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FinalSolutions

# FinalSolutions - UNIVERSITY OF MICHIGAN Official...

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Unformatted text preview: UNIVERSITY OF MICHIGAN Official Examination Booklet Name of Student: Date of Examination: AErﬂ 20, 2006 Course: CEESll — Structural Dxnarnics The University of Michigan Honor Code The Honor Code outlines standards for ethical conduct for graduate and undergraduate students, faculty members, and administrators of the Coilege of Engineering at the University of Michigan. Policies When Taking Exam The instructor need not monitor the exam The instructor will announce the time of the exam and the instructor's whereabouts will be communicated to the class Students will ailow at ieasi: one empty seat between themselves and their neighbor The instructor will inform the class prior to the exam if aids are allowed during the exam Students must write and sign the Honor Pledge on their exams The Honor Pledge is: I acknowledge that I have neither given nor received aid on this examination nor have I concealed any violation of the Honor Code. (Signed) QUESTION 1: STRUCTURAL ANALYSIS FOR EARTHQUAKES (30 pts) Consider a five story lumped—mass shear building with the following properties: 0 The mass matrix, [M], is diagonal with masses mg,- = 1 kip—seczlin for each story 0 The structure is assumed to have no damping o The structure is subjected to an earthquake, whose displacement response spectrum (for an undamped system) is assumed given by the following function of the system’s natural period, T: on” r s is s, (T) ,—_ B T >13 in which a: 5 ill/\$2 and ,6: 5 in. o The response is dominated by the first two modes of the structure. The first two modes are as follows: 1.0 1.0 1.7 1.3 ¢,= 2.3 ¢,= 0.9 2.9 41.3 3.2 — 1.3 o The foilowing properties have been deteimined already: {¢}1M1{¢}{T¢11K1{¢} ki -sec 2lin ki/in __— (a) Estimate the fundamental frequencies for the first two modes. (b) Express the corresponding response spectral acceleration, Sa(T), as a function of period, T. Estimate the peak ground acceleration of the earthquake. (c) Estimate the maximum absolute displacement at the top of the structure and the maximum absolute interstory drift between the first and second modes. @ LEE-Mi 5n QHEE? @5377 =' R =3 535? “45.333 gﬁﬁﬁgﬁ q A? f 1?? \$535313? “ :‘i‘ﬂ ﬁHEEEE \$324333? a: 4/, =.é,27_% Q 023,252. IQi.i"?“‘gf/&'.'_ .- ”‘ka WW W W W ”£505 65 M3\$% i 55f? \$EW¥E£W§S \$312442 "gm \$§~§§5§W§ 288‘ SﬁEﬁTS 22324 M 22-"! 5% {aﬁﬂﬁiﬁﬁ‘ (QWFFWEUH QUESTION 2: PASSIVE STRUCTURAL CONTROL SYSTEM (30 pts) A very high profile skyscraper in NYC (Citicorp Building) is experiencing some very unsettiing motions due to harmonic wind motions pushing on the structure. The owner hires you to solve the problem since inhabitants of the building are feeling “qweezy” with motion sickness. As lead structural engineer, you first seek to quantify the nature of the wind motions. Based upon instrumented structures in the vicinity of the Citicorp Building, you are confident the wind motion on the structure consists of two major portions superimposed. The first is a static push with a mild harmonic load superimposed. Recognizing the harmonic component of the loading is what is inducing the nausea of the building occupants, you consider the wind load as if it is harmonic with frequency, to (i.e., pa) = posinar). To determine a course of action, you need to conduct a quick yet crude approximation of the structure to make “back of the envelope” calculations. Let us assume, you simplify the Citicorp Building into a simple single degree—of—freedom structure defined by a mass (m) and spring (k). Of course, you select m to be the total superstructure weight of the structure and then select it to attain what you think is the first period of the structure. You also assume for the time being, the structure is undamped (not very realistic, but convenient). posinwt m 9 u One potential solution is a technology called the “tuned mass damper” which has proven very effective in numerous Japanese buildings over the past decade. Essentially, a tuned mass damper is a huge mass (ma) installed in the roof of the building. This mass rolls on rollers and is attached to the structure via some springs with total stiffness, kn. (a) Write an equation of motion of the structure with the TMD attached at the roof. (b) Provided the harmonic nature of the wind load, select an appropriate ratio of kaa such that the structural motion is zero (u = 0). It is acceptable for the tuned mass damper to move, but the structure must remain still. FYI: A picture of the actual TMD installed in the Citicorp Building is shown: W§1ﬁ2::6é”i§ "Naﬁ .. . / ﬁﬁi ‘i\ r / 4/ to gﬁfg 43% 5433333741 x\$wﬁsgam9wcmﬁs~ “\ ‘ ,/ ,- . \ Y] / ’/ m; xv am r I if d w? QAMPING MECHANISM LR? DAMI’ING MECHaNiSM STRUCTURE Citicorp Center [19?8), New York, N.Y. {Architectsz Hugh Ssubbjns, in association with Emery Roth and Sons» {Hugh Stubhins and Associate-5)] Tuned mass damper (TMD) reduces buiiding sway Caused by winds. TMD utilizes 4OG-lon conctete mass which moves in opposite direction from building 10 reduce buiiding sway by some 40 percent. ya aa‘m . P9. “mama 5MB? t QUESTION 3: EIGBNVALUE'ANALYSIS (25 pts) You are asked to analyze the following structure: (2) Write the mass and stiffness matrices of the structure using the notation shown. (b) Determine the fundamental modal frequencies of the structure. (c) Determine the mass normalized first mode ((1) Assuming damping in the structure to be 5% and 2% of critical in the first and second modes, respectively, determine the damping matrix of the structure (use Rayleigh Damping) __.!2_. -12 I I 13r5”w2 L. -__<Wm a» a X - (ee-wlumxw) - I44 go 2 45¢, - +455“)? Hﬁ , :44. a o :4 BArl — +9.53 m = w 4— 40). 5on 'sz : 3.55:. .' 449a). -—- 2.563 “as; w“? -. ‘iirlﬁ? {—M’BAE-Z. Mo): = 4.419%;4... l - 3 I i 5:53 SEE-JEETE» 'z‘l-iiE'i-ETS Q-‘i £12 ”3&1? SHEETS 22W 33 in; m \$22-\$13? 2’. N'ﬁﬁr ~ 3‘ \12 "SW a; mfai‘é‘iwﬁﬁ éo) (m—mmmks _- é ' . . \$3-) \ g f”.- .152. . L70 E 277'- [2%,13 \$37me "DA’ﬁP/Méi . a? 0'05. _ 7329:1592 3. I56 6.42 g 3,347 2.3%] 339;" 5m“ 04, = 0,299 I M: '0wa [C13on+d\K ”giorag o ] " 0.1033 .0. 0.2% _ "QIWE. [C]V;W oww 0 GM 0 2532 QUESTION 4: GENERAL QUESTIONS (15 pts) a. Write out the expression for Rayleigh’s Quotient: w 2 *. K é: a ' 4',- ﬂ :15.- b. You are provided with the following 3 mode shape matrices. Please identify which one is not a possible mode shape matrix. Circle your choice. 0.1048 —0.9643 —0.2431 [CD]: 0.2450 —0.2119 0.9461 . 0.9638 0.1587 —0.2141 2 M07513 0.3197 [<13]: 2 0.0988 —0.8105 2 0.6525 0.4908 c. What is the origin of the term “consistent” mass matrix? Roughly draw the ﬁrst two modes of the 20 story building shown below. Based upon the mode shapes drawn above, if you are provided with three accelerometers to measure the lateral acceleration of the structure, Where would you install the accelerometers and Why? I. Em Cm—ewmwﬁﬁ) 3. NF/m— 13 (mogmv‘ZLZ-J ...
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