A new look at reliability of reinforced concrete columns

L . ~ ¸ ~ = ;

ELSEVIER

Structural Safety Vol. 18, No. 2/3, pp. 123-150, 1996 Copyright © 1996 Elsevier Science Ltd.

Printed in The Netherlands. All rights reserved PII: S0167-4730(96)00015,X 0167-4730/96 $15.00 + .00

A new look at reliability of reinforced concrete columns

Dan M. Frangopol a.., Yutaka Ide b, Enrico Spacone a, Ichiro Iwaki a Department of Civil, Environmental, and Architectural Engineering, University of Colorado, Boulder, CO 80309-0428,

USA b Building Design Department, Takenaka Corporation, Chuo-ku, Tokyo 104, Japan

c Department of Civil Engineering, Tohoku University, Aoba, Aramaki, Sendai 960, Japan

Abstract

This paper presents an investigation on reliability of reinforced concrete columns. For short columns, the fiber model is used for generating failure surfaces and strain and stress histories of both steel and concrete fibers under proportional and sequential loads. Two failure criteria, one based on the collection of peak-load points, the other based on prescribed maximum concrete strains are presented. For slender columns, failure surfaces are generated using a method proposed in 1991 by Ba~ant et al. ( ACI Structural Journal, 1991, 88, 391-401). The reliability estimation of short and slender columns under random loads is formulated by Monte Carlo simulation in the load space. In this space, isoreliability contours for both deterministic and nondeterministic columns under different load paths and load correlations are plotted. It is demonstrated that these factors may have substantial effects on the reliability of reinforced concrete columns. Therefore, the results of this study can be used to support the consideration of load path and load correlation in the development of improved evaluation and design specifications for reinforced concrete columns. Copyright © 1996 Elsevier Science Ltd.

Keywords: Codes; Interaction diagram; Constitutive relations; Failure; Failure surfaces; Fiber model; Load combination; Load correlation; Load-path-dependent response; Monte Carlo method; Reinforced concrete; Short column; Slender column; Structural reliability

1. Introduct ion

Whi le several s tudies ([ 1 -6] , a m o n g others) have been devo ted to the evalua t ion o f the safety o f

re inforced concre te (RC) structures, little in format ion exists to quant i fy the effects o f both load

corre la t ion and load path on R C co lumn reliabili ty. Mos t o f the studies reported so far have been

pe r fo rmed for a f ixed load eccentr ici ty. Consequen t ly , load path and load correla t ion effects were

ignored. These assumpt ions , wh ich are also usual ly impl ied in present des ign and evaluat ion

specif icat ions , m a y lead to underdes ign or ove rdes ign o f R C co lumns . For this reason, reliabili ty

fo rmula t ions for R C co lumns based on load-pa th - independen t response (i.e. structural response is

independen t o f load his tory) and perfect load correla t ion, m a y result in overes t imat ing or underest i-

mat ing the probabi l i ty o f failure o f the co lumn. Therefore , dec is ions based on these formula t ions are

* Corresponding author.

123

1 2 4 D.M. Frangopol et al.

usually inappropriate for the rational use of limited funds available at all levels of design, evaluation, inspection, repair and strengthening of RC columns.

The present study, which explains and further develops some ideas and results presented in [7], is an attempt to address and quantify the effects of various factors including load-path-dependency and load correlation on the reliability of both short and slender RC columns. Awareness of these effects may contribute to improved design and evaluation specifications for RC columns. This paper presents methods for generating the failure surfaces of short and slender columns using the fiber model and evaluates the reliability of both deterministic and nondeterministic RC columns under random loads. The reliability estimation is formulated by Monte Carlo simulation in the load space. In this space, isoreliability contours associated with different load paths and load correlations are plotted. The investigation shows that these factors, among others, may have substantial effects on the reliability of RC columns. Therefore, the results of this study may be used to support the incorporation of load path and load correlation effects in the development of improved evaluation and design specifications for RC columns.

2. Failure surfaces for short columns

In this section, a deterministic analysis is used to obtain failure surfaces for short RC columns. The columns are discretized into several longitudinal layers with history-dependent properties. Three different load paths are examined as follows: simultaneously applied axial load and bending moment, sequentially applied axial load and bending moment with the axial load applied first, and sequentially applied axial load and bending moment with the bending moment applied first. The analysis considers rectangular RC columns subjected to uniaxial bending.

Although various closed form solutions are available when the cross section and its material properties are simple, computer programs constructed on these assumptions are inevitably limited. A convenient formulation used for the nonlinear static and dynamic analysis of concrete frame structures is based on the fiber model [8-10]. In this formulation, the cross section is divided into steel and concrete layers (or fibers) with perfect bond, and the section force-deformation relation is established by integration of the uniaxial stress-strain behavior of the layers, assuming that plane sections remain plane and normal to the longitudinal axis of the column. The fiber model used in this study has been implemented in the general purpose finite element program FEAP, described in [11], and all the analyses presented herein were obtained using FEAP.

Fig. 1 shows the section discretization, the strain distribution and the stress distribution, where A s = area of tension reinforcement; A' s --- area of compression reinforcement; C c = compression force in concrete; C s = compression force in steel; M = bending moment; P = axial load; T = tension force in steel; b and d = cross-sectional dimensions of the column; c n = distance from section neutral axis to concrete surface in compression; es = strain in tension reinforcement; e~ = strain in compression reinforcement; and e u --- maximum concrete compression strain in the section.

The nonlinear behavior of the fiber RC section derives entirely from the nonlinear behavior of its fibers. Since the effect of bond-slip is neglected, only two material models are required: one for the concrete and one for the reinforcing steel. The constitutive models used in the present study offer a good compromise between simplicity and accuracy. The concrete fiber was assumed to have no tensile resistance and after its peak in compression it has a decaying branch. The concrete

D . M . F r a n g o p o l et al. 125

(a)

d

i

(b) EL__ 7-

A'~ 8's / /

/

As /~ Es

(c)

>T

Fig. 1. Reinforced concrete section: (a) section discretization; (b) strain distribution; and (c) stress distribution.

stress-strain relationship was adapted from a general smeared crack model proposed in [12]. After an initial linear elastic line, the stress-strain law becomes parabolic. After reaching the peak compressive strength, the concrete response decays exponentially to zero stress for very large compression strains. In the present study, the concrete initial modulus is E c = 24,100 MPa, and the peak compressive stress is f" = -24 .1 MPa at a strain ~c = -0 .0025. The steel constitutive model is elastic-perfectly plastic. In the present study, the initial linear modulus is E S = 200,000 MPa and the yield stress is fy = 344.8 MPa.

The RC cross section used in the following analyses is shown in Fig. 2. The cross section is

30.48 c m

20 concrete fibers 2 steel fibers

i 5 ' , ; ' , 1ol

Ill 6.35 c m

50.8 c m

]i e

2O

L. -E 6.35 c m

% Fig. 2. Reinforced concrete reference section [ 16]: geometry and section discretization.

126

Table 1 Proportional loading: eccentricity, load path,

D.M. Frangopol et al.

and failure point (see Figs. 3 and 4)

Eccentricity (cm) Load path Failure point

0.00 Path 1-1 A 3.81 Path 1-2 B 8.89 Path 1-3 C

16.51 Path 1-4 D 21.60 Path 1-5 E 25.40 Path 1-6 F 35.56 Path 1-7 G 50.80 Path 1-8 H

Path 1-9 I

discretized into 20 concrete layers and four steel fibers with the stress-strain constitutive laws previously discussed. The column is reinforced with four No. 9 bars of area 6.45 cm 2 each, one in each comer as shown in Fig. 2 [16]. For the case of proportional loading, the load P is applied to the section with an eccentricity e (see Fig. 2).

2.1. Proportional loading

Assuming proportional loading and the nine eccentricities in Table 1, Fig. 3 shows the associated points on the failure surface (or interaction diagram, i.e. ultimate axial force P versus ultimate moment Pe). For the same case of proportional loading considered in Fig. 3 (see also Fig. 4a), Fig. 4b shows the axial force versus strain in the extreme compression concrete fiber and Fig. 4c depicts the axial force versus both load eccentricity e and neutral axis position c n (see Fig. lb). When the eccentricity is zero (point A on the failure surface), the neutral axis position is at infinity. Conversely,

P

1-1 e=0.0 PATH 1-2

¢=-3.81 cm

/ / ~ ~ ~ l'~0cm

, / / ~ ~ ~ . ~ j ~ 8 ~'76 cm

50 0om I

0 1-9 M

Fig. 3. Proportional loading: nine points associated with different eccentricities on the failure surface (see also Table I).

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12 8 D.M. Frangopol et al.

0.002

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0 . ~ I ~ ~ ~ ~

. . . . . . . . . . . . . . . . . . . . . . ]0

15 / ) , . /

,.0.elm / /

AXIAL FORCE (kN)

Fig. 5. Proportional loading: strain histories in selected concrete and steel fibers along path 1-2 (e = 3.81 cm).

when the eccentricity is at infinity (point I), the neutral axis is near the extreme compression concrete fiber (fiber 20).

The strain distributions in seven different fibers are shown in Fig. 4d. Since the model assumes that plane sections always remain plane, the strains in all interior fibers are bounded by those associated with the two extreme concrete fibers. Stresses in concrete and steel fibers are shown in Fig. 4e and Fig. 4f, respectively. Even though the strains in the interior fibers are bounded by the strains associated with the two extreme fibers, the stresses in the extreme concrete fibers are not always the bounds of the stresses in all inside fibers. For example, as shown in Fig. 4e, the stress in fiber 15 dominates a large portion of the failure surface.

In Fig. 4f, fiber 21 is yielding in tension between the points F and I on the failure surface. Clearly, the failure point F is very close to the balanced failure point. On the other hand, fiber 22 is yielding in compression between points A and G. This is because the yield strain Cy in the compression steel fiber is reached before the strain in concrete associated with the peak stress (¢u = -0.0025) is attained.

Strain histories of selected fibers along the path 1-2 (eccentricity = 3.81 cm) are shown in Fig. 5. It should be emphasized that in the post-peak (strain-softening) portion of the axial force-strain relation, the section curvature changes at a faster rate than in the pre-peak regime.

Stress histories of selected concrete and steel fibers along the same path 1-2 are shown in Fig. 6a and Fig. 6b, respectively. As the peak axial force is reached, the stress in fiber 20 has already passed its peak (point A in Fig. 6a) and the stress in fiber 15 just attains its peak (point B). The stress in fiber 10 reaches its peak (point C) after the peak of axial force is attained. Fiber 5 does not reach the concrete peak stress and the stress in fiber 1 becomes zero after the peak of axial force is attained. As shown in Fig. 6b, the stress in steel fiber 22 reaches its peak before the peak of axial force is attained, and keeps its peak value both before and after the axial force reaches its peak.

D.M. Frangopol et al. 1 2 9

10 0 t TF.NSION

~ I \ ' ~ ,

-30

,a, i

AXIAL FORCE (kN)

g ,

AXIAL FORCE (kN)

Fig. 6. Proportional loading: stress histories along path 1-2 (e ~ 3.81 cm) for (a) selected concrete fibers; and (b) steel fibers.

2.2. Sequential loading

Fig. 7 shows the failure surface of a short RC column and three different load paths: proportional loading (path 1), sequential loading with axial force followed by bending moment (path 2), and sequential loading with bending moment followed by axial force (path 3). For the proportional loading case, three lines representing three different load eccentricities (e t < e 2 < e 3) are also shown in Fig. 7. The proportional load case was discussed in the previous section. Consider now the two


Pa

~ COMPRESSION VA~/ ~ . FA~t~

~ T E N S l O N FAILURE

. I ~ Mo

Fig. 7. Three load paths.

cases of sequential loading represented by paths 2 and 3 applied to the short RC column section of Fig. 2.

Fig. 8 shows the failure surface generated by applying proportional loading (dashed line) and contains several failure points associated with sequential paths 2 (i.e. paths 2-1 to 2-7) and 3 (i.e. path 3-1). The coordinates of these failure points are given in Table 2. For sequential paths, the last applied load (i.e. bending moment for path 2 and axial force for path 3) never goes beyond the failure surface associated with the proportional loading case. Load paths 1 and 2 have identical failure surfaces. It may be noted that load path 3 has the failure surface enclosing the smallest safety domain. This is

Z

. 1

i " - . LOADING PATH 1 (PROPORTIONAL LOADING)

~PATH 2-1 "-

PATH 2-2 i " /t

i

PATH 3-1 " , ,

PATH 2-3 ",

PATH 24

) PATH 2-5

PATH 2.6 , • j~"

PATH 2 - 7

PATH 3-1 A j , n i

1 0 0 2 0 0 3 0 0

MOMENT (~lm)

/

4 0 0

Fig. 8. Failure surface for proportional load path, and failure points associated with sequential load paths 2 and 3 in Fig. 7,


Table 2 Sequential loading: load path and failure point (see Fig. 8)

131

Failure point

Load path Axial force (kN) Moment (kNm)

Path 2-1 3731.0 146.8 Path 2-2 2946.8 260.7 Path 2-3 2122.6 349.5 Path 2-4 1765.0 380.7 Path 2-5 1522.1 384.6 Path 2-6 982.6 348.3 Path 2-7 569.8 285.3 Path 3-1 3732.6 141.6

clearly evidenced by the fact that when the moment is applied first it cannot exceed the moment associated with point A in Fig. 8.

Fig. 9a shows strain histories of selected fibers along the sequential load path 2-1 in Fig. 8. During the first stage, when only the axial force is applied, the strains in all the fibers are identical. In the second stage, during the application of the moment, the compression strains in fibers 1, 5 and 21 are reduced and those in fibers 10, 15, 20 and 22 are increased. Stress histories in concrete and steel fibers along the same sequential load path (i.e. 2-1) are shown in Fig. 9b and c, respectively. In Fig. 9b, the stress in fiber 15 dominates at failure. Fiber 22 has to reach yielding at failure. However, as shown in Fig. 9c, the stress in fiber 22 is close but does not reach the yield stress. This is due to the lack of stability of the global solution algorithm.

Fig. 10a shows strain histories of selected fibers associated with sequential load path 3-1 in Fig. 8. The corresponding stress histories in the concrete and steel fibers are shown in Fig. 10b and c, respectively. During the f'trst stage, under bending moment only, fiber 20 is in compression and other concrete fibers (i.e. 1, 5, 10 and 15) are not stressed. In the second stage, during the application of the axial force, the stresses in all fibers reach the compressive domain at different values of the axial force.

2.3. S impl i f ied p r o g r a m

In the previous sections the failure surface was defined on the basis of a theoretically consistent stability-based definition of failure. This surface was obtained as the collection of the peak-load stability points and corresponded to the collection of all ultimate compressive forces and ultimate bending moments for a short RC column section. In reliability analysis there is often the need for using a simpler method which does not sacrifice the accuracy too much. Such a method was used by Pytte [13], Ide [14] and Iwaki [15], among others.

After fixing the ultimate strain in the extreme compression concrete fiber, eu, the neutral axis position, c,, is assumed. By using e , and c n, the curvature of the section, K, and the strain in each fiber, e;, can be calculated as follows:

= ( 1 )

1 3 2 D.M. Frangopol et aL

-o.~1

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[--

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o

.Io

t~

[- t~

.20

~ "i ] (a) ! : 21

i . \ / ! FIBER !

l \\\ I ~o.~,~,~ 2° \\~ .......................... ~,-~

STRERGTH STRAIN :

373111 ' 146.11

MOM'F24rr OraNm) AXIAL FORCE (kN) AXIAL FORCE (kN)

)> /

i : /

__c°_" _c~__~__~Y2_ _ ! ........ 3731.11 146.1

MOMENT (kNm)

]~,! ~ . i(c> " x ALL STEEL i

Ells . I

~ - ~

• 4u ~er~n.0 l~.S \

i " , m ~,o 3 , ~ o ~ ,6 AXIAL FORCE (kN) MOMENT (l~m)

Fig. 9. Sequential loading with axial force followed by moment (load path 2-I in Fig. 8): (a) strain histories of selected concrete and steel fibers, (b) stress histories of selected concrete fibers; and (c) stress histories of steel fibers.

~i= K(c° -y;) , (2)

where y; = distance from the centroid of the i-th fiber to a conveniently chosen reference axis (i.e. concrete surface at which the strain is ~u)- From the fiber strain and the selected concrete and steel


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$ ~ ' TENSION

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MOMENT (kNm) AXIAL FORCE (kN)

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-200

-300

~ (c) I J

\ TENSION

$41 1N 141.15 0 1000 2000 30QO 3?'52.6 MOMENT (kNm) AXIAL FORCE (kN)

Fig. 10. Sequential loading with moment followed by axial force (load path 3-1 in Fig. 8): (a) strain histories of selected concrete and steel fibers; (b) stress histories of selected concrete fibers; and (c) stress histories of steel fibers.

stress-strain relations the fiber stresses, m o m e n t are obtained as fo l lows:

o- i, are computed. The resultant sect ion axial force and

P = E A i o ' i , ( 3 ) i

13 4 O,M, Frtt~pi~! et at.

OFEAP

4000 0 ~- ~ ~.-'~.~.K ~ -- - - 0.0035 • i ~ ~ ! "<~.

~N 3°°°'° i

.< )

O0 . . . . . . . . . . . . . . . . . . . . . . . . . ~ ' ' .

Fig. 11. Proportional loading: Corapafi~ofi ~sf failure surfaces obtained with the simplified method and with the fiber model of program FEAP.

M= Y'.Aio'i(yi- Yc), (4) i

where A; = area of ith fiber and Yc = distance from the section centroid to the reference axis. For the case of proportional loading, a comparison between the results obtained using the

simplified method and the fiber section implemented in program FEAP are shown in Fig, 11. This figure shows: (a) three failure surfaces obtained with the simplified method (associated with three ultimate strains, ~u, at the extreme concrete compression fiber, -0.0025, -0.01)30 and -0.0035); and (b) discrete results obtained with the fiber element in program FEAP. For the simplified method, 50 concrete and two steel fibers were used, while 20 concrete and two steel fibers were used for the analyses with FEAP. While the results of the simplified method are not identical to the results obtained with FEAP, most of the results produced by the simplified method are acceptable.

3. Failure surfaces for slender columns

Many RC columns in present-day practice fall in the short column category [16], but with the increasing availability of high strength concrete [6], more slender columns will be designed in the future. The strength of a slender column is lower than that of the corresponding short column, since collapse is initiated by buckling. Accordingly, the slenderness effect must be considered in the reliability evaluation of slender columns in addition to the material strengths, the dimensions and shape of the cross section, and the type of loading.

It has long been known that for a slender column under an eccentric axial compressive force, the moment at midheight increases at a faster rate than the load due to the so-called P - 6 effect. For the


NEUTRAL AXIS POSITION, Cn

MIDDLE SECTION

P

wl L

M=P(e+w l

P

Fig. 12. Cross section and elevation of a slender column.

column hinged at both ends shown in Fig. 12, the moment which is developed at midheight, M, is divided into two parts: the moment M 0 = Pe due to the eccentricity of the axial load, and the secondary moment Pw~ due to the force P and the lateral deflection of the column w.

In order to include this secondary moment effect into the analysis, an approximate expression for the lateral deflection at midheight is necessary. For simplicity, it is customary to assume that the deflected curve is sinusoidal. The method used in this paper is similar to that presented by Ba~ant et al. [17]. The lateral deflection of the column, w(x), is assumed to be described by the sinusoidal curve

w( x) = w 1 sin( Tr x / L ) , (5)

where L = effective length of the column; w~ = lateral deflection at midheight of the column; and x = coordinate measured from the bottom of the column. By differentiating Eq. (5), the curvature at the. midheight of the column, K, is computed as

K = ,n-2wJL 2, (6)

and the moment applied to the midheight of the column is

M = P ( e + w l ) . (7)

An iterative procedure is followed to find the moment at midheight. Assuming an initial value for w 1, a trial curvature is computed. By assuming c n (neutral axis position), the strain distribution and resultant stresses are calculated. The stresses are integrated to find the section resisting moment and axial force. The corresponding w~ is then found from Eq. (7). If this does not coincide with the trial w~, the analysis is repeated and the new w I becomes the trial wl.

For the undeflected initial state of a perfect column (e = 0), the tangent modulus load, Pt, is used. This is obtained by solving the next two equations [17]:

77" 2

Pt = --'if[ E[Ic + E' ls], (8)

136 D.M. Frangopol et al.

50 concrete fibers 2 steel fibers

6 0 4 0

H ~ , , , , l l I , , , l ~ ,

55.88cm ;111:1;; 11:11:;

!i!iiiii! ::::::: ) [ l l l r l r l , l l l l l l

I;:O:: ::l:::i ' ! ! ' ~ ! ! ! ! ! ! ! ! ! !

7.62 cm

30 20 10

I ,E , , , , H ~ l , ~ l , I ' , l , , J ~

iiikiii!',:iiiiiiil ' iiiiiii I J i ~1 I J I I I I l l I I r I I ~ i l ~ I

iilliiii!!!ii!!ili ii!ii!i ~ , 1 , , H H * l , , I , l , ~ , l ~ , l l l

iiiLiiii!!iii!ii!i iiiii!! i ~ , q l , * , l i l i I H H H , * l l ,

:::I::::H::::::: ' , :::::::

iiiliiii!!iiiiiiil :::::: '

7.62 cm

e

% Fig. 13. Fiber model of a square section [17] with 50 concrete fibers and two (51 and 52) steel fibers.

Pt= OcA~ + trsAs, (9)

where E c' = tangent modulus of concrete; E l = tangent modulus o f steel; I c = concrete moment of inertia with respect to the section centroid; I s = steel moment of inertia with respect to the section centroid; o- c = concrete stress; ~ = steel stress; Ac = concrete area; and A s = steel area. As the compression strain increases, Pt obtained from Eq. (8) becomes smaller and, conversely, Pt obtained from Eq. (9) gets larger. Using a simple iteration process, Pt can be easily computed.

The RC column cross section considered herein is identical to that considered by Ba~ant et al. [17]. It has a square cross section, b = 55.88 cm, a symmetric reinforcement with a reinforcing steel ratio of p = 0.03, a concrete cover such that the bar centroid is at 7.62 cm from the surface, and is hinged at both ends. Three slenderness ratios h are considered: A = 23.6, 70.8 and 94.5. These slenderness ratios are associated with effective column lengths o f about 3.8, 11.4 and 15.1 m, respectively.

Fig. 13 shows the fiber discretization for the section used in this example. The steel yield stress is 413.7 MPa and the modulus o f elasticity is 2 × l0 s MPa. The concrete compressive strength is 34.5 MPa and the tensile strength is neglected. In order to maintain continuity o f the tangent modulus, the original constitutive laws were modified as follows: (a) the steel bilinear law was modified into an exponential function, and (b) the concrete law was modified into a quadratic function with peak compressive stress of 34.5 MPa at a strain ec = - 0 . 0 0 2 5 .


z

0.~ (a)

S ~ \ 40 30

-0.002 I ~ i 20

\ ¢

. . . . . . . . . . . . . . .

-0.OO3 O 1OO0O 15OO0 AXIAL FORCE (kN)

(b)

-10 MAXIMUM I

0 5O00 I0000 ISO00 AXIAL FORCE (kN)

r~ r~

o ~ (c)

NN ~ F ~ E R $2

m E R S I \ ',

. . . . . . . . . . . . . . . . . . . . . . . i

i AXIAL FOROg (N~I)

Fig. 14. RC column with slendemess ratio A ffi 23.6 and load eccentricity e = 2.54 cm: (a) strain histories of selected concrete and steel fibers; (b) stress histories of selected concrete fibers; and (c) stress histories of steel fibers.

Fig. 14a shows the strain his tor ies in selected fibers fo r the case where the s lenderness ratio is

A = 23.6 and the eccent r ic i ty is e = 2 .54 cm. Strains in all f ibers are b o u n d e d by those o f ex t reme

fibers 1 and 50 (see f iber n u m b e r i n g in Fig. 13). Figs. 14b and c show the co r re spond ing stress

histories o f concre te and steel fibers, respect ively .

13 8 D.M. Frangopol et al.

-0.003

.0.002

-0.001 L

0.000

(-0.0025)

23.6

15000

10000

50OO

23.6

50 100 150 X

Fig. 15. Strain at failure and tangent modulus load, Pt, versus slenderness ratio, 3..

15000.0 ,

Z

m

0

el0.0 cm

2 .54m

U 0.0 60,0 80.0

SLENDERNESS RATIO, 100.0

Fig. 16. Maximum axial force versus slenderness ratio for different eccentricities.


SECTION FAILURE LINE

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' , I l

~') 12.70 X\\

II// //.,o'/',

0 0 500 1 ~ 1 5 ~ 0.0 L0 Z0 3.0 0 0 ~ 1 ~ 15 MOMENT (kNm) DEFLECTION (era) MOMENT (kNm) DEFLECTION (era)

SECTION FAILURE LINE

IM~ ~ / / , r [ ~

} , , - , " , / / 1

/ ,,~ ',I ~ I ~ , [

i / 2'70 i ]

0 ~ 1000 1500 0 10 20 MOMENT (kNm) DEFLECTION (cm)

Fig. 17. Axial force versus moment and axial force versus midheight deflection for various eccentricities: ( a ) ) t = 23.6; (b) h = 70.8; and (c) h = 94.5.


Fig. 15 shows both the strain at failure and the column strength versus the slenderness ratio h when e = 0 (i.e. P = Pt)- The larger the slenderness ratio, the greater the difference between the strain at failure and the concrete peak strain (8 c = -0.0025). The effects of both slenderness and eccentricity on the maximum axial force are clearly shown in Fig. 16.

For the case h = 23.6, Fig. 17a shows load-versus-moment and load-versus-deflection at midheight. The section failure surface is represented by a dashed line. The strength for the case of pure compression, Pt, is shown by the horizontal solid line. Figs. 17b and c show the same diagrams for slenderness ratios h = 70.8 and 94.5, respectively.

From the above analyses on slender columns we can conclude that an increase in the column slenderness yields: (a) a decrease in the failure load; and (b) a substantial increase in the deflection at failure.

4. Refiability analysis

In this study, the Monte Carlo-based reliability analysis software MCREL [18] is used to obtain the probability of failure of RC columns under random loads. Ide [14] developed and implemented the link between MCREL and the RC fiber model used in FEAP, and also between MCREL and the simplified method presented in Section 2.3. This latter link is shown in Fig. 18. The purpose is to obtain isoreliability contours /3 = /3 * in the load space.

The notation in Fig. 18 is as follows: M, P = moment and axial force generated by Monte Carlo simulation; eload = M/P = load eccentricity; M u, Pu = moment and axial force at failure obtained in RC column analysis; e = MJP~ = eccentricity; Cu = ultimate compression strain in concrete; c n = neutral axis position; v = median load ratio = Mm/M ~ = Pm/Pu where Pm and M m a r e median values of axial force and moment used in random load generation; N = number of failure points

CHANGE v ~ / M o n t e Carlo Simulation

RC ~ ~yas

Fig. 18. Flow-chart for reliability analysis of deterministic columns under random loads.

D,M. Frangopol et al. 1 41

obtained in Monte Carlo simulation; COV = coefficient of variation; COV < COV * criterion of convergence for Monte Carlo simulation, where COV * = prescribed upperbound of the coefficient of variation of the estimated probability of failure Pf, est; ~ = rib-l[1-Pf.est] =est imated reliability index; and /3 * = prescribed reliability index. For a detailed description of the reliability analysis as shown in Fig. 18, and of the link between the RC column structural analysis program based on the fiber model [9,10] and the Monte Carlo simulation program MCREL [18], the interested readers are referred to [ 14].

5. Isoreliability contours for deterministic columns

The short column example of Fig. 2 and the slender column of Fig. 13 are studied herein from a reliability viewpoint. The RC column reliability is investigated for both concurrently and sequentially applied random loads. The loads are assumed to follow a log-normal distribution and their coefficients of variation are C O V ( P ) = COV(M) = 0.15. The column properties are assumed to be deterministic.

5.1. Effect of concrete strength

In order to quantify the effect of the concrete strength on the reliability of the RC column of Fig. 2 under axial force and bending moment, two values of the concrete strength are considered herein: 24.1 and 27.6 MPa. They both occur at the same strain s = -0 .0025 . The axial force P and the moment M are assumed to be independent log-normal variates acting simultaneously.

Fig. 19 shows three isoreliability contours in the median load space for each of the two specified concrete strengths. Naturally, the safety regions associated with higher reliability levels /3 are smaller

so0¢

24.1 M P a

0 ! ' - -

Z M~ b I "

MEDIAN MOMENT, M m (kNm) 300

Fig. 19. lsoreliability contours of the cross section of Fig. 2 for two concrete strengths.

1 4 2 D . M . F r a n g o p o l et al.

than those associated with lower values of /3. In the tensile failure region, the isoreliability contours associated with the same /3 but different concrete strengths are almost identical. Since in this region yielding of the steel initiates failure, the compression strength of concrete has little impact on the

Z

2ooe

Z

1000

(a)

.. .... ,/ 13=1.5

I " / / ] / / "

: (- /"" . f J"

0 0 100 200 300

MEDIAN MOMENT, M~ (kNm)

(b)

Z E3000

L~

.¢

Z .¢ 1000

4000

0 0 100 200 300

i

///~ "

MEDIAN MOMENT, M s (kNm)

Fig. 20. Ef fec t o f load correlation on the safety domain: (a) p ffi 0 .0 and 1.0; (b) p ffi 0 .0 and - 1.0.

to K

f~

r~

O

<

ME

DIA

N

LO

AD

RA

TIO

, v,

for

[3=

1.5

~g

II b~

b~

h-" II k

Jl

ME

DIA

N

LO

AD

RA

TIO

, v,

for

[$=

3.5

t~

° !.

ME

DIA

N

LO

AD

RA

TIO

, v,

for

13=

2~5

eh X


isoreliability contour. Conversely, in the compression failure region, where failure occurs through overstraining of the concrete, the concrete compression strength has a substantial influence on the safety domain.

5.2. Effect of load correlation

In this section the concrete strength is assumed to be 24.1 MPa, and the effect of the variation of the correlation coefficient between P and M on the isoreliability curves is analyzed.

Fig. 20a shows three isoreliability contours (/3 = 1.5, 2.5 and 3.5) for p = 0.0 and 1.0. In the case of perfect positive correlation between loads (i.e. p = 1.0), load combinations generated by Monte Carlo simulation are distributed along a line where the values of P and M tend to be both large or small simultaneously. Conversely, in the case of statistical independence between loads (i.e. p = 0.0), the load combinations generated are distributed around the median values of P and M. Therefore, when p = 0.0, in the tension failure range, random load combination points are more likely to be outside the safety domain than those in the compression failure range.

Fig. 20(b) shows the isoreliability contours for p = - 1.0 and 0.0. In the compression failure range, the contours associated with perfect negative correlation (i.e. p = - 1.0) are enclosing a larger safety region than that associated with the independent load case ( p = 0.0). Conversely, in the tension failure range, the isoreliability contours associated with the load independent case enclose a larger safety region. This is due to the fact that load combinations associated with p = - 1.0 are distributed along a line where the values of P tend to be large when the values of M are small, and vice versa. Therefore, in the tension failure range, random load combination points associated with p = - 1.0 are more likely to be out of the safety domain than in the case of a correlation larger than - 1.0.

v= Pm/Po = M m / M u , used for generating log-normal Fig. 21 shows the median load ratio,

4000 I !

il / [~=1.5

I I I

PATH 1, 2 t ""

1000 PATH3 ,"

I ', ) / i /

f £ " .

0 0 100 200 300

MEDIAN MOMENT, M,~ (id~m)

Fig. 22. Effect of load path on safety domain.


distributed loads, versus the ultimate load ratio M u / P u, for the cases associated with fl = 3.5, 2.5 and 1.5, respectively. In the compression failure range (e.g. M u / P u < 20 cm), the smaller the load correlation coefficient p, the larger the median load ratio v. On the other hand, in the tension failure range (e.g. M u / P u > 20 cm), the smaller the load correlation coefficient p, the smaller the median load ratio v. Theoretically, in the case of perfect positive load correlation (i.e. p = 1.0), the median load ratio has to be constant over the entire failure range. However, as shown in Fig. 21a-c, in the extreme compression failure range (e.g. M u / P u < 10 cm) this condition is not fully satisfied. This is due to the fact that in this range, the assumed iteration tolerance was not small enough.

5.3. Effect o f load path

In this section, a reliability analysis incorporating load path dependency is performed. Isoreliability contours are plotted for concurrent loading (path 1 *) and two sequential loadings with axial load applied first (path 2) and bending moment applied first (path 3). The RC column section is that of Fig. 2 with P and M log-normally distributed and COV(P) = COV(M) = 0.15. The reliability analysis formulations for deterministic RC columns under concurrent or sequential loadings are presented in Pytte [ 13]. These formulations are applied herein by assuming that the axial load and bending moment are statistically independent (i.e. p = 0.0). The notation "path 1" " for concurrent-independent loading is used herein in order to differentiate the notation from that of "path 1" (i.e. concurrent-proportional loading) in Fig. 7.

Fig. 22 shows the isoreliability contours for both concurrent loading (path 1 *) and sequential loading (paths 2 and 3). The contours associated with paths 1 * and 2 are identical. However, path 3 encloses a smaller safety domain than those associated with paths 1 * and 2. This is due to the fact that

Z 8oo0'

O

4o00

Z .¢

N 2ooo N

~=l.S

0 ~

MEDIAN MOMENT, M. (kNm)

Fig. 23. Effect of reliability level on the safety domain of a slender column.

146

Table 3 Probabilistic characteristics of random variables


Random variable Nominal value Mean value Standard deviation Type of distribution

Concrete peak strength 3" (MPa) 24.1 24.8 3.97 Steel yield stress fy (MPa) 344.8 384.1 27.2 Steel modulus of elasticity E s (MPa) 200,000 200,000 6,600 Dimensions of cross section b (cm), d (cm) Nominal +0.152 0.635 Concrete cover (cm) Nominal + 0.762 0.422 Model error 1.00 1.00 0.07

Axial force P (kN) F 0.15 F

Bending moment M (kNm) M 0.15 M

Log-normal Log-normal Log-normal Normal Normal Normal

Log-normal

Log-normal

the probability of failure is dominated in the first stage by the bending moment which may fail the column before any axial force can be applied.

5.4. Slender column

The slender column of Fig. 12 with the cross section defined in Fig. 13 is used hereafter in a reliability-based context. The slenderness ratio A = 70.8 is considered. The axial force and the moment are assumed log-normal variates with the coefficients of variation C O V ( P ) = C O V ( M ) = 0.15. Using the fiber model and Monte Carlo simulation, the isoreliability contours for/3 = 1.5 and 2.5 in the median load space are obtained. They are both shown in Fig. 23. Due to the slenderness effect, it may be noted that the isoreliability contours enclose a much smaller safety domain than that associated with the corresponding short column (see Fig. 17b). Also, as expected, a higher reliability level decreases the safety domain.

CALCULATE P~ Mu

i (Re COLUMN ANALYSIS)

\ MONTE CARLO SIMULATION

Fig. 24. Flow-chart for reliability analysis of nondeterministic columns under random loads.

D.M. Frangopol et al. 147

4000

Z

U

0 2 ~

<

1000

t~

(a)

,,~.. ~

"'~,"'-. 1~1.5

~ , , ;'J/ ~" . ; / "

i t / / ~ ' ' " I r 108 150 200 1,50 300

MEDIAN MOMENT, M~ (kNm) 0

0 350

4~ r (b)

/_2 " ' - . ~ - . ~- ~

Z

0 0 100 200 300

MEDIAN MOMENT, Mm,0CNm) Fig. 25. Effect of load correlation on isoreliability contours for deterministic (CASE 1) and nonde~rministic (CASE 2) columns under random loads: (a) p ~ 1.0 and (b) p = 0.0.


6. Isoreliability contours for nondeterminist ic columns

In the previous section the applied loads are considered random variables and the RC column dimensions and material properties are assumed fixed (i.e. deterministic column under random loading). In reality, however, both dimensions and material properties are random. Additionally, uncertainty associated with (imperfect) modeling is also present. Accordingly, a realistic reliability analysis should include all these uncertainties associated with column strength and modeling. Such analysis is presented in this section.

Table 3 summarizes the probabilistic characteristics of the random variables considered herein. Some of these characteristics were reported by Mirza and MacGregor [19,20], Mirza and Skrabek [21], Diniz [6], Ide [14] and Iwaki [15], among others.

To compute the probability of failure of a short RC column based on the probabilistic characteristics given in Table 3, Monte Carlo simulation is linked to a column analysis program as shown in Fig. 24. The only difference with the procedure outlined in Fig. 18 is that the column strength is calculated using generated values of the random variables shown in Table 3 and that a model error coefficient (also shown in Table 3) is introduced. The method can also include correlation between random variables.

The RC short column of Fig. 2 is considered herein using the random variables in Table 3. Three reliability levels /3 are considered (/3 = 1.5, 2.5 and 3.5). Figs. 25a and b compare isoreliability contours for two cases: CASE 1 corresponds to a deterministic column under random loading and

4O0O

z

g~

0 20OO

Z t-~ 1O00

i

0 0

~=3.5

A I~1.5

~=o.o ~ ~ - - ' "

50 IQ0 150 300 350 MEDIAN MOMENT, M. (kNm)

Fig. 26. Effects of reliability level and load correlation on isoreliability contours associated with a nondeterministic column (CASE 2) under random loads.

D.M. Frangopol et al. 149

CASE 2 corresponds to a nondeterministic column under random loading (all random variables are shown in Table 3). The axial force and bending moment are assumed to be perfectly positive correlated (Fig. 25a) and independent (Fig. 25b). The isoreliability contours for CASE 2 enclose smaller safety regions when compared with those associated with CASE 1. The reason is that both randomness in materials and dimensions and uncertainty associated with modeling increase the probability of failure. It should also be emphasized that in the compression failure region the assumption of perfect positive load correlation leads to a smaller safety domain. This is clearly shown in Fig. 26 which compares, for CASE 2, isoreliability contours associated with load independence ( p = 0.0) and perfect positive correlation between axial force and moment ( p = 1.0). The opposite is true in the tension failure region.

7. Conclusions

Based on the results presented and discussed, the following conclusions may be drawn. 1. For both proportional and sequential load applications, the column failure surfaces may be

obtained using a simple though powerful fiber model implemented in a general purpose finite element program. In this case the failure surface is defined rigorously using a stability-based criterion for failure. In general, the codes of practice are not adopting the rigorous stability-based criterion for failure. They usually assume proportional loading, impose a constant value for the strain corresponding to the peak stress (e.g. c u = -0 .003) , select the neutral axis, and repeat the calculations for successive choices of the neutral axis to establish the curve defining the failure surface. A simplified method based on the above assumptions was presented and compared with the more rigorous fiber model. For slender columns, a method similar to that by Ba~ant et al. [17] was used for generating failure surfaces. Unloading effects were neglected since their impact on the failure surface is generally very small.

2. The reliability of RC columns can be computed by linking a structural analysis program based on the fiber model to a Monte Carlo simulation software. Material and geometric nonlinearities and load path and load correlation effects can be included in the reliability analysis.

3. The reliability of RC columns may be load-path-dependent. For example, the safety domain associated with a sequential loading case where the bending moment is applied first, followed by application of the axial force, is smaller than that associated with a proportional loading case or that associated with a sequential loading case where the axial force is followed by the moment.

4. The reliability of RC columns depends on the load correlation. The numerical results presented in this paper clearly show that for a RC column under random loads whose correlation is less than 1.0, the assumption of perfect positive load correlation is conservative (i.e. leads to overdesign by underestimating the actual reliability) in most of the compression failure region. Conversely, this assumption is unconservative (i.e. leads to underdesign by overestimating the actual reliability) in most of the tension failure region. Consequently, the assumption of proportional loading used in most building codes for RC columns may not be conservative in the tension failure region.

5. In order to develop a new, more uniform and consistent reliability-based analysis and design format for RC columns incorporating load path and load correlation effects work should continue for the development of efficient reliability-based techniques for load-path-dependent structural response [7,22], especially in the case of slender columns.


Acknowledgements

Part ia l suppor t f r o m the Nat iona l Sc ience Founda t ion under Gran t s

9506435 is g ra te fu l ly a c k n o w l e d g e d .

M S M - 9 0 1 3 0 1 7 and C M S -

References

[1] Tich~, M. and Vorlicek, M., Safety of eccentrically loaded reinforced concrete columns. Journal of the Structural Division, ASCE, 1962, 88(ST5), 1-10.

[2] Israel, M., Ellingwood, B. and Corotis, R. B., Reliability-based code formulation for reinforced concrete buildings. Journal of Structural Engineering, ASCE, 1987, 113(10), 2235-2252.

[3] Mirza, S. A., Probability-based strength criteria for reinforced concrete slender columns. ACI Structural Journal, 1987, 84(6), 459-466.

[4] Mirza, S. A. and MacGregor, J. G., Slenderness and strength reliability of reinforced concrete columns. ACI Structural Journal, 1989, 86, 428-438.

[5] Ruiz, S. E. and Aguillar, J. C., Reliability of short and slender reinforced concrete columns. Journal of Structural Engineering, ASCE, 1994, 120(6), 1850-1865.

[6] Diniz, S. M. C., Reliability evaluation of high-strength concrete columns. Ph.D. thesis, Department of Civil, Environmental, and Architectural Engineering, University of Colorado, Boulder, CO, 1994.

[7] Frangopol, D. M., Ide, Y., Pytte, J. E. and Iwaki, I., Load-path-dependent reliability of structural systems. In Proc. of the Asian-Pacific Symp. on Structural Reliability and its Applications, APSSRA'95, Tokyo, Japan, 1996 (in press).

[8] Kaba, S. and Mahin, S. A., Interactive computer analysis methods for predicting the inelastic cyclic behavior of structural sections. EERC Report 83-18, Earthquake Engineering Research Center, University of California, Berkeley, CA, 1983.

[9] Spacone, E., Filippou, F. C. and Taucer, F. F., Fiber beam-column model for nonlinear analysis of R / C frames I: Formulation. Earthquake Engineering and Structural Dynamics, 1996, 25(7), 711-725.

[10] Spacone, E., Filippou, F. C. and Taucer, F. F., Fiber beam-column model for nonlinear analysis of R / C frames II: Applications. Earthquake Engineering and Structural Dynamics, 1996, 25(7), 727-742.

[11] Zienkiewicz, O. C. and Taylor, R. L., The Finite Element Method, Vols. 1 and 2. McGraw-Hill, London, 1989, 1991. [12] Lotfi, H. R. and Shing, P. B., An appraisal of smeared crack models for masonry shear wall analysis. Computers and

Structures, 1991, 51, 413-425. [13] Pytte, J. E., Load-path and load definition effects on structdral system safety. MSc. thesis, Department of Civil,

Environmental and Architectural Engineering, University of Colorado, Boulder, CO, 1994. [14] Ide, Y., Reliability of reinforced concrete columns under random loading. MSc. thesis, Department of Civil,

Environmental, and Architectural Engineering, University of Colorado, Boulder, CO, 1995. [15] Iwaki, I., Reliability of reinforced concrete bridge piers under seismic loads. MSc. thesis, Department of Civil,

Environmental and Architectural Engineering, University of Colorado, Boulder, CO, 1996. [16] Nilson, A. H. and Winter, G., Design of Concrete Structures. McGraw-Hill, New York, 1991. [17] Ba~ant, Z. P., Cedolin, L. and Taggara, M. R., New method of analysis of slender columns. ACI Structural Journal,

1991, 88, 391-401. [18] Lin, K. Y. and Frangopol, D. M., MCREL--Monte-Carlo RELiability analysis, Version 1.1. Department of Civil,

Environmental and Architectural Engineering, University of Colorado, Boulder, CO, 1995. [19] Mirza, S. A. and MacGregor, J. G., Variations in dimensions of reinforced concrete members. Journal of the Structural

Division, ASCE, 1979, 105(ST4), 751-766. [20] Mirza, S. A. and MacGregor, J. G., Variability of mechanical properties of reinforcing bars. Journal of the Structural

Division, ASCE, 1979, 105(ST5), 921-937. [21] Mirza, S. A. and Skrabek, B. W., Reliability of short composite beam--column strength interaction. Journal of

Structural Engineering, ASCE, 1991, 117, 2320-2339. [22] Wang, W., Corotis, R. B. and Ramirez, M. R., Limit states of load path-dependent structures in basic variable space.

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A new look at reliability of reinforced concrete columns