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Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
Basis of Structural Design[EN1990 – 02]
Prof. Dr.-Ing. Jürgen Grünberg
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
Introduction of myself
Name
Date of birth
Present position
Key qualifications
Contribution to Code Writing
Jürgen Grünberg
18 May 1944
Professor of concrete structures and director of the Institute of Concrete Construction, University of Hannover
Consulting engineer for structural design, testing and supervision in structural engineering, Hamburg
Reliability analysis in structural engineeringMaterial models for RC and UHPC structuresAnalysis of young concrete during the hydration processFatigue design of concrete structuresStructural design (e.g. towers, bridges, offshore structures)
EN 1990 (in Germany: DIN 1055-100)EN 1991 (in Germany: DIN 1055-1 to 10)EN 1992 (in Germany: DIN 1045-1)
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
Scope:
Principles and requirements for safety, serviceability, and durability.
Direct application for buildings and civil engineering works in conjunction with EN 1991 to 1999.
Guidelines relating to safety, serviceabilty and durabilty for designing structures out of the scope of EN 1991 to 1999, to serve as reference document, e.g. for product codes.
Basis of structural design [EN 1990 – 02]
Application also for the structural appraisal of existing construction, in developing the design of repairs, alterations or in assessing changes of use.
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
Content:
Annex B (informative) Management of structural reliability for construction worksAnnex C (informative) Basis for partial factor design and reliability analysis
Basis of structural design [EN 1990 – 02]
Annex D (informative) Design assisted by testing EN 1992 to 1999
EN1990 – Main text
Foreword1 General2 Requirements3 Principles of limit state design4 Basic variables5 Structural analysis and design assisted by testing
Principles andrequirements
Annex A1 (normative) Application for buildings EN 1991-1Annex A2 (not published) Application for bridges EN 1991-2
Direct application
6 Verification by the partial factor method
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
Topics:
1. Bases of safety concept(Principles and requirements; explanation of terms and definitions)
2. Combinations of actions(Verification by the partial factor method according to the different limit states and design situations)
3. Basis for partial factor design and reliability analysis (probabilistic analysis)
Basis of structural design [EN 1990 – 02]
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
1 Bases of safety concept
To assure the structural safety the following measures are required:
1. Measures to avoid human errors (Assumptions and preconditions for structural design),
2. Measures to warrant a sufficient safety margin between action effect and structural resistance (Basic requirements for design and execution of structures),
3. Measures to prevent potential causes of failure and/or reduce their consequences (Limiting or avoiding of potential damage).
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
1.1 Measures to avoid human errors (Assumptions and preconditions for structural design),
Human errors are not covered by the safety margins defined in the design codes!
1. The choice of the structural system and the design of the structure is made by appropriately qualified and experienced personnel.
2. Execution is carried out by personnel having the appropriate skill and experience.
3. Adequate supervision and quality control is provided during execution of the work, i.e. in design offices, factories, plants, and on site
4. The construction materials and products are used as specified in EN 1990 or in ENs 1991 to 1999 or in the relevant execution standards or reference material or product specifications.
5. The structure will be adequately maintained.
6. The structure will be used in accordance with the design assumptions.
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
1.2 Basic requirements for structures
The basic requirements for structures are established in the
Interpretative Document
„Mechanical Resistance and Stability"
associated to the Construction Product Directive
published by the European Community at 21-12-1988
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
1.2 Basic requirements for structures
A structure shall be designed and executed in such a way that it will, during its intended life, with appropriate degrees of reliability and in an economical way :
sustain all actions and influences likely to occur during execution and use,
and remain fit for the use for which it is required.To reach a sufficient reliability, a structure shall be designed to have adequate:
structural resistance,
serviceability,
and durability.
To assure structural resistance, the following events are not allowed to occur
collapse of the entire structure or of one structural element,
or large deformations exceeding the limits of failure.
A structure shall not be damaged by events such as
explosion, impact, and the consequences of human errors,
to an extent disproportionate to the original cause.
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
1.3 Limiting or avoiding of potential damage
Furthermore, actions are possible which have not been considered in design, as they are resulting
• from insufficient knowledge and wrong activities of persons, e.g. the users of the
structure who have not been informed about the loading limits
• from errors which were not detected although systematic inspections were performed,
• from the stochastic coincidence of extreme events,
• from exceeding the loading limits during the working life,
• from hazards which are caused by persons or nature (e.g. explosions),
In spite of these two strategies –
• Measures to avoid human errors
• Measures to warrant a sufficient safety margin
errors cannot be excluded completely !
There is a remaining risk.
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
To assure structural safety, the third strategy is to reduce the consequences of failure and, especially, to avoid injuring and even killing of people.
Therefore, potential damage shall be avoided or limited by appropriate choice of one or more of the following :
• avoiding, eliminating or reducing the hazards to which the structure can be subjected;
• selecting a structural form which has low sensitivity to the hazards considered;
• selecting a structural form and design that can survive adequately the accidental removal of an individual member or a limited part of the structure, or the occurrence of acceptable localised damage;
• tying the structural members together.
• avoiding as far as possible structural systems that can collapse without warning;
1.3 Limiting or avoiding of potential damage
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
1.4 Principles of limit state design
Serviceability criterion (permissible stresses,
crack widths, deformations)
Design value of resistance (stabilising actions, material strengths,
cross area resistances)
Resistance
Design value of action effects (stresses, crack widths,
deformations)
Design value of action effects (destabilising actions,
internal forces)
Action effects
Rare or characteristicFrequent
Quasi-permanent
Persistent and transientAccidental
Seismic
Design situations
Stress limitationCrack propagation
Deformations Vibrations
Loss of static equilibriumFailure by strength limitation
Loss of stabilityFailure by fatigue
Verification criteria
Functioning of the structureComfort of people
Appearance of construction
Safety of peopleSafety of the structure
RequirementsServiceabilityUltimateLimit state
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
1.5 Representative values
Characteristic values of actions ( Fk ):
Action codes (EN 1991)
Characteristic values of material properties ( Xk ):
construction specific design codes (EN 1992 to EN 1999)
according material codes (EN 206 etc.)
Characteristic values of actions
The characteristic values of permanent actions Gk
generally are their mean values.
The characteristic values of variable actions Qk
generally are their 98 %-quantiles for the reference period of 1 year.
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
Other representative values of variable actions
… shall be defined as products of a characteristic value Qk
and a combination factor i ( 1,0 ).
1. Combination value: Qrep,0 = 0 Qk
The factors 0 are chosen such, that the failure probabilities for the action effect resulting from combination of actions and from a single action are adequate.
2. Frequent value: Qrep,1 = 1 Qk
with a limited duration or frequency of being exceeded within the reference period.
3. Quasi-permanent value: Qrep,2 = 2 Qk
determined as the value averaged on the reference period.
In case of fatigue other representative values may be considered.
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
Comparison of representative values of a variable action
Q
Design value Qd = Q Qk
Characteristic value Qk
Combination value 0 Qk
Frequent value 1 Qk
Quasi-permanent value 2 Qk
t
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
Characteristic values for material properties
… generally are defined as quantiles of a statistical distribution, for instance:
• as 5 %-quantiles of material strength parameters,
• as mean values of structural stiffness parameters,
• as upper nominal values for determination of indirect actions.
1.5 Representative values
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
1.6 Design values
Design values of actions
Frep represents either Gk, Qk or Qrep.
d Ed f rep F repF F F
Design values of material properties
ork k
dRd m M
X XX
k
dM
XX
The conversion factor takes into account volume and scale effects,effects of moisture and temperature, etc.
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
Relations between individual partial factors
Uncertainty of representative values of actions
Model uncertainties
Uncertainty of material properties
Actions and action effects
Structural resistances
f
Ed
Rd
m
F
M
1.6 Design values
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
Design values of geometrical data
or
Nominal values anom
Deviations a ( e.g. in case of geometrical imperfections )
d noma a d noma a a
Design values of action effects
The action effects (E) are the answers of the structure to the actions (F), depending on the geometrical data (a) and the material properties (X).
General format:
d d,1 d,2 d,1 d,2 d,1 d,2E E F ,F ,...,a ,a ,...,X ,X ,...
1.6 Design values
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
Applying partial factors, the following formats can be derived:
1. General format:
d Ed g,1 k,1 g,2 k,2 q,1 rep,1 q,2 rep,2E E G , G ,..., Q , Q ,...
2. Formats for combination of actions in non-linear analysis:
2.1. The action effect Ed increases more than the leading action Qk,1:
d G,1 k,1 G,2 k,2 Q,1 k,1 Q,2 rep,2E E G , G ,..., Q , Q ,...
2.2. The action effect Ed increases less than the leading action Qk,1:
G,1 G,2 Q,2d Q,1 k,1 k,2 k,1 rep,2
Q,1 Q,1 Q,1
E E G , G ,...,Q , Q ,...
3. Format only to be used in linear-elastic structural analysis:
d G,1 Gk,1 G,2 Gk,2 Q,1 Qrep,1 Q,2 Qrep,2E E E ... E E ...
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
EQd,1
Q1
HQd,1 > Q,1 HQk,1
(Arch structure)
NQd,1 = Q,1 NQk,1
(Suspension bridge)
linear
a) above proportionality
b) below proportionality
Qk,1 Qd,1 = Q,1 Qk,1
NQk,1
HQk,1
1.6 Design values
Formats for combination of actions in non-linear analysis
Predominant action effect EQd,1 = E (Qk,1; Q,1) in non-linear structural analysis
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
Design values of resistances
The resistances (R) depend on the geometrical data (a) and the material properties (X).
General Format:
d d,1 d,2 d,1 d,2R R a ,a ,... X ,X ,...
1.6 Design values
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
1. Format applying divided partial factors:
k,1 k,21 2 nom,1 nom,2
Rd m,1 m,2
d1 X X
R R , ,...,a ,a ,...
2. Format applying integrated partial factors:
k,1 k,2d 1 2 nom,1 nom,2
M,1 M,2
X XR R , ,...,a ,a ,...
3. Format applying on partial factor R for structural resistance:
Rd 1 k,1 2 k,2 nom,1 nom,2
R M,1 M,2
R1R R X , X ,...,a ,a ,...
• Application: e.g. non-linear structural analysis of reinforced concrete structures
Applying partial factors, the following formats can be derived:
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
1.7 Verification of limit statesby the partial factor method
It shall be verified that,
in all relevant design situations,
no relevant limit state is exceeded
when the design values for actions or action effects and resistances are used in the design models.
For the selected design situations and the relevant limit states the individual actions for the critical load cases should be combined using the
characteristic values or other representative values in combination with
partial factors (F; M) and other factors (e.g. combination factors i).
However, actions that cannot occur simultaneously should not be considered together in combinations.
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
1.7 Verification of limit states
Verification formats for Ultimate Limit states (ULS)
The following ultimate limit states shall be verified as relevant:
a) EQU: Loss of static equilibrium of the structure or any part of itconsidered as a rigid body
b) STR: Internal failure or excessive deformation of the structure, one of its members or the foundation, where the strength of construction materials governs
c) GEO: Failure or excessive deformation of the soil where the strengths of the soil or rock are significant in providing resistance
d) FAT: Fatigue failure of the structure or structural elements(Note: For fatigue design see EN 1992 to EN 1999)
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
1.7 Verification of limit states
Verification formats for Ultimate Limit states (ULS)
• Limit state of static equilibrium (EQU)(e.g. overturning, buoyancy, lifting off)
Verification of a structure considered as a rigid body:
stb,ddst,d EE
Ed,dst Design value of the effect of
destabilising actions
Ed,stb Design value of the effect of
stabilising action (= gravity resistance)
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
• Limit state of structural failure (STR) (rupture, excessive deformation)
Verification of a structural cross area, member or joint:
dd RE
Ed Design value of the effect of actions (internal forces, stresses)
Rd Design value of the structural resistance (bearing capacity)
1.7 Verification of limit states
• Limit state of static equilibrium involving the resistance of anchoring structural members
Furthermore, the limit state of structural failure has to be verified with respect to the anchoring structural member
G,STR,sup G,EQU,sup d,dst d,stb d,anch/ E E R
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
dd CE
Ed Design value of the effects of actions
(e.g. deformation, stress)
Cd Limiting design value of the effects of actions
specified in the serviceability criterion (e.g. limiting values of deformations, stresses, etc.)
1.7 Verification of limit states
Verification formats for Serviceability Limit states (SLS)
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
2 Combinations of actions
2.1 Single actions for buildings
AEdSeismic actions
AdAccidental actions
Qk,
Qk,H
6. Indirect actions, caused by uneven settlements
Gk,H4. Fluid pressure, permanent 5. Fluid pressure, variable
Gk,E3. Earth pressure Qk,T4. Thermal actions
Qk,W3. Wind loadsPk2. Prestressing
Qk,S2. Snow and ice loads
Qk,N1. Imposed loads, life loadsGk1. Self-weights
QkiVariable actionsGkj; PkPermanent actions
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
• Generally, the self-weights of the structure and of the fixed equipment, as permanent loads, may be united to one common single action Gk.
• In case of a limit state of static equilibrium, the permanent actions have to be subdivided into their unfavourable and their favourable parts ( Gk,dst,j and Gk,stb,j).
• Generally, all the imposed loads and life loads within one building
coming from different categories of use appearing there
are assembled to one multi-component action QN,k.
2.1 Single actions for buildings
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
2.2 Ultimate Limit States (ULS)
Persistent and transient design situations (fundamental combinations)
• General format
(special formats in non-linear structural analysis, see 1.6)
G,j k,j p Q,1 k,1 Q,i 0,i k,id kj 1 i 1
E E G P Q Q
• Format used only in linear-elastic structural analysis
Leading variable action effect:
G,j Gk,j p Pk Q,1 Qk,1 Q,i 0,i Qk,idj 1 i 1
E E E E E
Q,1 0,1 Qk,1 Q,i 0,i Qk,i1 E Max. 1 E
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
Alternatively for STR and GEO limit states,
the less favourable of the following formats may be applied:
• Alternative format, general
d G,j k,j P k Q,i 0,i k,ij 1 i 1
E E G P Q
a)
d G,j j k,j P k Q,1 k,1 Q,i 0,i k,ij 1 i 1
E E G P Q Q
b)
• Alternative format, used only in linear-elastic structural analysis
d G,j Gk,j P Pk Q,i 0,i Qk,ij 1 i 1
E E E E
a)
b) d G,j j Gk,j P Pk Q,1 Qk,1 Q,i 0,i Qk,ij 1 i 1
E E E E E
j Reduction factor for unfavourable permanent actions Gk,j
(j = 0,85 indicative)
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
Accidental design situations
• General Format
dA GA,j k,j pA k d 1,1 k,1 2,i k,ij 1 i 1
E E G P A Q Q
• Format only used in linear-elastic structural analysis
Leading variable action effect:
dA GA,j Gk,j PA Pk Ad 1,1 Qk,1 2,i Qk,ij 1 i 1
E E E E E E
1,1 2,1 Qk,1 1,i 2,i Qk,iE Max. E
2.2 Ultimate Limit States (ULS)
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
Seismic design situations
• General Format
• Format only used in linear-elastic structural analysis:
dE k,j k Ι Ed 2,i k,ij 1 i 1
E E G P A Q
dE Gk,j Pk Ι AEd 2,i Qk,ij 1 i 1
E E E E E
2.2 Ultimate Limit States (ULS)
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
2.3 Serviceability Limit States (SLS)
Formats for linear-elastic structural analysis (normal case)
Rare (characteristic) combination
normally used for irreversible limit states (e.g. remaining deformations):
d,rare Gkj Pk Qk1 0i Qkij 1 i 1
E E E E E
Leading variable action effect:
01 Qk1 0i Qki1 E Max. 1 E
∙
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
Frequent combination
normally used for reversible limit states (e.g. corrosion attack on reinforcement in cracked concrete):
d,frequ Gkj Pk 11 Qk1 2i Qki
j 1 i 1
E E E E E
Leading variable action effect:
11 21 Qk1 1i 2i QkiE Max. E
Quasi-permanent combination
Normally used for long-term effects and the appearance of the structure (e.g. deformations of the structure):
d,perm Gkj Pk 2i Qkij 1 i 1
E E E E
2.3 Serviceability Limit States (SLS)
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
2.4 Fatigue Limit State (FLS)
The level of the design values of actions – including the relevant numbers of load cycles – corresponds to the Serviceability Limit State (SLS).
The level of the design values of material resistances – depending on the numbers of load cycles – corresponds to the Ultimate Limit State (ULS).
For fatigue design, the combinations of actions depend on the kind of material and, therefore, are given in EN 1992 to EN 1999.
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
00,50,6Temperature (non-fire) in buildings 00,20,6Wind loads
00,20,5 Sites located at altitude H ≤ 1000 m above sea level 0,20,70,7 Sites located at altitude H > 1000 m above sea level
Snow and ice loads 000
Category H:
roofs
0,30,50,7 Category G:
traffic areas,30 kN < v. weight 160 kN
0,60,7 0,7 Category F:
traffic areas, vehicle weight 30 kN
0,80,91,0 Category E:
storage areas
0,60,70,7 Category D:
shopping areas
0,60,70,7 Category C:
congregation areas
0,30,50,7 Category B:
office areas
0,30,50,7 Category A:
domestic, residential areas
Imposed loads in buildings (see EN 1991-1-1)210
Action
2.5 factors for buildings (recommended values)
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
1,00 -A accidental
1,00 1,30 Q Variable, unfavourable
1,00 1,00 G permanentC) Failure of the soil ground failure or loss of stability of a slope (GEO)
1,00-Aaccidental
1,001,50Qvariable, unfavourable
1,001,00G,inf favourable
1,001,35G,supPermanent, unfavourable B) Failure of the structure,
one of its members or of the foundation (STR)
1,00-A accidental
1,001,50Q variable, unfavourable
0,950,95G,infsmall deviations
1,001,05G,supin case of
0,950,90G,inf favourable
1,001,10G,sup permanent, unfavourable A) Loss of static equilibrium (EQU)
A P/T
SituationSymbolActionsUltimate Limit State (ULS)
2.6 Partial factors F applied to actions (recommended
values)
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
Differentiation of design values of permanent actions
Loss of static equilibrium (EQU)
The characteristic values of all the permanent actions are separated into two parts:
• all the parts acting unfavourably are multiplied by the factor G,sup;
• all the parts acting favourably are multiplied by the factor G,inf.
Failure of the structure, one of its members, or of the foundation (STR) All the characteristic values of one independent (single) permanent action Gk are multiplied by one unique factor G:
• by G,sup, if the resulting effect of Gk is unfavourable,
• by G,inf, however, if the resulting effect of Gk is favourable.
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Basis of Structural Design
Design of structural members (footings, piles, basement walls, …) (STR)
Approach 1
Applying design values according to Limit State B (STR) as well as to Limit State C (GEO) – in two separate calculations – to the geotechnical actions as well as to the other actions on/from the structure.
involving geotechnical actions and the resistance of the ground (GEO)
Approach 2
Applying design values only according to Limit State B (STR) to the geotechnical actions as well as to the other actions on/from the structure.
Approach 3
Applying design values according to Limit State C (GEO) to the geotechnical actions and, simultaneously, design values according to Limit State B (STR) to the other actions on/from the structure.
The use of approaches, either 1 or 2 or 3, is chosen in the National Annex.
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
Design of structural members (footings, piles, basement walls, …) (STR)
involving geotechnical actions and the resistance of the ground (GEO)
Advantage of Approach 2:
The limit states STR and GEO are clearly separated.
So the structural and geotechnical verifications can be performed independently.
Structural verification:Applying design values only according to Limit State
B) Failure of the structure (STR) to the geotechnical actions as well as to the other actions on/from the structure.
Geotechnical verification:The limit state C) Failure of the soil (GEO)
– e.g. ground failure or loss of stability of a slope – should be verified in accordance with EN 1997.
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
3 Basis for partial factor design and reliability analysis
3.1 Overview of reliability methods
Historical methods
Empirical methods
First Order Reliability Method FORM (Level II)
Full probabilistic methods (Level III)
Semi-probablistic methods(Level I)
Partial factor design
CalibrationCalibration Calibration
Method a
Method bMethod c
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
Most of the partial factors and -factors established in the present Eurocodes are generated by calibration (c) of the partial factor method (Level ) to the traditional procedures for verification (a).
In both the Level and Level methods the measure of reliability should be identified with the survival probability Ps:
Ps = () = (1 – Pf),
is the cumulative distribution function of the standardised Normal distribution
is the reliability index
3.1 Overview of reliability methods
where Pf is the failure probability for the considered failure mode and within an appropriate reference period.
Pf = (– )
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
Relation between und Pf
If the calculated failure probability is higher than the target value n:
Pf > (– n), then the structure is considered unsafe!
Pf10-1 10-2 10-3 10-4 10-5 10-6 10-7
1,28 2,32 3,09 3,72 4,27 4,75 5,20
3.1 Overview of reliability methods
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
Target values of reliability index for structural members
1,53,0Serviceability (irreversible)
1,5 to 3,8 2)Fatigue
3,84,7Ultimate (RC 2)
50 (n = 50 years) 1) 1 (1 year)
Target reliability indexLimit state
2) Depends on degree of inspectability, reparability and damage tolerance
1)
3.1 Overview of reliability methods
nn
s,n n 1 s,1P P
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
Reliability differentiation in ultimate limit states (see Annex B)
50 (n = 50 years) 1) 1 (1 year)
Target reliability indexReliability class(Consequences Class)
4,35,1RC 3 (CC 3)
3,84,7RC 2 (CC 2)
3,34,3RC 1 (CC 1)
1)
3.1 Overview of reliability methods
nn
s,n n 1 s,1P P
Partial factors given in EN 1990 to 1999 are based on RC 2
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
Definition of consequences classes (see Annex B)
Consequences for loss of human life, or economic, social or environmental consequences
Consequences Class
Low: agricultural buildings, green housesCC 1
Medium: Residential and office buildingsCC 2
High: Grandstands, public buildings, concert hallsCC 3
3.1 Overview of reliability methods
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
Quality assurance (see Annex B)
3.1 Overview of reliability methods
Reliability Class
Consequences Class
Design supervision level
Inspection level
RC 1CC 1 Self-checking Self inspection
RC 2CC 2Checking by
different personsSpecified inspection
procedures
RC 3CC 3 Third party checking Third party inspection
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
3.2 R-E-Model
R = structural resistance
E = resulting action effect
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
r
e
fR,E(r,e)fR(r)
fE(e)
Distribution densities of R and E:
3.2 R-E-Model
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
fR(r)
fE(e)
r
e
fR,E(r,e)
If E and R are stochastically independent, then:
fR,E(r,e) = fR(r) · fE(e)
3.2 R-E-Model
Distribution densities of R and E:
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
r
e
fR,E(r,e)
fR(r)
fE(e)mE
mR
E
R Limit state function:
Z = r – e = 0
Failure part:
Z < 0
Failure probability:
Pf = ∫Z<0
fR,E(r,e) · de ∙ dr = ∫Z<0
fR(r) · fE(e) · de ∙ dr
3.2 R-E-Model
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
Distribution densities and limit state straight line (in the standardised space)
Precondition: e and r are stochastically independent and standard normally distributed
Survival part Failure part:
Pf = ∫Z<0
fR( ) · fE( ) · d ∙ d
Limit state straight line: Z = β – αR∙ + αE∙
Design point: yd
R
R
σmr
r
E
E
σme
e
βαe Ed
βαr Rd
f ( , ) =
fR ( ) fE ( ) = const.
r e
r e
r ree
r e
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
Sensitivity factors 2E
2R
EE2
E2R
RR
σσ
σα;
σσ
σα
Reliability index R E
2 2R E
m m
3.2 R-E-Model
Design values (in the original space)
βασmeandβασmr EEEdRRRd
Reliability parameters
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
3.3 Approach for calibration of design values
Resulting action effect ed and structural resistance rd are separated.
Survival part:Z > 0
Failure part: Z < 0
Design point: yd
min
max
Limit state straight line
R
E
σσ
min
R
E
σσ
max
E
E
σme
e
βαe E
βαr R
R
R
σmr
r
According sensibility factors E andR are assessed by fixed values with respect to the limit state expressed in standardized coordinates.
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
For limit state straight lines within the intervalR
E
R
E
R
E
σσ
maxσσ
σσ
min
Therefore, the design values can be determined as follows:
1 1d R R Rr F F 0,8
1 1d E E Ee F F 0,7
the sensibility factors are fixed by: R = – 0,8 and E = + 0,7
Then, the partial factors can be defined, each in relation to the according characteristic values:
E = ed / ek
R = rk / rd
3.3 Approach for calibration of design values
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
Partial factors for variable actions (Gumbel distributions)
3.3 Approach for calibration of design values
0,00
0,50
1,00
1,50
2,00
2,50
0,00 0,10 0,20 0,30 0,40
Wind and Snow LoadsImposed Loads
Q
VQ (VN; VS; VW)
S; W
N
E 50 = 0,7
3,8
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
Sensibility factors E1 and E2 in case of two simultaneous actions (e1, e2)
Survival part: Z > 0
Failure part: Z < 0
Design points: ed1; ed2
E
= 22,5
= 22,5
1,077E
E1 = 0
E2 = 0
E1 = E2
E2
E222 σ
mee
βα4,0e Edi
βαe Ed
E1
E111 σ
mee
βαe Ed
βα4,0e Edi
Limit state straight line for 1σσ
0E2
E1
Limit state straight line
for 0σσ
1E1
E2
3.3 Approach for calibration of design values
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
In this case, the global sensibility factors E und R are multiplied by the
accompanying sensitivity factors Ei und Ri.
Design values on the safe side result, if
E1 = R1 = 1,0 is used for the leading value, and
Ei = Ri = 0,4 is used for the accompanying value
Design values of accompanying basic variables:
i i i
1 1di R R R Rr F F 0,32
i i i
1 1di E E E Ee F F 0,28
3.3 Approach for calibration of design values
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
Combination of two actions (e1, e2)
Design value of the leading action:
1 1d E E Ee F F 0,7
Combination factor: 0 = edi / ed
Design value of the accompanying action:
1 1
i i i
N N1 1di E E Ee F ' F 0,4 '
where ‘ is the reliability index referred to the basic time interval T1
and N1 is the number of basic time intervals during the design working life
3.3 Approach for calibration of design values
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
Design working life
3.3 Approach for calibration of design values
Indicative design working life (EN 1990, 2.3)
2 10-25 Replaceable structural parts, e.g. gantry girders, bearings
3 15-30 Agricultural and similar structures
4 50 Building structures and other common structures
5 100 Monumental building structures, bridges, and other civil engineering structures
Design working life category
Indicative design working life (years)
Examples
1 10 Temporary structures (1)
1) Structures or parts of structures that can be removed with a view to being re-used should not be considered as temporary.
Prof. Dr.-Ing. Jürgen GrünbergUniversität Hannover
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Basis of Structural Design
Combination factors 0,i for variable actions Qi
3.3 Approach for calibration of design values
0,00
0,10
0,20
0,30
0,40
0,50
0,60
0,70
0,80
0,90
1,00
0,00 0,10 0,20 0,30 0,40 0,50 0,60
50 years5 years3 months1 month12 days3 days
0,i
VQ
Design working life: T = 50
E = 0,7
50 = 3,8N1 = 1
N1 = 10
N1 = 200
N1 = 600
N1 = 1500
N1 = 6000