Truncatjon Errors Numerical Integratjon Multjple Support Excitatjon - PowerPoint PPT Presentation

Truncatjon Errors Numerical Integratjon Multjple Support Excitatjon Giacomo Boffj htup://intranet.dica.polimi.it/people/boffj‐giacomo Dipartjmento di Ingegneria Civile Ambientale e Territoriale Politecnico di Milano April 2, 2020

Truncatjon Part I Num. Integra‐ tjon Support Exc. Giacomo Boffj How many eigenvectors? Introductjon Modal partecipatjon factor Dynamic Introductjon magnifjcatjon factor Statjc Modal partecipatjon factor Correctjon Dynamic magnifjcatjon factor Statjc Correctjon

Truncatjon Num. Integra‐ Sectjon 1 tjon Support Exc. Giacomo Boffj Introductjon Introductjon Modal partecipatjon factor Introductjon Dynamic magnifjcatjon factor Modal partecipatjon factor Statjc Correctjon Dynamic magnifjcatjon factor Statjc Correctjon

How many eigenvectors? Truncatjon Num. Integra‐ tjon Support Exc. Giacomo Boffj Introductjon Modal partecipatjon To understand how many eigenvectors we have to use in a modal factor analysis, we must consider two factors, the loading shape and the Dynamic magnifjcatjon excitatjon frequency. factor Statjc Correctjon

Introductjon Truncatjon Num. Integra‐ tjon Support Exc. Giacomo Boffj Introductjon In the following, we’ll consider only external loadings whose dependance Modal partecipatjon on tjme and space can be separated, as in factor Dynamic 𝐪(𝐲, 𝑢) = 𝐬 𝑔(𝑢), magnifjcatjon factor Statjc so that we can regard separately the two aspects of the problem. Correctjon

Multjplicatjon of 𝐍 and division of ̈ 𝑣 𝑕 by 𝑕 , acceleratjon of gravity, serves to show a dimensional load vector multjplied by an adimensional functjon. 𝑦, 𝑢) = 𝑕 𝐍 ̃ 𝑣 g (𝑢) 𝑞( ⃗ ⃗ 𝑠 ⃗ 𝑕 = 𝐬 g 𝑔 g (𝑢) ̃ ̈ Introductjon Truncatjon Num. Integra‐ It is worth notjng that earthquake loadings are precisely of this type: tjon Support Exc. 𝑦, 𝑢) = 𝐍 ̃ Giacomo Boffj 𝑞( ⃗ ⃗ 𝑠 ̈ ⃗ 𝑣 g Introductjon where the vector ̃ 𝑠 is used to choose the structural dof’s that are excited by the ⃗ Modal partecipatjon ground motjon component under consideratjon. factor ⃗ 𝑠 is an incidence vector, ofuen simply a vector of ones and zeroes where the Dynamic magnifjcatjon ones stay for the inertjal forces that are excited by a specifjc component of the factor earthquake ground acceleratjon. Statjc Correctjon

̈ ̃ Introductjon Truncatjon Num. Integra‐ It is worth notjng that earthquake loadings are precisely of this type: tjon Support Exc. 𝑦, 𝑢) = 𝐍 ̃ Giacomo Boffj 𝑞( ⃗ ⃗ 𝑠 ̈ ⃗ 𝑣 g Introductjon where the vector ̃ 𝑠 is used to choose the structural dof’s that are excited by the ⃗ Modal partecipatjon ground motjon component under consideratjon. factor ⃗ 𝑠 is an incidence vector, ofuen simply a vector of ones and zeroes where the Dynamic magnifjcatjon ones stay for the inertjal forces that are excited by a specifjc component of the factor earthquake ground acceleratjon. Statjc Correctjon Multjplicatjon of 𝐍 and division of ̈ 𝑣 𝑕 by 𝑕 , acceleratjon of gravity, serves to show a dimensional load vector multjplied by an adimensional functjon. 𝑦, 𝑢) = 𝑕 𝐍 ̃ 𝑣 g (𝑢) 𝑞( ⃗ ⃗ 𝑠 ⃗ 𝑕 = 𝐬 g 𝑔 g (𝑢)

Truncatjon Num. Integra‐ Sectjon 2 tjon Support Exc. Giacomo Boffj Modal partecipatjon factor Introductjon Modal partecipatjon factor Introductjon Dynamic magnifjcatjon factor Modal partecipatjon factor Statjc Correctjon Dynamic magnifjcatjon factor Statjc Correctjon

on the characteristjcs of the tjme dependency of loading, 𝑔(𝑢) , on the so called modal partecipatjon factor Γ 𝑗 , Γ 𝑗 = 𝝎 𝑈 Γ 𝑗 = 𝑕 𝝎 𝑈 𝐬/𝑁 𝑗 = 𝝎 𝑈 𝑗 𝐬 g /𝑁 𝑗 𝑗 𝐬/𝑁 𝑗 or 𝑗 𝐍 ̂ Note that both the defjnitjons of modal partecipatjon give it the dimensions of an acceleratjon. ̈ Modal partecipatjon factor Truncatjon Under the assumptjon of separability, we can write the 𝑗 ‐th modal Num. Integra‐ tjon equatjon of motjon as Support Exc. Giacomo Boffj 𝝎 𝑈 𝑗 𝐬 𝑁 𝑗 𝑔(𝑢) Introductjon 𝑟 𝑗 + 𝜕 2 𝑟 𝑗 + 2𝜂 𝑗 𝜕 𝑗 ̇ 𝑗 𝑟 𝑗 = � = Γ 𝑗 𝑔(𝑢) 𝑕 𝝎 𝑈 𝑗 𝐍 ̂ 𝐬 Modal 𝑔 g (𝑢) partecipatjon 𝑁 𝑗 factor Dynamic with the modal mass 𝑁 𝑗 = 𝝎 𝑈 𝑗 𝐍𝝎 𝑗 . magnifjcatjon factor It is apparent that the modal response amplitude depends Statjc Correctjon

Note that both the defjnitjons of modal partecipatjon give it the dimensions of an acceleratjon. ̈ Modal partecipatjon factor Truncatjon Under the assumptjon of separability, we can write the 𝑗 ‐th modal Num. Integra‐ tjon equatjon of motjon as Support Exc. Giacomo Boffj 𝝎 𝑈 𝑗 𝐬 𝑁 𝑗 𝑔(𝑢) Introductjon 𝑟 𝑗 + 𝜕 2 𝑟 𝑗 + 2𝜂 𝑗 𝜕 𝑗 ̇ 𝑗 𝑟 𝑗 = � = Γ 𝑗 𝑔(𝑢) 𝑕 𝝎 𝑈 𝑗 𝐍 ̂ 𝐬 Modal 𝑔 g (𝑢) partecipatjon 𝑁 𝑗 factor Dynamic with the modal mass 𝑁 𝑗 = 𝝎 𝑈 𝑗 𝐍𝝎 𝑗 . magnifjcatjon factor It is apparent that the modal response amplitude depends Statjc Correctjon on the characteristjcs of the tjme dependency of loading, 𝑔(𝑢) , on the so called modal partecipatjon factor Γ 𝑗 , Γ 𝑗 = 𝝎 𝑈 Γ 𝑗 = 𝑕 𝝎 𝑈 𝐬/𝑁 𝑗 = 𝝎 𝑈 𝑗 𝐬 g /𝑁 𝑗 𝑗 𝐬/𝑁 𝑗 or 𝑗 𝐍 ̂

̈ Modal partecipatjon factor Truncatjon Under the assumptjon of separability, we can write the 𝑗 ‐th modal Num. Integra‐ tjon equatjon of motjon as Support Exc. Giacomo Boffj 𝝎 𝑈 𝑗 𝐬 𝑁 𝑗 𝑔(𝑢) Introductjon 𝑟 𝑗 + 𝜕 2 𝑟 𝑗 + 2𝜂 𝑗 𝜕 𝑗 ̇ 𝑗 𝑟 𝑗 = � = Γ 𝑗 𝑔(𝑢) 𝑕 𝝎 𝑈 𝑗 𝐍 ̂ 𝐬 Modal 𝑔 g (𝑢) partecipatjon 𝑁 𝑗 factor Dynamic with the modal mass 𝑁 𝑗 = 𝝎 𝑈 𝑗 𝐍𝝎 𝑗 . magnifjcatjon factor It is apparent that the modal response amplitude depends Statjc Correctjon on the characteristjcs of the tjme dependency of loading, 𝑔(𝑢) , on the so called modal partecipatjon factor Γ 𝑗 , Γ 𝑗 = 𝝎 𝑈 Γ 𝑗 = 𝑕 𝝎 𝑈 𝐬/𝑁 𝑗 = 𝝎 𝑈 𝑗 𝐬 g /𝑁 𝑗 𝑗 𝐬/𝑁 𝑗 or 𝑗 𝐍 ̂ Note that both the defjnitjons of modal partecipatjon give it the dimensions of an acceleratjon.

Partecipatjon Factor Amplitudes Truncatjon Num. Integra‐ tjon Support Exc. Giacomo Boffj For a given loading 𝐬 the modal partecipatjon factor Γ 𝑗 is proportjonal to Introductjon the work done by the modal displacement 𝑟 𝑗 𝝎 𝑈 𝑗 for the given loading 𝐬 : Modal partecipatjon if the mode shape and the loading shape are approximately equal factor (equal signs, component by component), the work (dot product) is Dynamic magnifjcatjon maximized, factor Statjc if the mode shape is signifjcantly difgerent from the loading Correctjon (difgerent signs), there is some amount of cancellatjon and the value of the Γ ’s will be reduced.

Consider also the external, assigned load shape vector 𝐬 ... ̂ Example Truncatjon Num. Integra‐ tjon Support Exc. Consider a shear type Giacomo Boffj building, its fjrst 3 Introductjon eigenvectors as sketched Modal partecipatjon above, with mass factor distributjon approximately Dynamic magnifjcatjon constant over its height and factor its earthquake load shape Statjc Correctjon 𝑕𝐍 ̂ vector 𝐬 𝐬 𝝎 1 𝝎 2 𝝎 3 𝐬 = {1, 1, … , 1} 𝑈 𝐬 ≈ 𝑛𝑕{1, 1, … , 1} 𝑈 . → 𝑕 𝐍 ̂

̂ Example Truncatjon Num. Integra‐ tjon Support Exc. Consider a shear type Giacomo Boffj building, its fjrst 3 Introductjon eigenvectors as sketched Modal partecipatjon above, with mass factor distributjon approximately Dynamic magnifjcatjon constant over its height and factor its earthquake load shape Statjc Correctjon 𝑕𝐍 ̂ vector 𝐬 𝐬 𝝎 1 𝝎 2 𝝎 3 𝐬 = {1, 1, … , 1} 𝑈 𝐬 ≈ 𝑛𝑕{1, 1, … , 1} 𝑈 . → 𝑕 𝐍 ̂ Consider also the external, assigned load shape vector 𝐬 ...

Example, cont. Truncatjon Num. Integra‐ tjon For EQ loading, Γ 1 is Support Exc. relatjvely large for the fjrst Giacomo Boffj mode, as loading Introductjon components and Modal partecipatjon displacements have the factor same sign, with respect to Dynamic magnifjcatjon other Γ 𝑗 ’s, where the factor oscillatjng nature of the Statjc Correctjon higher eigenvectors will lead 𝑕𝐍 ̂ 𝐬 𝐬 𝝎 1 𝝎 2 𝝎 3 to increasing cancellatjon. On the other hand, consider the external loading, whose peculiar shape is similar to the 3rd mode. Γ 3 will be more relevant than Γ 𝑗 ’s for lower or higher modes.

Premultjplying by 𝝎 𝑈 𝑘 the above equatjon we have a relatjon that enables the computatjon of the coeffjcients 𝑏 𝑗 : 𝑏 𝑗 = 𝝎 𝑈 𝑗 𝐬 𝝎 𝑈 𝑗 𝐬 = 𝝎 𝑈 𝑗 � 𝐍 𝝎 𝑘 𝑏 𝑘 = � 𝜀 𝑗𝑘 𝑁 𝑘 𝑏 𝑘 = 𝑏 𝑗 𝑁 𝑗 → 𝑁 𝑗 𝑘 𝑘 Modal Loads Expansion Truncatjon Num. Integra‐ We defjne the modal load contributjon as tjon Support Exc. Giacomo Boffj 𝐬 𝑗 = 𝐍 𝝎 𝑗 𝑏 𝑗 Introductjon and express the load vector as a linear combinatjon of the modal Modal partecipatjon contributjons factor 𝐬 = � 𝐍 𝝎 𝑗 𝑏 𝑗 = � 𝐬 𝑗 . Dynamic magnifjcatjon factor 𝑗 𝑗 Statjc Correctjon