Elements of seismic analysis methods - Elastic response spectra

Elastic response spectrum analysis for the design of structures. A brief
structural and earthquake engineering lecture to study.

1. Lecture No. 07 Elements of seismic analysis methods

2. CONTENT Elastic Dynamic Analysis of SDOF Structures Elastic Earthquake Response Spectra “Exact” Dynamic Analysis of Linear MDOF Structures Spectral Analysis of Linear MDOF Structures Nonlinear Static Analysis of MDOF Structures Nonlinear Dynamic Analysis of MDOF Structures Incremental Dynamic Analysis References

3. 1. Elastic Dynamic Analysis of SDOF Structures Mathematical Model F(t) = a time varying dynamic load x(t) = time varying relative displacement of the mass with respect to the base corresponding to F(t) m = mass of the system (weight/g) k = stiffness of the system (static force required to induce a unit displacement). For a building subjected to lateral loads, k represents the lateral stiffness of the structure c = equivalent viscous damping coefficient of the system (dynamic force required to produce a unit velocity). This viscous damping is introduced to represent the energy dissipated during the vibrations of the system. This energy dissipation arises from many sources which are not easy to separate. The viscous damping model is retained because it is mathematically convenient, as we will see later.

4. 1. Elastic Dynamic Analysis of SDOF Structures Example of Lateral Stiffness Coefficient Column fixed at both ends Column simply supported at the base and fixed at the floor level.

5. 1. Elastic Dynamic Analysis of SDOF Structures Example of Lateral Stiffness Coefficient Diagonal bracing members simply supported at both ends. Cantilever Wall. (V =Poisson’s ratio)

6. 1. Elastic Dynamic Analysis of SDOF Structures Equation of Motion Arbitrary dynamic loading

7. 1. Elastic Dynamic Analysis of SDOF Structures Equation of Motion Base Excitation (Earthquake Problem) where = Ground Acceleration  

8. 1. Elastic Dynamic Analysis of SDOF Structures Equivalency of Seismic problem

9. 1. Elastic Dynamic Analysis of SDOF Structures Free vibrations (undamped) The simplest dynamic response of an SDOF system occurs when the system is in free vibrations. Free vibrations are the results of mere initial conditions (displacement or velocity) without external dynamic excitation. The free vibrations response is very important to find the fundamental characteristics of a system: the natural period of vibrations. Undamped systems. The equation of motion for an undamped SDOF system in free vibrations is written as follows: It can also be written as: Where

10. 1. Elastic Dynamic Analysis of SDOF Structures The general (homogenous) solution for above equation is the following : in which the integration constants A and B depend on the initial conditions of the system. Let us suppose that the initial displacement is x0 and that the initial velocity is x . It yields : The general solution is then written as:

11. 1. Elastic Dynamic Analysis of SDOF Structures Free vibration (undamped)

12. 1. Elastic Dynamic Analysis of SDOF Structures Free Vibrations (damped) Damped systems . In reality, the damping of a structure in free vibrations tends to shorten the amplitude of the vibrations with time. There are different types of damping: viscous damping, where the damping force is proportional to the velocity; friction damping, where the damping force is constant; internal damping, where the force is proportional to the displacement’s amplitude. In civil engineering structural dynamics, viscous damping is almost always used because of its mathematical convenience

13. 1. Elastic Dynamic Analysis of SDOF Structures Free Vibrations (damped) in this case, the equation of motion for a linear, damped SDOF system in free Vibration is: OR if we define: angular frequency It yields: (4.28) critical damping ratio  

14. 1. Elastic Dynamic Analysis of SDOF Structures Free Vibrations (damped) The general (homogenous) solution of equation 4.28 is: (4.29) Substituting equation 4.29 into 4.28, yields: Which is a quadratic equation in with the following roots: (4.31) According to equation 4.31, the solution depends of the sign of the square root. There are three possible solutions : Overdamped system Critically damped system Underdamped system (no practical significance for civil engineering structures)

15. 1. Elastic Dynamic Analysis of SDOF Structures Free Vibrations (damped) Underdamped System ( <1) In this case , the roots of equation 4.31 are complex values. Where  

16. 1. Elastic Dynamic Analysis of SDOF Structures Free Vibrations (damped) the solution of equation 4.28 is: Using Euler’s relation, the solution is simplified to: To evaluate constant A and B, the initial condition are set (at t=0) It then yields: (4.44)

17. 1. Elastic Dynamic Analysis of SDOF Structures Free Vibrations (damped) Damped angular frequency Damped natural frequency Damped natural period

18. 1. Elastic Dynamic Analysis of SDOF Structures Free Vibrations (damped) in practice, not necessary to make a distinction between undamped and damped natural frequencies

19. 1. Elastic Dynamic Analysis of SDOF Structures Forced vibrations.

20. 1. Elastic Dynamic Analysis of SDOF Structures Forced vibrations. Vibration Under a Short Impulse. To develop a general method to solve equation 4.64 under an arbitrary dynamic load, we first consider the response of a system under a short impulse (impact)

21. 1. Elastic Dynamic Analysis of SDOF Structures Forced Vibrations. If the contact duration t1 is very short compared to natural period of system T, then the impulse can be treated as a sudden change of velocity. In other words, the system does not have time to react during the time interval t1. when the system is at rest at t=0, the fundamental equation relating the impulse to the momentum is: After the end of impulse (t>t1), the system is in free vibration with the following initial conditions: The response of the system is given by equation 4.44: (4.67)

22. 1. Elastic Dynamic Analysis of SDOF Structures Forced vibrations. Vibrations Under an Arbitrary Dynamic Load; Duhamel’s Integral. The response of a SDOF system under a short impulse can be used to deduce the response of the same system under an arbitrary load, F(t) . All dynamic arbitrary loads can be represented by a succession of impulses. Now, let us consider a specific impulse, ending at time τ after the beginning of the load application. This impulse has an infinitesimal duration dτ . The area under the impulse corresponds to F(τ) dτ . This specific impulse produces a unit response in free vibrations at time t given by equation 4.67 :

23. 1. Elastic Dynamic Analysis of SDOF Structures Forced vibrations. The total response of the system at time t is obtained by superposing (integrating) the unit responses until time t : Equation 4.69 is called Duhamel’s integral. It is important to realize that the principle of superposition is used to deduce it. Therefore, equation 4.69 is valid only for a linear system. If a dynamic load, F(t) , is represented by a mathematical function, equation 4.69 is then integrated directly. In practice, and especially in the case of a seismic excitation at the base, it is often necessary to use a numerical integration.

24. 2. Elastic Earthquake Response Spectra Definition. The response of a SDOF structure, caused by an earthquake accelerogram , is obtained by simply replacing the dynamic load, F(t) , in equation 4.69 by a fictitious dynamic load equal to - m Usually, the accelerogram can not be defined by a simple mathematical expression. Therefore, equation 4.94 requires a numerical integration. The response of a SDOF structure, to a given accelerogram , is a function of the damping, ζ , and the natural frequency, ω . For a given structure ( ω and ζ ), the absolute value of the maximum response is evaluated using equation 4.94 for a specific accelerogram (earthquake) and it is plotted on a graph. Many coordinates are obtained on the graph by changing ω and ζ . The resulting graph is called : a relative displacement response spectrum. In general, the response spectrum is a fundamental characteristic of the accelerogram in terms of the maximum response it produces on a linear SDOF system.  

25. 2. Elastic Earthquake Response Spectra Definition.

26. Example No.01 Calculate the lateral stiffness for the frame shown in the following fig, assuming the elements to be axially rigid.

27. Example No.01 Solution. Here we are using the definition of stiffness influence coefficients to solve the problem. The system has the three DOFs shown in Fig. To obtain the first column of the 3×3 stiffness matrix, we impose unit displacement in DOF u 1, with u 2 = u 3 = 0. The forces ki 1 required to maintain this deflected shape are shown in Fig. The elements ki 2 in the second column of the stiffness matrix are determined by imposing u 2 = 1 with u 1 = u 3 = 0; see Fig. Similarly, the elements ki 3 in the third column of the stiffness matrix can be determined by imposing displacements u 3 = 1 with u 1 = u 2 = 0. Thus the 3 × 3 stiffness matrix of the structure is known and the equilibrium equations can be written.

28. Example No.01 For a frame with Ib = Ic subjected to lateral force fs , they are From the second and third equations, the joint rotations can be expressed in terms of lateral displacement as follows:

29. Example No.01 Substituting Eq. (b) into the first of three equations in Eq. (a) gives The lateral stiffness of the frame is

30. Example No.02 A small one-story industrial building, 20 by 30 ft in plan, is shown in Fig. E1.2 with moment frames in the north–south direction and braced frames in the east–west direction. The weight of the structure can be idealized as 30 lb / ft2 lumped at the roof level. The horizontal cross bracing is at the bottom chord of the roof trusses. All columns are W8 × 24 sections; their second moments of cross-sectional area about the x and y axes are Ix = 82 . 8 in4 and Iy = 18 . 3 in4, respectively; for steel, E = 29 , 000 ksi . The vertical cross bracings are made of 1-in.-diameter rods. Formulate the equation governing free vibration in ( a ) the north–south direction and ( b ) the east–west direction.

31. Example No. 02

32. Example No. 02 Solution. The mass lumped at the roof is Because of the horizontal cross-bracing, the roof can be treated as a rigid diaphragm.

33. Example No. 02 North–south direction. Because of the roof truss, each column behaves as a clamped–clamped column and the lateral stiffness of the two moment frames (Fig. E1.2b) is. And the equation of motion is (a)

34. Example No. 02 (b) East–west direction. the lateral stiffness of a braced frame can be estimated as the sum of the lateral stiffnesses of individual braces. The stiffness of a brace (Fig. E1.2d) is k brace = (AE/L) cos2 θ . This can be derived as follows. We start with the axial force–deformation relation for a brace: (b) By statics f S = p cos θ , and by kinematics u = δ/ cos θ . Substituting p = f S / cos θ and δ = u cos θ in Eq. (b) gives where (c)

35. Example No. 02 For the brace in Fig. E1.2c, cos θ = 20 / 122 + 202 = 0 . 8575, A = 0 . 785 in2, L = 23 . 3 ft , and Considering the two frames, And the equation of motion is Observe that the error in neglecting the stiffness of columns is small: k col = 2 × 12 E I y /h 3 = 4 . 26 kips / in . versus k brace = 59 . 8 kips / in

36. Example No.03 A uniform rigid slab of total mass m is supported on four columns of height h rigidly connected to the top slab and to the foundation slab (Fig. E1.7a). Each column has a rectangular cross section with second moments of area I x and I y for bending about the x and y axes, respectively. Determine the equation of motion for this system subjected to rotation u gθ of the foundation about a vertical axis. Neglect the mass of the columns

37. Example No.03

38. Example No.03 Solution The elastic resisting torque or torsional moment f S acting on the mass is shown in Fig. E1.7b, and Newton’s second law gives. Here uθ is the rotation of the roof slab relative to the ground and I O = m(b 2 + d 2 )/ 12 is the moment of inertia of the roof slab about the axis normal to the slab passing through its center of mass O .

39. Example No.03 The torque f S and relative rotation u θ are related by where k θ is the torsional stiffness. To determine k θ , we introduce a unit rotation, u θ = 1, and identify the resisting forces in each column (Fig. E1.7c). For a column with both ends clamped

40. Example No.03 The torque required to equilibrate these resisting forces is Substituting Eqs. (c), (d), and (b) in (a) gives This is the equation governing the relative rotation uθ of the roof slab due to rotational acceleration of the foundation slab.  

41. Assignment Starting from the basic definition of stiffness, determine the effective stiffness of the combined spring and write the equation of motion for the spring–mass systems shown in Figs. P1.1 to P1.3.

42. Assignment 2) Develop the equation governing the longitudinal motion of the system of Fig. The rod is made of an elastic material with elastic modulus E ; its cross-sectional area is A and its length is L . Ignore the mass of the rod and measure u from the static equilibrium position.

43. Assignment 3) Assuming the beam to be massless, each system has a single DOF defined as the vertical deflection under the weight w . The flexural rigidity of the beam is E I and the length is L .