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7.2 Active prestressing of a simply supported beam Passive prestressing The concept of introducing an initial stress in a structure to offset the stressproduced by the design loading is

Chapter 7

Quasi-static Control Algorithms

7.1 Introduction to control algorithms

Referring back to Fig 6.1, an active structural control system has 3 main

components: i) a data acquisition system that collects observations on the excitation and response, ii) a controller that identifies the state of the structure and decides on a course of action and iii) a set of actuators that apply the actions

specified by the controller The decision process utilizes both information abouthow the structure responds to different inputs and optimization techniques toarrive at an “optimal” course of action When this decision process is based on aspecific procedure involving a set of prespecified operations, the process is said to

be algorithmic, and the procedure is called a “control algorithm” A non-adaptivecontrol algorithm is time invariant, i.e., the procedure is not changed over thetime period during which the structure is being controlled Adaptive controlalgorithms have the ability to modify their decision making process over the timeperiod, and can deal more effectively with unanticipated loadings They also canupgrade their capabilities by incorporating a learning mechanism This text isconcerned primarily with time invariant control algorithms which are wellestablished in the control literature Adaptive control is an on-going research areawhich holds considerable promise but is not well defined at this time A briefdiscussion is included here to provide an introduction to the topic

The topic addressed in this chapter is quasi-static control, i.e., where the

structural response to applied loading can be approximated as static response.Since time dependent effects are neglected, stiffness is the only quantity availablefor passive control Active control combines stiffness with a set of pseudo-staticcontrol forces The quasi-static case is useful for introducing fundamentalconcepts such as observability, controllability, and optimal control Bothcontinuous and discrete physical systems are treated.

The next chapter considers time- invariant dynamic feedback control ofmulti-degree-of freedom structural systems A combination of stiffness, damping,and time dependent forces is used for motion control of dynamic systems Thestate-space formulations of the governing equations for SDOF and MDOFsystems are used to discuss stability, controllability, and observability aspects ofdynamically controlled systems Continuous and discrete forms of the linearquadratic regulator (LQR) control algorithm are derived, and examplesillustrating their application to a set of shear beam type buildings are presented.The effect of time delay in the stability of LQR control, and several other linearcontrol algorithms are also discussed

7.2 Active prestressing of a simply supported beam

Passive prestressing

The concept of introducing an initial stress in a structure to offset the stressproduced by the design loading is known as prestressing This strategy has beenused for over 60 years to improve the performance of concrete structures,particularly beams The approach is actually a form of quasi-static control, wherethe variables being controlled are the stresses Figure 7.1 illustrates prestressing of

a single span beam with a single cable When the cable shape is parabolic, thetension introduced in the cable creates an “upward” uniform loading, wo, that isrelated to the tension by

(7.1)The initial moment distribution is parabolic, and the moment is negativeaccording to the conventional notation

w o L2

8d

- = T

Fig 7.1: Passive prestressing schemeSuppose the design loading is a concentrated force that can act at any point

on the span The maximum positive moment due to the force occurs when theforce acts at mid-span, and the resultant positive moment at mid-span is given by

(7.2)The initial mid-span moment is negative and equal to

(7.3)

If the prestress level is selected such that

(7.4)which requires

then the maximum positive and negative moments are equal The cross sectioncan now be proportioned for , which is 1/2 the design moment corresponding

to the case of no prestress This reduction is the optimal value; taking willincrease the initial moment beyond and result in the cross-section beingcontrolled by the initial prestress The limitation of this approach is the need to

apply the total prestress loading prior to the application of the actual loading.

Since the tension is not adjusted while the loading is being applied, the schemecan be viewed as a form of passive control The best result that can be obtainedwith prestressing for this example is equal design moment values for theunloaded and loaded states

Active prestressing

Suppose the cable tension can be adjusted at any time The equivalentuniform upward loading due to the cable action can now be considered to be anactive loading Deforming as the equivalent active loading and noting eqn

(7.1), the loading is related to the “active” tension force, T(t ), by

(7.7)Enforcing the constraint on the maximum moment, which occurs at mid-span,

(7.8)results in the following control algorithm,

No action needs to be taken until reaches , since the maximum moment isless than Above this load level, the active loading counteracts the differencebetween and With active prestressing, the constraint imposed on theinitial prestressing is eliminated Theoretically, the total applied load can becarried by the active system for this example This result is due to the fact that themoment distributions for the actual and active loadings have the same form.When these distributions are different, the effectiveness of active prestressingdepends on the difference between the distributions The following discussionaddresses this point.

Consider the case where the loading is a concentrated force that can act atany point on the span, and the prestressing action is provided by a single cable.The moment diagrams for the individual loadings are shown in Fig 7.2 Whenthese distributions are combined, there is a local positive maximum at point B, thepoint of application of the load, and possibly also at another point, say C.Whether the second local negative maximum occurs depends on the level ofprestressing As is increased, the positive moment at B decreases, and thenegative moment at C increases For a given position of the loading, the controlproblem involves establishing whether can be selected such that themagnitudes of both local moment maxima are less than the prescribed targetdesign value, , indicated in Fig 7.2 With passive prestressing, the optimalprestressing scheme produced a 50% reduction in the required design moment,i.e., it resulted in =0.5(PL/4) Whether an additional reduction can beachieved with active prestressing remains to be determined

Fig 7.2: Active prestressing scheme for a concentrated load

The net moment is given by

Region B-C-D

(7.11)Specializing eqn (7.10) for leads to

(7.12)The location of the second maxima is established by differentiating eqn (7.11) withrespect to x and setting the resulting expression equal to 0 This operation yields

(7.13)The value for M at has the following form:

(7.14)When , the maximum negative moment occurs outside the span, and

(7.16)When is greater than , the maximum positive moment, , is set equal to,

(7.17)Solving eqn (7.17) for leads to

(7.18)The last step involves checking whether for this value of , the maximumnegative moment, , exceeds

It is convenient to work with dimensionless variables for x and M.

(7.19)where

(7.20)

The factor, f, can be interpreted as the “reduction” due to prestressing No prestress corresponds to f=1; passive prestress for this loading and prestressing scheme corresponds to f=0.5 Using this notation, is given by

(7.21)The dimensionless form of eqn (7.18) is written as

(7.22)Lastly, the dimensionless peak negative moment is expressed as

(7.23)where

(7.24)The peak negative moment is a function of the position coordinate, , and

the design moment reduction factor, f Since must be less than 1 for all values

of between 0 and 0.5, the magnitude of f is constrained to be above a limiting value, f min Figure 7.3 shows plots of vs for a range of values f For this case, the limiting value of f is equal to 0.26 Therefore, with active prestressing, the

design moment can be reduced to 50% of the corresponding value for passiveprestressing The influence line for the cable tension required for the “optimal”active prestressing algorithm is plotted in Fig 7.4 Also plotted is the required

tension corresponding to f=0.5, the optimal passive value As expected, lowering

the cross-sectional design moment results in an increase in the required cabletension In order to arrive at an optimal design, the costs associated with thematerial (cross-section) and prestressing need to be considered

-=

a

Ma

Fig 7.3: Influence lines for the peak negative moment

Fig 7.4: Influence lines for the optimal cable tension

Active prestressing with concentrated forces

In this section, the use of concentrated forces to generate prestress momentfields is examined The design loading is assumed to be a single concentratedforce that can act anywhere along the span

Example 7.1: A single force actuator

Consider the structure shown in Fig (1) The active prestressing is provided

by a single force, F, acting at mid-span This loading produces 2 local momentmaxima, M1 and M2 The moment at mid-span may be negative for certaincombinations of and F, and therefore it is necessary to check both M1 and M2when selecting a value for the control force Adopting the strategy discussedearlier, the control algorithm is based on the following requirements

(1)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0

for all where

(2)

(3)and is the design moment for the cross-section

M+

L/2F

M+

M1

M2

Shifting to dimensionless variables,

(9)

Figures 2 and 3 show the variation of and with and f The limiting value

of f for active prestressing is 0.345; when f>0.345, the negative moment at

mid-span is greater than the design moment, For passive prestressing, the

optimal solution is f=0.5 Shifting from passive to active control results in an

additional 30 percent reduction in the allowable design moment

-=

M*

a

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0

F

a

Example 7.2: Two force actuators

The previous example showed that the effectiveness of a prestressingscheme depends on the difference between the moment distributions for theapplied loading and the prestressing forces In the case of a single control forceapplied at mid-span, the limiting condition occurs when the applied load is nearthe end support, where the difference in the moment distributions is a maximum.Increasing the number of control forces provides the capability to modify the

“shape” of the “prestress” moment distribution to conform better with theapplied moment field, and therefore increase the amount of prestressing that can

be applied This example illustrates the use of self-equilibrating control forcesystems which provide the maximum flexibility for adjusting the moment field

Consider the self-equilibrating force system shown in Fig 1 This forcesystem produces a bilinear moment field which is local, i.e., confined to theloaded region Therefore, perturbing the control force magnitude, F, has no effectoutside of this region

Figure 1Applying a set of these self-equilibrating systems results in a piecewizelinear moment distribution Figure 2 illustrates the case of 2 force systems locatedimmediately adjacent to each other The corresponding moment field is defined interms of 2 force parameters, F1 and F2

-Region A-B

(1)Region B-C

(2)Region C-D

(3)where

(4)

Figure 2Example 7.1 treated the case of a single actuator deployed on a simplysupported beam subjected to a single concentrated force that can act at any point

on the beam Suppose the control force system now consists of 2 local momentfields centered at the third points of the span Figure 3 shows the 2 loadingscenarios for this example The moments at B,C, and D corresponding to thedifferent loading scenarios are:

l

– (x x– A)

Region A-B (Fig 3a):

(5)

(6)

(7)Region B-C (Fig 3b):

Figure 3bLet represent the design moment Expressing as a fraction of themaximum moment for the case where there is no prestressing,

(11)and working with dimensionless moments,

(12)transforms eqns (5) thru (10) to the following:

using 2 of the constraints in eqn (19), and then adjusting f such that the third

constraint is also satisfied Figure 4 shows the variation in the moment measureswith , the coordinate defining the position of the load, corresponding to thefollowing choice of constraints:

Region A-B

(20)

Region B-C

(21)

The minimum value of f is equal to 0.228 This value is controlled by the

constraint on in the region A-B For the case of a single actuator applied at

mid-span, the optimum value for f was found to be 0.345 Applying 2 force

actuators leads to an additional reduction in the “permissible” design moment

Figure 4a

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0

Figure 4b

Figure 4c

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0

Figure 4d

A general active prestressing methodology

The discussion to this point has been concerned with a specific loading and

a specific prestressing scheme In what follows, a general methodology fordealing with the combination of an arbitrary design loading and prestressingscheme applied to a simply supported beam is described, and the controlalgorithm corresponding to a particular choice of error measure is formulated.This methodology is also applicable for displacement control which is discussed

in the following section

Let denote the moment due to the design loading, the

moment generated by the prestress system, and M(x) the net moment By

definition,

(7.25)The design objective is to limit the magnitude of to be less than , thedesign moment for the cross-section

(7.26)Equation (7.26) imposes a constraint on the magnitude of

(7.27)These limits establish the lower and upper bounds for Given , onegenerates the limiting boundaries and then decides on a “target” distribution for

Fig 7.5: Limiting prestress moment fieldsFigure (7.5) illustrates the process of establishing the desired distributionfor the control moment The curves shown in Fig (7.5b) correspond to

; allowable values of are defined by the shaded area Thedistribution corresponding to selecting the minimum allowable value of

at each x is plotted in Fig (7.5c) Assuming magnitude is the dominant measure,

-1

-2

10-1

this distribution represents the optimal “target” prestress moment field.

Let denote the desired “target” prestress moment field Suppose

the actual prestress moment field is a linear combination of r individual fields,

(7.28)where are moment amplitude parameters and are dimensionlessfunctions The error associated with a specific choice of moment parameters isrepresented by the difference, ,

(7.29)Ideally, one wants for However this goal cannot be achievedwhen is an arbitrary function, and it is necessary to work with anapproximate error condition established using collocation, the least squaremethod, or some other weighted residual scheme

The least square method is based on taking the integral of as ameasure of the accuracy of the approximation represented by eqn (7.28) Thisintegral is denoted as J

(7.30)

In general, J is a function of the r moment parameters Equations for these

parameters are generated by requiring J to be stationary

(7.31)Expanding eqn (7.31) results in the following linear matrix equation,

(7.32)where are r’th order matrices, and the elements of and are:

(7.33)

(7.34)Given ,one determines and then solves for This solution producesthe least value for J, for a particular set of ‘s A sense of convergence can be

obtained by expanding the set of basis functions, and comparing thecorresponding values of J It should be noted that the exact condition, e(x)=0 for0<x<1, is generally not satisfied by eqn (7.32).

This formulation works with continuous functions, and requires theevaluation of a set of integrals It is more convenient to work with vectors ratherthan functions, since the computation reduces to matrix operations Suppose the

moment is monitored at n points within the interval The desired

prestress moment vector is of order n

(7.35)Evaluating eqn (7.28) at these observation points leads to

(7.36)where is of order The error vector is taken to be the difference between and ,

(7.37)When and the individual prestress moment fields are linearlyindependent, is non-singular and it is possible to determine an that exactlysatisfies

(7.38)When , a least square procedure can be used to establish an approximatesolution for The error measure is taken as the norm of

(7.39)Requiring J to be stationary with respect to leads to an equation having thesame form as eqn (7.32), with and now given by

(7.40)(7.41)The following example illustrates the application of the discrete formulation

Example 7.3: Multiple actuators

Consider the design moment field shown in Fig (1) The longitudinal axis

is discretized with 10 equal segments, resulting in 11 (n=11) observation points.

Applying the criteria defined by eqn (7.27), and taking leads to thebounding curves for plotted in Fig (2) The problem now consists ofgenerating a prestress moment distribution which lies between these bounds

Suppose 4 self-equilibrating force system (r=4) that produce bilinear

moment fields are applied at equally spaced interior points The correspondingfunctions are shown in Fig (3a) and the typical field is plotted in Fig (3b) Sincethe prestress moment field is defined in terms of the moments at only 4 fixedpoints (3,5,7,9), and there are 9 interior points, the solution for the case of anarbitrary target distribution will be approximate Various solutions for aparticular target distribution are plotted in Fig (4) Curve (1) corresponds to

solution for the 4 actuator system defined in Fig (3a) Curve (3) is based on the 7actuator system defined in Fig (3c) The 3 additional actuators applied at points2,4, and 6 eliminate the error up to point 6 Incorporating 2 more actuators atpoints 8 and 10 would completely eliminate the error associated with the 4actuator system It produces a moment field that is fully contained within theallowable zone and has the lowest cost as measured by the sum of the actuatormoment magnitudes,

Figure 1: Moment due to design loading

-1-2

Figure 2: Upper and lower bounds on the prestress moment field

M c* 3

21

-1-2

-3-4

lower bound

Figure 3: Prestress moment fields for 4 and 7 actuators

Figure 4: Prestress moment fields

7.3 Quasi-static displacement control of beams

The previous section discussed how active prestressing can be applied tolimit the magnitude of the bending moment in a beam subjected to a quasi-staticloading In this section, a procedure for controlling the quasi-static displacementprofile of a beam is described Both prestress and displacement control are based

on the same least square error minimization algorithm which is generalized later

in the next section

Consider a beam subjected to a loading that produces the displacementfield Suppose the desired displacement profile is , and the difference