[lecture_1) Dynamics Of Structures Chapter 1

Dynamics of Structures Dr. Naik Muhammad Department of Civil Engineering BUITEMS, Quetta

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Textbook Dynamics of Structures: Theory and Applications to Earthquake Engineering, Anil K. Chopra, University of California at Berkeley, Prentice Hall, Fourth Edition.

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Course Contents

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Part I: Single-degree-of-freedom • Week 01: Chapter 1

Part IV: Multi-degree-of-freedom

• Week 02: Chapter 2

• Week 10: Chapter 9

• Week 03: Chapter 3

• Week 11: Chapter 9

• Week 04: Chapter 3

• Week 12: Chapter 10

• Week 05: Chapter 4 • Week 06: Chapter 5

Part II: Spectrum & Earthquake • Week 07: Chapter 6

• Week 13: Chapter 11 & Appendix A • Week 14: Chapter 12 • Week 15: Chapter 12 • Week 16: Chapter 13 • Week 17: Chapter 17

• Week 08: Chapter 8

• Week 09: Mid-term exam

• Week 18: Finial exam

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Purpose of structure dynamic analysis

Dynamic problems: • Structure vibration under earthquake • Long-span bridge and high-rise building vibration under wind load • Bridge and road vibration under vehicle or train load • Prevention work response under blast impact …..and so on

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Objective • Learning the basic theories of dynamics • Understanding the modeling of the structures • Learning the analytical methods to analyze the simple structures to understand the implications of the dynamics of structures • Learning the numerical methods to analyze the complex structures and understand the practical applications of the dynamics of structures • Analyze the structure dynamic responses in both time-domain and frequencydomain to understand the characteristics of the structures • To understand the earthquake response of structures

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Classification of structural dynamics problems How to classify structure dynamics problems? • By the number of degrees of freedom (DOF) • By the linearity of the governing equations • By the type of excitation • By the type of mathematical problem • By the type of energy dissipated mechanisms

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Classification of structural dynamics problems 1. By the number of degrees of freedom (DOF) • Single DOF

• Multiple DOF

Discrete (Finite number of DOF) Continuous (Infinite number of DOF)

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Classification of structural dynamics problems 2. By the linearity of the governing equations • Linear systems • Nonlinear systems

Nonlinear elastic (conservative) Nonlinear inelastic (non-conservative)

fs

1

k u

Nonlinear elastic

Nonlinear inelastic

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Classification of structural dynamics problems 10 3. By the type of excitation •

Free vibration Structural

•

Forced vibration Seismic

Harmonic Non-harmonic Deterministic Transient Stationary Random Non-stationary Periodic

4. By the type of mathematical problem • Static equation • Dynamic equation

Boundary value problem Free vibration Forced vibration

Eigenvalue problems Initial value problems Propagation problems

Classification of structural dynamics problems 11 5. By the type of energy dissipated mechanisms • Undamped • Damped

Viscous damping Hysteretic damping (structural damping) Friction damping (Coulomb’s) etc.

Viscous damping

Hysteretic damping

Viscoelastic damping

Chapter 1: Equation of motion, problem statement, and solution methods

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Chapter 1: Contents 1.

Simple Structures

2.

Single-Degree-of-Freedom System

3.

Force-Displacement Relation

4.

Damping Force

5.

Equation of Motion: External Force

6.

Mass-Spring-Damper System

7.

Equation of Motion: Earthquake Excitation

8.

Problem Statement and Element Forces

9.

Combining Static and Dynamic Responses

10. Methods of Solution of the Differential Equation

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1.1 Simple structures Simple structures: They can be idealized as a concentrated or lumped mass m supported by a massless structure with stiffness k in the lateral direction.

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1.1 Simple structures

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Unrealistic: Oscillations continue forever and these idealized systems would never come to rest (Figure 1.1.3 c). Real: Oscillate with ever-decreasing amplitude and eventually come to rest. The kinetic energy and strain energy of the vibrating system are dissipated by various damping mechanisms

1.2 Single-degree-of-freedom system

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Degrees of Freedom (DOFs): The number of independent displacements required to define the displaced positions of all the masses relative to their original position Single-degree-of-freedom (SDF) system: The one-story frame constrained to move only in the direction of the excitation has only 1 DOF—lateral displacement—for dynamic analysis if it is idealized with mass concentrated at one location.

1.3 Force–displacement relation Consider the system with a static force f S along the DOF u. The internal force resisting the displacement u is equal and opposite to the external force f S (Fig. 1.3.1b). To determine the relationship between f S and u is a standard problem in static structural analysis.

Inelastic system

fS = fS (u )

Linearly elastic system

f S = ku

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1.3.1 Linearly elastic systems The lateral stiffness k of the frame can readily be determined for two extreme cases: • Beam is rigid (Fig. 1.3.2b) EI b = ∞

k=

12 EI c EI c = 24  3 3 h h columns

• Beam has no stiffness (Fig. 1.3.2c) EI b = 0

3EI c EI c k=  =6 3 3 h h columns

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1.3.2 Inelastic Systems The force-deformation relation is path dependent, i.e., it depends on whether the deformation is increasing or decreasing. Thus the resisting force is an implicit function of deformation: fS = fS (u )

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1.3.2 Inelastic Systems

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The force–deformation relation can be determined in one of two ways. 1.

Use methods of nonlinear static structural analysis.

2.

Define the inelastic force–deformation relation as an idealized version of the experimental data.

1.4 Damping force

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Damping: The process by which free vibration steadily diminishes in amplitude. In damping, the energy of the vibrating system is dissipated by various mechanisms. Equivalent viscous damping: The damping coefficient c is selected so that the vibrational energy it dissipates is equivalent to the energy dissipated in all the damping mechanisms, combined, present in the actual structure. As shown in Figure 1.4.1, the damping force is related to the velocity across the linear viscous damper by:

f D = cu

1.5 Equation of motion: external force

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The idealized one-story frame (Figure 1.5.1a) introduced earlier subjected to an externally applied dynamic force p ( t ) in the direction of the DOF u.

Two methods to derive the differential equation governing the displacement u(t): 1.

Newton’s second law of motion

2.

Dynamic equilibrium

1.5.1 Using Newton’s second law of motion

The resultant force along the x-axis is p − f S − f D Newton’s second law of motion gives: p − f S − f D = mu or mu + f D + f S = p ( t ) Linearly elastic system: with f S = ku and f D = cu

mu + cu + ku = p ( t ) Inelastic system: with f S = f S ( u )

mu + cu + f S ( u ) = p ( t )

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1.5.2 Dynamic equilibrium

D’Alembert’s principle of dynamic equilibrium: This principle is based on the notion of a fictitious inertia force, a force equal to the product of mass times its acceleration and acting in a direction opposite to the acceleration.

Mass replaced by its inertia force: mu + f D + f S − p ( t ) = 0

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1.5.3 Stiffness, damping, and mass components 25 An alternative viewpoint: f I + f D + f S = p ( t ) • The state of the system: displacement u ( t ) , velocity u ( t ) and acceleration u( t ) • Visualize the system: (1) the stiffness f S relate to the displacement u ( t ) (2) the damping component f D relate to the velocity u ( t ) (3) the mass component f I relate to the acceleration u( t )

• The external force p ( t ) visualized as distributed among the three components of the structure, and f I + f D + f S must equal the applied force p ( t ) . This viewpoint is useful for complex systems

1.5.3 Stiffness, damping, and mass components 26

1.5.3 Stiffness, damping, and mass components 27

1.5.3 Stiffness, damping, and mass components 28

1.6 Mass–spring–damper system

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SDF system: Consider the spring and damper to be massless, the mass to be rigid, and all motion to be in the direction of the x-axis.

The classic SDF system is the mass–spring–damper system of Fig. 1.6.1a

1.6 Mass–spring–damper system

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Example 1.4

Derive the equation of motion of the weight w suspended from a spring at the free end of the cantilever steel beam shown in Fig. E1.4a. For steel, E = 29,000 ksi. Neglect the mass of the beam and spring.

1.6 Mass–spring–damper system

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Example 1.4 Solution

Figure E1.4b shows the deformed position of the free end of the beam, spring, and mass. The displacement of the mass u is measured from its initial position with the beam and spring in their original undeformed configuration. Equilibrium of the forces of Fig. E1.4c gives mu + f S = w + p ( t ) (a ) where f S = keu

(b)

The effective stiffness ke of the system remains to be determined. The equation of motion is:

mu + keu = w + p ( t )

(c)

The displacement u can be expressed as u = δ st + u (d) where δ st is the static displacement due to weight w and u is measured from the position of static equilibrium.

1.6 Mass–spring–damper system

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Example 1.4

Substituting Eq. (d) in Eq. (a) and noting that (1) u = u because δ st does not vary with time, and (2) keδ st = w gives mu + keu = p ( t )

(e)

The effective stiffness ke remains to be determined. It relates the static force f S to the resulting displacement u by f S = ke u

(f )

1.6 Mass–spring–damper system Example 1.4

The effective stiffness ke

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1.6 Mass–spring–damper system

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1.7 Equation of motion: earthquake excitation

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: Displacement of the ground : Total (or absolute) displacement of the mass : Relative displacement between the mass and ground

Linear elastic system

fI + fD + fS = 0

f I = mut = m ( ug ( t ) + u( t ) ) f D = cu ( t )

f S = ku ( t ) or

fS = fS (u )

Inelastic system

Effective earthquake force: =−

1.8 Problem statement and element forces

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Response: Displacement, velocity, or acceleration of the mass; also, an internal force or internal stress in the structure. Element forces: Once the deformation response history u ( t ) has been evaluated by dynamic analysis of the structure, the element forces—bending moments, shears, and axial forces—and stresses needed for structural design can be determined by static analysis of the structure at each instant in time.

This static analysis of a one-story linearly elastic frame: 1. From the known displacement and rotation of each end of a structural element, the element forces can be determined through the element stiffness properties; and stresses can be obtained from element forces. 2. Introduce the equivalent static force. At any instant of time t this force fS is the static external force that will produce the deformation u determined by dynamic analysis.

1.9 Combining static and dynamic responses

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In practical application, for a linear system the total forces can be determined by combining the results of two separate analyses: • Static analysis of the structure due to dead and live loads, temperature changes, and so on. • Dynamic analysis of the structure subjected to the time-varying excitation.

The analysis of nonlinear systems cannot be separated into two independent analyses: • The dynamic analysis of such a system must recognize the forces and deformations already existing in the structure before the onset of dynamic excitation. This is necessary to establish the initial stiffness property of the structure required to start the dynamic analysis.

1.10 Methods of solution of the differential equation 38 For a linear SDF system, the equation of motion is: mu + cu + ku = p ( t )

The initial displacement 0 and initial velocity specified to define the problem completely. Four methods of solution: • Classical Solution • Duhamel’s Integral • Frequency-domain Method • Numerical Methods

0 at time zero must be

1.10.1 Classical solution

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Example 1.8

+

=

Complete solution

=

= =

Particular solution

Complete solution

0 =0

+

Complementary solution

Constants

0 =0

Initial conditions

cos

+

sin

⁄

= 0 are determined by initial conditions

=−

=

1 − cos

Natural frequency =

⁄

1.10.4 Numerical methods

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Central difference method Newmark-β method Wilson-θ method …and so on These methods are also useful for evaluating the response of linear systems to excitation which is too complicated to be defined analytically and is described only numerically