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MarcAntonio Liotta
Design of
Reinforced Concrete Structuresat the LIMIT STATE
Design and check of srtuctural elements
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References
website:
dsg.uniroma1.it/liotta
email:
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Contents
Theoretical fundamentals
Probabilistic base
Design at Ultimate Limit State
Simple Flexure
Shear
Design at Exercise Limit State
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Design Objective
Gather an adequate protection for two limit conditions:
Ultimate Limit State A structural damage preluding collapse
It is preferred that the structrure undergoes to inelastic deformations
Exercise Limit State Damages occur to non structural emlements
It is preferred that the structrure remains elastic
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Instruments
Use of linear or nonlinear analysis methods static or dynamic indesign
Use of capacity design in the structure design and conception
Use of the Ultimate Limit State Method in the structureverification.
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The Limit State method
The fundamental problem of structural reliability:The safety equation.
Uncertainty of variables
Semistochastic approach Characteristic and design values
Th epartial (safety) factors.
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The Limit State method
Principal characteristics:
It is based on uncertainty of variables (actions, strengths)
Several aspects are considered (exercise, collapse) againstwhich the risk is differenciated.
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The Semi Probabilistic Limit State method
Definitions
Limit State The structure or its part cant fulfil its design requirements
Ultimate Limit State Collapse and human life losses can happen
Exercise Limit State Functionality loss can happen
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The fundamental problem
The relation between tho variables is studied
E: Effect of the action: Demand
R : Resistance (strength): Capacity of the section(element structure)
Force(Effect of loads)
Resistance
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The fundamental problem
E.G.
E(Effect of the Force F): bending momentME(F)
R (corresponding Resisting capacity of the beam): resisting moment
MRF
RME (F)
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The fundamental problem
E.G.
E: tensions due toME(F)
R : limit tensionf(of the material)
F
R
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The fundamental problem
It must be verified that
R E
We have safety if,
R E
We have collapse if
R < E Graficamente
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Il problema fondamentale
S
R
R >SR < S
F = 5
R = 10
F = 10
R = 5
safety
collapse
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Uncertainty: variable distribution exmple
Frequency istograms of wheight and height ofstudents in the classroom (year 2006-2007)
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Uncertainty: variable distribution exmple
Frequency istograms of wheight and height of students inthe classroom (present class)
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Uncertainty of demand (actions)
Eis a stochastic variable Characterized (if normal) by a probability density
fE(E)
Which is a function of two parameters:
Mean (average) value:mE
scattering:sE
E
fE(E)
mE
sE
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Uncertainty of capacity (strengths)
R is a stochastic variable Characterized (if normal) by a probability density
fR(r)
Which is a function of two parameters :
Mean (average) value :mR
scattering :sR
r
fR(r)
mR
sR
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Stochastic variables
The Mean Value
The standard deviation o average discrepancy
The Variation Coefficient
1
1 n
i
i
xn
m
21
n
i
i
x
x
n
m
s
CVs
m
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Aleatoriet: richiami di statistica - Deviazione standard
The standard deviation or average discrepancy
Is a discrepancy index It measures the variability of a population of variables
It measures the data dispersion around the mean value(expected value)
It has the same unit of the observed variables
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The equation describing a normal distribution is
Equation of the Normal distribution curve
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s
m
21
21
( ) 2
x
f x e
m
s
s
x
f(x)
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Uncertainty
Derives from uncertainies due to:
Intensity of actions and from the probability of their coesistency
Geometry of the structure
Resistences of materials
Divergence from calculated effects and those really induced onto thestructure.
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Collapse probability
s
rSafety Region
Collapse region
The events shownhave different
probability to occur
fR(r)fS(s) dr ds
Which is thecollapseprobability?
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Probabilit di collasso
the collapse (failure)probability is given by:
safety
collapse
0
0Pr
SR
SR
f
dsdrsfrf
SRp
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Nel caso di v.a. normali
The failure probabilitypfis simply calculated as:
Si noti che:
SesR esS diminuiscono,pf diminuisce
Se (mRmS) aumenta,pf diminuisce
ss
mm
220PrSR
SRf SRp
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A semi-probabilistic approach
s
rsafety
collapse
Searching for two
values, Rdand Sdfor which if wehave:
Rd> Sdthen:
pf
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A semi-probabilistic approach
s
rsafety
collapse
Searching for two
values, Rdand Sdfor which if wehave:
Rd> Sdthen:
pf
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The design values
The design valuesRdandEdallow to transforma a stochasticproblem:
Pr[R E] pf,ammin a deterministic problem:
Rd
Ed
That is, if:RdSdpfpf,amm
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The design values
The design valuesRdandEdare expressed as a function of
the characteristic valuesRkandEk
The characteristic valuesRkandEkare percentages of 5%
and 95% of the distributions ofR andE
The ratios gR=Rk/Rd andgS=Ed/Ekare the partial safety factors
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Ed,Ek and gE (in case ofoccurrencies)
E
fE(S)
mE
sE
EkEd
The characteristic value is :Ek= mE+ 1.64sE
Pr[E
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Rd,Rk and gR (in case ofoccurrencies) The characteristic value is:
Rk= mR1.64sR
Pr[R
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The safety equation
It is required that:
Rk /gREk gE In a more general form we have that:
R(fk /gm)E(Fk gE)
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The Codes
NORME TECNICHE PER LE COSTRUZIONI (DM 14/01/2008)
CIRCOLARE ESPLICATIVA (2/02/2009)
Vecchie normative ormai superate:
D.M. LL.PP. del 09/01/1996 Norme tecniche per il calcolo, l'esecuzione ed il collaudo delle strutture in cemento armato, normale e
precompresso e per le strutture metalliche
Circolare M. LL.PP. del 15/10/96, n. 252 Istruzioni per l'applicazione delDM 09/01/1996
D.M. LL.PP. del 16/01/1996 Norme tecniche relative ai Criteri generali per la verifica di sicurezza delle costruzioni e dei carichi e
sovraccarichi
Circolare M. LL.PP. del 04/07/96, n. 156 Istruzioni per l'applicazione delDM 16/01/1996
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Design Actions
The safety checks must be done for the Ultimate Limit Stateand the Exercise Limit State
The actions Combinations are specified in the paragraph2.5.3.
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Design Life Span
2.4.1. Nominal Life Span
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Limit States
Ultimate Limit States (ULS [SLU])
Corresponding to the reach of extreme conditions
Exercise Limit States (ELS, [SLE])
Corresponding to ordinary needs of usage and duration.
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Ultimate Limit States
Are reguarding:
Safety of the people
Safety of the structure
Safety of the content (in some cases)
The Limit States to check are:
Equilibrium loss of (part of) the structure as a rigid body
Collapse, excessive deformatione, tranformation inmechanism, stability loss
Fatigue failure.
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Exercise Limit States
Are reguarding :
Functionality of the structure
Comfort of the people
Appearence and urability (deformations, ctracking)
The Limit States to check are :
Excessive deformations
Premature or excessive crackings Decay, corrosion
Excessive vibrations.
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Limit State design
Based on the use of models Of hte structures
Of loads
No Limit state is overtaken when are used adequatevalues of: Actions
Material properties
Geometry
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Fulfilment of Requirements
The requrements are fulfilled by means of the use of the partialsafety factor g
The method foresees The introduction ofcharacteristic values
The transformation of characteristic bvalues in design values introducing th
epartial factors gm or gf
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The partial factors
The design capacities of materials are obtained dividing thecharacteristic values of materials by cefficients gm (>1)
The design actions are obtained multiplying the characteristicvalues of the actions by gf
>1 or 1 depending from the fact that they increase or
decrease safety
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Basic variables
Actions Classification
Characteristic values
Other representative values
Material and products propreties
Geometric data
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Actions (F) and their Effects (E)
Action F
Each cause (dead loads, live loads, impressed deformations,chemical physical agents...) able to induce limit states onto
the structure EffectE(internal force)
Eache internal force (Normal force, bending moment, shearforce, etc.) thato is caused in the structure by the actions
In general, E can also be a deformation or a crackopening, etc.
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Actions: NTC 2008
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2.5.1 Action classification:based on the way to esplicitate (chap. 2.5.1.1)
a. Directa. Concentrated Forced, fixed or mobile loads
b. Indirecta. Imposed dispacements, temperature and humidity variations,
shrinking, prestressing, restraints failure,
c. Decaya. Due to the material: natural alteration of the material in time;
b. Due to external factors: alteration delle caratteristiche dei materialicostituenti lopera strutturale, a seguito di agenti esterni.
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2.5.1 Action classification:based on the structural response (2.5.1.2)
Actions:
a. static: not producing significant accelerations of thestructure;
b. pseudo static: dynamic actions producing significantaccelerations that may be translated as static equivalentones;
c. dinamiche: dynamic actions producing significantaccelerations
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2.5.1 Action classification:based on their variation in time (2.5.1.3)
Based ontheir variation in time, actions are defined:
Permanent (G)
G1: structural loads
G2: non structural (dead loads) Variable (Q)
Loads on floors, wind, snow
Exceptional (A)
Hurricanes, vehicle crashes, explosions Seismic (E)
Deriving by earthquakes
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2.5.2 Action classification(refferring to their distribution)
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2.5.3 Action combination
For the ULS:
k303Q3k202Q2k1Q1P2G21G1
QQQPGG gggggg
where:G1, G2 are the permanent actions
P is the prestressing acionQik are the characteristic values of the n independent live loadsgi = partial safety factors
Y0i = contemporaneity factor.
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2.5.3 Action combination
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2.5.3 Action combination coefficients (tab. 2.5.I)
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2.6 Actions ULS check
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2.6 Actions:Partial safety factors
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3.1 live loads (1 of 2)(chap 3.1)
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3.1 Live loads (2 of 2)(NTC2008, chap 3.1)
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3.1.3.1 Internal partitions
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3.1.3.1 Internal partitions
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4. Design strength of materials
In general, the design strength is given by:
k
dm
f
f g
where:Xk characteristic value of the material or product
h Scale or environmental factorgm partial safety factor
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ULS and ELS check
ULS it must be checked that:
dd
RE
where:Ed design value of the effect of the action
Rd the corresponding capacity design valueCd the limit value of the exercise criterion
ELS it must be checked that:
dd CE
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Load analysis
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Basic elements
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