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Dynamics of Structures: Theory and applications to by Anil K. Chopra

By Anil K. Chopra

Designed for senior-level and graduate classes in Dynamics of buildings and Earthquake Engineering.

Dynamics of constructions includes many subject matters encompassing the idea of structural dynamics and the applying of this conception relating to earthquake research, reaction, and layout of buildings. No past wisdom of structural dynamics is believed and the way of presentation is adequately distinct and built-in, to make the publication compatible for self-study through scholars engineers.

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The resulting displacement of the mass also varies with time; it is denoted by u(t). 2 we derive the differential equation governing the displacement u(t) by two methods using (1) Newton’s second law of motion, and (2) dynamic equilibrium. 3. 1 Using Newton’s Second Law of Motion The forces acting on the mass at some instant of time are shown in Fig. 1b. These include the external force p(t), the elastic (or inelastic) resisting force f S (Fig. 1), and the damping resisting force f D (Fig. 1). The external force is taken to be positive in the direction of the x-axis, and the displacement u(t), velocity u(t), ˙ and acceleration u(t) ¨ are also positive in the direction of the x-axis.

2b); (2) the damping component: the frame with its damping property but no stiffness or mass (Fig. 2c); and (3) the mass component: the roof mass without the stiffness or damping of the frame (Fig. 2d). , from left to right. 2 fD + Displacement u Velocity u˙ Acceleration u¨ Chap. 1 fI + Velocity u˙ Acceleration u¨ (c) (d) (a) System; (b) stiffness component; (c) damping component; (d) mass component. The external force f S on the stiffness component is related to the displacement u by Eq. 1) if the system is linearly elastic, the external force f D on the damping component is related to the velocity u˙ by Eq.

3 k Abutment 2 Sec. 6 Mass–Spring–Damper System 19 is 123 ft2 . The weight of the bridge is idealized as lumped at the deck level; the unit weight of concrete is 150 lb/ft3 . The weight of the bents may be neglected. Each bent consists of three 25-ft-tall columns of circular cross section with I y = Iz = 13 ft4 (Fig. 3b). Formulate the equation of motion governing free vibration in the longitudinal direction. The elastic modulus of concrete is E = 3000 ksi. 45 kips/ft. 2 The longitudinal stiffness of the bridge is computed assuming the bridge deck to displace rigidly as shown in Fig.

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