Test Bank for Dynamics of Structures: Theory and Applications to Earthquake Engineering 5th Edition by Chopra Questions and 100% Verified Answers

***Download Test Bank Immediately After the Purchase. Just in case you have trouble downloading, kindly message me, and I will send it to you via Google Doc or email. Thank you*** The Test Bank for Dynamics of Structures: Theory and Applications to Earthquake Engineering 5th Edition by Chopra is a comprehensive study guide designed to aid students in mastering the concepts of structural dynamics. This test bank provides an extensive set of questions and answers that delve into the theory and practical applications of earthquake engineering. The Dynamics of Structures 5th Edition Test Bank offers detailed insights into the subject, helping learners gain a profound understanding of how structures respond to dynamic loads, particularly earthquakes. Anil K. Chopra's expertise in the field brings real-world scenarios to the fore, ensuring students are well-prepared to apply their knowledge in practical situations. This Chopra Earthquake Engineering 5th Edition resource is an invaluable tool for students aiming for success in their exams and future careers in earthquake engineering. Problem 1.1 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 Fig. P1.1. Figure P1.1 Solution: If ke is the effective stiffness, fS  keu u k1 u S k2 u S Equilibrium of forces: fS  (k1  k2 ) u Effective stiffness: ke  fS u  k1  k2 Equation of motion: mu  keu  p(t ) Problem 1.2 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 Fig. P1.2. Figure P1.2 Solution: If ke is the effective stiffness, fS  keu (a) u S If the elongations of the two springs are u1 and u2 , u  u1  u2 (b) Because the force in each spring is fS , fS  k1u1 fS  k2u2 (c) Solving for u1 and u2 and substituting in Eq. (b) gives fS  fS ke k1  fS k2  1  1  1  ke k1 k2 k  k1 k2 e k  k 1 2 Equation of motion: mu  keu  p(t ). Problem 1.3 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 Fig. P1.3. Figure P1.3 Solution: Figure P1.3a Figure P1.3b u Figure P1.3c This problem can be solved either by starting from the definition of stiffness or by using the results of Problems P1.1 and P1.2. We adopt the latter approach to illustrate the procedure of reducing a system with several springs to a single equivalent spring. First, using Problem 1.1, the parallel arrangement of k1 and k2 is replaced by a single spring, as shown in Fig. 1.3b. Second, using the result of Problem 1.2, the series arrangement of springs in Fig. 1.3b is replaced by a single spring, as shown in Fig. 1.3c: 1  ke k1 1  1  k2 k3 Therefore the effective stiffness is k  (k1  k2 ) k3 e k  k  k 1 2 3 The equation of motion is mu  keu  p(t ). Problem 1.4 Derive the equation governing the free motion of a simple pendulum that consists of a rigid massless rod pivoted at point O with a mass m attached at the tip (Fig. P1.4). Linearize the equation, for small oscillations, and determine the natural frequency of oscillation. Figure P1.4 Solution: 1. Draw a free body diagram of the mass. O 4. Determine natural frequency. n  mg sin  mg cos  m 2. Write equation of motion in tangential direction. Method 1: By Newton’s law. mg sin  ma mg sin  mL mL  mg sin  0 (a) This nonlinear differential equation governs the motion for any rotation . Method 2: Equilibrium of moments about O yields mL2  mgL sin or mL  mgsin  0 3. Linearize for small . For small , sin   , and Eq. (a) becomes mL  mg  0    g   0  L  (b) Problem 1.5 Consider the free motion in the xy plane of a compound pendulum that consists of a rigid rod suspended from a point (Fig. P1.5). The length of the rod is L, and its mass m is uniformly distributed. The width of the uniform rod is b and the thickness is t. The angular displacement of the centerline of the pendulum measured from the y-axis is denoted by θ(t). (a) Derive the equation governing θ(t). (b) Linearize the equation for small θ. (c) Determine the natural frequency of small oscillations. mL2  mgL sin  0 3 2 4. Specialize for small . For small , sin   and Eq. (a) becomes mL2   mgL  0 3 2   3 g   0 2 L 5. Determine natural frequency. (a) (b) Figure P1.5 Solution: 1. Find the moment of inertia about O. From Appendix 8, n  I0  1 mL2  m 12  L 2    2   1 mL2 3 2. Draw a free body diagram of the body in an arbitrary displaced position. 3. Write the equation of motion using Newton’s second law of motion.  M 0  I0 mg sin  1 mL2 2 3 Problem 1.6 Repeat Problem 1.5 for the system shown in Fig. P1.6, which differs in only one sense: its width varies from zero 3. Write the equation of motion using Newton’s second law of motion.  M 0  I0 at O to b at the free end. mg 2L sin  1 mL2 3 2 mL2   2 2mgL 3 sin  0 (a) Solution: Figure P1.6 4. Specialize for small . For small , sin  , and Eq. (a) becomes mL2  2mgL  0 2 3 1. Find the moment of inertia about about O. L or  4 g   0 3 L (b) I   r 2 d A x 5. Determine natural frequency. 0 L    r 2 (rdr ) n    L4  4 In each case the system is equivalent to the spring- mass system shown for which the equation of motion is  w   1 mL2 2   u    ku  0 2. Draw a free body diagram of the body in an arbitrary displaced position.