Điều khiển kết cấu - Chương 3 pps

The concept of equivalent viscous damping is introduced and is used to express viscoelastic, structural, and hysteretic damping in terms of theirequivalent viscous counterpart.. dampers

1 second For low damping ratio, the energy dissipated per cycle is small, andmany cycles are required before the input energy is eventually dissipated As isincreased, the energy dissipated per cycle increases, and the stored energy build

up is reduced Shifting from = 0.02 to = 0.1 reduces the peak stored energydemand by a factor of 3.7 for this particular system and earthquake excitation Itshould be noted that seismic accelerograms differ with respect to their frequencycontent and intensity, and therefore one needs to carry out energy time historystudies for individual excitations applied to a specific system For example, Fig3.2 shows the response of the same system for a typical Northridge accelerogram.The input energy build up for the Northridge loading is quite different than forthe El Centro loading

ξ

Fig 3.1: Energy Build Up, El Centro (S00E), Imperial Valley 1940

Fig 3.2: Energy Build Up, Arleta Station (90 DEG), Northridge 1994, = 2%Dissipation and absorption are attributed to a number of external andinternal mechanisms, including the following:

• Energy dissipation due to the viscosity of the material This processdepends on the time rate of change of the deformations, and is referred

to as material damping Viscoelastic materials belong to this category.

• Energy dissipation and absorption caused by the material undergoingcyclic inelastic deformation and ending up with some residualdeformation The cyclic inelastic deformation path forms a hysteresisloop which correspond to energy dissipation; the residual deformation

is a measure of the energy absorption This process is generally termed

hysteretic damping.

• Energy dissipation associated with overcoming the friction between

moving bodies in contact, such as flexible connections Coulomb damping refers to the case where the magnitude of the friction force is constant Structural damping is a more general friction damping

mechanism which allows for a variable magnitude of the friction force

• Energy dissipation resulting from the interaction of the structure withits surrounding environment Relative motion of the structure

generates forces which oppose the motion and extract energy from thestructure Fluid-structure interaction is a typical case The fluid exerts a

drag force which depends on the relative velocity and functions as an

equivalent viscous damping force

• Damping devices installed at discrete locations in structures tosupplement their natural energy dissipation/absorption capabilities.These mechanisms may be passive or active Passive mechanismsrequire no external energy, whereas active mechanisms cannot functionwithout an external source of energy Passive devices include viscous,friction, tuned mass, and liquid sloshing dampers Active damping isachieved by applying external control forces to the structure overdiscrete time intervals The magnitudes of the control forces are

adjusted at each time point according to a control algorithm.

• Passive damping removes energy from the response, and therefore cannot cause the response to become unstable Since active control involves

an external source of energy, there is the potential for introducing an

instability in the system The term “semi-active” refers to a particular

class of active devices that require a relatively small amount of externalenergy and apply the control force in such a way that the resultingmotion is always stable Chapter 6 discusses active control devices

In this chapter, the response characteristics for material, hysteretic, andfriction damping mechanisms are examined for a single degree-of-freedom

(SDOF) system The concept of equivalent viscous damping is introduced and is

used to express viscoelastic, structural, and hysteretic damping in terms of theirequivalent viscous counterpart Numerical simulations are presented todemonstrate the validity of this concept for SDOF systems subjected to seismicexcitation This introductory material is followed by a discussion of the influence

of distributed viscous damping on the deformation profiles of freedom (MDOF) systems The damping distribution is initially taken to beproportional to the converged stiffness distribution generated in the previouschapter, and then modified to allow for non-proportional damping Numericalresults and deformation profiles for a range of structures subjected to seismicloading are presented, and the adequacy of this approach for distributingdamping is assessed

multi-degree-of-Distributed passive damping can be supplemented with one or morediscrete damping devices to improve the response profile Discrete viscous

dampers inserted in discrete shear beam type structures are considered in thischapter; the basic theory for tuned mass dampers is presented in the next chapter.Subsequent chapters deal with base isolation, a form of passive stiffness/damping control, and active control.

3.2 Viscous, frictional, and hysteretic damping

Viscous damping

Viscous damping is defined as the energy dissipation mechanism where thedamping force is a function of the time rate of change of the correspondingdisplacement measure:

(3.1)where is the damping force and is the velocity in the direction of Thelinearized form is written as:

(3.2)where , the damping coefficient, is a property of the damping device Linear

viscous damping is convenient to deal with mathematically and therefore is thepreferred way of representing energy dissipation

In general, the work done on the device during the time interval, is given by

(3.3)Considering periodic excitation

(3.4)and evaluating eqn (3.3) for one full cycle under linear viscous damping leads to

(3.5)This term represents the energy dissipated per cycle by the damping device, asthe system to which it is attached undergoes a periodic motion of amplitudeand frequency Figure 3.3 shows the force-displacement relationship forperiodic excitation; the enclosed area represents

Fig 3.3: Viscous response - periodic excitation

Example 3.1: Viscous damper

Figure 3.4 shows a possible design for a viscous damping device The gapbetween the plunger and external plates is filled with a linear viscous fluidcharacterized by

(3.6)where and are the shearing stress and strain measures respectively and isthe viscosity coefficient Assuming no slip between the fluid and plunger, theshear strain is related to the plunger motion by

uˆ

–

cΩuˆ F

plunger

where is the thickness of the viscous layer Letting and represent theinitial wetted length and width of the plunger respectively, the damping force isequal to

(3.8)Substituting for results in

(3.9)Finally, eqn (3.9) is written as

(3.10)where represents the viscous coefficient of the device,

(3.11)The design parameters are the geometric measures and the fluid viscosity,

A schematic diagram of a typical viscous damper employed for structuralapplications is contained in Fig 3.5; an actual damper is shown in Fig 3.6 Fluid isforced through orifices located in the piston head as the piston rod position ischanged, creating a resisting force which depends on the velocity of the rod Thedamping coefficient can be varied by adjusting the control valve Variabledamping devices are useful for active control Section 6.4 contains a description of

a particular variable damping device that is used as a semi-active force actuator.This chapter considers only passive damping, i.e a fixed damping coefficient

Fig 3.5: Schematic diagram - viscous damper

Fig 3.6: Viscous damper - 450 kN capacity(Taylor Devices Inc http://www.taylordevices.com)

Equation (3.5) shows that the energy loss per cycle for viscous dampingdepends on the frequency of the excitation This dependency is at variance withobservations for real structural systems which indicate that the energy loss percycle tends to be independent of the frequency In what follows, a number ofdamping models which exhibit the latter property are presented

Friction damping

Coulomb damping is characterized by a damping force which is in phase with thedeformation rate and has constant magnitude Mathematically, the force can beexpressed as

(3.12)where denotes the sign of Figure 3.7 shows the variation of withfor periodic excitation The work per cycle is the area enclosed by the responsecurve

F = Fsgn( )u˙

u˙

( )

Fig 3.7: Coulomb damping force versus displacement

(3.13)Figure 3.8 shows a coulomb friction damper used with diagonal X bracing

in structures Friction pads are inserted at the bolt-plate connections Interstorydisplacement results in relative rotation at the connections, and the energydissipated is equal to the work done by the frictional moments during thisrelative rotation

Fig 3.8: Friction brace damperStructural damping removes the restriction on the magnitude of thedamping force, and considers the force to be proportional to the displacementamplitude The definition equation for this friction model has the form

(3.14)where is a pseudo-stiffness factor Figure 3.9 shows the corresponding cyclicresponse path The energy dissipated per cycle is equal to

(3.15)

u

u u

Fig 3.9: Structural damping force versus displacement

Hysteretic damping

Hysteretic damping is due to the inelastic deformation of the material composingthe device The form of the damping force-deformation relationship depends onthe stress-strain relationship for the material and the make-up of the device.Figure 3.10 illustrates the response path for the case where the material force-deformation relationship is elastic-perfectly plastic

The limiting values are , the yield force, and , the displacement atwhich the material starts to yield; is the elastic damper stiffness The ratio ofthe maximum displacement to the yield displacement is referred to as theductility ratio and is denoted by With these definitions, the work per cycle forhysteretic damping has the form

(3.16)

u u

Fig 3.10: Hysteretic damping force versus displacement

Figure 3.11 shows a bracing element that functions as an hysteretic damper(Watanabe, 1998) The element is composed of a core member fabricated withhighly ductile low strength steel (yield strength of 200 MPa., maximum percentstrain of 60%), a cylindrical jacket, and mortar placed between the core memberand the jacket The jacket functions as an additional bending element, and itscross sectional moment of inertia is selected such that the buckling load is equal tothe yield force This design feature allows the brace to be used for both tensile andcompressive loading Triangular plate hysteretic dampers that dissipate energythrough bending action have also been used for buildings (Tsai, 1993 and 1998)

Fig 3.11: Hysteretic damper brace element (Watanabe, 1998)

Example 3.2: Stiffness of a rod hysteretic damper

Consider a damping device consisting of a cylindrical rod of length and area

Suppose the material is elastic-perfectly plastic, as shown in Fig 3.12.

The relevant terms are

(3.17)(3.18)

Fig 3.12: Elastic-perfectly plastic damper device

Then,

(3.19)(3.20)(3.21)

Example 3.3: Stiffness of two hysteretic dampers in series

The device treated in Example 3.2 is modified by adding a second rod in series, asshown in Fig 3.13 The yield force for the second rod is assumed to be greaterthan the yield force for the first rod,

(3.22)Since the force is the same for both devices, the total elastic displacement is thesum of the individual contributions

Fig 3.13: Two rod hysteretic damping device.

(3.23)Specializing eqn (3.23) for the onset of yielding, one obtains

(3.24)(3.25)When two elements are used, one can vary both the yield force, , and the elasticyield deformation The energy dissipation increases with decreasing , for agiven deformation amplitude

3.3 Viscoelastic material damping

A material is considered to be elastic when the stresses due to an excitation areunique functions of the associated deformation Similarly, a material is said to beviscous when the stress state depends only on the deformation rates For simpleshear, these definitions translate to

Elastic

(3.26)

Viscous

(3.27)The stress-deformation paths for periodic strain are illustrated in Figs 3.14(a) and

3.14(b) There is no time lag between stress and strain for elastic behavior,whereas the stress is radians out of phase with the strain for viscousbehavior If these relations are linearly combined, one obtains the path shown inFig 3.14(c).

Fig 3.14: Stress-deformation relations

Materials that behave similar to Fig 3.14(c) are called viscoelastic The

properties of a linear viscoelastic material are determined by applying a periodicexcitation and observing the response, which involves both an amplification and

a phase shift The basic relations are expressed as

(3.28)(3.29)where is the storage modulus and is the loss modulus The ratio of the loss modulus to the storage modulus is defined as the loss factor,

(3.30)

An alternate form for eqn (3.29) is

(3.31)(3.32)The angle is the phase shift between stress and strain Delta ranges from forelastic behavior to for pure viscous behavior

Experimental observations show that the material properties and

vary with temperature and the excitation frequency Figure 3.15 illustrates thesetrends for ISD110, a 3M product The dependency on frequency makes it difficult

to generalize the stress-strain relationships based on periodic excitation to allowfor an arbitrary time varying loading such as seismic excitation This problem isaddressed in the next section

To determine the damping properties at the desired temperature and frequency from the data graph shown above, proceed as follows:

• Locate the desired frequency on the RIGHT vertical scale.

• Follow the chosen frequency line to the desired temperature isotherm.

• From this intersect, go vertically up and/or down until crossing both the shear (storage) modulus and loss factor curves

• Read the storage modulus and loss factor values from the appropriate LEFT hand scale.

Fig 3.15: Variation of 3M viscoelastic material, ISD110, with frequency and

temperature

The energy dissipated per unit volume of material for one cycle ofdeformation is determined from

(3.33)Substituting for and using eqns (3.28) and (3.29) results in

The corresponding expression for a pure viscous material is generated using eqn(3.27)

(3.35)This expression involves the frequency explicitly whereas the effect of frequency

is embedded in for the viscoelastic case

Example 3.4: Viscoelastic damper

A damper device is fabricated by bonding thin sheets of a viscoelastic material tosteel plates, as illustrated in Fig 3.16

Since the elastic modulus for steel is considerably greater than the shearmodulus for the sheet material, one can consider all the motion to be due toshear deformation of the sheets Defining as the relative displacement of theends of the damper device, the shearing strain is

(3.36)Given , one evaluates with the stress-strain relation and then using theequilibrium equation for the system

(3.37)Applying a periodic excitation

(3.38)

Fig 3.16: Viscoelastic damper device

and taking according to eqn (3.29), one obtains

τ

(3.39)(3.40)Equation (3.39) can also be written as

(3.41)(3.42)Finally, the energy dissipated per cycle is given by

(3.43)Typical polymer materials, such as Scotchdamp ISD110 (3M Company, 1993) have

in the range of and

Based on the result of the previous example, the expressions defining theresponse of a viscoelastic damper due to periodic excitation can be written in ageneralized form

(3.44)(3.45)(3.46)where depends on the geometric configuration of the device, is the storagemodulus, and is the material loss factor Figure 3.17 shows the variation ofwith over the loading cycle

F = fdGsuˆ[sinΩt+ηcosΩt]

Fig 3.17: Variation of with for viscoelastic material.

3.4 Equivalent viscous damping

The expression for the damping force corresponding to linear viscous damping isthe most convenient mathematical form, in comparison to the other dampingforce expressions, for deriving approximate analytical solutions to the forceequilibrium equations Therefore, one way of handling the different damping

models is to convert them to equivalent viscous damping models In what follows, a

conversion strategy based on equating the energy dissipated per cycle of periodic

excitation to the corresponding value for linear viscous damping is described

Linear viscous damping is defined by eqn (3.2):

(3.47)Specializing eqn (3.47) for periodic excitation

(3.48)leads to

(3.49)Also, noting eqn (3.5), the energy loss per cycle is

(3.50)The force and energy loss for the other models are expressed in terms of anequivalent damping coefficient,

(3.51)(3.52)Substituting for a particular damping model in eqn (3.52), and taking, one obtains the equivalent damping coefficient The coefficients for thevarious models are listed below:

as pseudo-viscous damping but require specifying a representative frequency,, and amplitude, In this case, eqns (3.54) and (3.55) are written as

Structural

(3.57)

Hysteretic

(3.58)where

(3.59)(3.60)Numerical simulations illustrating the accuracy of this approximation are

provided by the following examples.

Example 3.5: Structural and hysteretic damping comparison - seismic excitation

A 1 DOF shear beam having the following properties is considered:

The equivalent structural stiffness is generated using eqn (3.57), taking

and equal to the fundamental frequency The corresponding structuralstiffness is

(3.61)Results for this model subjected to Taft excitation are compared with thecorresponding results for the linear viscous model in Figs 3.18 and 3.19 Closeagreement is observed

Fig 3.18: Response of SDOF with structural damping.

Fig 3.19: Structural damping force versus deformation

The hysteretic model calibration defined by eqn (3.58) is not as straightforward since both the yield force and the ductility are involved For periodicmotion, the maximum displacement is known Then, one can specify thedesired ductility and compute the required force level and initial stiffness with

(3.62)(3.63)(3.64)For non-periodic motion, one needs to specify the limiting elasticdisplacement , and estimate the maximum amplitude This leads toestimates for the ductility ratio and the peak force Figures 3.20 and 3.21show the results based on taking equal to the peak amplitude observed for pureviscous damping, and a ductility ratio One can adjust and toobtain closer agreement Since the energy is dissipated only during this inelasticphase, hysteretic damping is generally less effective than either viscous orstructural damping for low intensity loading

Fig 3.20: Response of SDOF with hysteretic damping

T1 = 5.35s

ξ1 = 2%

Viscous dampingHysteretic damping

Fig 3.21: Hysteretic damping force versus deformation.

The calibration of the equivalent viscous damping coefficient was based onassuming a periodic excitation As discussed above, non-periodic excitationrequires some assumptions as to the response An improved estimate of theequivalent damping coefficient can be obtained by evaluating the actual workdone by the damping force Starting with

(3.65)and writing

(3.66)leads to

(3.67)Equation (3.67) can be used to evaluate the variation over time of the equivalentdamping ratio Taking = , the total duration of the response, provides anestimate of the effective damping ratio Figure 3.22 shows results generated for arange of seismic excitations and hysteretic damper yield force levels As expected,the effective damping increases with increasing excitation and decreases with

increasing yield force.

Fig 3.22: Equivalent viscous damping ratio vs yield force

The viscoelastic model calibration is more involved since the materialproperties are also frequency dependent Referring back to eqn (3.45), thedamping force for periodic excitation

(3.68)was expressed as

(3.69)where is a geometric factor defined by the geometry of the device Ourobjective is to express as

(3.70)where and are equivalent stiffness and damping terms Consideringperiodic excitation, eqn (3.70) takes the form

(3.71)One can obtain estimates for and with a least square approach Assuming

0 0.1 0.2

there are material property data sets, and summing the squares of the errors for and over the ensemble results in

(3.72)(3.73)Minimizing eqn (3.72) with respect to yields

(3.74)Similarly, minimizing eqn (3.73) with respect to results in

(3.75)The form of eqn (3.75) suggests that be expressed as

(3.76)Substituting for and leads to the definition equation for

(3.77)Note that depends only on the material, i.e it is independent of the geometry

of the device

With this notation, the equivalent viscous force-deformation relation for a linearviscoelastic damper is written as

(3.78)

Example 3.6: Determining for 3M ISD110 damping material

This example illustrates how the procedure discussed above can be applied tocompute the parameters for the 3M Scotchdamp ISD110 material Using Fig 3.15,data corresponding to five frequencies is generated Table 3.1 contains this data.Applying eqns (3.74), (3.75), and (3.77), one obtains

Table 3.1: Data for ISD110 Scotchdamp material (from Fig 3.15).

3.5 Damping parameters - discrete shear beam

Damping systems

This section extends the treatment of discrete shear beams to includedamping devices located between the floors Figure 3.23 illustrates 2 different

placement schemes of viscous type dampers for a typical panel Scheme a

combines the damper with a structural element and deploys the composite

element on the diagonal between floors Scheme b places the damper on a roller

support at the floor level, and connects the device to the adjacent floor with

structural elements An actual installation of a scheme b system is shown in Fig.

3.24 The structural elements are modelled as linear springs and therepresentation defined in Fig 3.25 is used

Fig 3.23: Damper placement schemes.

Fig 3.24: Viscous dampers coupled with chevron bracing.

Fig 3.25: Idealized models of structures with viscous dampers.

A differential story displacement generates a deformation of the damper,resulting in a damper force which produces the story shear, A subscript isused to denote quantities associated with the damper The total story shear is thesum of the “elastic” shear force due to elastic frame/brace action and the

“damper” shear force The former was considered in Chapter 2 This contribution

is written as

(3.79)where subscript refers to “elastic” frame/brace action The damper shear force

is a function of both the relative displacement and the relative velocity This term

is expressed in a form similar to eqn (3.78):

(3.80)where and are “equivalent” properties that depend on the makeup of thedamping system Various cases are considered in the following sections

Rigid structural members - linear viscous behavior

Consider first the case where the stiffness , of the structural memberscontained in the damping system is sufficiently large so that the extension of themember is negligible in comparison to the extension of the damper Defining

as the extension of the damper, and considering scheme a, the damper force for

linear viscous behavior is given by

(3.81)The corresponding shear force is

(3.82)The equivalent damping coefficient for story is obtained by summing thecontributions of the dampers present in story

(3.83)

Equation (3.83) also applies for scheme b; = 0 for this arrangement of structural

members and dampers Scheme b is more effective than scheme a (a factor of 2 for

bracing), and is more frequently adopted

The general spring-dashpot model shown in Fig 3.26 is useful forrepresenting the different contributions to the story shear force For this case, thedamper ( ) acts in parallel with the elastic shear stiffness of the frame/bracingsystem ( ) and is equal to the interstory displacement An extended version ofthis model is used to study other damping systems

V d i, = k d i, (u i–u i 1– )+c d i, (u˙ i–u˙ i 1– )

k d c d

k'

e d

F d = c d e˙ d = c d(u˙ i–u˙ i 1– )cosθ

V d i, = F dcosθ = c d(u˙ i–u˙ i 1– )cos2θ

i i

Fig 3.26: Spring and dashpot in parallel model.

Example 3.7: Example 2.15 revisited

Consider the 5 DOF shear beam defined in example 2.15 Taking theconstant nodal mass as 10,000 kg, and using the stiffness calibration based on

, results in the following values for the element shearstiffness factors, fundamental frequency, nodal mass, and damping:

e V K

Suppose the chevron brace scheme (scheme b) is used, and 2 dampers are

deployed per floor The “design” values for the dampers are obtained by dividingthe above results by a factor 2 For the uniform case, In order todesign the damper, one also needs to specify the peak value of the damper force.This quantity is determined with where is the maximumrelative velocity of the damper piston For this damper deployment scheme, therelative damper displacement is equal to the interstory displacement It followsthat for level is equal to

(5)

The nodal displacements for this 5 DOF model are considered to vary linearlywith height:

where is the modal amplitude Then

The peak amplitude is determined with

(7)One can estimate by assuming the response is periodic, with frequency

(8)Using the problem data,

(9)and the peak damper force is estimated as

Rigid structural members - linear viscoelastic behavior

Fig 3.27: Spring-dashpot model for viscoelastic damping

The case where the damping mechanism is viscoelastic is represented by themodel shown in Fig 3.27 Here, the damping force has an elastic component aswell as a viscous component Noting eqn (3.78), the damping force is expressed as

(3.84)where and (or ) are the equivalent stiffness and damping parameters for

the visco-elastic device, and is the interstory displacement This formulation

assumes the viscous device is attached to a rigid element so that all the

deformation occurs in the device The more general case is treated later Using eqn(3.84), one obtains

(3.85)When is small with respect to unity, the contribution of the visco-elasticdamper to the stiffness can be neglected

Example 3.8:

Consider a SDOF system having an elastic spring and a visco-elasticdamper modeled as shown in the figure Suppose , , and are specified, andthe objective is to establish values for the spring stiffness and damper properties

The governing equation has the form

(1)

By definition,

(2)

(3)Given and , is determined with eqn (3) The stiffness factors are related by

(4)Our strategy for dealing with a visco-elastic device is based on expressingthe equivalent damper coefficient as (see eqn (3.78)):

(5)where is a “material” property Example 3.6 illustrates how to evaluate for

a typical visco-elastic material The procedure followed here is to first determine, using eqns (3) and (5),

u p k

c

k1m

and then substitute for in eqn (4) This operation results in an equation for k.

(7)

Suppose =10,000 kg, = 2 rad/s, and Then,

Using a typical value for ,

leads to

For these parameters, the visco-elastic element contributes approximately 20% ofthe stiffness

Example 3.9: Example 3.7 revisited

Suppose visco-elastic dampers are used for the 5 DOF system considered

in example 3.7 The damper force is taken as

(1)where and depend on the device, and is the displacement of thedamper Consider the case where a chevron brace with 2 dampers is installed ineach floor and the damping distribution defined by eqn 4 in example 3.7 is used.The damper coefficients are determined by dividing the values listed in eqn 4 by 2(2 dampers per floor):

(2)(units of kN-s/m)

The damper stiffness is determined with

(3)

Assuming , the corresponding values of damper stiffness are:

(4)(units are kN/m)

The total story shear stiffness distribution is given by eqn (1) in example 3.7 Thisvalue is the sum of the elastic stiffness due to frame/brace action and the stiffnessdue to the 2 dampers

(5)Using (4) and the data from example 3.7, the frame/brace story shear stiffnessfactors for this choice of are

(6)(units are MN/m)

The contribution of the damper stiffness is about 14% of the total stiffness for thisexample

Example 3.10: Viscoelastic damper design

Referring back to eqn (3.74), the elastic stiffness of the damper depends on the

average storage modulus of the viscoelastic material and a geometric parameter

(1)Given and , one solves for

(2)

To proceed further, one needs to specify the geometry of the device The figurelisted below shows a system consisting of layers of a viscoelastic materiallocated between metal plates Considering the metal elements to be rigid withrespect to the viscoelastic elements, the shape factor is given by

corresponding shape factor is

Substituting for in eqn (3), the variables are related by

(6)

(7)Taking results in

(8)

Example 3.11: Hysteretic damper design - diagonal element

Equation (3.58) defines the equivalent viscous damping parameter for hystereticdamping Substituting the extension, , for the displacement measure , andsolving for the yield force, , results in

(1)

where and are representative extension and frequency values, and is givenby

(2)

where is the extension at which the diagonal material yields

The representative extension is a function of the representative transverse

shear deformation Taking equal to , the design level for , leads to

(3)and

(4)

f d 2n wL( d) = t d⋅ f d = 4.0t d (meters)