CE 3202 STRUCTURAL ENGINEERING

Module Reader: Mr. Noor M S Hasan

INTRODUCTION

Civil Structural Dynamics broadly covers:

Earthquake Engineering

Offshore Engineering

Wind Engineering

Civil Structural Dynamics broadly covers:

Civil Structural Dynamics broadly covers:

Blast and Impact Engineering

Vibration Engineering Building vibration due to external ground borne vibration

•Typically dealt via vibration isolation of the whole building or of the machinery

Vibration due to human-induced excitation

Definitions: Dynamics vs Vibration ?

Def: Dynamics is the study relating the forces to motion and the laws governing the motion are the well-known Newton’s laws

Dynamic load – any load of which the magnitude, direction or position varies with time

Def: Vibration is an omnipresent type of dynamic behaviour where the motion is actually an oscillation about a certain equilibrium position

Vibration - any motion that repeats itself after an interval of time

Newton’s Law of Motion:

• First Law: A body continues to maintain its state of rest or of uniform motion unless acted upon by an external unbalanced force.

• Second Law: Momentum mv is the product of mass and velocity. Force and momentum are vector quantities and the resultant force is found from all the forces present by vector addition. This law is often stated as, “F = ma: the net force on an object is equal to the mass of the object multiplied by its acceleration.”

• Third Law: To every action there is an equal and opposite reaction.

Distinctive features of a dynamic analysis

Time-varying nature of the excitation (applied loads) and the response (resulting displacements, internal forces, stresses, strain, etc.)

A dynamic problem does not have a single solution but a succession of solutions corresponding to all times of interest in the response history

A dynamic analysis is more complex and computationally intensive than a static analysis

Excitation

Response

Distinctive features of a dynamic analysis

Inertia forces when the loading is dynamically applied

Inertia is the property of matter by which it remains at rest or in motion at a constant speed along a straight line so long as it is not acted by an external force

Translation motion, the measurement of inertia is the mass m

Rotational motion, the measurement of inertia is the mass moment of inertia I0

M

V

F F (t )

M (t)

V (t)I n e r tia fo rc es

Dynamic vs Static analysis

In reality, no loads that are applied to a structure are truly static

Since all loads must be applied to a structure in some particular sequence during a finite period of time, a time variation of the force is inherently involved

When do we opt for dynamic analysis ?

When forces change as a function of time ?

No, but when the nature of the force is such that causes accelerations so significant that inertial forces can not be neglected in the analysis

Static analysis – when the loading is such that the accelerations caused by it can be neglected

The same load may be treated on one structure as dynamic whereas on the other is static

Why not doing dynamic analysis always ?

Dynamic analysis is considerably more expensive than the static analysis

More skills, knowledge, “feel” for the structural behaviour under various types of dynamic loading are required in order to deal with it both correctly and efficiently (a dynamic analysis is much more computational than a static analysis)

The skill of the analyst is to make a judgement if a dynamic analysis is necessary

Dynamic vs Static analysis

Dynamic vs Static analysis

Situations in which dynamic loading must be considered

response of bridges to moving vehicle

action of wind gusts, ocean waves, blast pressure upon a structure

effect on a building whose foundation is subjected to earthquake excitation

response of structures subjected to alternating forces caused by oscillating machinery

Dynamic analysis procedure

Main steps of a dynamic investigation:

Identification of the physical problem (existing structure)

identifying and describing the physical structure or structural component and the source of the dynamic loading

Definition of the mechanical (analytical) model

a set of simplifying assumptions (loading, boundary conditions, etc)

a set of drawings depicting the adopted analytical model

a list of design data – geometry, material properties, etc.

Definition of the mathematical model

a set of equations where the unknowns are the response sought

Having all this information available, investigation into dynamic behaviour can start

Flow-chart of a typical dynamic analysis

P H Y S IC A L P R O B L E M

M E C H A N IC A L M O D E L A ssum p tions on : * G eom etry * M a teria l law s * L oad ing * B oun da ry co nd ition s * E tc .

S O L U T IO N O F G O V E R N IN G D IF F E R E N T IA L E Q U AT IO N S* C o n tin uou s m o del: P ar tia l d ifferen tia l equ atio ns* D iscrete m ode l: O rd ina ry d iffe ren tia l eq ua tion s

IN T E R P R E TAT IO N O F R E S U L T S R E F IN E A N A LY S IS

IM P R O V E M E C H A N IC A L M O D E L

C H A N G E O F P H Y S IC A L P R O B L E M

D E S IG N IM P R O V E M E N T S S T R U C T U R A L O P T IM IZ AT IO N

Dynamic modelling of structures

Definition: Degree of freedom (DOF) – number of independent geometrical coordinates required to completely specify the position of all points on the structure at any instant of time

There are two types of geometrical coordinates:

Linear displacements (translations),

Angular displacements (rotations),

Three main procedure for the discretization of a structure: Finite number of DOF – discrete parameter (lumped) model

Infinite number of DOF – distributed parameter (continuous) model

Combination of these two – finite element (FE) model

)(t)(tx

Dynamic modelling of structures

Lumped mass model (discrete model)

The mass of the system is assumed to be concentrated (localized/lumped) in various discrete points around the system

Single-degree-of-freedom (SDOF) system - the entire mass m of the structure is localized at a single point

Multi-degree-of-freedom (MDOF) system - the mass m of the structure is localized at many points around the system

m

x (t)

m 3

m 2

m 1x 1 (t)

x 2 (t)

x 3 (t)

a . b . c .

Dynamic modelling of structures

Lumped mass model (discrete model)

x (t)

x (t)1

2

y (t) y (t) y (t)1 2 3

m 1 m 2 m 3

x (t)1

y (t)2

(t)

y(t)

y(t)

x(t)m

Dynamic modelling of structures

Distribute model (continuous model)

mass is considered uniformly distributed throughout the system

in reality, structures have an infinite number of degrees of freedom

using the continuous model, a better accuracy of the results

can be achieved in a dynamic analysis than by the lumped mass model

x

y

x (y,t)

m (y )

a . b .

Dynamic modelling of structures

Finite element (FE) model

Combination of the discrete and continuous model The structure is divided into elements, which are connected at

discrete points called nodes The nodes are allowed to displace in a prescribed manner to represent

the motion of the structure The sum of the displacements (translations and rotations) represents

the total number of DOF for the systemThe mass of the system is concentrated within each element

e le m e n t n o d e

Comparison between static degree of freedom and dynamic degree of freedom

The number of dynamic coordinates can be maximum equal to the number of static degrees of freedom of the system

An infinity of static DOF 6 static DOF

(3 for each node)

3 static DOF

(axial def neglected)

Comparison between static degree of freedom and dynamic degree of freedom

From dynamic point of view, the system can have:

An infinity of DOF

(mass distributed)

6 DOF 3 DOF

Comparison between static degree of freedom and dynamic degree of freedom

From dynamic point of view, the system can have:

1 DOF (SDOF)

Vibrations and classification of vibrations

Vibration is an omnipresent type of dynamic behaviour where the motion is actually an oscillation about a certain equilibrium position

Any motion that repeat itself after an interval of time – vibration or oscillation

Vibration can be classified in several ways

Classification of vibrations

Free and forced vibration

Free vibration - the structure vibrates freely under the effect of the initial conditions with no external excitations applied

Forced vibration - structure vibrates under the effect of external excitation

Undamped and damped vibration

Undamped vibration – no energy is lost or dissipated in friction or other resistance during oscillation

Damped vibration – any form of energy is lost during oscillation

x (t)

t

x (t)

t

Undamped vibration Damped vibration

Classification of vibrations

Periodic and nonperiodic vibration

Periodic vibration – repeats itself at equal time intervals called periods T . The simplest form of periodic vibration is the simple harmonic vibration

T

T

Nonperiodic vibration – any other vibration that can not be characterized as periodic

Periodic and nonperiodic vibration

Linear and nonlinear vibration

Linear vibration – all the components of a system (spring, mass and damping) behave linearly. The principle of superposition is valid. - eg. twice larger force will cause twice larger response - mathematical techniques for solving linear systems are much more developed than for the non-linear systems

Nonlinear vibration – any of the basic components behave nonlinearly. The superposition principle is not valid.

Which is which?

Linear-elastic material

Non-linear material

Non-linear elastic material

D eform ation

L oad

L o a d ing p

ath

U n lo a d ing p

a th

D eform ation

L oadL

oad in

g p a

th

Un lo

adin

g p a

t h

D eform ation

L oad

L o a d ing p

ath

U n lo ad ing p

a th

Deterministic and nondeterministic vibration

Deterministic vibration – the value (magnitude), point of application and time variation of the loading are completely known - eg. periodic vibration is a deterministic vibration

Nondeterministic (random) vibration – the time variation and other characteristics of the load are not completely known but can be defined only in a statistical sense

Sources of dynamic loading

Environmental – wind load, wave load, earthquake load Machine induced (in industrial installations) – rotating engines,

turbines, conveyer mechanisms, fans Vehicular induced – road traffic, railway Blast – explosive devices or accidental explosions

Components of a vibration system

k

c

x (t)

m

There are 3 key components of discrete systems:

Mass or inertia element

Spring element

Damping (dashpot) element

These interact with each other during the system’s motion

Therefore, it is very useful how each of the components behave

Mass or inertia element

Mass relates force to acceleration

Mass is assumed to behave as rigid body (does not deform)

The 2nd Newton’s law relates forces to accelerations via mass acting as a coefficient of proportionality

Inertia force “resisting” acceleration is developed and is acting in the direction opposite to the external loading

Units N/(m/s^2) or kg

F (t)i

x (t)..

slope = m

m

x(t)..

F (t)= m x (t)i...

F (t)i

Spring element

Spring relates force to displacement

Spring is assumed to have no mass and damping

An elastic force is developed whenever there is a relative motion between the ends of the spring

k – spring constant or spring stiffness - Units N/m

F e (t)

x (t)

slope = k

Spring element

n

iieq kk

1

Equivalent stiffness of springs in series

Equivalent stiffness of springs in parallel

n

ii

eq

kk

1

1

Spring element

Determine the equivalent spring stiffness of the following dynamic systems

mk 1 k 2

mk 1

k 3

k 2

21 kkkeq 321 kkkkeq

Spring are in parallel: Spring are in parallel:

Damping (dashpot) element

Dashpot relates force to velocity

Dashpot is assumed to have no mass and elasticity

A damping force is developed whenever there is a relative velocity between the ends of the dashpot

c – viscous damping coefficient - Units Ns/m

Introduction Features of a dynamic analysis Dynamic analysis procedure Dynamic modelling of structures Vibrations Components of a vibration system

SUMMARY