ULS Reinforced Concrete Design

8/2/2019 ULS Reinforced Concrete Design

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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:

marcantonio.liotta@uniroma1.it

MarcAntonio Liotta Il progetto agli Stati Limite 2/200marcantonio.liotta@uniroma1.it


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marcantonio.liotta@uniroma1.it 3/248

Contents

Theoretical fundamentals

Probabilistic base

Design at Ultimate Limit State

Simple Flexure

Shear

Design at Exercise Limit State

MarcAntonio Liotta Il progetto agli Stati Limite


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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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E.G.

E(Effect of the Force F): bending momentME(F)

R (corresponding Resisting capacity of the beam): resisting moment

MRF

RME (F)

MR (F)MarcAntonio Liotta Il progetto agli Stati Limite


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E.G.

E: tensions due toME(F)

R : limit tensionf(of the material)

F

R



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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


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

MarcAntonio progetto agli Stati Limite


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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:

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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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MarcAntonio Liotta Il progetto agli Stati Limite 328/200marcantonio liotta@uniroma1 it

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