1 Equivalent SDOF Approach. Sri Tudjono 1,*, and Patria Kusumaningrum 2

Size: px

Start display at page:

Download "1 Equivalent SDOF Approach. Sri Tudjono 1,*, and Patria Kusumaningrum 2"

Cornelius Franklin
5 years ago
Views:

1 MATEC Web of Conferences 159, (018) IJCAET & ISAMPE Dynamic Response of RC Cantiever Beam by Equivaent Singe Degree of Freedom Method on Eastic Anaysis A Review on Transformation Factors and Dynamic Magnification Factors Sri Tudjono 1,*, and Patria Kusumaningrum 1 Diponegoro University, Civi Engineering Department, Semarang, Indonesia ITB, Civi Engineering Department, Bandung, Indonesia Abstract. The response of muti-degree-of-freedom (MDOF) structure can be correated to the response of an equivaent singe-degree-of-freedom (SDOF) system, impying that the response is controed by a singe, unchanged mode shape. This equivaent SDOF method is eminent as an approximate method of dynamic anaysis. In this study, equivaent SDOF method anaysis is carried out on RC cantiever beam subjected to dynamic bast oading to review the transformation factors (TFs) provided by TM code. 1 Equivaent SDOF Approach This simpe approach of equivaent Singe Degree of Freedom (SDOF) method to anayze structura eements subjected to bast oads has first been performed by Mays and Smith [1]. A umped mass of an equivaent SDOF system is proposed to predict the dispacement of simpified structures subjected to bast by appying transformation factors of oad (K L), stiffness (K S), and mass (K M) to the SDOF anaysis. This method foows the TM US Army Code []. Based on this code, ampitude and time of peak response can be obtained from the ideaized charts based on natura period (T) of respective structure. SDOF method on one way eement was studied by Crawford et a. [3] for singe coumn. Big differences between the dispacement curves obtained from the numerica anaysis and the SDOF anaysis were reported. Study on two-way eements was carried out by Morison [] on was and sabs. TFs proposed by TM for SDOF anayses were reviewed using the yied ine theory of pates. Errors were found in some of TFs of two way eements some reaching as much as 50%. In this study, TFs provided by TM code were reviewed on RC cantiever beam. 1.1 Eastic Anaysis of Cantiever RC Beam * Corresponding author: tudjono@gmai.com The Authors, pubished by EDP Sciences. This is an open access artice distributed under the terms of the Creative Commons Attribution License.0 (

2 MATEC Web of Conferences 159, (018) IJCAET & ISAMPE Muti degree of freedom (MDOF) system can be represented using SDOF system by introducing equivaent dynamic systems. In equivaent dynamic systems, K L, K S, and K M are introduced into the system. Trianguar bast pressure (Fig. 1a) appied with respect to time is assumed to be uniformy distributed aong the height of beam (Fig. 1b). It is then converted into a atera oad (Fig. 1c) appied at the beam's tip after being mutipied by K L. Fe Me p M, K Ke (a) (b) (c) Fig. 1. (a) Trianguar Bast Pressure (b) MDOF System (c) Equivaent SDOF System The governing equation of the SDOF system may be given in terms of mass m, damping c, stiffness k, force F and dispacement u as foows. + + = () (1) whie the governing equation of equivaent SDOF system is given as + + = () () The equivaent terms of mass m e, stiffness k e and oad F e are as foows =. =. () =. () (3) K M, K S and K L refer to the mass, the stiffness and the oad transformation factors respectivey. The transformation factors, theoreticay, are vaues used to transform the φ MDOF probems into SDOF system by modeing the defection as a shape function of ( x). The defection functions of a cantiever RC beam are determined from the boundary conditions. The defection function assumed for the first mode is represented as π x φ ( x) = 1 cos () where x is the distance from fixed support whereas is the beam s ength. The equivaent mass, stiffness and force are as given in Eq. 5. This equation is equa to Eq. 3.

3 MATEC Web of Conferences 159, (018) IJCAET & ISAMPE e e e ( ) φ ( ) m = m x x dx '' ( ) φ ( ) k = EI x x dx ( ) φ ( ) F = q x x dx (5) Thus, K L, K S, and K M are theoreticay obtained as : K K S M π x 3 = = = 0.7 π φ ( x) dx 1 cos dx (6) '' πx πx ( ) cos dx φ x dx π = = = K L π x = = = π φ ( x) dx 1 cos dx (8) (7) The dispacement function is derived using the direct integration method of equivaent SDOF system [5]. It is obtained as a function of the dispacement and the veocity at time step i and the appied forces at time step i and i+1. = Δ + Δ + Δ + ( Δ + ) Δ + ( ) Δ + + () ( Δ ) + Δ + Δ (9) Simiary, by using the same procedures as the dispacement equation, the veocity function is derived as foows [5]: = Δ + Δ Δ + + Δ + Δ () Δ + Δ 1 (10) Dispacement Anaysis of the Cantiever RC Beam Subjected to Bast Load In this study, 35 cantiever RC beams of different sizes are investigated (Tabe 1) and subjected to uniformy distributed pressure varies from 1 MPa to 5 MPa, provided that the beam is sti in eastic condition. Loading duration t D is varied from 5 to 15 ms to obtain higher variation of t D/T ratios. K L of 0. based on TM is first appied to the mode []. 3

4 MATEC Web of Conferences 159, (018) IJCAET & ISAMPE Tabe 1. Mode Descriptions of Cantiever Beams B H L T B H L T (mm) (mm) (m) (ms) (mm) (mm) (m) (ms) It is shown in Fig. that for the case of a cantiever RC beam subjected to bast oad over a range of appied pressures, the peak dispacement which occurs at the tip of the beam potted against t D/T foows a power function. The resuts aso show that under eastic condition, for structures with the same t D/T ratio, a 5 ms onger of oading duration t D, resuts in higher peak dispacement at the cantiever tip of up to 5 times. dispacement (mm) 1.0 td=5 msec 0.8 td=10 msec 0.6 td=15 msec (a) dispacement (mm).5 td=5 msec.0 td=10 msec 1.5 td=15 msec (b)

5 MATEC Web of Conferences 159, (018) IJCAET & ISAMPE dispacement (mm) td=5 msec td=10 msec td=15 msec (c) (d) Fig.. Peak dispacements of beams subjected to bast pressures of (a) 1 MPa, (b) 3 MPa, (c) MPa and (d) 5 MPa dispacement (mm) td=5 ms td=10 ms td=15 ms.1 Dynamic Magnification Factor (DMF) DMF is the ratio of the dispacement under dynamic oading F(F 0,t) to the dispacement under the static oading F 0. The DMF vaue of a beam differs in time foowing trigonometric function. DMF equation is derived [5] and given, based on its intervas of time and oading duration t D as foows : u 1 t DMF = = 1 cosωet + sinωet, for 0 t td (11) u ω t t static e D D u 1 DMF = = { sinωet sinωe( t td) } cosωet, for t td u ω t static e D (1) As shown in Fig. 3, the maximum vaue of DMF depends upon the ratio of t D / T. The higher the ratio of, the DMF obtained is greater. Once the ratio of t D/T reaches a vaue arger than 1, the sope of the curve reduces unti the DMF reaches the maximum vaue of. The DMF anaysis can therefore be used as to vaidate the resuts obtained from the theoretica equivaent SDOF anaysis. Fig. 3 aso shows a comparison between the DMF vaue obtained from the structura dynamic theory and from SDOF anaysis. It is observed that the DMF of SDOF resuts are in good agreement with the theoretica vaues. DMF DMF_SDOF 0. DMF_St. Dynamic Theory Fig. 3. Maximum DMF 5

6 MATEC Web of Conferences 159, (018) IJCAET & ISAMPE Transformation Factor (TF) On the basis of oad transformation factor (K L), TM proposed a vaue of 0.0. It sighty overestimates the K L vaue derived by the structura dynamics theory which provides a unique KL vaue equa to (Fig. ). With regards to the mass transformation factors (K Ms), TM proposed a K M vaue of 0.6, again, higher than K M derived using structura dynamics theory which is given as 0.7. KL, KM KL_TM 5 KM_TM 5 KL_SD KM_SD Fig.. Transformation Factors 3 Concusion With respect to DMF, the DMFs obtained from the SDOF anaysis are found to be in good agreement with those obtained from the structura dynamics theory. On oad and mass transformation factors, TM provides an upper bound vaue, meanwhie the structura dynamics theory gives a ower bound vaue. The TM may use different approach with respect to defection function and may incude additiona safety factor of 10 to 15% for the purpose of bast resistant design. References 1. C. G. Mays, P. D. Smith, Bast effects on buidings (Thomas Teford Pubication, London, 1995). TM5-1300, Structures to resist the effects of the accidenta exposions (US Department of Army, New Jersey, 1990) 3. J. E. Crawford, J. L. Mavar, J. W. Wesevich, J. Vaancius, D. R. Aaron, Retrofit of RC structures to resist bast effects. ACI Struct J, 9, (1997). C. M. Morison, Dynamic response of was and sabs by singe-degree-of-freedom anaysis-a critica review and revision, Int J Impact Eng, 3, (006) 5. P. Kusumaningrum, Numerica Modeing of RC and ECC Encased RC Coumns Subjected to Cose-in Exposion (PhD Thesis, NUS, Singapore, 010) 6

In-plane shear stiffness of bare steel deck through shell finite element models. G. Bian, B.W. Schafer. June 2017

In-plane shear stiffness of bare steel deck through shell finite element models. G. Bian, B.W. Schafer. June 2017 In-pane shear stiffness of bare stee deck through she finite eement modes G. Bian, B.W. Schafer June 7 COLD-FORMED STEEL RESEARCH CONSORTIUM REPORT SERIES CFSRC R-7- SDII Stee Diaphragm Innovation Initiative