002059

Mahyar Kazemian, M.Sc.

2026-06-29

Calculation of Local and Structural Pressure Coefficients (cp,1 and cp,10) Using Time-Domain Extreme Value Analysis

Wind-induced pressure coefficients are fundamental parameters in the design of building envelopes, roof cladding systems, and structural components. Traditional wind standards provide tabulated values for local pressure coefficients (cp,1) and structural pressure coefficients (cp,10), which are generally derived from wind tunnel measurements and statistical processing. However, modern computational techniques allow these coefficients to be estimated directly from pressure time histories obtained from measurements, CFD simulations, or synthetic turbulence generation.

1. Introduction

Wind pressure acting on building surfaces is highly fluctuating due to atmospheric turbulence and flow separation phenomena. While instantaneous pressures may exhibit large peaks, structural design requires representative extreme values associated with specific tributary areas. Two commonly used coefficients are [1]:

* cp,1: Represents local peak pressures acting on small cladding elements with an effective area of approximately 1 m²

* cp,10: Represents pressures acting on larger structural regions with an effective area of approximately 10 m²

Because larger areas average out small-scale turbulence structures, cp,10 values are generally less severe than cp,1 values. The challenge lies in transforming highly fluctuating pressure signals into statistically meaningful design coefficients. This article presents a complete time-domain methodology for deriving cp,1 and cp,10 through:

Quasi-steady pressure modeling
Area averaging
Load-duration filtering
Extreme value extraction
Gumbel distribution fitting
Gust loading factor correction

The methodology follows the framework of pseudo-steady wind loading theory and provides a physically consistent procedure for converting transient pressure fluctuations into design pressure coefficients.

1 Cp,10 and Cp,1 Algorithm from Transient CFD

2 Input Parameters, Recommended Value Ranges, and Derived Quantities Used in the Extreme Pressure Coefficient Analysis

2. Quasi-Steady Wind Pressure Model

The starting point is the quasi-steady assumption, which relates pressure fluctuations directly to velocity fluctuations. The instantaneous wind velocity is:

V (t) = U + u^{'} (t)

U	Mean Wind Speed
u'(t)	Turbulent Fluctuation

Quasi-Steady Wind Pressure Model

The corresponding pressure is calculated as:

w (t) = 0.5 ρ V^{2} (t) \cdot {\bar{c}}_{p} = 0.5 ρ U^{2} \cdot {[1 + \frac{u^{'} (t)}{U}]}^{2} \cdot {\bar{c}}_{p}

cp,bar	Mean pressure coefficient
ρ	Air Density

Quasi-Steady Pressure

This formulation captures the nonlinear relationship between velocity fluctuations and aerodynamic pressure. For roof suction regions, cp,bar is typically negative.

3 Instantaneous Wind Velocity

3. Generation of Pressure Time Histories

A turbulent velocity signal may be obtained from:

Wind tunnel measurements
CFD simulations
Synthetic turbulence generators
Field measurements

The resulting pressure coefficient time history is cp(t). In practice, additional pressure fluctuations arise from:

Flow separation
Vortex shedding
Local corner vortices
Turbulent wake interactions

These effects are superimposed on the quasi-steady pressure component to generate a realistic pressure signal.

4 Quasi-Steady Pressure Time History

4. Area Averaging and Load Duration Effects

Wind loading standards recognize that pressure peaks become less severe as the loaded area increases. To account for this phenomenon, pressure coefficients are filtered over an equivalent load duration:

c_{p, A} (t) = \frac{1}{τ} \int_{0}^{τ} c_{p} (t - \bar{t}) d \bar{t}, (τ = \frac{4.5 \sqrt{A}}{V_{m}})

A	Tributary Area
Vm	Mean Wind Speed

Filtered Pressure Coefficients

Typical values are commonly defined based on the tributary area of the loaded surface, where cp,1 represents the pressure coefficient associated with an effective area of 1 m², while cp,10 corresponds to an effective area of 10 m². As the tributary area increases, the associated load duration and averaging time also increase, resulting in a stronger smoothing effect on the pressure signal. Consequently, the larger averaging time associated with cp,10 filters out a greater portion of the high-frequency pressure fluctuations caused by local turbulence and flow separation. This reduction in short-duration pressure peaks leads to lower extreme suction values compared to cp,1, making cp,10 generally less severe and more representative of the pressures acting on primary structural elements rather than local cladding components.

Moving-Average Filtering of Pressure Coefficients for the evaluation of cp,1 and cp,10

5 Moving-Average Filtering of Pressure Coefficients for Evaluation of cp,1 and cp,10

6 Effect of Load Duration on Pressure Coefficient Time Histories

5. Extraction of Extreme Pressures

After area averaging, the filtered pressure records are divided into blocks of equal duration. For each block, the minimum pressure coefficient is extracted cp,min. These block minima represent statistically independent extreme events. The procedure transforms thousands of pressure samples into a manageable set of extreme values suitable for probabilistic analysis.

7 Block Extreme Suction Values Extracted

6. Extreme Value Theory and Gumbel Distribution

Wind pressure extremes are commonly modeled using the Gumbel distribution. For a sample set of extreme values: X=−cp. The Gumbel parameters are estimated from the sample mean and standard deviation. The cumulative distribution function is:

F (x) = \exp [- \exp (- \frac{x - μ}{β})]

Cumulative Distribution Function

where:

μ = location parameter
β = scale parameter

The Gumbel model provides a continuous representation of the probability of extreme pressure events.

7. Evaluation of the 78% Fractile

Wind engineering standards often define characteristic pressure coefficients using the 78% fractile of the extreme value distribution. The characteristic extreme value becomes:

c_{p, (78)} = μ + 1.4 β

Characteristic Extreme Value

The 78% fractile provides a robust estimate of design-level pressure peaks while avoiding excessive sensitivity to isolated outliers.

8 Gumbel Distributions of Extreme Pressure Coefficients and Corresponding 78% Fractile Values

8. Gust Loading Factor

The gust loading factor accounts for amplification due to atmospheric turbulence. The velocity peak factor and the gust loading factor are obtained from:

g_{u} = \sqrt{2 \ln (ν_{u} T)} + \frac{0.5772}{\sqrt{2 \ln (ν_{u} T)}}

G_{w, l i n} (A) = 1 + 2 g_{u} I_{u} \sqrt{B^{2} (A)}

νu	Mean up-crossing rate
T	Averaging Period
Iu	Turbulence Intensity
B²(A)	Background Factor

Gust Loading Factor

This formulation incorporates turbulence characteristics and spatial averaging effects.

9. Calculation of Final Pressure Coefficients

The final pressure coefficients cp,1 and cp,10 are obtained by dividing the characteristic peak pressure coefficients by the corresponding gust factor. This removes the amplification caused by wind turbulence and converts the peak pressures into equivalent pseudo-steady pressure coefficients. The resulting values represent static aerodynamic loading effects that can be directly used for structural design and code-based wind load calculations.

c_{p, 1} = \frac{c_{p, (78), 1}}{G_{w, lin} (A_{1})}, c_{p, 10} = \frac{c_{p, (78), 10}}{G_{w, lin} (A_{10})}

Final Pressure Coefficients

The resulting coefficients represent equivalent static pressure coefficients suitable for structural design.

9 Final Pseudo-Steady Pressure Coefficients ∣cp,1∣ and ∣cp,10∣ Obtained from the Extreme Value Analysis

10. Equivalent Wind Load

The final equivalent wind load acting on a tributary area is

W_{l i n} = \frac{1}{2} ρ U^{2} c_{p} G_{w, l i n} A

Equivalent Wind Load

This equation transforms pressure coefficients into design loads directly applicable within structural analysis models.

11. Interpretation of cp,1 and cp,10

The methodology naturally reproduces the well-known area reduction effect:

cp,1

Sensitive to local vortices
Higher suction peaks
Used for cladding and fastener design
Short averaging duration

cp,10

Represents larger loaded regions
Reduced peak suction
Used for primary structural design
Longer averaging duration

As the tributary area increases, high-frequency pressure fluctuations are progressively filtered out, reducing extreme pressure coefficients.

12. Advantages of the Time-Domain Methodology

Compared with traditional code-based approaches, the presented procedure offers several advantages:

Applicable to CFD-generated pressure fields
Applicable to wind tunnel measurements
Explicit treatment of turbulence intensity
Physical representation of area effects
Consistent extreme value statistics
Suitable for arbitrary tributary areas
Direct generation of pseudo-steady design loads

The methodology bridges the gap between transient aerodynamic pressure data and practical structural design parameters.

Conclusion

A comprehensive time-domain framework has been presented for deriving local and structural pressure coefficients, cp,1 and cp,10, from transient wind pressure records. The methodology combines quasi-steady wind loading theory, area averaging, extreme value statistics, and gust factor corrections to generate physically meaningful design coefficients. The approach is particularly valuable for CFD-based wind engineering, where complete pressure time histories are available and traditional code coefficients may not adequately represent complex geometries. By integrating load-duration effects and Gumbel extreme value analysis, the procedure provides a rigorous pathway from transient pressure fluctuations to equivalent static design loads.

Author

Mahyar is responsible for product development and marketing at Dlubal Software, with a particular focus on RWIND 2. He combines technical development with the enhancement of product communication.

References

Picozzi, V., et al. (2022). The relationship between wind pressure and pressure coefficients for the definition of wind loads on buildings. Buildings 12(2). p. 225.

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