Introduction to Structural Dynamics & Earthquake Engineering

a home base to excellence Mata Kuliah Kode SKS : Dinamika Struktur & Pengantar Rekayasa Kegempaan : CIV 308 : 3 SKS Single Degree of Freedom System General Dynamic Loading Pertemuan - 4

a home base to excellence TIU : Mahasiswa dapat menjelaskan tentang teori dinamika struktur. Mahasiswa dapat membuat model matematik dari masalah teknis yang ada serta mencari solusinya. TIK : Mahasiswa mampu menghitung respon sistem struktur sederhana akibat beban luar sembarang

a home base to excellence Sub Pokok Bahasan : Eksitasi Sembarang

a home base to excellence Duhamel s Integral Undamped System An impulsive loading is a load which is applied during a short duration of time. The corresponding impulse of this type of load is defined as the product of the force and the time of its duration. Using Duhamel s Integral, the displacement response at time t due to continuous F( ) is : F( ) t 1 F( ) d ( ) t ( ) ( ) 0 = (1) u F sin t d n m n t + d

a home base to excellence Response to Step Forces p(t) p(t) po u(t) p(t) = po m p ( ) o = u t st 0 k k SDoF System Without Damping Maximum displacement : ( )o 2 t = ( ) ( t ) ( o ) ( = ) 2 u u (2) = 1 1 u u cos t u cos o st st n st o T n SDoF System With Damping ( ) ( t ) (3) = + t 1 u u e cos t sin t n st D D o 2 1

a home base to excellence Response to Step Force With Finite Rise Time p(t) ( ) t t p t / t t t ( ) t p(t) = p o r r po p u(t) o r Maximum displacement : m t If tr< Tn/4 : uo 2(ust)o If tr> 3Tn: uo 2(ust)o If tr/Tn= 1,2,3, uo = (ust)o k tr sin t t ( ) ( t ) (4) For t < tr = n u u st o t t r n r (5) 1 ( ) ( t ) ( ) = For t > tr 1 u u sin t sin t t st n n r o t n r

a home base to excellence Example The elevated water tank in Fig weighs 45,5 tons when full with water. The tower has a lateral stiffness of 145 kg/mm. Treating the water tower as an SDoF system, estimate the maximum lateral displacement due to each of the two dynamic forces shown without any exact dynamic analysis. Neglect damping. P(t) W k P(t) 25 tons t (dt) 0,2 P(t) 25 tons t (dt) 4

a home base to excellence Response to Pulse Excitations We next consider an important class of excitations that consist of essentially a single pulse Air pressure generated on a structure due to aboveground blasts or explosions is an example of pulse force. Force Time

a home base to excellence Response to Rectangular Pulse Force p(t) p(t) po t p t t u(t) ( ) t = p o d 0 t m d t k td ( ) t 2 u t For t < td (6) = = 1 1 cos t cos ( ) n u T st n o ( ) t u ( ) = For t > td cos t t cos t (7) ( ) n d n u st o

a home base to excellence Response to Rectangular Pulse Force The maximum deformation is : p ( ) = = (8) o u u R R o st d d o k With deformation response factor : = 2 t 2 for 1 2 d sin t T (9) d n R T d n for 1 2 t T d n The maximum value of the equivalent static force is : = = f ku p R (10) So o o d

a home base to excellence Example A one-story, idealized as a 3,6 m high frame with two columns hinged at the base and a rigid beam, has a natural period of 0,5 sec. Each column is a wide-flange steel section W8x18. Its properties for bending about its major axis are Ix = 2,570 cm4, S = Ix/c = 249 cm3; E = 200 GPa. Neglecting damping, determine the maximum response of this frame due to a rectangular pulse force of amplitude 1,800 kgf and duration td = 0,2 sec. The response quantities of interest are displacement at the top of the frame and maximum bending stress in the column.

a home base to excellence Response to Half Cycle Sine Pulse Force p(t) ( ) t p(t) sin p t/t t t ( ) t = p o d d po u(t) 0 t d m t k td ( ) t 1 T 2 u t t = 2 n sin sin For t < td ( ) ( ) 2 u t t T 1 2 T t st d d n o n d For td/Tn ( ) ( ) ( ) t 1 T t cos 2 t / 1 T t u t = For t > td 2 n d d n d sin ( ) ( ) 2 2 u T T T t st n n o n d

a home base to excellence For td/Tn= ( ) t 2 2 2 1 u t t t For t < td = sin cos ( ) 2 u T T T st n n n o ( ) t 1 u t For t > td = 2 cos ( ) 2 2 u T st n o

a home base to excellence Response to Symmetrical Triangular Pulse Force p(t) p(t) po u(t) m t k td td/2