ﺗﺨﻤﯿﻦ ﻋﺪم ﻗﻄﻌﯿﺖ در ﻣﺪل ﺳﺎزي ﺳﺎزه ﻫﺎي ﻓﻮﻻدي ﺗﻮﺳﻂ ﺗﯿﺮ ﺗﯿﻤﻮﺷﻨﮑﻮ

Uncertainty Quantification in Modeling of Steel Structures Using Timoshenko Beam

اﯾﻦ ﻣﻘﺎﻟﻪ ﺑﻪ ﮐﻤﯽ ﺳﺎزي ﻋﺪم ﻗﻄﻌﯿﺖ ﻧﺎﺷﯽ از ﻣﺪل ﺳﺎزي ﺳﺎزه ﻫﺎي ﻓﻮﻻدي ﺗﻮﺳﻂ ﺗﯿﺮ ﺗﯿﻤﻮﺷﻨﮑﻮ ﻣﯽ ﭘﺮدازد. ﯾﮑﯽ از روش ﻫﺎي ﻣﺮﺳﻮم در ﺗﺤﻠﯿﻞ دﯾﻨﺎﻣﯿﮏ ﺳﺎزه ﻫﺎ، ﻣﺪل ﮐﺮدن ﺳﺎزه ﻫﺎي ﺳﺎﺧﺘﻤﺎﻧﯽ ﺗﻮﺳﻂ ﻣﺪل ﻫﺎي ﺗﯿﺮ ﭘﯿﻮﺳﺘﻪ اﺳﺖ. اﺳﺘﻔﺎده از ﺗﯿﺮ ﺗﯿﻤﻮﺷﻨﮑﻮ، ﺑﻪ ﻋﻨﻮان ﯾﮏ ﮐﻼس از ﻣﺪل ﻫﺎي ﺗﯿﺮ ﭘﯿﻮﺳﺘﻪ، ﺑﻪ ﺟﺎي ﯾﮏ ﻣﺪل اﺟﺰاي ﻣﺤﺪود ﺑﻪ ﺗﺴﺮﯾﻊ ﻣﺤﺎﺳﺒﺎت ﻣﻨﺠﺮ ﻣﯽ ﺷﻮد. اﯾﻦ ﮐﺎﻫﺶ ﻣﺤﺎﺳﺒﺎت، اﻣﮑﺎن ﺷﻨﺎﺳﺎﯾﯽ اﺣﺘﻤﺎﻻﺗﯽ ﺑﺮﺧﻂ ﺳﺎزه ﻫﺎي ﺳﺎﺧﺘﻤﺎﻧﯽ و ﺗﺸﺨﯿﺺ ﺳﺮﯾﻊ ﺧﺮاﺑﯽ در آن ﻫﺎ را ﭘﺲ از ﯾﮏ روﯾﺪاد ﻟﺮزه اي ﻓﺮاﻫﻢ ﻣﯽ ﺳﺎزد. ﻻزﻣﻪ ﺷﻨﺎﺳﺎﯾﯽ ﺳﺎزه ﻫﺎ ﺑﻪ اﯾﻦ روش، ﮐﻤﯽ ﺳﺎزي ﻋﺪم ﻗﻄﻌﯿﺖ ﻧﺎﺷﯽ از ﺗﻘﺮﯾﺐ ﺳﺎزه ﺑﺎ ﯾﮏ ﺗﯿﺮ ﭘﯿﻮﺳﺘﻪ اﺳﺖ ﮐﻪ ﻫﺪف اﺻﻠﯽ اﯾﻦ ﭘﮋوﻫﺶ را ﺗﺸﮑﯿﻞ ﻣﯽ دﻫﺪ. ﺑﺪﯾﻦ ﻣﻨﻈﻮر ﺑﺎ ﺗﻮﺳﻌﻪ ﯾﮏ ﺳﯿﺴﺘﻢ ﺧﻮدﮐﺎر، ﺗﻌﺪاد 1000 ﻣﺪل اﺟﺰاي ﻣﺤﺪود ﺳﺎزه ﺳﺎﺧﺘﻤﺎﻧﯽ ﻃﺮاﺣﯽ ﺷﺪه و ﺗﺤﺖ ﺗﺤﻠﯿﻞ ﻣﻘﺪار وﯾﮋه ﻗﺮار ﻣﯽ ﮔﯿﺮﻧﺪ ﺗﺎ ﺑﺴﺎﻣﺪ و ﺷﮑﻞ ﻣﻮدي ﻃﺒﯿﻌﯽ آن ﻫﺎ ﺗﻌﯿﯿﻦ ﺷﻮد. ﺑﺮاي ﻫﺮ ﺳﺎزه، ﺗﯿﺮ ﺗﯿﻤﻮﺷﻨﮑﻮي ﻣﻌﺎدل آن ﺗﻌﯿﯿﻦ ﻣﯽ ﺷﻮد. ﺳﭙﺲ، ﺧﻄﺎي ﻧﺎﺷﯽ از ارﺿﺎ ﻧﺸﺪن ﻣﻌﺎدﻟﻪ ﻣﺸﺨﺼﻪ ﺗﯿﺮ ﺗﯿﻤﻮﺷﻨﮑﻮ ﺑﻪ ازاي ﺑﺴﺎﻣﺪ و ﺷﮑﻞ ﻣﻮدي ﻧﺎﺷﯽ از ﻣﺪل دﻗﯿﻖ اﺟﺰاي ﻣﺤﺪود ﻣﺤﺎﺳﺒﻪ ﻣﯽ ﺷﻮد. ﺑﺎ ﺗﺤﻠﯿﻞ آﻣﺎريِ اﯾﻦ ﺧﻄﺎﻫﺎ ﺑﺮاي ﺳﯿﺴﺘﻢ ﻫﺎي ﻣﺨﺘﻠﻒ، ﻣﺸﺨﺼﺎت ﺗﻮزﯾﻊ اﺣﺘﻤﺎل ﺧﻄﺎ، از ﺟﻤﻠﻪ ﺷﮑﻞ ﺗﻮزﯾﻊ، ﻣﯿﺎﻧﮕﯿﻦ و اﻧﺤﺮاف ﻣﻌﯿﺎر آن ﺗﻌﯿﯿﻦ ﻣﯽ ﺷﻮد. ﻧﺘﺎﯾﺞ ﻧﺸﺎن ﻣﯽ دﻫﺪ ﮐﻪ ﻣﯿﺰان ﺧﻄﺎ ﺣﺴﺎﺳﯿﺖ اﻧﺪﮐﯽ ﺑﻪ ﻃﻮل، ﻋﺮض و ارﺗﻔﺎع ﺳﺎزه دارد، اﻣﺎ ﺑﻪ ﺳﯿﺴﺘﻢ ﺳﺎزﻫﺎي ﺑﺴﯿﺎر واﺑﺴﺘﻪ اﺳﺖ. ﺑﺮ اﺳﺎس اﯾﻦ ﻧﺘﺎﯾﺞ، اﻧﺤﺮاف ﻣﻌﯿﺎر ﺧﻄﺎ ﺑﺮاي ﺳﺎزه ﻫﺎﯾﯽ ﺑﺎ ﺳﯿﺴﺘﻢ ﻗﺎب ﺧﻤﺸﯽ ﻓﻮﻻدي 3/5 ﺑﺮاﺑﺮ ﮐﻤﺘﺮ از ﺳﺎزه ﻫﺎﯾﯽ ﺑﺎ ﺳﯿﺴﺘﻢ ﻣﻬﺎرﺑﻨﺪي ﻓﻮﻻدي اﺳﺖ. ﺑﻨﺎﺑﺮاﯾﻦ، ﮐﻼس ﻣﺪل ﺗﯿﺮ ﺗﯿﻤﻮﺷﻨﮑﻮ ﺗﻘﺮﯾﺐ ﺑﻬﺘﺮي از ﺳﯿﺴﺘﻢ ﻫﺎي ﻗﺎب ﺧﻤﺸﯽ ﻓﻮﻻدي اراﺋﻪ ﻣﯽ ﮐﻨﺪ.

This paper quantifies the uncertainty emanated from modeling steel structures using a Timoshenko beam. Using continuous beams to model building structures is a conventional approach in structural dynamic analyses. The use of Timoshenko beams, as a class of continuous beam models, in lieu of finite element models significantly expedites the computations. In turn, the reduction in the computational cost facilitates probabilistic, online identification and rapid damage detection of building structures in the aftermath of a seismic event. The prerequisite for structural identification using this method is the quantification of uncertainties that arise as a result of approximating the structure with a continuous beam, which is the primary objective of this research. To this end, an automated system is developed and subsequently employed to design 1000 finite element models of building structures followed by eigen-analyses to determine their natural frequencies and mode shapes. Thereafter, the residual of the eigen-equation of the Timoshenko beam given the frequency and mode shape from the exact finite element model is computed for each structure. Statistical analysis of these residuals for various structures leads to the distribution type, mean, and standard deviation for the probability distribution of the model error. The results show that the error is nearly insensitive to the height and dimensions of the plan, but is heavily dependent on the structural system. The results also show that the error standard deviation for steel moment-resisting frame systems is 3.5 times smaller than that of the steel concentrically-braced systems. This indicates that the Timoshenko beam model class provides a better estimation of steel moment-resisting frame systems.