Erratum for “Multivariate Extreme Wind Loads: Copula-Based
                    Analysis”

Ji, Xiaowen

doi:10.1061/JENMDT.EMENG-7299

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Jul 11, 2023

Erratum for “Multivariate Extreme Wind Loads: Copula-Based Analysis”

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Author: Xiaowen Ji, Ph.D. [email protected]Author Affiliations

Publication: Journal of Engineering Mechanics

Volume 149, Issue 9

https://doi.org/10.1061/JENMDT.EMENG-7299

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The following correction should be made to “Multivariate Extreme Wind Loads: Copula-Based Analysis” by Xiaowen Ji, https://doi.org/10.1061/(ASCE)EM.1943-7889.0002174:

In the section “Monte Carlo Simulation of Annual Maximum Wind Loads,” the first sentence in Step 2c should be corrected to:

c. In total,

Δ \times T

sample vectors are simulated via 2a or 2b.

In Appendix IV, the inverse functions

h^{- 1} (u_{i}, u_{j | i}; θ)

for partial derivatives of the Clayton, AMH, and Frank families are incorrect in the original paper. The corrected table is included herein.

Appendix IV. Partial Derivatives and Their Inverse Functions for Representative Copula Families

Family	$u_{j \| i} = h (u_{i}, u_{j}; θ)$	$u_{j} = h^{- 1} (u_{i}, u_{j \| i}; θ)$
Clayton	${[\frac{C (u_{i}, u_{j}; θ)}{u_{i}}]}^{θ + 1}$	${(u_{j \| i}^{\frac{- θ}{θ + 1}} u_{i}^{- θ} + 1 - u_{i}^{- θ})}^{- 1 / θ}$
AMH	$\frac{u_{j} - θ u_{j} (1 - u_{j})}{{[1 - θ (1 - u_{i}) (1 - u_{j})]}^{2}}$	$1 - \frac{2 θ (1 - u_{i}) u_{j \| i} - θ - 1 + {({[2 θ (1 - u_{i}) u_{j \| i} - θ - 1]}^{2} - 4 θ [θ {(1 - u_{i})}^{2} u_{j \| i} - 1] (u_{j \| i} - 1))}^{1 / 2}}{2 θ [θ {(1 - u_{i})}^{2} u_{j \| i} - 1]}$
Frank	$\frac{e^{- θ u_{i}} (e^{- θ u_{j}} - 1)}{(1 - e^{- θ u_{i}}) (1 - e^{- θ u_{j}}) - (1 - e^{- θ})}$	$- \frac{1}{θ} \ln [1 - \frac{1 - e^{- θ}}{(u_{j \| i}^{- 1} - 1) e^{- θ u_{i}} + 1}]$
GH	$\frac{C (u_{i}, u_{j}; θ)}{u_{i}} {[\frac{\ln u_{i}}{\ln C (u_{i}, u_{j}; θ)}]}^{θ - 1}$	No analytical expression
Galambos	$\frac{C (u_{i}, u_{j}; θ)}{u_{i}} [1 - {(\frac{Ψ}{{\tilde{u}}_{i}})}^{θ + 1}]$	No analytical expression
Gaussian	$Φ_{1} (\frac{{\overset{⌢}{u}}_{j} - θ {\overset{⌢}{u}}_{i}}{\sqrt{1 - θ^{2}}})$	$Φ_{1} (θ {\overset{⌢}{u}}_{i} + \sqrt{1 - θ^{2}} {\overset{⌢}{u}}_{j \| i})$
FGM	$θ (2 u_{i} - 1) u_{j}^{2} + [1 - θ (2 u_{i} - 1)] u_{j}$	$\frac{θ (2 u_{i} - 1) - 1}{2 θ (2 u_{i} - 1)} + sgn {θ (2 u_{i} - 1)} {[{(\frac{θ (2 u_{i} - 1) - 1}{2 θ (2 u_{i} - 1)})}^{2} + \frac{u_{j \| i}}{θ (2 u_{i} - 1)}]}^{1 / 2}$

Note: $u_{i | j} = h (u_{j}, u_{i}; θ)$ is obtained by exchanging $u_{i}$ and $u_{j}$ in the functions.

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Journal of Engineering Mechanics

Volume 149 • Issue 9 • September 2023

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Received: Apr 6, 2023

Accepted: May 24, 2023

Published online: Jul 11, 2023

Published in print: Sep 1, 2023

Discussion open until: Dec 11, 2023

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Xiaowen Ji, Ph.D. [email protected]

Associate Professor, Key Laboratory of Urban Security and Disaster Engineering of Ministry of Education, Beijing Univ. of Technology, Beijing 100124, China. Email: [email protected]

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Erratum for “Multivariate Extreme Wind Loads: Copula-Based Analysis”

Appendix IV. Partial Derivatives and Their Inverse Functions for Representative Copula Families

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