factors

a mathematical miracle occurs

..

m1 0 0 y1 k1 0 0

y1 r1

..

0 m2 0 y2 0 k2 0

y 2 - r2 üg (t)

..

0 0 m3 y3 0 0 k3

y3 r3

Response spectrum method considering more than

one mode of vibration (Multimodal Analysis)

- Each mode is being excited by the earthquake
- per mode k there is one modal mass mk
- amplitude of vibration of mode described by yk

- scaling of excitation by participation factor

rk

From the system of differential equations, we get

a system of decoupled diff. Equations for n

modes of vibrations with the frequency fk (or

circular frequency ?k)

..

.

mk yk ck yk mk ?k2 yk

-rk üg (t)

kk

Generalisized mass generalized

stiffness of k-th modal (modal mass)

vibration

rk (fT M ex)k

(e.g. for excitation in x-direction direction

of u)

Maximum response for mode k

max yk rk Sa(fk)

..

Sa(T)

for k1, i.e. 1st mode

q

Sa(T1)

Spectral value of reponse acceleration in mode 1.

T3

T1

T2

n

..

..

..

Back - Transformation u S fkyk F y

k1

Superposition of maximum responses in all single

modes Normally, the maximum responses in the

different modes will not occur at exactly the

same point of time. Hence, they need not to be

added arithmetically. This also holds true for

Deformations, accelerations and cross sectional

forces .

max y1

for random processes which are Independent from

each other max y² max y1² max y2² max

y3² max yn² max y

v(Syi²) (Pythagoras, in earthquake

engineering known as SRSS-Rule (Square Root of

Sum of Squares)

y1(t)

t

max y2

y2(t)

t

y3(t)

max y

max y1

t

max y3

max y2

Combination of modal member forces and

deformations

the responses due to modal vibrations with the

periods Ti and Tj may be regarded as independent

from each other, if the values of the periods

differ significantly from each other Aaccording

to EC 8 if

then

(SRSS-Rule)

(Square Root of Sum of Squares)

with EE seismic action effect under

consideration (force, displacement,

etc.) EEi value of this seismic action effect

due to the vibration mode i

Further remarks on the Response Spectrum Method

considering multiple modes

- All modes of vibration which significantly

contribute to the global vibration - behaviour should be considerd.
- In order to reach this goal
- all individual modes which have an effective

modal mass of 5 of the total mass of the

structure should be considered, - the sum of the effective modal masses for all

considered modes should be 90 of the total

mass of the structure,

Combination of forces for arbitrary /more than

one directions of earthquake input

Method A SRSS-Regel

V2

V1(Ex) V2(Ex) V3(Ex)

V1

V3

Ex

Ey

V1(Ey) V3(Ey)

Procedure Calculate force of interest, e.g. V1,

for each direction (x or y) of input

acceleration separately, then superimpose

results using SRSS-Rule

Combination of forces for arbitrary /more than

one directions of earthquake input

Method B Arithmetic combination with 30 for

transverse direction

V2

V1(Ex) V2(Ex) V3(Ex)

V1

V3

Ex

Ey

V1(Ey) V3(Ey)

Procedure Calculate each force , e.g. V1, for

each direction of earthquake excitation (x or y)

separately, then perform addition 100 in main

direction 30 perpendicular If main

direction is not known check 2 combinations

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Non Structural Elements

Partition walls, machinery, other components

Ta

Value of spectral acceleration for nonstructural

elements

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Alternatively possible more refined calculation

/ modeling, e.g. using floor response spectrum

or integral model of building including the

nonstructural elements of interest

Sa, Etage

aEtage

Tbuilding

T

Recommendations / hints for simplified

calculations and plausibility checks

- determination of modal frequencies
- mode shapes

- Practical estimation of modal frequencies and

mode shapes - Modal frequencies fk Hz or modal periods of

vibration Tk s 1 / fk - from rule of thumb formulae
- by simplified methods, (e.g. Rayleigh-

Quotient) - calculation using Computerprograms (FE,

Truss/Frame, with dynamics-capabilities) - Mode shapes
- the first mode shape can be estimated
- approximately in many cases rather
- simply when regarding the support
- conditions as an approximation
- sinus
- quadratic parabola

Approximation linear

Approximate formulae for the first (fundamental)

vibration period T1

Typical R/C-frame structure T1 in Seconds

number of storeys n / 10 Example 3 storey

commercial building T1 3 / 10 0,3

s or, respectively f1 1/T1 1 /

(0,3 s) 3,33 Hz

The same the other way around f1 Hz 10 /

n in Example f1 10 / 3 3,33 Hz

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Generally applicable Rayleigh Quotient for

equivalent bar structure

mn

with mj storey mass at height hj dj

horizontal displacement at height hj due

to the storey forces Fj Fj storey

force at height hj ideally consider

forces proportional to mass

mode shape! Approximation dead load

applied horizontally 11

m2

m1

Even more simplified

with T1 fundamental period of vibration of the

structure, in s d horizontal deflection of top

of structure, in m, due to the

gravity loads, assumed in horizontal direction

Equivalent static force method (Lateral Force

Method)

- Basis
- application of equivalent static forces to

account for inertia forces - considering most important mode shape, which is

the 1st mode shape

Fb M Sa

spectral acceleration

Mass of buildilng

Base shear force of building (Base-Shear)

correction factor for lower modal mass, if 2

storeys

EC 8

T1 2 TC

bei

in all other cases

ordinate of design response spektrum (including

division by q-factor) Index d Design

Fb

Simplified Response Spectrum Method

Application for buldings with regular layout,

if T1 4 TC

Distribution of lateral forces over the height

Lateral force acting at storey i

n

Fi

si

i

Fi lateral force acting at storey i Fb base

shear force si, sj displacements of masses mi,

mj in the fundamental mode of

vibration mi, mj storey masses of storeys i and

j

j 1

1. mode shape

Fb

Further simplification 1. mode shape assumed as

linear, i.e. proportional to height

distribution of lateral forces across the height

Lateral force acting at storey i

n

Fi

si

i

Fi lateral force acting at storey i Fb base

shear force zi, zj heights of masses mi, mj

mi, mj storey masses of storeys

i and j

j 1

zi

Fb

Torsion

For 3-dimensional Model systematic effects of

mass and stiffness eccentricities in plan are

being considered automatically If planar models

are used for each direction

(possible for structures being regular in

plan) special simple method for determintion

of torsional actions according to position and

stiffness of bracing system elements and mass

in EC 8

Torsion

The motion of the ground not only contains

translational components (in x, y, or

z-directions) but also rotational components.

These excite a building, even if ideally

symmetrical, so that rotations around the

vertical axis will result. Thus, the lateral

stiffening system of the structure gets torsion.

In reality, the masses will not be distributed

exactly as intended. The torsional moment

(action effect) accounting for these effects

M1i e1i Fi with M1i the torsional

moment (load) of storey i e1i 0,05

Li accidental eccentrizity of storey mass mi for

all relevant directions Fi lateral load

acting at storey i Li floor dimension

perpendicular to the direction of the seimic

action

Calculation of displacements according to EC 8

Simplifying assumption Hypothesis of equal

displacements Calculated on the base of linear

elastic behaviour Elastic displacements ds of

structural system, Obtained from elastic

response spectrum Se(T) In practice, one uses

the reduction of forces by the behaviour factors

(q-factors) for the determination of member

forces. For this, the Design Response Spectrum

Sd(T) will be used. It contains, implicitly, a

division by q. In order to obtain the same

displacements as for linear elastic behaviour,

the calculated displacements de shall be

multiplied by q.

Design spectrum accounting for nonlinear /

ductile behaviour

Hypothesis of equal displacement

F

Elastic Response Spectrum Se(T)

/ q

u

/ q

Design-Response Spectrum Sd(T)

Modal Period T s

F

/ q

Behaviour Factor q according to material /type

of construction q 1,0 8

u

de ds

How to ensure ductile system behaviour The

principle of Capacity Design

Electrical electrical circuit with

fuse Engineering limitation of electrical

current Amperes Structural limitation of

member forces kN, kNm Engineering in regions

with possible brittle failure by plastic

mechanisms (e.g.plastic hinges), Design of

other members/regions for this capacity

..

m u

vorh. M (x) Mpl

brittle duktile

x

M

Mpl

Mpl

?

..

Mpl

ug

Capacity Design for Frames

Avoiding brittle failure of column by Capacity

Design S MRd, Column ?Rd S Mpl, Beam

Resistance of brittle member gt Capacity of

ductile member

Confinement of beam column joint Canary Bird -

Test ?

Plastic hinge mechanisms

d

d

Soft storey - Mechanism

Lateral Displacement mechanism

Requires small plastic rotations

Requires big plastic rotations

Should be avoided !

Seismic Isolation of Buildings

Character of Loading due to Earthquake More an

enforced deformation than a clearly defined force

!

- Possible Experimental Methods for the
- Simulation of Earthquake Behaviour of Building

Structures - Shaking Table Tests close to reality, real time,

but expensive, size of test specimens

limited - Pseudodynamic in slow motion, simulation of

inertia forces by hydraulic jacks /

forces big specimen sizes and forces

possible (e.g. ELSA Reaction-Wall

facility at JRC of EU in Ispra (I) h 16

m )

Methods for Experimental Simulation of Seismic

Loading

Shaking Table Reaction- Wall Pseudo- Dyna

mic Tests

Pseudodynamische Methode

Reaktions- wand

Reaction floor / Aufspannfeld

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Displacement based Analysis

T 2p / ?

o

o

T 2p / ?

eq

eq

with d displacement H Horizontal Force aresp

response- acceleration of SDOF with period

T ? circular frequency ? 2p f k stiffness T m

odal period

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Response spectrum in modified presentation

(maximum acceleration)

(maximum displacement)

Influence of Energy Dissipation equivalent

viscous damping

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(Envelope / back bone curve)

Capacity curve of building

total horizontal force (Base shear) kN

Top displacement mm

from SIA 2018

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