Introduction to structural dynamics and earthquake engineering

University of Engineering and Technology Peshawar, Pakistan CE-409: Introduction to Structural Dynamics and Earthquake Engineering MODULE 1: FUNDAMENTAL CONCEPTS RELATED TO THE EARTHQUAKE ENGINEERING Prof Dr Akhtar Naeem Khan Mohammad Javed drakhtarnaeem@nwfpuet.edu.pk & Prof Dr Why to carry out dynamic analysis ? CE-409: MODULE ( Fall 2013) Importance of dynamic analysis Concepts discussed in courses related to structural engineering that you have studied till now is based on the basic assumption that the either the load (mainly gravity) is either already present or applied very slowly on the structures This assumption work well most of the time as long no acceleration is produced due to applied forces However, in case of structures/ systems subjected to dynamics loads due to rotating machines, winds, suddenly applied gravity load, blasts, earthquakes, using the afore mentioned assumption provide misleading results and may result in structures/ systems with poor performance that can sometime fail This course is designed to provide you fundamental knowledge about how the dynamic forces influences the2013) structural/systems response CE-409: MODULE ( Fall Sources of Dynamic Excitation Impact Blast Machine vibration CE-409: MODULE ( Fall 2013) Sources of Dynamic Excitation Wind CE-409: MODULE ( Fall 2013) Ground motion Static Vs Dynamic Force A dynamic force is one which produces acceleration in a body i.e dv/dt ≠ where v = velocity of body subjected to force A dynamic force always varies with time Examples of dynamic forces are: forces caused by rotating machines, wind forces, seismic forces, suddenly applied gravity loads e.t.c dv/dt≠0 v t CE-409: MODULE ( Fall 2013) Static Vs Dynamic Force A static force is one which produces no acceleration in the acting body A static force usually does not vary with time A force, even if it varies with time, is still considered static provided the variation with time is so slow that no acceleration is produced in the acting body e.g., dv/dt = v slowly applied load on a specimen tested in a UTM A static force can be considered as special case of dynamic force in which dv/dt =0 CE-409: MODULE ( Fall 2013) t Static Vs Dynamic Force H.A What will be the effect of truck (load) on bridge and response of bridge (structure)?, when: 1)Truck is not moving and present on bridge all the times 2)Moving on the bridge 3) Truck entering in to the bridge through a speed breaker 4)A truck with a capacity of 100 tonnes crosses the bridges half a million times while carrying a load which is 60% of its capacity CE-409: MODULE ( Fall 2013) Implications of dynamic forces CE-409: MODULE ( Fall 2013) Dynamic forces exerted by rotating machines (Harmonic loading) A common source of dynamic forces is harmonic forces due to unbalance in a rotating machines (such as turbines, electric motors and electric generators, as well as fans, or rotating shafts) Unbalance cloth in a rotating drum of a washing machine is also an harmonic force When the wheels of a car are not balanced, harmonic forces are developed in the rotating wheels If the rotational speed of the wheels is close to the natural frequency of the car’s suspension system in vertical direction , amplitude of vertical displacement in the car’s suspension system increases and violent shaking occur in car A Single degree of freedom system?(SDOF) respond harmonically till motion cease after the removal of force (irrespective of the type of dynamic load).CE-409: MODULE ( Fall 2013) 10 Nodal forces acting on the structure in nth mode, fn First mode Vbn { f n } = [ m ]{ φ n } Ln  f1n  φ1n   f11  φ11    Vb1     Vb1   [ m] φ 2n  ⇒  f 21  = [ m] φ 21   f 2n  =  f  L1 φ   f  L1 φ   31   31   3n   3n   f11  12 0   0.45 129.0  645         12 0 80 = 229  f 21  =        f  (2.25 *12)  0 12 1.00   286.7       31  CE-409: MODULE (Fall-2013) 34 Nodal forces acting on the structure in nth mode, fn Second mode Vbn { f n } = [ m ]{ φ n } Ln  f12  φ12  12 0   + 1.00  72.1    Vb      [ m] φ 22  = 12 + 0.45  f 22  =      f  L2 φ  (0.65 *12)  0 12  − 0.80     32   32   f12   + 110.8      f 22  =  + 49.8   f   − 88.6    32   CE-409: MODULE (Fall-2013) 35 Nodal forces acting on the structure in nth mode, fn Third mode Vbn { f n } = [ m ]{ φ n } Ln  f13  φ13  12 0   + 0.80 10.7    Vb      [ m] φ 23  = 12 − 1.00  f 23  =      f  L3 φ  (0.25 *12)  0 12  + 0.45     33   33   f13   + 34.2      f 23  =  − 42.8  f   + 19.3    33   CE-409: MODULE (Fall-2013) 36 Nodal forces acting on the structure in nth mode, fn { f n } = { f1n f 2n 129.0 + 110.8 + 34.2   f 3n } =  229.3 + 49.8 − 42.8  286.7 − 88.6 + 19.3  CE-409: MODULE (Fall-2013) 37 Nodal forces acting on the structure in nth mode, fn 88.6 k 286.7 k 229.3 k i1 49.8 k 129.0 k j1 110.8 k 645.0 kips Mode 19.3 k i2 42.8 k i3 j2 34.2 k j3 72.0 kips 10.7 kips Mode Mode CE-409: MODULE (Fall-2013) 38 Combination of Modal Maxima The use of response spectra techniques for multi-degree of freedom structures is complicated by the difficulty of combining the responses of each mode It is extremely unlikely that the maximum response of all the modes would occur at the same instant of time When one mode is reaching its peak response there is no way of knowing what another mode is doing The response spectra only provide the peak values of the response, the sign of the peak response and the time at which the peak response occurs is not known CE-409: MODULE (Fall-2013) 39 Combination of Modal Maxima { u} max ≠ [ Φ ]{ q} max Therefore and, in general { u} max ≤ [ Φ ]{ q} max The combinations are usually made using statistical methods CE-409: MODULE (Fall-2013) 40 Combined Response ro Let rn be the modal response quantity (base shear, nodal displacement, inter-storey drift, member moment, column stress etc.) for mode n The r values have been found for all modes (or for as many modes that are significant) Most design codes not require all modes to be used but many require that the number of modes used is sufficient so that the sum of the Effective Weights of the modes reaches, say, 90% of the weight of the building Checking the significance of the Participation Factors may be useful if computing deflections and rotations only CE-409: MODULE (Fall-2013) 41 Absolute sum (ABSSUM) method The maximum absolute response for any system response quantity is obtained by assuming that maximum response in each mode occurs at the same instant of time Thus the maximum value of the response quantity is the sum of the maximum absolute value of the response associated with each mode Therefore using ABSSUM method N ro ≤ ∑ rno n =1 This upper bound value is too conservative Therefore, ABSSUM modal combination rules is not popular is structural design applications CE-409: MODULE (Fall-2013) 42 Square-Root-of-the Sum-of-the-Squares (SRSS) method The SRSS rule for modal combination, E.Rosenblueth’s PhD thesis (1951) is ( N r ≅ ∑r o n =1 no developed in ) 1/ The most common combination method and is generally satisfactory for 2-dimensional analyses is the square root of the sum of the squares method The method shall not be confused with the rootmean-square of statistical analysis as there is no denominator CE-409: MODULE (Fall-2013) 43 Square-Root-of-the Sum-of-the-Squares (SRSS) method This method was very commonly used in design codes until about 1980 Most design codes up to that time only considered the earthquake acting in one horizontal direction at a time and most dynamic analyses were limited to 2-dimensional analyses CE-409: MODULE (Fall-2013) 44 Three Dimensional Structures In three-dimensional structures, different modes of freevibration in different directions may have very similar natural frequencies If one of these modes is strongly excited by the earthquake at a given instant of time then the other mode, with a very similar natural frequency, is also likely to be strongly excited at the same instant of time These modes are often in orthogonal horizontal directions but there may be earthquake excitation directions where both modes are likely to be excited In these cases the Root-Sum-Square or SRSS combination method has been shown to give non-conservative results for the likely maximum response In such cases some other methods such as CQC, DSC are used CE-409: MODULE (Fall-2013) 45 Modal combination of responses Consider nodes i & j of the frame for which R.S.modal analysis was carried out on previous slides Using SRSS Ai1 Ai2 Ai3 i1 Mi1 i2 Mi2 i3 Mi3 method A i = A j = A i12 + A i 2 + ( − A i3 ) M i = M i12 + M i 2 + ( − M i3 ) j1 j2 Mj1 j3 Mj2 Mj3 Aj1 Aj2 Aj3 Mode Mode Mode ( ) M j = M j12 + M j2 + − M j3 CE-409: MODULE (Fall-2013) 46 Caution It must be stressed that what ever response item r that the analyst or designer requires it must be first computed in each mode before the modal combination is carried out If the longitudinal stress is required in a column in a frame, then the longitudinal stress which is derived from the axial force and bending moment in the column must be obtained for each mode then the desired combination method is used to get the maximum likely longitudinal stress It is NOT correct to compute the maximum likely axial force and the maximum likely bending moment for the column then use these axial forces and bending moments, after carrying out their modal combinations, to compute the longitudinal stress in the column CE-409: MODULE (Fall-2013) 47 Home Assignment No M9 A story R.C building as shown below is required to be designed for a design earthquake with PGA=0.25g, and its elastic design spectrum is given by Fig 6.9.5 (Chopra’s book) multiplied by 0.25) It is required to carry out the dynamic modal analysis by using the afore mentioned design spectrum Take: m3 • Story height = 10ft •Total stiffness of first stories = 2000 kips/ft m2 • Total stiffness of top floor = 1500 kips/ft • Mass of first floors = 5000 slugs • Mass of top floor = 6000 slugs CE-409: MODULE (Fall-2013) m1 k3 k2 k1 48 ... is produced due to applied forces However, in case of structures/ systems subjected to dynamics loads due to rotating machines, winds, suddenly applied gravity load, blasts, earthquakes, using...Why to carry out dynamic analysis ? CE-409: MODULE ( Fall 2013) Importance of dynamic analysis Concepts discussed in courses related to structural engineering that you have... common source of dynamic forces is harmonic forces due to unbalance in a rotating machines (such as turbines, electric motors and electric generators, as well as fans, or rotating shafts) Unbalance