Structural Safety, 5 (1988) 159-168 159 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

RELIABIL ITY A N A L Y S I S OF D E F L E C T I O N - D R I F T L IMITED S T R U C T U R E S *

Dan M. Frangopol and Rachid Nakib

Department of Civil Engineering, University of Colorado, Boulder, CO 80309-0428 (U.S.A.)

(Received May 26, 1987; accepted February 18, 1988)

Key words: building codes; deflection; drift; limit states; probability theory; reliability analysis; serviceability; steel frames; stiffness; system reliability.

ABSTRACT

Deflections, structural deteriorations, floor vibrations and other serviceability criteria are naturally random with respect to their magnitudes. A realistic analysis of serviceability limit states therefore requires the consideration of risk of unserviceability. Isosafety functions are proposed here for analyzing the influence of different parameters on the reliability levels with respect to limit states associated with deflection and drift conditions.

INTRODUCTION

Structural systems are typically multifunctional in nature, required to perform a combination of ultimate and serviceability limit state constraints. While this basic design philosophy has been accepted for some time, the area of serviceability limit state research has not enjoyed the same development as the ultimate limit state area. The Ad Hoc Committee on Serviceability Research, Committee on Research of the Structural Division [1] recognized that the problems in the area of serviceability limit states are more difficult to define and analyze, they are not necessarily safety related, the basic data are more available to the practitioner than to the researcher, and the current code treatment of serviceability issues is limited or inadequate. It is also important to recognize that during the past decade the desire to reduce structural weight and rigidity has been a strong driving force behind the development of serviceability limit states procedures. Today, the need for reliability-based structural optimization via weight reduction provides further motivation for the development of serviceability procedures that are consistent with the concepts of probability-based limit states design.

* Originally presented at the Fifth International Conference on Applications of Statistics and Probability in Soil and Structural Engineering, Vancouver, Canada, May 1987.

0167-4730/88/$03.50 © 1988 Elsevier Science Publishers B.V.

160

The papers by Turkstra and Reid [2] and Galambos and Ellingwood [3] deserve special attention in view of the information on probabilistic design for serviceability. Turkstra and Reid [2] recognized the problems of subjectivity, discretization, load- t ime history, and optimization and developed a general approach to analyze these problems for codification purposes. Galam- bos and Ellingwood [3] treated deflection and drift as random variables using the first-order second-moment reliability theory. Both of these papers indicate that in reality there are not distinct sets of serviceability limit states; they depend on subjective factors (e.g., the perceptions of the building occupants). However, as pointed out by Galambos and Ellingwood [3], there are indicators providing some guidlines as to what behavior might be expected at different static deformations. For example, at a deflection limit of 1/300 of the span of a flexural member, or a drift limit of 1/300 of the story height, visible damages to ceiling and flooring, and cracking in reinforced walls are expected.

From the broad spectrum of common serviceability problems discussed by Galambos and Ellingwood [3] as well as the Ad Hoc Committee on Serviceability Research [1], this paper deals with two limit states related to excessive static deformation: deflection and drift. These serviceability requirements are simultaneously accounted for in the reliability analysis process. The deflection and drift limit states for a structure acted on by random loads are defined here as isosafety functions in the space of the mean values of the loads (i.e., isosafety loading functions). The space of the mean values of deformations is also used for defining isosafety functions against unserviceability by excessive deflection and drift (i.e., isosafety deformation functions). A parametric analysis is conducted to study the effect of various parameters on both the isosafety loading and deformation functions.

RELIABILITY WITH RESPECT TO DEFLECTION AND DRIFT

The problem considered in this paper is to find the reliability of a deterministic structure acted on by random service loads with respect to both deflection and drift. In particular, aspects of the problem such as the influence of the input parameters on the reliability associated with deflection and drift, and the individual contribution of loads to the global safety with respect to deformation are also considered.

The following assumptions are made: (a) deflection and drift computations correspond to an idealized elastic deterministic structure with fixed geometry and rigidity; (b) the service static loads acting on the structure are random variables with known statistical properties for a given serviceability reference period (e.g., eight years according to Ref. [3]); (c) the positions of loads are deterministic; (d) the allowable (vertical) deflection, Aav lj°w, and the allowable (horizontal) drift, A~ l°w, are deterministic limits specified by the designer; (e) the statistical dependencies between the service loads are accounted for through correlation coefficients; (f) consistent with a first-order second-moment reliability theory, the necessary statistical information is given in terms of service load vectors (mean values and coefficients of variation) and service load correlation matrix (coefficients of correlation among loads); (g) on the same basis, of a first-order second-moment reliability approach, regardless of the probability distributions of the individual loads, the reliability of the structure with regard to (vertical) deflections is given as

flv= l/V(gv) (1) and with regard to (horizontal) drift is given as

/3 h = 1 / V ( g h ) (2)

161

where Bv and Bh are safety indices,

gv ~--- Aay ° w - Av (3)

A all°w -- A h (4) gh ~ *"*h

are the reserve safety margins (also termed performance functions) with respect to deflection and drift, respectively, and V(gv) and V(gh) are the coefficients of variations of the safety margins gv and gh, (Av and A h in eqns. (3) and (4) are the (vertical) deflection and the (horizontal) drift due to service static loads); and (h) the global safety of a structure with regard to both deflection and drift (i.e., deformation serviceability) is given as

~system = 1 / V ( gsystem ) (5)

where the global safety (generalized system reliability) index is defined as the reciprocal of V(gsystem ) = the coefficient of variation of the unserviceability mode expression (i.e., gsystem ~-- reserve safety margin of the system with respect to deformation) including both deflection and drift.

The safety indices (1), (2) and (5) provide an exact probability of unserviceability with respect to deflection, drift, and system deformation, respectively, if the loads are normally distributed. In this case

Pf,v = O ( - f l v ) (6)

Pf.h : (I)(-- ~ h ) (7)

Pf,system ---- (1) ( - ~system ) (8)

where cI,(.) is the standard normal distribution function. If a structure under a given service load combination has m critical sections with respect to

vertical deflections and n critical sections with respect to lateral deflections (drift), the occur- rence of Aav u°w or Aah n°w or both in any of these sections will constitute loss of deformation serviceability of the system. The probability of system unserviceability, therefore, is

Pf,system = P(A a U . . . U A i U . . . U A m U B 1 k . ) . . . U B j U . . . O B n ) (9)

where the unserviceability events A i and Bj represent the occurrence of Aay °w and "-'hAan°w in the critical sections i and j , respectively, and the symbol U represents the union of the individual unserviceability events. Exact evaluation of eqn. (9) is usually impractical or impossible because of numerical difficulties. However, because the system failure probability (9) is based on the "weakest-link" concept, boundary techniques similar to those used for calculating the overall collapse probability of plastic structures could be used (see, for example, the computer program described in Frangopol and Nakib [4]).

SAFETY INDEX WITH RESPECT TO UNSERVICEABILITY OF RIGID FRAME

As an example, reliability analysis of a rigid frame with respect to flexural deflection and drift was performed. The steel frame shown in Fig. 1 has fixed geometry ( l = 10 m, h = 4 m) and uniform flexural rigidity (EI = 20525.6 kN m2), and is acted on by two random service loads with a given serviceability reference period.

162

H

•¢'//; ~.,r/ A

I..

&"i

z//2

a = 8 m , , j~ b = 2 m . i

E "'" i ~=lOm

Fig. 1. Rigid portal frame under random loads.

/k h II.......11,[

The coefficients of variation of the loads are V(H)= 0.15 and V(P)= 0.20; the loads are assumed to be independent P(P, H) = 0. The two critical sections of the frame with respect to loss of serviceability due to excessive flexural deflection, A v, and drift, A h, are the sections 1 and 2, respectively. Both the (vertical) deflection A v and the (horizontal) drift A h have two components as follows:

av = + a'v ( lo )

(11)

where Aev = the component of the (vertical) deflection due to P, A" v = the component of the (vertical) deflection due to H, A/~ = the component of the (lateral) drift due to H, and A~ = the component of the (lateral) drift due to P.

75

[- 9 v ~

7

0 .50 6 0 9 0 I ~ 0 1 5 0

M E A N L A T E R . A _ I , L O A D H ( k N )

Fig. 2. Deflection and drift safety indices.

1 5

--P-80kN × -P-7OkN M

=60kN - - 9

M ~ 6 <

0 .~0 6 0 9 0 1 2 0 1 5 0

~ E A N L A T E R A L L O A D H (kN) Fig. 3. Global deflection-drift safety indices.

163

The influence of different combinat ions of the mean values of service loads (P , H ) on the safety indices flv (see eqn. 1) and flh (see eqn. 2) is shown in Fig. 2. The allowable deflection and drift limits are AavU°W = 1/300 = 3.33 cm and A~hn°W = hZ200 = 2 cm. In Fig. 2 the results are given for four different values of the mean vertical load P (50, 60, 70 and 80 kN) while the mean lateral load H is increased f rom 0 to 150 kN. The decrease in the drift safety index, flh, with increasing H a n d / o r P is due to the fact that H and P have the same effect on A h (i.e., A~ > 0 and A~ > 0). On the other hand, the increase in the deflection safety index, fly, with increasing a n d / o r decreasin__g P is due to the fact that for this part icular posi t ion of the vertical load (i.e., a = 8 m) H and P have opposi te effects on A v " n e (1.e.,___A v < 0 and A v > 0). As expected, it appears that changes in the mean value of the vertical load, P, show considerable influence on the safety index associated with vertical deflection fly and almost no influence on the safety index associated with drift flh- This is because I a vl >> I a"vl in eqn. (10), and A~_<< A~ in eqn. (11).

The influence of different combinat ions of the mean values of loads (P , H ) on the global safety index ~system (see eqn. 5) of the frame shown in Fig. 1 is i l lustrated in Fig. 3. The global safety index in Fig. 3 corresponds to independent deflection and drift safety margins, p(g~, gh) = 0. It is importa__nt to observe that flsymm increases as P decreases. However, ~system b e c o m e s

less sensitive to P than to changes in H when the unreliability with respect to drift becomes dominan t (i.e., flh <flv); it is clear (see Fig. 2) that this refers to large values of H (e.g., H > 110 kN).

ISOSAFETY LOADING FUNCTIONS IN MEAN LOAD SPACE

Each combinat ion of mean values of service loads acting on a structure represents a point in the mean service load space. An incremental mean loading technique is used to obtain d i f fe ren t isosafety loading functions in the space of mean values of service loads. Each isosafety loading function with respect to def lec t ion-dr i f t corresponds to a specified value of the global reliability index ~system (see eqn. 5). Therefore, each point on such a funct ion represents a combinat ion of

164

~., 2 6 0 Z

I1~ 2 1 2

4 0

1 5 6

.< o

~ 104

M >

Z < 52

h/200, ~/100

[Pj, system = 10 2 ]

h/200, £/200 \

I I I I

O 5 2 1 0 4 1 5 6 2 1 2

M E A N

260

L A T E R A L LOAD H (kN)

Fig. 4. Mean load space: Influence of allowable deflection and drift limits.

the mean values of the service loads leading to a specified value of flsystem (i.e., ~system = fls~stem)" The set of mean service load combinations leading to a prescribed value of fls~stem forms an isosafety function against deflection-drift. Using eqn. (8), it is clear that

Pf,*system -~" (I) ( -- ~system ) (12)

Therefore, we may view each isosafety loading function against deflection-drift, as the set of mean service load combinations leading to a prescribed value of the global probability of deflection-drift unserviceabliltiy, Pf,*system" In general, an isosafety loading function cuts the mean load space into t w o regions: the safe loading region (where flsystem > flsystern or Pf,system < ef,*system) and the u n s a f e loading region (where flsystem < flsystem or Pf,system > * Pf,system). The set of mean

1 5 0

o

< o

>

Z

1 2 0

9 0

6O p , -1 I , s ~ J t e m =

i;; o , , ' ° - ' , l ; i

0 5 0 6 0 9 0 1 2 0 1 5 0

MEAN LATERAL LOAD H (kN)

Fig. 5. Mean load space: Influence of global reliability level.

165

120

I!~ 9 6

~3

0 '-] 7 2

O

~ 4a

2; 2 4 <

P(P. H) =i

I I I I

0 3 0 6 0 9 0 1 2 0

M E A N L A T E R A L L O A D H

150

( k N )

Fig. 6. Mean load space: Influence of load correlation.

service load combinations that satisfy the equation ~system ~-~s;stem (or Pf,system = Pf,*system) are points on the isosafety loading function against unserviceability.

Examples of isosafety loading functions in the space of mean loads (P, H) of the portal frame shown in Fig. 1 are illustrated in Figs. 4-7. Figures 4 and 5 show the sensitivity of the isosafety loading functions against deformation unserviceability to changes in the allowable deflection-drift limits and in the global probability of system failure, respectively; the isosafety functions in these figures correspond to independent (i.e., p(g,,, g h ) = 0) normal distributed deflection and drift safety margins. Figures 6 and 7 show the effects of load correlation and methods for global reliability evaluation, respectively, on the isosafety loading functions corresponding to a specified value of the probability of deformation unserviceability of the frame Pf, system* = 10 -2 (or ~system* ----

~,, 120 Z

gL

I:1 < 0 M

,2

lJ

N >

Z

M

9 6 LOWER BOUND

48 I I / 0 i i ~ I

0 ,30 6 0 9 0 1 2 0

M E A N L A T E R A L L O A D

Fig. 7. Mean load space: Influence of method for

1 5 0

( k N ) global reliability evaluation.

166

3O

2 4 Z o [.-, r.j

1 8

12

U

P 6

>

Z

[ P],,~,t~ = 10-2 ]

2 . 0 5 . 8 9 . 2 ! 2 . 8 ~ 6 . 4 2 0 . 0

M E A N L A T E R A L D E F L E C T I O N ( m m )

Fig. 8. Isosafety deflection-drift function in mean deformation space.

2.32). It is important to observe that the isosafety loading function is sensitive to the correlation between loads (Fig. 6) and almost insensitive to the correlation between deflection and drift safety margins; the lower and upper bounds in Fig. 7 correspond to perfectly correlated (i.e., #(gv, gh) = 1) and independent (i.e., p(gv, gh) = 0) safety margins, respectively.

ISOSAFETY DEFORMATION FUNCTIONS IN MEAN DEFORMATION SPACE

Each combination of mean values of vertical (deflection) and lateral (drift) deformations of a structure represents a point in the mean deformation space. An incremental mean deformation

E v

Z 0 7~ (..) ;d

,< U

P ~d >

Z <

3 0

2 4

18

12

= 1 0 - ~

10-2m~

- i

o

i i 2 . 0 5 6 9 2 1 2 . 8 1 6 . 4 2 0 . 0

M E A N L A T E R A L D E F L E C T I O N ( m m )

Fig. 9. Mean deformation space: Influence of global reliability level.

30

24 Z o

{J

~d

12

U

• 6 N

Z

P(P, H)=1

P(P, H) =0

I PT,,y,t,m = 10-2 I

1 r T l :~ 2 . 0 5 . 8 9 . 2 1 2 . 8 1 6 . 4 2 0 . 0

MEAN LATERAL DEFLECTION (mm)

Fig. 10. Mean deformation space: Influence of load correlation.

167

technique, similar to the incremental technique used for the mean service load space, is used to obtain different isosafety deformation (deflection-drift) functions in the mean deformation space. Each point on such a function represents a combination of the mean values of deforma- tion leading to specified values of flsystem (i.e., flsystem = /~s;stem) and Pf,system (i.e., Pf,system ~ ef~Lvstem)"

Examples of isosafety deformation functions in the space of mean deformations (A v, A h) of the portal frame acted on by the normal distributed loads P and H shown in Fig. 1 are illustrated in Figs. 8 -11 . The allowable deflection and drift limits are •/200 = 3.33 cm and h / 2 0 0 = 2 cm, respectively. The reliability level against unserviceability is the same in Figs. 8, 10 and 11: Pf,*system = 10 -2. The influence of the prescribed reliability level on the isosafety deformation functions is shown in Fig. 9. Figures 10 and 11 present the sensitivity of the

Z o r~ L)

,-.l a,

Q

,-I

L}

:>

Z

30

24

18

12

UPPER BOUND

0

, i i 2 . 0 5 . 6 9 . 2 1 2 8

LOWER BOUND

614 M E A N L A T E R A L DEFLECTION

20 .0

( r a m ) Fig. 11. Mean deformation space: Influence of method for global reliability evaluation.

168

isosafety deformation functions to changes in correlation between loads and methods for global reliability evaluation, respectively.

CONCLUSION

Isosafety loading and deformation functions of structural systems against unserviceability with respect to both excessive deflection and drift are briefly examined in this paper. From the results it can be concluded that the proposed isosafety functions allow deflection and drift conditions to be considered simultaneously in the reliability analysis process. This makes it possible to approach the serviceability limit states related to excessive static deformations in a much more integrated manner than is possible with the conventional approach in which deformation unserviceability conditions are checked independently.

ACKNOWLEDGMENTS

This research was partially supported by the Graduate School of the University of Colorado and by the National Science Foundation under Grants MSM-8618108 and ECE-8609894. This support is gratefully acknowledged.

REFERENCES

1 Ad Hoc Committee on Serviceability Research, Committee on Research of The Structural Division, Structural serviceability: A critical appraisal and research needs, J. Struct. Eng. ASCE, 112 (1986) 2646-2664.

2 C.J. Turkstra and S.G. Reid, Structural design for serviceability, In: M. Shinozuka and J.T.P. Yao (Eds.), Probabilistic Methods in Structural Engineering, ASCE, New York, NY, 1981, pp. 81-101.

3 T.V. Galambos and B. Ellingwood, Serviceability limit states: Deflection, J. Struct. Eng. ASCE, 112 (1986) 67-84. 4 D.M. Frangopol and R. Nakib, Isosafety loading functions in system reliability analysis, Comput. Struct., 24(3)

(1986) 425-436.