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The conventional structural design process proportions thestructure based on strength requirements, establishes the corresponding stiffnessproperties, and then checks the various service

Chapter 1

Introduction

1.1 Motivation for structural motion control

Limitations of conventional structural design

The word, design, has two meanings When used as a verb it is defined as the act

of creating a description of an artifact It is also used as a noun, and in this case, isdefined as the output of the activity, i.e., the description In this text, structuraldesign is considered to be the activity involved in defining the physical makeup

of the structural system In general, the “designed” structure has to satisfy a set of

requirements pertaining to safety and serviceability Safety relates to extreme

loadings which are likely to occur no more than once during a structure’s life Theconcerns here are the collapse of the structure, major damage to the structure andits contents, and loss of life Serviceability pertains to moderate loadings whichmay occur several times during the structure’s lifetime For service loadings, thestructure should remain fully operational, i.e the structure should suffernegligible damage, and furthermore, the motion experienced by the structure

should not exceed specified comfort limits for humans and motion sensitive

equipment mounted on the structure An example of a human comfort limit is therestriction on the acceleration; humans begin to feel uncomfortable when theacceleration reaches about 0.02g A comprehensive discussion of human comfort

criteria is given by Bachmann and Ammann (1987).

Safety concerns are satisfied by requiring the resistance (i.e strength) of theindividual structural elements to be greater than the demand associated with theextreme loading The conventional structural design process proportions thestructure based on strength requirements, establishes the corresponding stiffnessproperties, and then checks the various serviceability constraints such as elasticbehavior Iteration is usually necessary for convergence to an acceptable

structural design This approach is referred to as strength based design since the

elements are proportioned according to strength requirements

Applying a strength based approach for preliminary design is appropriatewhen strength is the dominant design requirement In the past, most structuraldesign problems have fallen in this category However, a number ofdevelopments have occurred recently which have limited the effectiveness of thestrength based approach

Firstly, the trend toward more flexible structures such as tall buildings andlonger span horizontal structures has resulted in more structural motion underservice loading, thus shifting the emphasis from safety toward serviceability Forinstance, the wind induced lateral deflection of the Empire State Building in NewYork City, one of the earliest tall buildings in the United States, is several incheswhereas the wind induced lateral deflection of the World Trade Center tower isseveral feet, an order of magnitude increase This difference is due mainly to theincreased height and slenderness of the World Trade Center in comparison to theEmpire State tower Furthermore, satisfying the limitation on acceleration is adifficult design problem for tall slender buildings

Secondly, some of the new types of facilities such as space platforms andsemi-conductor manufacturing centers have more severe design constraints onmotion than the typical civil structure In the case of microdevice manufacturing,the environment has to be essentially motion free Space platforms used tosupport mirrors have to maintain a certain shape to a small tolerance in order for

the mirror to properly function The design strategy for motion sensitive structures

is to proportion the members based on the stiffness needed to satisfy the motionconstraints, and then check if the strength requirements are satisfied

Thirdly, recent advances in material science and engineering have resulted

in significant increases in the strength of traditional civil engineering materials

such as steel and concrete, as well as a new generation of composite materials.Although the strength of structural steel has essentially doubled, its elasticmodulus has remained constant Also, there has been some increase in the elasticmodulus for concrete, but this improvement is still small in comparison to theincrement in strength The lag in material stiffness versus material strength hasled to a problem with satisfying the serviceability requirements on the variousmotion parameters Indeed, for very high strength materials, it is possible for theserviceability requirements to be dominant Some examples presented in thefollowing sections illustrate this point.

Motion based structural design and motion control

Motion based structural design is an alternate design process which ismore effective for the structural design problem described above This approachtakes as its primary objective the satisfaction of motion related designrequirements, and views strength as a constraint, not as a primary requirement.Motion based structural design employs structural motion control methods todeal with motion issues Structural motion control is an emerging engineeringdiscipline concerned with the broad range of issues associated with the motion ofstructural systems such as the specification of motion requirements governed byhuman and equipment comfort, and the use of energy storage, dissipation, andabsorption devices to control the motion generated by design loadings Structuralmotion control provides the conceptional framework for the design of structuralsystems where motion is the dominant design consideration Generally, one seeksthe optimal deployment of material and motion control mechanisms to achievethe design targets on motion as well as satisfy the constraints on strength

In what follows, a series of examples which reinforce the need for analternate design paradigm having motion rather than strength as its primaryfocus, and illustrate the application of structural motion control methods tosimple structures is presented The first three examples deal with the issue ofstrength versus serviceability from a static perspective for building typestructures The discussion then shifts to the dynamic regime A single-degree-of-freedom (SDOF) system is used to introduce the strategy for handling motionconstraints for dynamic excitation The last example extends the discussionfurther to multi-degree-of-freedom (MDOF) systems, and illustrates how to dealwith one of the key issues of structural motion control, determining the optimalstiffness distribution Following the examples, an overview of structural motioncontrol methodology is presented

1.2 Motion versus strength issues for building type structures

Building configurations have to simultaneously satisfy the requirements of site(location and geometry), building functionality (occupancy needs), appearance,and economics These requirements significantly influence the choice of thestructural system and the corresponding design loads Buildings are subjected to

two types of loadings: gravity loads consisting of the actual weight of the structural

system and the material, equipment, and people contained in the building, and

lateral loads consisting mainly of wind and earthquake loads Both wind and

earthquake loadings are dynamic in nature and produce significant amplificationover their static counterpart The relative importance of wind versus earthquakedepends on the site location, building height, and structural makeup For steel

buildings, the transition from earthquake dominant to wind dominant loading for a

seismically active region occurs when the building height reaches approximately Concrete buildings, because of their larger mass, are controlled byearthquake loading up to at least a height of , since the additional gravityload increases the seismic forces In regions where the earthquake action is low(e.g Chicago in the USA), the transition occurs at a much lower height, and thedesign is governed primarily by wind loading

When a low rise building is designed for gravity loads, it is very likely thatthe underlying structure can carry most of the lateral loads As the buildingheight increases, the overturning moment and lateral deflection resulting fromthe lateral loads increase rapidly, requiring additional material over and abovethat needed for the gravity loads alone Figure 1.1 (Taranath, 1988) illustrates howthe unit weight of the structural steel required for the different loadings varieswith the number of floors There is a substantial weight cost associated withlateral loading

100m

250m

Fig 1.1: Structural steel quantities for gravity and wind systems

To illustrate the dominance of motion over strength as the slenderness ofthe structure increases, the uniform cantilever beam shown in Fig 1.2 isconsidered The lateral load is taken as a concentrated force applied to the tip ofthe beam, and is assumed to be static The limiting cases of a pure shear beam and

a pure bending beam are examined

Fig 1.2: Building modeled as a uniform cantilever beam

0 20 40 60 80 100 120 140

Gravity loads Lateral loads

Example 1.1: Cantilever shear beam

The shear stress is given by

(1.1)

where is the cross sectional area over which the shear stress can be considered

to be constant When the bending rigidity is very large, the displacement, , atthe tip of the beam is due mainly to shear deformation, and can be estimated as

(1.2)

where is the shear modulus and is the height of the beam This model iscalled a shear beam The shear area needed to satisfy the strength requirementfollows from eqn (1.1):

As

serviceability

p G

more emphasis on the motion constraint since it corresponds to a decrease in theallowable displacement, Furthermore, an increase in the allowable shearstress, , also increases the dominance of the displacement constraint.

Fig 1.3: Plot of versus for a pure shear beam

Example 1.2: Cantilever bending beam

When the shear rigidity is very large, shear deformation is negligible, and thebeam is called a “bending” beam The maximum bending moment in thestructure occurs at the base and equals

(1.6)The resulting maximum stress is

(1.7)

where is the section modulus, is the moment of inertia of the cross-sectionabout the bending axis, and is the depth of the cross-section (see Fig 1.2) Thecorresponding displacement at the tip of the beam becomes

u∗τ∗

(1.8)The moment of inertia needed to satisfy the strength requirement is given by

(1.11)

Figure 1.4 shows the variation of with for a constant value of theaspect ratio ( for tall buildings) Similar to the case of the shearbeam, an increase in places more emphasis on the displacement since itcorresponds to a decrease in the allowable displacement, , for a constant Also, an increase in the allowable stress, , increases the importance of thedisplacement constraint

For example, consider a standard strength steel beam with an allowable

aspect ratio of The value of at which a transition from strength

- E

σ∗

- d H

≈200

=

H u⁄ ∗>200 r>1

strength steel is utilized ( and )

(1.13)

and motion essentially controls the design for the full range of allowabledisplacement

Fig 1.4: Plot of versus for a pure bending beam

Example 1.3: Quasi-shear beam frame

This example compares strength vs motion based design for a single bay frame ofheight and load (see Fig 1.5) For simplicity, a very stiff girder is assumed,resulting in a frame that displays quasi-shear beam behavior Furthermore, thecolumns are considered to be identical, each characterized by a modulus ofelasticity , and a moment of inertia about the bending axis

The maximum moment, , in each column is equal to

The lateral displacement of the frame under the load is expressed as

Fig 1.5: Quasi-shear beam example

The serviceability requirement constrains the maximum displacement to

be less than the allowable displacement

1.3 Design of a single-degree-of freedom system for dynamic loading

The previous examples dealt with motion based design for static loading Asimilar approach applies for dynamic loading once the relationship between theexcitation and the response is established The procedure is illustrated for thesingle-degree-of-freedom (SDOF) system shown in Fig 1.6

Response for periodic excitation

The governing equation of motion of the system has the form

Fig 1.6: Single-degree-of-freedom systemwhere , , are the mass, stiffness, and viscous damping parameters of thesystem respectively, is the applied loading, is the displacement, and is the

independent time variable The dot operator denotes differentiation with respect

to time Of interest is the case where is a periodic function of time Taking to

be sinusoidal in time with frequency ,

(1.24)the corresponding forced vibration response is given by

(1.25)where and characterize the response They are related to the system andloading parameters as follows:

(1.31)

The term is the displacement response that would occur if the loadingwere applied statically; represents the effect of the time varying nature of theresponse Figure 1.7 shows the variation of with the frequency ratio, , forvarious levels of damping The maximum value of and correspondingfrequency ratio are related to the damping ratio by

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

(1.37)

When the forcing frequency, , is close to the natural frequency, , theresponse is controlled by adjusting the damping Outside of this region, dampinghas less influence, and has essentially no effect for and

Differentiating twice with respect to time leads to the acceleration, ,

(1.38)Noting eqn (1.26), the magnitude of can be written as

(1.39)where

(1.40)

The variation of with for different damping ratios is shown in Fig 1.8 Notethat the behavior of for small and large is opposite to The maximumvalue of is the same as the maximum value for , but the location (i.e thecorresponding value of ) is different They are related to by

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

(1.45)Note that for periodic response, the acceleration is related to the displacement bythe square of the forcing frequency One can also work with instead of

Design criteria

The design problem differs from analysis in that one starts with the mass ofthe system, , and the loading characteristics, and , and determines andsuch that the motion parameters, and , satisfy the specified criteria In general,one has limits on both displacement and acceleration

(1.46)(1.47)where and are the target design values In this case, since and arerelated by

(1.48)one needs to determine which constraint controls If , the acceleration

limit controls and the optimal solution will be

In what follows, both cases are illustrated

Methodology for acceleration controlled design

One works with eqn (1.39) Expressing the target design acceleration as afunction of the gravitational acceleration,

(1.53)and defining as

(1.54)the design constraint takes the form

(1.55)where is the weight of the system

The totality of possible solutions is contained in the region below Figure 1.9 illustrates the region for For low damping, the intersection of

and the curve for a particular value of , , establishes twolimiting values, and Permissible values of for thedamping ratio , are

Fig 1.9: Possible values of

(1.56)The second region does not exist when

Noting eqn (1.40), the expressions for and are

(1.57)

These functions are plotted in Fig 1.10 for representative values of

The limiting values of for reduce to

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

+−

H2∗

- 2–

-=

ξ∗

Fig 1.10: Plot of and versus andNoting eqn (1.29), one can express eqn (1.56) in terms of limiting values ofstiffness By definition,

and (1.62)

Given , one specifies a value of , computes with eqn (1.57), andselects a value for which satisfies the above constraints on stiffness Thedamping parameter is determined from

(1.63)

Example 1.4: An illustration of acceleration controlled design

Figure 1.10 shows that damping has a negligible effect for this value of ; thedesign is essentially controlled by stiffness Taking , and using eqn (1.58)results in

To illustrate the other extreme, and is considered.Here, The two allowable regions for k corresponding to differentvalues of are obtained by applying equations (1.57), (1.60), and (1.62)

Methodology for displacement controlled design

The starting point is eqn (1.26) Noting equations (1.40) and (1.59), eqn(1.26) can be written as

Various values for are considered.