Slender-Square FRP-confined RC columns

Construction and Building Materials 151 (2017) 370–382

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Structural reliability of biaxial loaded Short/Slender-Square FRP-confined RC columns Osama Ali Civil Engineering Department at Faculty of Engineering, Aswan University, Egypt

h i g h l i g h t s
Reliability of FRP confined RC columns was studied using first order reliability method.
The mechanical model includes a three dimensional finite difference model based geometrical and material nonlinearity.
Sensitivity factors of the reliability index with respect to random variables were evaluated.
FRP-confining gives valuable enhancement of columns reliability with small eccentricities.

a r t i c l e

i n f o

Article history: Received 25 February 2017 Received in revised form 4 June 2017 Accepted 5 June 2017

Keywords: Slender/short RC columns FRP concrete confinement Finite difference method Structural reliability

a b s t r a c t This paper presents a structural reliability analysis of RC square columns confined with FRP laminates. The structural analyses were carried out using the Finite Difference Method FDM considering material and geometrical nonlinearity. Both FRP laminates and steel stirrups confinement effects were considered. Different values and types of eccentricities (single & biaxial) were taken into account. An experimental dataset containing 32 confined/unconfined RC columns were collected from literature and used to examine the accuracy of the FDM model. Reliability analyses were performed using First Order Reliability Method FORM. The reliability analyses were investigated including geometrical, material and loading variables. Loads and resistance factors were considered according to ACI provisions. Results of the reliability analysis have confirmed that confining the RC section with FRP material can improve the reliability by 0.1 ? 0.5 depending on value and type (uniaxial and/or biaxial) of the eccentricity. The study includes parametric analysis reflecting the relation between the thickness of FRP materials and both reliability index & % increases in live load at different values of slenderness ratio and eccentricity. Results indicate that % increases in live load is linearly related to FRP thickness while the relation between reliability index and FRP thickness depends on eccentricity and slenderness values. An additional safety factor of a value equal to 0.65 proposed to be considered in design rules in of case of high eccentricity (Normal force outside core of the section). As results of FORM algorithm, the sensitivity factors were provided which demonstrate that live load and model error are variable of the greatest importance. Furthermore, within material and geometrical variables considered, concrete compressive strength is the variable of the first importance followed by column dimension and steel strength, while all other variables can be treated as deterministic variables. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction Externally bonded (EB) Fiber Reinforced Polymers (FRP) composite materials (laminates or sheets) have been used for over two decades to strengthen reinforced concrete RC structures. This relatively new strengthening technique is widely accepted strengthening alternative around the world. The rise of FRP composites in strengthening applications of RC structures can be

E-mail address: [email protected] http://dx.doi.org/10.1016/j.conbuildmat.2017.06.032 0950-0618/Ó 2017 Elsevier Ltd. All rights reserved.

attributed to their high strength, resistance to corrosion, light weight, and ease of application [1]. It is well proven that EB-FRP strengthening technique can improve flexural and shear strength of RC members [1–9]. Mukhrjee & Joshi [10] in an experimental work justify the significant enhancement of yield load, stiffness and energy dissipated capacity. Additionally, EB-FRP strengthening technique can be used to enhance the axial capacity of RC columns by means of wrapping the FRP laminates around the RC column [10]. Such RC column strengthened using the later EB-FRP

371

O. Ali / Construction and Building Materials 151 (2017) 370–382

strengthening technique is, also, known as FRP confined RC column. The growth of strengthening RC columns by means of FRP confining effect requires a more and more accurate analysis procedure such as nonlinear analysis theories. From other side, codes design procedures requires, just, fulfilling a certain number of mathematical expressions which does not match accurately the complicity of the strengthened structure. In order to overcome such this common shortcoming – i.e. considering the structural response independent of the load history – the present study proposes to develop a numerical method involving both material and geometrical nonlinearity in the analysis. A three dimensional Finite difference model FDM which was used in different previous studies [e.g. 25–28], were constructed and used to predict the structural response of the FRP confined RC columns. On the other hand, the variability inherited in material properties, member geometry and applied loads give rise to uncertainties in safety evolution. As a result, probabilistic analysis is used to guarantee and measure an acceptable safety margin. Structural reliability of RC columns subjected to axial load and bending moment had been studied in many previous studies. As an example, Mirza [11] performed a study in order to calculate the crosssection resistance safety factor of RC columns. The author consider loads as random variables while column strength was considered as one random global variable regardless the level of randomness inherited in each design variable such as column dimensions, concrete and steel strengthens . . .etc. Fiber sectional analysis was used to determine the ultimate axial load and curvature for section at the mid-height of the column. A second order parabola has been suggested to represent the shape function curvature line between the mid-height and the ends of the column. The author tends to use a simplified mathematical formula in order to calculate the lateral deflection at mid-height. The study proposed a resistance factor of varies linearly from 0.7 to 0.9 as axial load decreases from balanced stage load to zero. Frangopol et al. [12] studied the effect of various factors on the reliability of RC columns. Such these factors are concrete strength, load correlation (axial load and bending moment), load path dependency and slenderness. Reliability analysis was performed using Monte-Carlo simulation. Second order effect was included using an approximate expression for the lateral deflection at mid height via sinusoidal shape function between the deflections at the ends of the column and the section at the mid-height. Unlike the above mentioned study, a more accurate probabilistic representation of reliability problem was introduced as the authors considered many design variables - as random variables - such as material properties, column dimensions and loads. In a more recent study [13], Frangopol developed the previous study by including the long term effect - creep – of sustained loads. Both normal and high strength concrete were considered. Results indicated that ACI procedure for slender column design is more conservative for high strength concrete than for normal strength concrete. In addition, longitudinal steel reinforcement plays an important role in the columns reliability especially in case of large eccentricities. However, the above mentioned studies [11–13] lacking to take into account the concrete confinement effect due to lateral steel reinforcement. Furthermore, a more accurate structural analysis is required, in order to provide accurate structural responses, such as finite element or finite difference method. Mohamed et al. [14] proposed a partial safety factors format. Structural analysis was performed using finite element method considering both material and geometrical nonlinearity. Reliability analyses were performed using Monte-Carlo simulation. Due to complicity of defining an explicit failure surface, the authors build up the failure surface using quadratic response surface method. Material properties,

column dimensions and loads were considered as random variables. Results of the considered numerical applications lead to a better representation of strength safety factors than those proposed by the Euro code provisions. The present study aims to carry out a reliability analysis of square FRP confined RC columns under an eccentric or a biaxial eccentric load. A three dimensional numerical procedure using finite difference method FDM was established in order to provide accurate structural responses. Confinement effects due to FRP material and lateral steel reinforcement were considered. In addition, material and geometrical nonlinearities were included in the analysis. Reliability analysis was carried out using First Order Reliability method FORM. 2. Stress-strain relationships of materials The stress-strain relationship of unconfined concrete in compression is described by the nonlinear relationship represented in the following equation [15]: 0

f ¼ fc

" 2ec

eco




ec eco



2 #

ð1Þ

where fc is the concrete compressive strength, eco;=2fc/Ec, indicates the strain corresponding to the peak stress of the unconfined concrete (see Fig. 1) and Ec represents the initial tangent modulus of the concrete; Ec = 4700(fc)0.5 in MPa [16]. The stress strain for confined concrete is divided into two ascending parts as shown in Fig. 1; the first is similar to that of unconfined concrete starting from zero to a stress equal to fc while the second part is linear until the ultimate strength of confined concrete stress fcc. The ultimate strength of confined concrete fcc and its corresponding strain ecc are given, respectively, by [17]: 0

0




0

f cc ¼ f c þ ðf cco  f c Þ

1 1 þ e=t

ecc ¼ ecu þ ðeccu  ecu Þ






1 1 þ e=t

ð2Þ  ð3Þ

where e is the eccentricity value and t is the cross-section dimension in the same direction of eccentricity. The terms fcco and eccu are as follow;

f cco ¼

0 fc

eccu ¼ eco

f 1 þ 2 0l fc

!

ð4Þ

f 1:75 þ 10 0l fc

! ð5Þ

where fl is the lateral confining pressure and is expressed as:

f l ¼ f l;FRP þ f l;st

ð6Þ

f fcc confined

fc unconfined

0.2f c

Ec

ε co

ε cu

Fig. 1. Concrete stress-strain curves.

εcc ε

372

O. Ali / Construction and Building Materials 151 (2017) 370–382

where fl,FRP and fl,st are the lateral pressure due to FRP laminates and steel stirrups confinement effects respectively. FRP lateral confining pressure fl,FRP for rectangular section is expresses as [17]:

f l;FRP ¼ ks

2f FRP tFRP 0:5ðb þ hÞ

ð7Þ

where fFRP and tFRP are the rupture strength and thickness of FRP confinement material respectively. b and h are width and height of RC section. ks is shape factor that is equal to the ratio of the effective confined area to the gross concrete area enclosed by FRP and is expresses as:

ks ¼

Ac  Aie Ac

ð8Þ

where Ac is gross concrete area enclosed by FRP and Aie is the unconfined concrete area as shown in Fig. 2. Maalej et al. [18] derived an analytical formula for Aie as:

8 2 ðwx þ w2y Þ=3 if wx 6 2h > > > > > 2 < X R w =2 2 0 x ð1Þiþ1 ð2h  yi Þdx Aie ¼ > > i¼1 > > > Rx w2 : þ 3y  2 0 1 ðy2  y1 Þdx if wx > 2h

ð9Þ

where y1 = x2/wx+(2h-wx)/4, y2 = (wx-2h)/4-x2/wx, wx = b-2rc, wy = h-2rc. rc is the corner radius. While the lateral confining pressure due to steel stirrups fl,st was calculated according to the models proposed by Mander et al. [19] as the average summation of the effective lateral confining pressures in the x and y directions, noted respectively as fl,st,x and fl,st,y, and was taken equal to,

f l;st ¼ ðf l;st;x þ f l;st;y Þ=2

ð10Þ

where,

Ast;x shc

ð11Þ

unconfined concrete

ax

ke ¼

ð12Þ

    nst;x a2y þnst;y a2x 0 0 1 1  2bs c 1  2hs c 6bc hc

2.1. Structural modeling of FRP-confined RC columns It is well known that, numerical method provides a robust tool to predict the full behavior of RC structures. Such one of these methods is the Finite Difference Method FDM which was used in a number of previous studies [e.g. 20–23]. In the present study, FDM was used to perform a full three dimensional nonlinear analysis. Both material and geometrical nonlinearity were considered. The FDM is, basically, to be derived under the following assumptions [21];

confined concrete area

unconfined concrete area o

FRP composites

h wy

x

ay bc

wx b

section A-A stirrups confined RC section

section A-A FRP confined RC section

unconfined concrete confined concrete core

A

A

ð13Þ

1  qcc

where qcc is the longitudinal steel ratio to the area of core enclosed by stirrups. nst,x and nst,y are numbers stirrup braches in the x and y directions respectively. s’ is clear vertical spacing between stirrups and taken equal to s’ = s-/v; as /v is the diameter of stirrups. ax and ay are stirrups braches spacing in the horizontal plane in the x and y directions respectively. Concrete is assumed to carry no tensile stresses in tension regions. The stress-strain relationship for steel is assumed to be elastic-perfectly plastic under both compression and tension strains. FRP laminates stress-strain relationship is assumed to be linear until failure point.

stirrups

hc

Ast;y sbc

where fyv is the stirrups yield strength. s is the stirrups spacing, bc and hc are the concrete core dimension enclosed by steel stirrups in x and y directions, respectively, as shown in Fig. 2, Ast,x and Ast, y are the total area of transverse bars running in the x and y directions respectively. ke is confinement effeteness coefficient, ratio of confined area to total area enclosed by stirrups, and can be represented as [19]:

confined concrete core

s

f l;st;x ¼ ke f yv

f l;st;y ¼ ke f yv

Column elevation Fig. 2. Geometric representation of effectively confined concrete areas.

373

O. Ali / Construction and Building Materials 151 (2017) 370–382


Plane section remains plane before and after loading.
Deflections are small with respect to column dimensions.
Length and sectional area of RC column is to be divided into a number of segments and sub-areas, respectively, as shown in Fig. 3a.
The RC column has no initial deformations.
Secondary effects such as creep, shrinkage and shear deformation are to be neglected. Refering to the first assumption, the strain of a certain sub-area k, with respect to the centroid - reference point - coordinates x & y, can be expressed as:

ek ¼ eo þ /x y þ /y x

ð14Þ

where eo is the axial stain at the reference point of the section. /x and /y are the curvatures about x and y axes respectively. These sectional strains can be written in a vector form as D = {eo /x /y}T. The corresponding resultants of the internal forces Fc, can be expressed in term of the ek as;

8 9 n X > > > > > Ek ek Ak > > > > > > > 8 9 > > k¼1 > > > > > > n < Pc > = > > > : ; > > > k¼1 > > Myc > > > > n > > X > > > > > > E e A x > k k k k> : ;

ð15Þ

k¼1 n X

Ek Ak yk xk

k¼1

3 n X Ek Ak xk 7 78 9 k¼1 7> eo > n 7< = X Ek Ak xk yk 7 7 /x > 7> k¼1 7: /y ; 7 n X 5 Ek Ak x2k

k¼1

B12 ¼ B21 ¼

n X Ek Ak yk ; k¼1

B13 ¼ B31 ¼

n X Ek Ak xk ; k¼1

B22 ¼

n X Ek Ak y2k ; k¼1

B33 ¼

n X Ek Ak x2k ; k¼1 n X Ek Ak yk xk :

By noting dx and dy as the deflection of the column in x and y directions respectively; and ey are the eccentricities in x and y directions respectively (see Fig. 3). The second order effect can be included in the analysis by reformulating Eq. (16) as;

8 > < > :

Pc Pc ðey þ dy Þ Pc ðex þ dx Þ

9 > = > ;

2

B11

6 ¼ 4 B21 B13

B12 B22 B32

38 9 B13 > < eo > = 7 B23 5 /x > : > ; /y B33

DL2 dxðiþ1Þ 2dxðiÞ þdxði1Þ D L2

¼ ðux ÞðiÞ

ð18Þ

¼ ðuy ÞðiÞ

where DL is the length of the segment (i). Substitute Eq. (18) into Eq. (17) leads to,

Load (P)

ex ey

X ex xk

P

d x,i-1

d x,i+1

d y,i

yk X

O

ΔL

dx d y,i+1 dy Y P

ey

Ak

d x,i

d y,i-1

ð17Þ

The FDM is used to relate the column deflections (dx & dy) and curvatures (/x & /y) at the different segments across the longitudinal direction of the column; i.e. for any two adjacent segments the following expressions can be applied: dyðiþ1Þ 2dyðiÞ þdyði1Þ

ð16Þ

k¼1

L

k¼1

k¼1 n X Ek Ak y2k

n X E k Ak ;

k¼1

where Pc, Mxc and Myc are axial force, bending moment about x-axis and bending moment about y-axis, respectively, at an iteration c. Ak and Ek are the area and its corresponding tangent modulus of the sub-area k respectively. Eq. (14) and Eq. (15) can be rearranged in the following form; n X Ek Ak yk

B11 ¼

B23 ¼ B32 ¼

k¼1

2 X n Ek Ak 6 8 9 6 k¼1 6 P > > n < c = 6X 6 Ek Ak yk Mxc ¼ 6 > > : ; 6 k¼1 6 Myc 6X 4 n Ek Ak xk

Let,

ey

ex Fig. 3. Dividing of column and cross-sectional area.

X

374

8 > < > :

O. Ali / Construction and Building Materials 151 (2017) 370–382

Pc Pc ðey þ dyðiÞ Þ

9 > =

> ; Pc ðex þ dxðiÞ Þ 9 2 38 eo B11 B12 B13 > > < = 2 6 7 ¼ 4 B21 B22 B23 5 ðdyðiþ1Þ  2dyðiÞ þ dyði1Þ Þ=DL > > : ; B13 B32 B33 ðdxðiþ1Þ  2dxðiÞ þ dxði1Þ Þ=DL2

ð19Þ

Expanding of Eq. (19),

DL 2

8 > <

Pc

9 > =

Pc ðey þ dyðiÞ Þ > > : ; Pc ðex þ dxðiÞ Þ 2 2 DL B11ðiÞ B12ðiÞ 6 2 ¼6 4 DL B21ðiÞ B22ðiÞ DL2 B31ðiÞ B32ðiÞ 8 9 eoðiÞ > > > > > > > > > dyði1Þ > > > > > > > > > > > > < dxði1Þ > =  dyðiÞ > > > > > dxðiÞ > > > > > > > > > > > d > > yðiþ1Þ > > > > : ; dxðiþ1Þ

B13ðiÞ

3

B13ðiÞ

2B12ðiÞ

2B13ðiÞ

B12ðiÞ

B23ðiÞ

2B22ðiÞ

2B23ðiÞ

B22ðiÞ

7 B23ðiÞ 7 5

B33ðiÞ

2B32ðiÞ

2B33ðiÞ

B32ðiÞ

B33ðiÞ

Fig. 4. Computational flow of FORM based FDM procedure.

ð20Þ

Table 1 Details of columns used in FDM validation [17,23–27]. kr=

Model

L mm

b mm

h Mm

cov mm

rc mm

fc MPa

As

fy MPa

Asv

fyv MPa

tFRP mm

fFRP MPa

ex mm

ey mm

Pu,exp KN

Pu,FDM KN

C1a C2a C3a

1219 1219 1219

76 76 76

76 76 76

12.7 12.7 12.7

– – –

36.1 39.9 38.2

4u10 4u10 4u10

400 400 400

2.05@75 2.05@75 2.05@75

455 455 455

– 0.33 0.66

– 965 965

50.8 50.8 50.8

50.8 50.8 50.8

35.6 43.6 45

38.3 41.7 41.2

0.93 1.05 1.09

Z2b Z3b Z4b Z5b

1350 1350 1350 1350

250 250 250 250

250 250 250 250

25 25 25 25

20 20 20 20

24.5 24.5 24.5 24.5

4u16 4u16 4u16 4u16

403.7 403.7 403.7 403.7

6.5@250 6.5@250 6.5@250 6.5@250

na na na na

0.16 0.16 0.16 0.16

3399 3399 3399 3399

35 55 75 115

– – – –

1400 1175 1000 650

1404 1151 951.9 648.8

1 1.02 1.05 1

FW-e1c FW-e2c FW-e3c FW-e4c UW-e1c UW-e2c UW-e3c UW-e4c PW-e1c PW-e2c PW-e3c PW-e4c

1200 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200

125 125 125 125 125 125 125 125 125 125 125 125

125 125 125 125 125 125 125 125 125 125 125 125

15 15 15 15 15 15 15 15 15 15 15 15

10 10 10 10 10 10 10 10 10 10 10 10

28.5 28.5 28.5 28.5 28.5 28.5 28.5 28.5 28.5 28.5 28.5 28.5

4u10 4u10 4u10 4u10 4u10 4u10 4u10 4u10 4u10 4u10 4u10 4u10

550 550 550 550 550 550 550 550 550 550 550 550

6@125 6@125 6@125 6@125 6@125 6@125 6@125 6@125 6@125 6@125 6@125 6@125

na na na na na na na na na na na na

0.381 0.381 0.381 0.381 – – – – 0.381P 0.381P 0.381P 0.381P

894 894 894 894 – – – – 894 894 894 894

37.5 54 71 107.5 37.5 54 71 107.5 37.5 54 71 107.5

– – – – – – – – – – – –

295 205 157 95 215 165 145 92 275 200 150 93

294.5 223.9 171.8 98.9 241.2 179.5 141.6 93.87 286 216.2 167.6 98.3

1 0.92 0.91 0.96 0.89 0.92 1.02 0.98 0.96 0.93 0.89 0.95

K9d K10d K11d

2000 2000 2000

355 355 250

355 355 500

23 23 23

30 15 30

39.1 37.7 37.7

8u14 8u14 8u14

560 560 560

8@140 8@140 8@140

560 560 560

0.6 0.6 0.6

780 780 780

– – –

– – –

5810 5140 4990

5950 5780 5600

0.98 0.89 0.89

QR1Re QR1Ce QR2Ce QR3Ce

750 750 750 750

150 150 151 154

150 150 151 154

25 20 20 20

– – 20 38

24.6 24.6 24.6 24.6

8u6 8u6 8u6 8u6

587 587 587 587

3@100 3@100 3@100 3@100

500 500 500 500

– 0.35 0.35 0.35

– 3983 3983 3983

– – – –

– – – –

608 766 1194 1417

598.5 875.3 1128 1319

1.02 0.88 1.06 1.07

900 300 300 30 30 18.9 4u20 439 10@180 365 – – – – 2127 Cs0f CS2f 900 300 300 30 30 18.9 4u20 439 10@180 365 2  1.27 375 – – 2525 CS6f 900 300 300 30 30 18.9 4u20 439 10@180 365 2  1.27 375 – – 4025 SR0f 900 450 300 30 30 18.9 8u20 439 10@180 365 – – – – 3268 CR2f 900 450 300 30 30 18.9 8u20 439 10@180 365 6  1.27 375 – – 3598 CR6f 900 450 300 30 30 18.9 8u20 439 10@180 365 6  1.27 375 – – 4494 a, b, c, d, e and f reported in [23], [24], [17], [25], [26] and [27] respectively. P, means partially confined with FRP width 65 mm and spacing 105 mm. na, means not available and a default value of 365 MPa is used.

2180 2880 4151 3344 4019 5290

0.98 0.88 0.97 0.98 0.9 0.85 0.962 0.065

lk rk

Pu,exp Pu,FDM

L length of the column; As and fy area and yield strength of longitudinal steel respectively; cov concrete cover; Pu,exp ultimate axial load of tested column. Pu,FDM ultimate axial load based FDM. Asv and fyv area and yield strength of transverse steel respectively.

375

O. Ali / Construction and Building Materials 151 (2017) 370–382 Table 2 Probabilistic parameters of random variables. Variable

Distribution

Units

Nominal

Biasa(mean)

CoVb(stdc)

Source

f’c fy Es As b h s tFRP fFRP Dead Load PDL Live Load PLL Model error cm

Log-normal Log-normal Normal Normal Normal Normal Normal Log-normal Weibull Normal Extreme type I Normal

MPa MPa GPa mm2 mm mm mm mm MPa KN KN –

21.25 400 200 Asn bn hn 200 tFRP,n 687.5 PDL,n PLL,n 1

(25) (460) 1 (0.97Asn) (bn + 1.5) (hn + 1.5) 1 1 (900) (1.05 PDL,n) (PLL,n) 1

(0.16) (0.1) 0.033 (0.024) (6.35) (6.35) 0.1 0.05 0.15 0.1 0.25 0.025 + 0.018 h/ey  0.11

[13] [13] [13] [4,29] [13] [13] [30] [4,29] [4,29] [4,13,29] [13] [31]

a

Bias; mean value/nominal value. CoV; coefficient of variation. std; standard deviation.

b c

8 9 1 > > > > > > > > > > þ d e > > y yð1Þ > > > > > > 2 > ex þ dxð1Þ > Hð2Þ > > > > > > > > 6 Gð3Þ > > . > > 6 > > . > > 6 . > > > > 6 > > > > > > 1 > > 6 < = 6 6 DL2 P c ey þ dyðiÞ ¼ 6 > > 6 > > > ex þ dxðiÞ > > 6 > > > 6 > > > > 6 > > > > 6 > > ... > > 4 > > > > > > > > > > 1 > > > > > > > > > > ey þ dyðnÞ > > > > > > : ; ex þ dxðnÞ

12φ 16mm 2cm

φ 10

3cm

4φ 12mm

φ8

Column C2

Column C1

Fig. 5. Columns configuration considered in the analysis.

Rearranging of Eq. (20),

DL2

8 > > < > > :

Pc P c ðey þ dyðiÞ Þ

9 > > =

> > ; P c ðex þ dxðiÞ Þ 2 0 B12ðiÞ B13ðiÞ 6 6 ¼ 6 0 B22ðiÞ B23ðiÞ 4 0 B32ðiÞ B33ðiÞ 8 9 0 > > > > > > > > > > > > > > dyði1Þ > > > > > > > > > > d > > xði1Þ > > > > > > > > > e > oðiÞ > > > > < =  dyðiÞ > > > > > > dxðiÞ > > > > > > > > > > > > > > 0 > > > > > > > > > > > > > dyðiþ1Þ > > > > > > > : ; dxðiþ1Þ

DL2 B11ðiÞ

2B12ðiÞ

2B13ðiÞ

0 B12ðiÞ

DL2 B21ðiÞ

2B22ðiÞ

2B23ðiÞ

0 B22ðiÞ

DL2 B31ðiÞ

2B32ðiÞ

2B33ðiÞ

0 B32ðiÞ

B13ðiÞ

3

7 7 B23ðiÞ 7 5 B33ðiÞ

ð21Þ

3 3 DL2 B11ðiÞ 2B12ðiÞ 2B13ðiÞ 0 B12ðiÞ B13ðiÞ 6 7 Let, GðiÞ ¼ 4 0 B22ðiÞ B23ðiÞ 5; HðiÞ ¼ 4 DL2 B21ðiÞ 2B22ðiÞ 2B23ðiÞ 5 2 0 B32ðiÞ B33ðiÞ DL B31ðiÞ 2B32ðiÞ 2B33ðiÞ 2

2

Assuming that ends of the column are pinned, Therefore, deflections at the ends of the columns dx(i) = dy(i) = dx(n+1) = dy(n+1) = 0. Expanding Eq. (21) for all segments of the column results in:

Gð2Þ Hð3Þ .. .

3 Gð3Þ .. . GðiÞ

..

.

HðiÞ .. .

GðiÞ .. . Gðn1Þ

..

.

Hðn1Þ GðnÞ

7 7 7 7 7 7 7 7 7 7 7 7 7 Gðn1Þ 5 HðnÞ

8 9 eoð2Þ > > > > > > > > > dyð2Þ > > > > > > > > > > dxð2Þ > > > > > > > > > > > . > > > . > . > > > > > > > > > > > > eoðiÞ > > < =  dyðiÞ > > > > dxðiÞ > > > > > > > > > > > > .. > > > > > > . > > > > > > > > > > e > > oðnÞ > > > > > > > dyðnÞ > > > > > > > : ; dxðnÞ ð22Þ

where the left side of Eq. (22) is the applied load, while the right side involves the overall tangential stiffness matrix and the unknown deformation vector. It is worth mentioning that Eq. (22) is a highly nonlinear system of equations, it must be solved in an incremental-iterative procedure; a very small increment of Pc is applied, then, the corresponding deformation increment is to be found iteratively. The overall tangential stiffness matrix is to be updated at each iteration. Considering the symmetry of the structure and loading, thus system of nonlinear equations given by Eq. (22) can be reduced to one have. The model and analysis control options are fully detailed in [21]. The proposed finite difference procedure can be formulated as load control or displacement control. Herein, it is proposed to be used as a load control model. The above described procedure was programmed using Matlab/package. It is assumed that ultimate axial load is to be reached when; & Concrete reaches its maximum strain at the corner point of core enclosed by stirrups in case of RC section. & Concrete reaches its maximum strain at the corner point in case of FRP confined RC section.

376

O. Ali / Construction and Building Materials 151 (2017) 370–382 400

a) axial load vs. deflection

b) Ultimate axial loads

e x/bt=e y /ht=0.1

350

700 300

600 500

FRP confined RC conlumn RC column

200

Pu (tons)

Pu (tons)

250

400 300

150

200 100

100

e x/bt=e y /ht=0.5

50 0

0

20

40

60

80

100

0

120

0.1

0.2

0.3

e x/bt

140

0.4

0.5

deflection d or d (mm) x

y

0.2

0.1

0

0.3

0.5

0.4

e /bt y

Fig. 6. Ultimate axial loads of Column C2 (k = 75).

4.9

4.9

confined r=25mm

4.1

3.7

4.5

4.1

3.7

0.2

0.1

0.3

0.4

0

0.5

0.2

0.1

x

4.1

3.7

0.2

0.1

0.3

0.4

3.7

0.2

0.1

0.3

0.4

0.1

0.2

0.3

0.4

0.5

0.2

0.1

0.3

0.4

0.5

0

0.1

0.2

0.3

0.4

0.5

ex /b

ex /b

0.2

0.1

0.3

0.5

ex /b 4.9

=75, ey =e

4.5

4.1

3.7

x

4.5

4.1

3.7

3.3

3.3 0

0.4

3.3 0

Reliability index

Reliability index 0

0.5

3.7

x

3.3

3.3

0.4

4.1

=50, e y =e

3.7

0.5

y

4.9

4.1

0.4

4.5

ex /b

4.5

0.3

=75, e =e x /2

3.7

x

3.7

0.2

0.1

ex /b

4.1

=25, ey =e

4.1

0

0.5

4.5

0.5

4.9 x

4.5

0.4

4.9

ex /b

4.9

0.3

3.3 0

3.7

x

4.1

0.5

4.1

=50, e y =e /2

4.5

ex /b =0, ey =e

0.2

0.1

4.9

3.3

3.3

4.5

ex /b

Reliability index

Reliability index

4.5

=75, e =0 y

3.3 0

0.5

=25, e =e /2 y x

=0, e =e /2 y

Reliability index

0.4

4.9

4.9

Reliability index

0.3

ex /b

ex /b

0

3.7

Reliability index

0

4.1

3.3

3.3

3.3

4.5

Reliability index

4.5

Reliability index

confined r=0

4.9

=50, e =0 y

=25, e y =0

RC section

Reliability index

=0,ey =0

Reliability index

Reliability index

4.9

0

0.1

0.2

0.3

0.4

0.5

ex /b

0

0.1

0.2

0.3

ex /b

Fig. 7. Reliability index b vs. eccentricity ratio (ex/b and/or ey/h) of column C1.

&

The RC column produces very high values of lateral deflection corresponding to a very small load increment; i.e. the tangent of load-lateral deflection relationship becomes almost horizontal. Such step can be reached mathematically as the tangent stiffness matrix in Eq. (22) is transformed in singular matrix.

2.2. Validation of FDM with experimental data In order to validate results of the FDM described above, 32 experimental dataset were, collected from [17,23–27]. The dataset includes FRP-confined/non-confined RC columns. In addition, columns having no/one/two eccentricities were evolved. Table 1 presents all material properties, details of the RC columns and the ultimate axial load obtained experimentally Pu,exp & numerically Pu,FDM. The bias ratio kr, which measure the deviation between Pu,exp and Pu,FDM is also reported in the Table 1. Results indicate that the finite difference method can accurately predict the ultimate axial capacities of FRP-confined RC columns as it gives a mean of

the bias ratio, lk, value approaches the unity and a corresponding standard deviation of the bias ratio rk equals to 0.065. Similar statistics for bias ratio between Pu,exp and Pu,FDM of 44 experimental test of slender RC columns were achieved by Metwally et al. [20]. As, their tests data give lk = 1.002 and rk = 0.0735. Also, similar conclusions of using FDM were reported in [21–23]. Therefore one can conclude that the FDM is valid and efficient to analyze FRP-confined columns having a rectangular section under eccentric axial loading. 2.3. Basic reliability aspects The reinforced Concrete column performance can be expressed in the term of the ultimate limit state G as:

GðXÞ ¼ cm R  S ¼ cm Pu ðX 1 ; X 2 ; :::::X n Þ  ðPDL þ PLL Þ

ð23Þ

where, cm is the model error which reflects the uncertainty in numerical FDM evaluation of the ultimate resistance R of RC

377

O. Ali / Construction and Building Materials 151 (2017) 370–382 4.9

4.9

4.9

=0,e =0

3.7

4.5

4.5

4.1

3.7

4.5

Reliability index

4.1

=75, e =0 y

y

Reliability index

4.5

Reliability index

Reliability index

4.9

=50, e =0

=25, e =0 y

y

4.1

3.7

4.1

RC section

3.7

confined r=0 confined r=50mm

3.3

3.3 0.2

0.3

0.4

0.5

0

0.2

0.1

ex /b 4.9

0.5

0

3.7

0.2

0.1

0.3

0.4

4.5

4.1

3.7

0.2

0.1

ex /b =0, e =e y

0.3

0.4

4.9

Reliability index

4.1

3.7

4.1

3.7

0.2

0.3

ex /b

0.4

0.5

0.1

0.2

0.3

0.4

0.1

0.2

0.3

0.4

0.5

0.4

0.5

0.4

0.5

x

4.1

3.7

0

0.5

0.2

0.1

0.3

ex /b 4.9

=50, e =e y x

4.5

4.1

3.7

=75, e =e y

x

4.5

4.1

3.7

3.3

3.3 0

0.5

3.3

4.9

4.5

0.4

4.5

ex /b

3.3

3.3 0.1

3.7

0

=25, e =e y

0.3

=75, e =e /2 y

4.1

x

4.5

0

4.9

ex /b

x

0.2

0.1

ex /b

4.5

0.5

Reliability index

4.9

0

3.3 0

0.5

0.5

x

3.3

3.3

0.4

=50, e =e /2 y

Reliability index

4.1

0.3

4.9

=25, ey =ex /2

4.5

0

0.2

0.1

ex /b

4.9

=0, e =e /2 y

Reliability index

Reliability index

0.4

ex /b

x

Reliability index

0.3

Reliability index

0.1

3.3

Reliability index

0

3.3

0

0.1

0.2

0.3

0.4

0.5

ex /b

ex /b

0

0.1

0.2

0.3

ex /b

Fig. 8. Reliability index b vs. eccentricity ratio (ex/b and/or ey/h) of column C2.

column. Herein, the resistance is expressed as the ultimate eccentric load Pu that is function of a vector of random variables X; {x1, x2, x3 . . .xN}T. S is the applied eccentric axial loads which include axial dead load PDL and live load PLL. The measure of the reliability index must be performed in a standard Gaussian space [28]. Thus, the limit state G(X) must be rewritten in the standard Gaussian space H(U) instead of the physical space G(X). U represents a vector of standard normal variables {u1, u2, u3 . . .uN}T that corresponds to X vector. Relation between X and U is to be determined using isoprobabilistic transformation T such that; U = T(X) ? H(U)=G(T1(U)). First Oder Reliability Method FORM was used to estimate the reliability index (b) which can be determined from the solution of constrained optimization as:

0vffiffiffiffiffiffiffiffiffiffiffiffi1 u n uX b ¼ minimize @t u2i A; under the constraint HðUÞ 6 0

ð24Þ

to drive such these derivatives directly. Therefore, it proposed in the present study to carry out sectional analysis, after the final increment, for the section at mid-height - most critical section of the RC column based on the defamations eo, /x and /y obtained using FDM. The sectional analysis involves reformulation of the axial force Pu presented in Eq. (26). Thus, a full analytical derivatives of Pu or H(U) w.r.t. each random variable ui can easily be formulated.

Pu ¼

n X

Ek ek Ak

ð26Þ

k¼1

Table 2 presents the statistical moments and density distribution type of all random variables considered in the reliability analysis. 2.4. Cases of study

i¼1

where, U⁄ vector of basic random variables in the standard normal space at the most probable design point. The minimization problem given in Eq. (24) can be solved iteratively using Hasofer–Lind–Rack witz–Fiessler (HL–RF) algorithm with the gradient projection method [28]. At each iteration U, X, H, b . . .etc are to be updated. Within the framework of calculation procedure of FORM, the limit state gradient {rH(ui)} can be obtained. Accordingly, normalizing the gradient with respect to its norm ||rH(U)|| results in the sensitivity factor ai of the reliability index with respect to the variable i;

fag ¼

frHðU  Þg jjrHðU  Þjj

ð25Þ

Sensitivity factors have two major purposes. First, they show the contributions of the random variables to the reliability index. Second, the sign of the sensitivity factor reflects wither the variable under consideration can negatively or positively affects the safety. Fig. 4 presents the sequences and the general steps of computational flow of the FORM-based-FDM. Partial derivatives of the limit state function H(U) with respect to each random variable ui requires an explicit formula of the derivatives of Pu w.r.t. ui. According to finite difference Equations described above it very is difficult

Two RC columns were considered (250 mm  250 mm) and (500 mm  500 mm) with reinforcement details shown in Fig. 5. Each of these columns were studied at six eccentricities ratios ex/b or/and ey/h - 0.05, 0.1, 0.2, 0.3, 0.4 and 0.5 at x or/and ydirections respectively. Nominal dead and live load were estimated according to ACI 318M-11 provisions [32]. Ratio of dead load to the total load was taken equal to 0.5. The analysis goal is to assess the effect of load eccentricities, slenderness ratio k on the reliability of FRP-confined-RC columns. Four values of slenderness ratio were studied; k = 0, 25, 50 and 75. Fig. 6a and b shows samples of ultimate load results Pu obtained for columns C2. It could be reported that effect of FRP confinement depends on, to some extent, the eccentricity ratio; i.e.% increase of Pu at ex/b (=ey/h) equals to 0, 0.1 and 0.5 is 8.07, 6.92 and 4.07% respectively. This could conclude that % increase in Pu, is inversely proportional to eccentricity ratio when the same slenderness ratio is adopted. This can be explained by the fact that, FRP confinement effect could improve the axial capacity of concrete and consequently the overall axial capacity of the RC columns Pu which is the dominate objective in case of RC columns with small eccentricities. Unlike the case of high eccentricity – e.g. ex/b = ey/h = 0.5 – where FRP confinement of RC column is

O. Ali / Construction and Building Materials 151 (2017) 370–382

Senstivity factors

a- C1 & e x=0.05b~0.5b & e y =0

i

-0.4

0.5

Senstivity factors

i

378

-0.6

min value max value

-0.8 f

f

c

y

b

h

A

s

P

m

DL

P

LL

E t

f

s FRP FRP

b- C2 & e x=0.05b~0.5b & e y =0

0

-0.5 fc

s

fy

b

h

i

c- C1 & e x=0~0.5b & e y =e x/2

0.5

Senstivity factors

i

Senstivity factors

0.5

0

fc

fy

b

h

As

0

fc

PDL PLL Es t FRP f FRP s

m

fy

b

h

As

m

PDL PLL Es t FRP f FRP s

random variables f- C2 & e =0~0.5b & e =e

e- C1 & e x=0~0.5b & e y =e x

x

y

x

i

0.5

Senstivity factors

i

PDL PLL Es t FRP f FRP s

d- C2 & e x=0~0.5b & e y =e x/2

random variables

Senstivity factors

m

-0.5

-0.5

0.5

As

random variables

random variables

0

-0.5

0

-0.5

f

f

c

y

b

h

A

s

m

P

DL

P

LL

E t s

FRP

f

FRP

s

f

random variables

c

f

y

b

h

A

s

m

P

DL

P

LL

E t s

FRP

f

FRP

s

random variables

Fig. 9. Minimum and maximum values for sensitivity factor of random variables (column C1 & C2; k = 0 ? 75).

4.6

4.6

=0

FRP

3.4

FRP FRP

3

FRP FRP

0

0.4

0.8

1.2

1.6

2

2.4

=1 =0.875 =0.75

Reliability index

Reliability index

3.8

3.8 3.4 3

=0.625

x

2.6

0

0.4

y

0.8

1.2

1.6

2

2.4

2.8

2.6

30

=0

30

=50

15 10 5

20 15 10 5

0.8

1.2

1.6

t FRP (mm)

0.8

2

2.4

2.8

0

1.2

1.6

2

2.4

2.8

2

2.4

2.8

=75

25

% Increase in LL

20

0.4

0.4

t FRP (mm)

25

0

0

t FRP (mm)

% Increase in LL

% Increase in LL

3

y

x

25

0

3.4

e =0.05h & e =0

t FRP (mm) 30

3.8

e =0.3h & e =0

=0.5 2.8

=75

4.2

Reliability index

4.2

4.2

2.6

4.6

=50

20 15 10 5

0

0.4

0.8

1.2

1.6

t FRP (mm)

2

2.4

2.8

0

0

0.4

0.8

1.2

1.6

t FRP (mm)

Fig. 10. Profiles of reliability b and % increases in live load PLL vs. FRP thickness tFRP, (column C1, ex > 0 & ey = 0).

379

O. Ali / Construction and Building Materials 151 (2017) 370–382 4.6

4.6

=0

3.8 FRP

3.4

FRP FRP

3

FRP FRP

0

0.4

=0.875 =0.75

3.8 3.4 3

=0.625

0.8

1.2

1.6

2

2.4

2.6

2.8

3

y

x

0

0.4

y

0.8

1.2

1.6

2

2.4

2.8

2.6

t FRP (mm)

30

=0

20 15 10 5

0.4

0.8

1.2

1.6

2

2.4

20 15 10

0

2.8

0.4

0.8

1.2

1.6

2

2.4

2.8

2

2.4

2.8

2

2.4

2.8

2

2.4

2.8

t FRP (mm) =75

25 20 15 10 5

5 0

0

30

=50

25

% Increase in LL

% Increase in LL

x

e =0.05h & e =ex/2

25

0

3.4

e =0.3h & e =ex/2

=0.5

t FRP (mm)

30

3.8

% Increase in LL

2.6

=1

=75

4.2

Reliability index

4.2

Reliability index

Reliability index

4.2

4.6

=50

0

0.4

0.8

1.2

t FRP (mm)

1.6

2

2.4

0

2.8

0

0.4

0.8

1.2

1.6

t FRP (mm)

t FRP (mm)

Fig. 11. Profiles of reliability b and % increases in live load PLL vs. FRP thickness tFRP, (column C1, ex > 0 & ey = ex/2). 4.6

4.6

4.6

3.4

FRP FRP

3

FRP FRP FRP

0

0.4

=1 =0.875

Reliability index

Reliability index

Reliability index

3.8

2.6

4.2

4.2

4.2

3.8

3.4

=0.75

3

=0.625

x x

1.2

1.6

2

2.4

2.8

2.6

0

0.4

y

0.8

1.2

30

1.6

2

2.4

2.8

2.6

15 10

0.4

0.8

1.2

t

FRP

1.6

2

2.4

2.8

FRP

1.6

(mm)

=75

20 15 10

0

1.2

25 20 15 10 5

5

5

0.8

30

% Increase in LL

% Increase in LL

20

0.4

t

25

0

0

=50

25

% Increase in LL

3

y

t FRP (mm)

=0

0

3.4

e =e =0.05h

t FRP (mm)

30

3.8

e =e =0.3h

=0.5

0.8

=75

=50

=0

0

0.4

0.8

1.2

(mm)

1.6

t FRP (mm)

2

2.4

2.8

0

0

0.4

0.8

1.2

1.6

t FRP (mm)

Fig. 12. Profiles of reliability b and % increases in live load PLL vs. FRP thickness tFRP, (column C1, ex > 0 & ey = ex).

ineffective as the column requires flexural strengthening (i.e. EBFRP composites need to be bonded to the longitudinal direction of the RC member) which is out the scope of this study. 3. Results 3.1. Effect of eccentricity ratio on the reliability index of FRP confined RC column The Computational flow described in Fig. 4 was used to evaluate the reliability index of columns C1 and C2 which were confined

with FRP material of thickness equals to 2  0.11 mm and 4  0.11 mm respectively. Corner radius was taken equal to 25 and 50 mm for columns C1 and C2 respectively. The goal of this section is to compare the reliability index of FRP-confined RC and unconfined (as a reference case) RC column considering the same nominal applied load. In other words, at each considered case (ex, ey and k), the nominal load of FRP-confined RC columns were considered equal to that of RC columns. Figs. 7 and 8 show the results obtained considering different values of slenderness ratios and eccentricities. Results are valid for eccentricity and slenderness ratios up to 0.5 and 75 respectively. According to Figs. 7 and 8, the

380

O. Ali / Construction and Building Materials 151 (2017) 370–382 4.6

4.6

=0

=75

3.8

3.4

FRP FRP FRP

3

FRP FRP

0

0.8

=1 =0.875 =0.75

4.2

Reliability index

4.2

Reliability index

Reliability index

4.2

2.6

4.6

=50

3.8

3.4

3

=0.625

x x

2.4

3.2

4

4.8

2.6

5.6

0

0.8

y

1.6

2.4

25

4

4.8

2.6

5.6

15 10 5

25

=50

20

0.8

1.6

2.4

3.2

0.8

1.6

4

4.8

15 10

0

5.6

2.4

3.2

4

4.8

5.6

4

4.8

5.6

4

4.8

5.6

4

4.8

5.6

=75

20 15 10

5

0

0

t FRP (mm)

% Increase in LL

=0

% Increase in LL

% Increase in LL

3.2

t FRP (mm)

20

0

3

y

e =0.05h & e =0

t FRP (mm) 25

3.4

e =0.3h & e =0

=0.5

1.6

3.8

5

0

0.8

1.6

2.4

t FRP (mm)

3.2

4

4.8

0

5.6

0

0.8

1.6

t FRP (mm)

2.4

3.2

t FRP (mm)

Fig. 13. Profiles of reliability b and % increases in live load PLL vs. FRP thickness tFRP, (column C2, ex > 0 & ey = 0).

4.6

4.6

=0

=1 FRP

3.4

FRP

=0.75 FRP

3

FRP

2.6

=0.875

0.8

3.8

3.4 3

=0.625

1.6

2.4

3.2

4

4.8

5.6

2.6

=0

0

0.8

1.6

2.4

3.2

4

4.8

5.6

10 5

1.6

2.4

3.2

t FRP (mm)

0.8

1.6

4

4.8

5.6

20 15 10

0

2.4 FRP

3.2

(mm)

=75

25

5 0.8

0

t

% Increase in LL

15

0

2.6

=50

25

% Increase in LL

% Increase in LL

3

t FRP (mm)

20

0

3.4

ex =e y=0.3h

t FRP (mm) 25

3.8

ex =e y=0.05h

=0.5 FRP 0

Reliability index

3.8

=75

4.2

4.2

Reliability index

Reliability index

4.2

4.6

=50

20 15 10 5

0

0.8

1.6

2.4

3.2

t FRP (mm)

4

4.8

5.6

0

0

0.8

1.6

2.4

3.2

t FRP (mm)

Fig. 14. Profiles of reliability b and % increases in live load PLL vs. FRP thickness tFRP, (column C2, ex > 0 & ey = ex/2).

most characteristic Feature of these curves is that the reliability b profiles (i.e. b vs. ey/h and/or ex/b) remains constant or increases slightly until eccentricity ratio ranges between 0.2 ? 0.3, then after the profiles decreases with noticeable values. This range corresponds to the point at which the section transforms from small eccentricity (axial load inside the core) to high eccentricity (axial load outside the core). In addition it can be noted that confining RC columns using FRP material can increase the reliability index by 0.1 ? 0.5. Another feature characterize the reliability profile is that rounding the corner can causes a very slight increase of the

reliability index in case of single eccentricity while a quite noticeable increase when subjected to bi-axial eccentricities. However, in most cases, RC columns could gain reliability indices more three which is the target value recommended by ACI codes. The first row of Figs. 7 and 8 presents the effect of slenderness ratio k on the reliability index b at a single eccentricity (ex > 0 & ey = 0). It can be noted that the reliability index b increases as k increases at eccentricity ex<(0.3 ? 0.4) while after b decreases. Evidently increasing slenderness k leads to a decrease in column capacity due to the additional deformations – as a result of

381

O. Ali / Construction and Building Materials 151 (2017) 370–382 4.6

4.6

=0

3.8

3.4

FRP FRP FRP

3

FRP

2.6

FRP

0

0.8

=1 =0.875

3.8

3.4

=0.75

3

=0.625

x x

2.4

3.2

4

4.8

5.6

2.6

0

25

=0

0.8

y

1.6

2.4

3.2

4

4.8

5.6

2.6

25

=50

10

15 10

0 1.6

2.4

3.2

1.6

4

4.8

5.6

2.4

3.2

4

4.8

5.6

4

4.8

5.6

=75

15 10 5

5

5

0.8

0.8

20

% Increase in LL

15

0

0

t FRP (mm)

20

% Increase in LL

% Increase in LL

3

y

t FRP (mm)

20

0

3.4

e =e =0.05h

t FRP (mm) 25

3.8

e =e =0.3h

=0.5

1.6

=75

4.2

Reliability index

4.2

Reliability index

Reliability index

4.2

4.6

=50

0

0.8

1.6

t FRP (mm)

2.4

3.2

4

4.8

5.6

0

0

0.8

t FRP (mm)

1.6

2.4

3.2

t FRP (mm)

Fig. 15. Profiles of reliability b and % increases in live load PLL vs. FRP thickness tFRP, (column C2, ex > 0 & ey = ex).

geometrical nonlinearity considered – which come from a single eccentricity ex. Thus, consequently, a decrease in the applied factored nominal load (Pn = 1.2PDL + 1.6PLL according to ACI code). However, both nominal load and column capacity decrease, but the decrease in nominal load is much greater leading to an increase in reliability index of column. On the other side, the reliability profiles of columns under biaxial bending are presented in the second and third rows of Figs. 7 and 8. Herein, additional strains are excessively increased in the column sections due to the biaxial bending effects. This excessive increase in strains results in a decrease in nominal load much smaller than the decrease in column capacity, thus, consequently lower values of reliability index are achieved. 3.2. Sensitivity factors Refering to the sensitivity factor described in §5 and given by Eq. (25). Fig. 9 presents the sensitivity factors for all the cases considered in §7.1. Live load PLL records the highest sensitivity factor. Unlike Live load, Deal load PDL could be considered as a variable of small importance. Also it could be noted that, within all of the material and geometrical random variables considered it can be concluded that concrete compressive strength is the variable of the first importance followed by column dimension b -and/or- h. Although steel strength comes of more importance than concrete compressive strength in case of sections subjected to pure bending moment [2–4]. However, herein, it comes of less importance and depends on number of eccentricities considered in the analysis; single or double eccentricities. Furthermore, the sensitivity factor of the model error cm gains a high value in case of double or high eccentricities. Such variable reflects error included in numerical results which considered as deterministic variables in many previous studies. Thus, based on the obtained results it strongly recommended to be considered as random variable. While all other variables have a very small sensitivity factor and can be considered as deterministic variables.

3.3. Effect FRP safety factor /FRP In order to quantify the effect of additional reduction factor of FRP materials /FRP on the reliability index of FRP confined RC columns, the columns cross-sections described above in § 6 were analyzed considering multiple values of tFRP and /FRP. Herein, tFRP ranges between 0.2 ? 4 mm and 0.4 ? 6 mm for C1 and C2 respectively, while, five values were suggested to /FRP 0.5, 0.625, 0.75, 0.875 & 1. For each combination of tFRP and /FRP, % increases in live load and reliability index were determined. The nominal load Pn was determined according to ACI 318M-11 provisions as: / Pn = 1.2PDL + 1.6PLL, where / is the global safety factor which is expressed as:

8 es 6 esy > < 0:65 u ¼ 0:65 þ 0:25ðes  esy Þ=ð0:005  esy Þ if esy 6 es 6 0:005 > : 0:90 es P 0:005 ð27Þ where es is the steel tensile stain. esy is the yield strain of steel reinforcement. FRP confinement contribution will be added by factoring the term fFRP given in Eq. (7). Results were presented in Figs. 10–15 for different values of slenderness ratios (k: 0, 50 and 75) and eccentricities. High eccentricity (ex/b or/and ey/h = 0.3). Overall, results indicate that, % increases in live load is linearly related to tFRP. However, for high values of /FRP and tFRP the relation becomes bilinear as the effectiveness in increasing PLL changes. There are two possible explanations for this change. First, at high values of k, bilinear profile takes place as steel strain es exceeds 0.005, thus the global safety factor / is limited to a fixed value equals to 0.9 as recommend by ACI 318M-11 provisions (see Eq. (27)). Second, for low values of k, bilinear profile takes place as tensile steel changes from non-yielded to yielded state. In what concerns the reliability index b, evidently it depends on PLL changes. Generally, b decrease as tFRP increases especially in case of biaxial loaded columns or uniaxial

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O. Ali / Construction and Building Materials 151 (2017) 370–382

with high slenderness ratio k  75. Neverless for lower values of k a slight increase in b is recorded when columns confined with small amount with FRP laminates. Eventually, design equations could not maintain b over the target value (bT = 3), thus, additional safety factor must be considered in design and suggested to be equal to 0.65. Small eccentricity (ex/b or/and ey/h = 0.05). In all cases it can be observed that increasing tFRP can improve both the reliability index b and % increase in live load PLL. Therefore, it can conclude that the FRP-confined RC columns with small eccentricity can be design taking the additional reduction factor equal to unity. Comparison of the findings obtained in the present study with ACI 440.2R-08 recommendations for additional strength reduction factor, the study propose that /FRP to be variable depending on the eccentricity and slenderness ratio and not unique value - /FRP = 0.95 - as recommended by ACI 440.2R-08. 4. Summary and conclusions In this paper structural reliability of biaxial loaded Short/ Slender-Square FRP-confined RC columns is evaluated. First Order Reliability Method FORM was coupled with the mechanical model which was build using a three dimensional Finite Difference Model FDM in order to provide the ultimate eccentric axial load of the column. The Mechanical model takes into account material and geometrical nonlinearities. Full analytical derivatives of the structural response, eccentric axial load, with respect to random variables were provided leading to robust convergent FORM procedure. Within the considered 12 random variables sensitivity results have shown that FRP strength, FRP thickness, stirrups spacing, longitudinal steel modulus and steel area have a negligible effect on the reliability analysis and can be considered as deterministic variable. Results shown in the present study indicate that FRP confinement could improve the reliability of the RC section depending on the thickness of FRP laminates, nevertheless rounding the corner can causes a very slight increase of the reliability index. As a result it could be neglected in design procedures especially in case of high eccentricities. It could be emphasized that additional safety factor /FRP must be considered in design and it is suggested to be equal to 0.65 and 1 in case of high eccentricity (ex/b or/and ey/h = 0.3) and small eccentricity (ex or/and ey = 0.05) respectively. Such these findings may contradict ACI 440.2R-08 recommendations, as /FRP for ACI 440.2R-08 proposed /FRP = 0.95 regardless eccentricity of the section. Thus, more analyses and results are needed, in a perspective work, to validate these findings considering a wide range of combinations of concrete strengths, FRP properties and steel strength. References [1] ACI Committee 440.2R, Guide for the Design and Construction of Externally Bonded FRP Systems for Strengthening Concrete Structures, American Concrete Institute, 2008. [2] A.A. Atadero, V.M. Karbhari, Calibration of resistance factor for reliability based design of externally-bonded FRP composites, Compos. B 39 (4) (2007) 665– 679. [3] D. Bigaud, O. Ali, Time-variant flexural reliability of RC beams with externally bonded CFRP under combined fatigue-corrosion actions, Reliab. Eng. Syst. Safety 131 (2014) 257–270. [4] O. Ali, D. Bigaud, E. Ferrier, Comparative durability analysis of CFRPstrengthened RC highway bridges, Constr. Build. Mater. 30 (2012) 629–642. [5] T. El-Maaddawy, Behavior of corrosion-damaged RC columns wrapped with FRP under combined flexural and axial loading, Cem. Concr. Compos. 30 (2008) 524–534.

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Osama Ali is lecturer in the civil engineering department, faculty of engineering, Aswan University, Egypt. He holds his BSc degree in civil engineering from faculty of engineering, Aswan University. He awarded his MSc degree from structural engineering department, faculty of engineering, Cairo University (Egypt). The thesis entitled: Elastic analysis and Design of Prestressed composite girder. He received his PhD from Angers University, France, in 2012. PhD title is: Timedependent reliability of FRP strengthened reinforced concrete beams under coupled corrosion and changing loading effects. His research activities in the areas of structural reliability of RC structures, finite element analysis of RC structures, strengthening and repair of RC structures using FRP composites, structural optimization, corrosion of steel reinforcement embedded in concrete and performance of RC structures under seismic loads.