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CE 424/CE 524St ruc t ura l St ee l DesignChapter 2

Design Philosophies: LRFD/ASD

Dr. Mahmoud Reda Taha, P. Eng.Department of Civil Engineering, University of New Mexico

Dr. M. M. Reda Taha, 2007.

CE 424/524 - Chapter 2 Slide Number 2

Design philosophies

Probabilistic Basis for LRFD

Reliability index

Determining load and resistance factors

AISC load and resistance factors

LRFD

ASD

Tab le o f con ten t s

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Design Phi losophies

Allowable Stress Design (ASD)

Plastic Design (PD)

Load and Resistance Factor Design (LRFD)


Al low able St ress Des ign Service loads are calculated as expected during service life.

Linear elastic analysis is performed.

A factor of safety (FOS) of the material strength is assumed

(usually 3-4)

Design is satisfactory if (maximum stress < allowable stress)

Limitations

Case specific, no guarantee that our design covers all cases

Arbitrary choice of FOS?! Can this be resolved ?!

FOS

StrengthMaterialStressAllowable =

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The new AISC code introduces a new trend for ASD

na

RR

New vers ion of ASD

The service loads shall be computed considering all possible load

combinations and this can result in determining the required

strength Ra

A new factor of safety is introduced. The new factor of safetyis > 1.0 and is derived using probabilistic methods

By dividing the nominal strength Rn by the factor of safety,the concept of allowable stress is satisfied.


Plast ic Design Service loads are factored by a load factor.

The structure is assumed to fail under these loads, thus,

plastic hinges will form under these loads Plastic Analysis.

The cross section is designed to resist bending moments and

shear forces from the plastic analysis.

Members are safe as they are designed to fail under these

factored loads while they will only experience service loads.

Limitations

No FOS of the material is considered, neglecting the uncertainty in

material strength!

Arbitrary choice of overall FOS?!

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Load and Resistanc e Fact or Design (LRFD)

LRFD is similar to plastic design in that it performs design with the assumption offailure!

Service loads are multiplied by load factors () and linear elastic analysis isperformed.

Material strength is reduced by multiplying the nominal material strength by a

resistance factor () The design rule is

Where Rn is the nominal strength and Q is the load effect for the ith limit state

Advantages of LRFD

Non-case specific, statistical calculations guarantee population behavior.

Uniform factor of safety as both load and material factors are tied by reliability analysis.

niii RQ This rule shall be attained

for all limit states!!


Probabi l is t ic B as is for LRFD

The basic statistical information we can get are the mean and the

standard deviation

Mean of a sample population

Standard deviation of a sample population

=n

i

ixn

1

=n

i

ix

n

2)(1

We can also calculate the coefficient of variation (V)

=V

(n-1) for samples of less than 30 observations

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=2

2

2

)x(

e2

1)x(P

Probability density

function (PDF)

Cumulative probability

ba

=

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The mean and the standard deviations for the test results are:

From the previous graphwe might assume normaldistribution of the testresults

Thus by using the normalGaussian distributionwe get this curve

kips29.1

kips9.86

=

=

Probability density function (PDF)

=2

2

2

)x(

e2

1)x(P



)0x(PPOFthenQ

Rlnxifthus)0

Q

Rln(P =

=

Materials and load distributions have been shown by researchers to follownormal or log-normal distributions

If normal distribution is used, then

If log-normal distribution is used, then

)0x(PPOFthenQRxifthus)QR(P ==

x = zero

POF = Area left to the line

Mean()

Normal or Log-normal distribution

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==

Q

Rln

x

TQ

Rln

x

The reliability index is tied to this distribution.

It simply represents how far the mean from the critical point as multiples ofthe standard deviation

If we consider log-normal distribution for example

x = zero

M

ean()

xT

Each corresponds to aspecific probability of failure


Probabi l is t ic bas is for LRFD

Frequency(%)

Load effect Q

Resistance R

R - Q

QR

R

Q

RQ

Looking at the two distributions of resistance and load effect

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If we have the probability distribution of the load effect (Q) and thematerial resistance (R) then:

The probability of failure can be represented by observing the probability ofthe function (R-Q)

The probability of failure PF can be represented as the probability that Q R:

Probability

of failure


Probabi l is t ic B as is for LRFD We can view the relation between these probabilities from another

prospective

QQ

CosQRN

45)( ==

R

Q

45)( CosQR

N

o

45

The higher the , the bigger the safety radius

On this line Q = R,

we are interested in

Q> R

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Rel iabi l i ty Index

The reliability index is a function of both The load effect Q and the resistance R and their probability distributions.

It represents how confident are we in our decision that the resistance of thematerial is higher than the load effects.

Q

N

=

Q

Q

R CosQR

=

)(

22cosQR

R

+=

22

)(

QR

QR

+

=

For lines inclined by an angle we can prove


Rel iabi l i ty Index

=

)ln(

)ln(

Q

R

mQ

R

By considering the previous graph

The higher the parameter , the lower the probability of failure PF The parameter is known as The reliability index

The reliability index is a function of both The load effect Q and the resistance R and their probability distributions.

By targeting a specific Reliability index for all the design elements, a consisttent level of safety in design can be achieved

22

QR

mm QR

+

=OR

Lognormal variables Normal variables

The formula we will use

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Determin ing load and res is tance fac t ors

RV

n

m e

R

R 55.0

=

As the load effect and resistance distributions can be determined bymeasurements, the reliability index (probability of failure) for anycombination of loads and materials can be determined.

However, determining the probability of failure for a specific load andmaterial combination is not the design target. The target is to determine theload and resistance factors that can achieve a specific probability of failure

The following equation can be used for determining the resistance factorfor a specific reliability index :

VRis the coefficient of variation of the resistance

The load factor can be determined : see next example


Exam ple 1

For the shown connection:

The 50 years wind records are used to calculate the maximum load effect

Experimental results of the connection resistance are recorded

WP

22.310

24.99

26.28

25.3722.26

25.35

26.54

243

28.12

23.51

R (kips)Test #

19.110

14.89

18.48

21.27

19.86

16.95

16.34

18.43

14.12

14.81

Pw (kips)Record #

Determine

The probability of failure

Load and resistance factors for a

probability of failure of 0.01%

kipsRn 4.25=

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Exam ple 1

To determine the probability of failure we need to determine the

probability that Qm>Rm

First: Determine the statistical parameters for Q and R

136.0

36.2

38.17

=

=

=

Q

Q

m

V

kips

kipsQ

075.0

87.1

83.24

=

=

=

R

R

m

V

kips

kipsR


Exam ple 1

The probability of failure represents the probability that Qm>Rm

22

QRTmT QR +=

This is relatively high POF, we need to determine the load and

resistance factors to achieve a specific POF (POF

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AISC LRFD Desig n

Design using AISC will target a specific probability of failureReliability index to achieve a consistent design at thedifferent design load combinations and limit states

4.54.54.5Connections

1.752.53.0Members

D+L+ED + L + WD + (L or S)

Loading Conditions

Where

Dead loads (D)

Live loads (L)

Wind Loads (W)

Earthquakes (E)

AISC Reliability index ()


Load Designat ion

Dead loads (D)

Live loads (LL)

Occupancy load (L)

Roof load (Lr)

Snow load (S)

Rain loads (R)

Wind Loads (W)

Earthquake load (E) Lateral earth pressure (H)

Fluid pressures (F)

Self-restraining force (T)

Based on definitions by ASCE document on load and load combinations (2002),

AISC considers the following loads for designing of Steel structures.

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LRFD: Load and Resis t anc e Fact ors

AISC considers the following load combinations in design

niii RQ )(5.06.12.12 RorSorLLD r++

D4.11

)8.0(5.0)(6.12.13 WorLRorSorLD r ++

SLED 2.05.00.12.15 ++

)0.13.1(9.06 EWorD

)(5.05.06.12.14 RorSorLLWD r+++

00.175.0 =

ii Q

ni R for yield = 0.9 and for bolt shear = 0.75

For garages, load factor for L in load combinations 3,4 and 5 shall be 1.0 and not 0.5 ( L = 100 psf)


ASD: Load and Resis tan ce Fac t ors AISC considers the following load combinations in design

na

RR

LD2 +

D1

)RorSorL(D3 r+

)RorSorL(75.0L75.0D4 r++

5.1=

aR

for yield = 1.67 and for bolt shear = 2.0

)E7.0orW(D5

)E7.0orW(D6.07

)RorSorL(75.0L75.0)E7.0orW(75.0D6 r+++

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Exam ple 2


References

Segui, W. T., LRFD Steel Design, Fourth Edition, 2007, Thompson,

Brooks/Cole, USA.

Manual of Steel Construction, Load and Resistance Factor Design, American

Institute of Steel Construction (AISC), 13th Edition. 2005

McCormac, J. C. and Nelson, J. K., Structural Steel Design: LRFD Method,

3rd Edition, 2003, Prentice Hall, NJ, USA

Kulak, G. L. and Gilmor, M. I., Limit State Design in Structural Steel, 6th

Edition, 1998, Canadian Institute of Steel Construction, Alliston, Ontario,

Canada.

Loov, R. E., Structural Steel Design: Lecture Notes, 1997, Calgary, Canada.

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