EAS 663/4 Dynamics and Stability of Structures EAS 663/4 Dinamik dan Kestabilan Struktur
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1 UNIVERSITI SINS MYSI 1 t. Semeter Examination 004/005 cademic Seion eperikaan Semeter 1 Sidang kademik 004/005 October 004 ES 663/4 Dynamic and Stability of Structure ES 663/4 Dinamik dan Ketabilan Struktur Duration: 3 hour Maa : 3 jam Intruction to candidate: 1. Enure that thi paper contain TEN (10) printed page included appendice. Sila patikan kerta peperikaan ini mengandungi SEUUH (10) muka urat bercetak termauk lampiran ebelum anda memulakan peperikaan ini.. Thi paper contain FIVE (5) quetion. nwer (5) quetion. Kerta ini mengandungi FIVE (5) oalan. Jawab KESEMU IM (5) oalan.. 3. ll quetion CN E anwered in Englih or ahaa Malayia or combination of both language. Semua oalan boleh dijawab dalam ahaa Inggeri atau ahaa Malayia ataupun kombinai keduadua bahaa. 4. Each quetion carry equal mark. Tiap-tiap oalan mempunyai markah yang ama. 5. ll quetion MUST E anwered on a new heet. Semua jawapan MESTIH dijawab pada muka urat yang baru. 6. Write the anwered quetion number on the cover heet of the anwer cript. Tulikan nombor oalan yang dijawab di luar kulit buku jawapan anda. /-
2 - - [ES 663/4] 1. (a) it two characteritic that ditinguih tructural dynamic problem from tatic one. (a) Senaraikan dua ciri yang membezakan maalah truktur dinamik daripada maalah tatik. (4 mark) (b) Define vicou damping. Sketch the diplacement repone, (v) veru (t) of undamped and damped SDOF ytem for free vibration. Doe the natural period of vibration, T, change with the preent of damping? (b) Takrifkan redaman likat. akarkan ambutan anjakan, v, melawan maa (t) untuk item Kebebaan Satu Darjah yang tanpa redaman dan dengan redaman. dakah kala getaran, T, berubah dengan kehadiran redaman? (6 mark) (c) Figure 1.0 how a model of pring-ma SDOF ytem that i ubjected to a harmonic excitation, p(t) = 50 co 10t N. The weight of the ma block i 150 kn and the pring tiffne, k = 7000 N/m. ume the damping of the ytem i equal to 5% of the critical damping. Determine the total diplacement repone of the ytem which i given by the following equation: t v t) V co( t ) e ( co t in ) V ( 1 d dt v ( 1 r ) (r r ) where : natural circular frequency of the ytem and v 0 : tatic diplacement due to p o. (c) Rajah 1.0 menunjukkan atu item jiim- pega dengan Kebebaan Satu Darjah yang dikenakan atu getaran harmonik, p(t) = 50 co 10t N. erat jiim blok ialah 150 kn dan kekukuhan pega, k = 7000 N/m. nggap bahawa redaman dalam item beramaan 5% daripada redaman kritikal. Kira nilai ambutan anjakan item terebut yang diberi dalam peramaan berikut : t v( t) V co( t ) e ( 1 co dt indt) vr V ( 1 r ) (r) iaitu ialah frekueni bulatan tabii item dan v 0 ialah anjakan tatik diebabkan oleh p o. (10 mark) v c p(t) m k Figure 1.0 3/-
3 - 3 - [ES 663/4]. (a) Duhamel Integral i normally ued for the evaluation of a linear SDOF ytem ubjected to arbitrary time varying force. Define the underlined term with the help of a graph Force, () veru (t). (a) Kamilan Duhamel biaanya digunakan untuk penilaian item Kebebaan Satu Darjah yang dikenakan daya ubah waktu embarangan. Takrifkan perkataan yang bergari dengan bantuan rajah daya, ()v, melawan maa (t). (5 mark) (b) Figure.0 how a pring-ma model for DOF ytem under free vibration. Derive the equation of motion for the ytem. (b) Rajah.0 menunjukkan atu model jiim-pega dengan Kebebaan Dua Darjah yang dikenakan getaran beba. Terbitkan peramaan gerakan untuk item terebut. (5 mark) k 1 u 1 (t) m 1 k u (t) m Figure.0 (a) (c) The water tower a hown in Figure.0 (b) weigh 700kN when filled with water i ubjected to tep force with rie time [Figure.0 (c)]. It i oberved that a horizontal jack force of 30kN i required to diplace the tower top by a ditance of 0mm. Etimate the maximum lateral diplacement repone due to dynamic force. The contant phae i given by the following equation: 1 v ( t) v0 1 in ( t tr ) tr in tr 1 co tr (in tr ), tan (1 co t ) where : natural circular frequency of the ytem, v 0 : tatic diplacement due to p o, v max : maximum repone and T n : natural period of vibration. plot of R d (= v max / v 0 ) veru t r / T n i hown in Figure.0 (d). Comment on the effect of ratio t r / T n on R d, without carrying out any exact dynamic analyi. r 4/-
4 - 4 - [ES 663/4] (c) Menara air yang ditunjukkan dalam Rajah.0 (b) mempunyai berat 700kN apabila penuh dengan air yang dikenakan daya langkah dengan maa kenaikan yang terhingga [Rajah.0 (c)]. Diberi bahawa atu daya ufuk ebear 30kN diperlukan untuk menganjakkan bahagian ata menara air ebanyak 0mm. nggarkan ambutan anajakan makima yang diebabkan oleh atu daya dinamik. eramaan ambutan anjakan emaa faa malar adalah eperti berikut: 1 v ( t) v0 1 in ( t tr ) tr in tr 1 co tr (in tr ), tan (1 co tr ) iaitu : frekueni bulatan tabii item, v 0 : anjakan tatik diebabkan oleh p o, v max : ambutan makimum dan T n : kala tabii getaran. Satu plot R d (= v max / v 0 ) lawan nibah t r / T n ditunjukkan dalam Rajah.0 (d). eri ulaan tentang kean nibah t r / T n ke ata R d tanpa menjalankan ebarang analii dinamik ebenar. (10 mark) p(t) p(t), kn 0 m 00 Figure.0 (b).00 Figure.0 (c) t, ec Figure.0 (d) 5/-
5 - 5 - [ES 663/4] 3. (a) y uing an axially loaded and perfectly traight column with both end pinned, explain the concept of table, untable and neutral equilibrium. (6 mark) (b) Figure 3.0 how an initially traight column ubjected to an axial load which act at an eccentricity e from the centroidal axi of the column. Obtain the following relation between mid-height deflection and ratio / E where E : Euler buckling load = / : e ec 1 E Sketch a plot of / E veru for three different value of e. Y Figure 3.0 Sketch alo on the ame plot the graph repreenting the behaviour of an initially traight column with e=0. aed on the graph, dicu the effect of imperfection of load on the behaviour of an axially loaded column. (14 mark) 4. (a) Derive the following fourth order differential equation for beam-column : y iv k y '' 0, k where y : lateral diplacement of beam-column, : axial force acting at both end of beam-column, : flexural rigidity and ( ) =d( )/dx. Next, explain how the above fourth order differential equation i ued to determine the critical load of beam-column with different end condition. You are required to pecifically point out in your explanation how tarting from the fourth order differential equation, one can arrive at the eigenvalue problem which can be ued to olve for the critical load of beam-column with different end condition. (10 mark) 6/-
6 - 6 - [ES 663/4] (b) imple two-bar frame i hown in Figure 3.0. load act at end of vertical member. oth upport and C are fixed. Obtain the effective length e for the two-bar frame by uing the following equation for an elatically retrained column: (1 1 1 ) in ( 1 )co 0 where 1 =/( 1 ), =/( ), =k, k =/, : flexural rigidity, : length of column, 1, : rotational tiffne of end 1 and of column being tudied, repectively. Jutify your olution for the effective length obtained by uing information provided in Table 1.0 (ee ppendix 1). (10 mark) C x Figure 3.0 7/-
7 - 7 - [ES 663/4] 5. (a) Figure 4.0 (a) and (b) how braced and unbraced frame, repectively. For each frame, ketch the buckling mode correponding to the lowet critical load. Uing uitable eigenvalue analyi, it can be hown that effective length factor K for i. column in braced frame i K < 1.0 and ii. column in unbraced frame i K > 1.0 Jutify the above concluion by referring to behavior of column with other tandard end condition. (6 mark) (a) raced frame (b) Unbraced frame Figure 4.0 (b) Slope deflection equation for a beam-column are given a follow : M M ii ji ij jj where ii, ij (= ji ), jj are tability function and M, M, and are a hown in Figure 5.0. M M = contant Figure 5.0 8/-
8 - 8 - [ES 663/4] Making ue of the previou et of lope-deflection equation and the following aumption for a member in a braced frame: i. ll member are primatic and behave elatically ii. The axial force in the beam are negligible iii. ll column in a torey buckle imultaneouly iv. t a joint, the retraining moment provided by the beam i ditributed among the column in proportion to their tiffne v. t buckling, the rotation at the near and far end of the beam are equal and oppoite how the proce of deriving the following eigenvalue problem for the determination of effective length of a column in a braced frame : ii G ij jj ij G 0 0 where G, G are defined a follow : G I column of I beam of column tiffnemeetingat end beam tiffnemeetingat end G I column of I beam of column tiffnemeetingat end beam tiffnemeetingat end You are required: i. to how the equation involved in arriving at the eigenvalue problem ii. to how a uitable ketch howing the column in a braced frame iii. to tate how the aumption lited above are ued in the derivation (14 mark) 9/-
9 - 9 - [ES 663/4] Supplement : x y, v - ooo O ooo -
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