Basics of Design and Structural Modeling

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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
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Measures of Earthquake -Strength

• Magnitude Richter-Scale logarithmic, for
  total Energy of an Earthquake
• Intensity measures local Effect,
• oriented towards description by observing
  persons

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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
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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)
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Effect of irregular layout in plan
S
uy
uy due to rotation
ux
Total deformation at distant building corner
Example L-shape in plan
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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
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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

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Criteria for Regularity according to EC 8
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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)
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Reality Engineering model
d displacement
mass m to stiffness k kN/m
ag
ground acceleration ag horizontal,
vertical, (rotations)
g ground
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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
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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
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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
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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

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vibration Response
Earthquake Excitation Response Spectrum
maximum value (absolute)
Base acceleration
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Response Spectrum
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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)
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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
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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)
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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
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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
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.
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
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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)
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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
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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
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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)
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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
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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
engineeringknown 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
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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
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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,

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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
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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
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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
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Recommendations / hints for simplified
calculations and plausibility checks

• determination of modal frequencies
• mode shapes

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• 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
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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 hjdj
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
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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
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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
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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
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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
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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
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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.
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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
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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
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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
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Confinement of beam column joint Canary Bird -
Test ?
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Plastic hinge mechanisms
d
d
Soft storey - Mechanism
Lateral Displacement mechanism
Requires small plastic rotations
Requires big plastic rotations
Should be avoided !
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Seismic Isolation of Buildings
Character of Loading due to Earthquake More an
enforced deformation than a clearly defined force
!
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• 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 )

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Methods for Experimental Simulation of Seismic
Loading
Shaking Table Reaction- Wall Pseudo- Dyna
mic Tests
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Pseudodynamische Methode
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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)
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Influence of Energy Dissipation equivalent
viscous damping
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(Envelope / back bone curve)
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Capacity curve of building
total horizontal force (Base shear) kN
Top displacement mm
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from SIA 2018
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