Adjusted Equivalent Static Wind Loads for non-Gaussian ...Adjusted Equivalent Static Wind Loads for non-Gaussian Linear Static Analysis Nicolas Blaise, Vincent Deno el University of

Adjusted Equivalent Static Wind Loads fornon-Gaussian Linear Static Analysis

Nicolas Blaise, Vincent Denoël

University of Liège (Belgium)

14th International Conference on Wind Engineering

ICWE 201521-26 June 2015, Porto Alegre

N. Blaise, V. Denoël | Adjusted Equivalent Static Wind Loads for non-Gaussian linear static analysis | ICWE 2015

Context

Le Nouveau Vélodrome Marseille Stade de Lille Métropole

→ Equivalent Static Wind Loads ?

N. Blaise, V. Denoël | Adjusted Equivalent Static Wind Loads for non-Gaussian linear static analysis | ICWE 2015

Academic Example

Considered frame

36.6 m

57.2

m

Wind

. Well-known wind pressure field [Main 2006]

. Limitations of existing ESWLs

. Linear & static structural behaviour

Non Gaussian pressure field !

N. Blaise, V. Denoël | Adjusted Equivalent Static Wind Loads for non-Gaussian linear static analysis | ICWE 2015

Academic Example: Pressure Field

-1.5 -1 -0.5 0 0 0.2 0.4 0.6 0.8 1

-2 -1.5 -1 -0.5 0 0.5 0 2 4 6 8

Mean Cp Standard deviation of Cp

Skewness coefficient of Cp Excess coefficient of Cp

N. Blaise, V. Denoël | Adjusted Equivalent Static Wind Loads for non-Gaussian linear static analysis | ICWE 2015

Academic Example: Response in Frame #2

20 kNm

10 kNm

0.6

2

4 61 kNm

Mean Bending Moment

Standard Deviation of Bending Moment Excess of Bending Moment

Skewness of Bending Moment

Example of Cp

Peak Factor (Nonsymmetric) Extreme/Design valueEnvelope

N. Blaise, V. Denoël | Adjusted Equivalent Static Wind Loads for non-Gaussian linear static analysis | ICWE 2015

Equivalent Static Wind Load

. Conditional Sampling technique [Holmes 1988]

Average

Equivalent Static Wind Load

Block 1 Block 2 Block 3 Block 4 Block 5 Block 7Block 6

Bending moment @ left support

Time

0 0.5 1 1.5Cp

Bending moments under ESWL Envelope overshooting

N. Blaise, V. Denoël | Adjusted Equivalent Static Wind Loads for non-Gaussian linear static analysis | ICWE 2015

Equivalent Static Wind Load. Load-Response Correlation (LRC) [Kasperski 1992]

Equivalent Static Wind Load

0 0.5 1 1.5

Bending moments under ESWL Envelope overshooting

Cp

N. Blaise, V. Denoël | Adjusted Equivalent Static Wind Loads for non-Gaussian linear static analysis | ICWE 2015

Equivalent Static Wind Load

. Non-Gaussian Load-Response Correlation: a bi-cubic model

, : Two correlated normal R.V.

, , , , , , : 7 parameters

Examples of PDFs generated with the bi-cubic model

N. Blaise, V. Denoël | Adjusted Equivalent Static Wind Loads for non-Gaussian linear static analysis | ICWE 2015

Equivalent Static Wind Load: Comparison

Bending Moments under ESWL Envelope overshooting

0 0.5 1 1.5

Equivalent Static Wind Load

Conditional Sampling

Load-Response Correlation

Non Gaussian Load-Response Correlation

• LRC: Severe over-estimation of the envelope• Non-G. LRC: Slight over-estimation of the envelope

N. Blaise, V. Denoël | Adjusted Equivalent Static Wind Loads for non-Gaussian linear static analysis | ICWE 2015

Two important properties & Adjustment

1. the Envelope value condition... The ESWL associated with a given response should return thedesign value for that response ...

2. the Non-overestimation condition... The responses under a given ESWL should not exceed the targetenvelope ...

→ 2-step adjustmentp̃(e) = β◦

(αp(e)

)α: load scaling coefficientβ : local adjustment coefficient

N. Blaise, V. Denoël | Adjusted Equivalent Static Wind Loads for non-Gaussian linear static analysis | ICWE 2015

Illustration of Adjustment

α β β ◦ αcp

1.13

a a

1.00

b b

1.1

c

0 0.5 1 1.5 0 0.5 1 1.5

Conditional Sampling

Load-Response Correlation

Non Gaussian Load-Response Correlation

25 %

38 %

39 %

Unadjusted ESWL Scaled ESWL Adjusted ESWL

α β β ◦ αcp

α β β ◦ αcp

• α = 1 for the LRC method• β is obtained with a constrained optimization algorithm

(as close as 1 as possible)

N. Blaise, V. Denoël | Adjusted Equivalent Static Wind Loads for non-Gaussian linear static analysis | ICWE 2015

Envelope Reconstruction

[%]

Design envelope Response under adjusted ESWL

Number of load cases

Enve

lope

reco

nstru

ctio

n

• Adjustment → faster reconstruction• Conditional Sampling, LRC, nG-LRC perform equally if adjusted

N. Blaise, V. Denoël | Adjusted Equivalent Static Wind Loads for non-Gaussian linear static analysis | ICWE 2015

Perspectives & Conclusions

Proposition of a Non Gaussian version of the LRC. bi-cubic model. regularly extends the LRC for non Gaussian pressure field/responses

2-Step Adjustment of Equivalent Static Wind Load to meet:. the Envelope Value Condition. the Non-Overstimation Condition

N. Blaise, V. Denoël | Adjusted Equivalent Static Wind Loads for non-Gaussian linear static analysis | ICWE 2015

Thank you ...

Vincent Denoël, Université de LiègeStructural & Stochastic Dynamicswww.ssd.ulg.ac.be

Read more about this topic:I Blaise N., Denoël V. (2013). Principal static wind loads. Journal of Wind

Engineering and Industrial Aerodynamics 113, 29-39.

I Blaise N., Canor T., Denoël V. (to appear). Reconstruction of the envelope ofnon-Gaussian structural responses with principal static wind loads.

I Kasperski M., (1992). Extreme wind load distributions for linear and nonlineardesign. Engineering Structures 14, 27-34

I Holmes J.D., (1988). Distribution of peak wind loads on a low-rise building.Journal Of Wind Engineering and Industrial Aerodynamics 29, 59-67

Available @ www.orbi.ulg.ac.be

N. Blaise, V. Denoël | Adjusted Equivalent Static Wind Loads for non-Gaussian linear static analysis | ICWE 2015

www.ssd.ulg.ac.bewww.orbi.ulg.ac.be

ContextAppendix