Strengthening of steel-reinforced concrete structural elements by externally bonded FRP sheets and evaluation of their load carrying capacity

Composite Structures 118 (2014) 377–384

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Strengthening of steel-reinforced concrete structural elements by externally bonded FRP sheets and evaluation of their load carrying capacity D. De Domenico a, P. Fuschi a, S. Pardo b, A.A. Pisano a,⇑ a b

Dept. PAU – University Mediterranea of Reggio Calabria, via Melissari, I-89124, Reggio Calabria, Italy Dept. DIBIMEF – University of Palermo, Via Divisi 81, I-90133, Palermo, Italy

a r t i c l e

i n f o

Article history: Available online 7 August 2014 Dedicated to Professor Castrenze Polizzotto on the occasion of his 90th birthday. Keywords: FRP-strengthening systems RC slabs limit design FE-based 3D analysis Peak loads and failure modes

a b s t r a c t The paper proposes a preliminary design tool for reinforced concrete (RC) elements strengthened by fiber-reinforced-polymer (FRP) sheets to be used in civil engineering applications and in particular in medical buildings. The design strategy is based on limit analysis theory and utilizes a numerical procedure which provides a direct method to determine peak load, failure mode and critical zones of the structural elements of interest. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction The present study is oriented to applications of engineering interest concerning the strengthening of existing steel RC members to face changed (higher) design load conditions or to situate cables, pipes and other devices within openings to be realized in the existing structures. In this context, one possible field of application is that of hospital rooms intended for magnetic resonance imaging (MRI) and computed tomography (CT) equipments. As in fact, in such circumstances the service loads are very high (about 2500 daN/m2) and, if such equipments have to be installed within an existing building, the strengthening as well as the geometrical updating (creation of openings) of the existing structures becomes compulsory. Externally bonded steel plates, steel or concrete jackets or external post-tensioning cables are traditional remedies, however steel can interfere with magnetic fields and then with all the equipments where magnetic neutrality is required. In such a context, the externally bonded FRP sheets (see e.g. [1,9]), on taking into account the high tensile strength and the magnetic neutrality of FRPs, represent a very effective solution. Experimental investigations confirm that, after the application of such FRP techniques [8,19], a significant increase in flexural/shear capacity of the RC elements (up to about 125%) is achieved. Many experiments, [28], show also the enhancement of the confinement ⇑ Corresponding author. Tel.: +39 0965 385218; fax: +39 0965 385219. E-mail address: [email protected] (A.A. Pisano). http://dx.doi.org/10.1016/j.compstruct.2014.07.040 0263-8223/Ó 2014 Elsevier Ltd. All rights reserved.

effect exerted on concrete by the FRP laminates, resulting in shifting the failure mode of the strengthened elements from a brittle to a more ductile one. In fact, the FRP strengthening system mitigates crack development and, as a result, increases the overall ductility of the RC element. Finally, the major cost of fibers and resins is largely offset from the lower cost of installation due either to the lightweight of the sheets or to the possibility to conform them to the existing structural geometries. Grounding on the above considerations, a limit state design carried out on the base of a plastic approach seems in this context both acceptable and effective, even if the viability of such an approach needs a validation which is the main goal of the present work. To this purpose, a limit analysis methodology, concerning RC structural elements and belonging to the so-called direct methods [7,29], is here promoted to predict the load carrying capacity of RC-elements strengthened by FRP bonded sheets. The procedure, by acting on each of the constituent materials of the structural element, allows to catch the behavior at collapse of the whole element by simultaneously taking into account the crushing of concrete, the possible yielding of steel bars as well as the collapse of FRP sheets. A Menétrey–Willam-type yield criterion, endowed with cap in compression, is employed for the RC members modeling, a Tsai–Wu-type yield criterion is used to tackle the FRP sheets, a von Mises yield criterion is used for the steel re-bars. The numerical limit analysis procedures is carried out within a 3D FE-layered formulation.

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of concrete, a non-associated flow rule is postulated for the adopted M–W-type yield surface.

It is worth noting that all the phenomena arising just after a state of incipient collapse, such as delamination [5], debonding [16], damage in a wider sense, are not treatable and this consistently with the spirit of a limit analysis approach. Accurate treatment of such post-elastic phenomena is prosecutable only with more accurate step-by-step FE nonlinear analyses. Aware of such limitation, the methodology here proposed should then be viewed only as a preliminary design tool to gain a quick insight into the bearing capacity evaluation of the analyzed elements by determination of the peak load value, the prediction (but not the description) of failure mode as well as the detection of critical zones within the addressed FRP-strengthened RC elements. In order to validate the effectiveness of the promoted approach, a comparison of the numerical predictions against experimental findings on large-scale slabs prototypes [3,4] is presented.

 ðrij Þ is the von Mises effective stress and f y is the yield where r strength. Since in the FE-model the steel reinforcement are modeled by 1D truss elements, a uniaxial stress condition is considered in the following and Eq. (4) applies in the simpler shape rr ¼ f y ; rr being the stress in the re-bar longitudinal direction.

2. Constitutive assumptions

2.3. Tsai–Wu-type yield criterion for FRP sheets

The adopted limit analysis methodology takes into account the actual behavior, at a state of incipient collapse, of all the constituent materials the RC elements are made with, namely: concrete, steel bars and FRP laminates. The pertinent constitutive assumptions are given next.

Finally, the FRP strengthening plates are modeled as orthotropic laminates in plane stress conditions obeying a Tsai–Wu-type yield criterion [30]. For a unidirectional lamina in plane stress case the Tsai–Wu polynomial criterion has the following form:

2.1. Menétrey–Willam-type yield criterion for concrete

where 1 and 2 denote the principal directions of orthotropy in plane stress case (the fibers are directed along the material axis 1) and r6  s12 in the contracted notation. The coefficients F i and F ij (i; j ¼ 1; 2; 6) entering Eq. (5) are functions of the strength parameters of the unidirectional lamina:

Concrete is assumed as an isotropic, nonstandard material obeying a plasticity model derived from the Menétrey–Willam (M–W) failure criterion [17]. The latter provides a three parameter failure surface having the following expression:

" #2 " # pffiffiffiffiffiffiffi q q n f ðn; q; hÞ ¼ 1:5 0 þ m pffiffiffi 0 r ðh; eÞ þ pffiffiffi 0  1 ¼ 0 fc 6f c 3f c

ð1Þ

where

r ðh;eÞ ¼

  4 1  e2 cos2 h þ ð2e  1Þ2 2ð1  e2 Þ cosh þ ð2e  1Þ½4ð1  e2 Þ cos2 h þ 5e2  4e 02

m :¼ 3

1=2

ð2aÞ

02

fc ft e : 0 0 fcft eþ1

ð2bÞ

Eq. (1) is expressed in terms of the three stress invariants n; q; h known as the Haigh–Westergaard (H–W) coordinates (i.e. hydrostatic and deviatoric stress invariants and Lode angle); m is the friction parameter of the material depending, as shown in Eq. (2b), on 0 0 the compressive strength f c , the tensile strength f t as well as the eccentricity parameter e. The eccentricity e, whose value governs the convexity and smoothness of the elliptic function r ðh; eÞ, describes the out-of-roundness of the M–W deviatoric trace and it strongly influences the biaxial compressive strength of concrete. The failure surface (1) is open along the direction of triaxial compression; therefore, to limit the concrete strength in high hydrostatic regime, a cap in compression closing the surface (1) is adopted. The cap is formulated in the H–W coordinates as follows [18]: cap

q ðn; hÞ ¼ 

qMW ðna ; hÞ  ðna  nb Þ2

2

n  2na ðn  nb Þ 

n2b



( for

nb 6 n 6 na 0 6 h 6 p3 ð3Þ

MW

where q ðn; hÞ is the explicit form of the parabolic meridian of the M–W surface easily obtainable from Eq. (1). The values na and nb entering Eq. (3), namely the hydrostatic stress values corresponding to the intersection of the cap surface with the M–W surface and the hydrostatic axis, respectively, locate the cap position and can be calibrated according to experimental results [11]. Due to the dilatancy

2.2. von Mises yield criterion for steel re-bars Steel is modeled as an isotropic, perfectly plastic material obeying the well-established von Mises yield criterion. For a multi-axial loading scenario the von Mises yield condition is expressed as:

 ðrij Þ  f y ¼ 0 f ðrij Þ ¼ r

ð4Þ

F 11 r21 þ F 22 r22 þ F 66 r26 þ 2 F 12 r1 r2 þ F 1 r1 þ F 2 r2 ¼ 1;

1 1 1 1 1 þ ; F 2 :¼ þ ; F 11 :¼  ; Xt Xc Yt Yc Xt Xc p ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1 1 1 F 11 F 22 ; :¼  ; F 66 :¼ 2 ; F 12 :¼  Yt Yc 2 S

ð5Þ

F 1 :¼

ð6aÞ

F 22

ð6bÞ

with: X t , X c the lamina longitudinal tensile and compressive strengths, respectively; Y t , Y c the lamina transverse tensile and compressive strengths, respectively; S the shear strength of the lamina. In the expressions (6) the compressive strengths X c and Y c have to be considered intrinsically negative. Also FRP composite plates are considered as nonstandard material and, therefore, a nonassociated flow rule is postulated for their constitutive behavior.

3. Limit analysis approach: multi-yield-criteria formulation Limit analysis approaches can be really useful from an engineering point of view because allow the direct evaluation of the load bearing capacity of a structure [12,13,20]. In its classical formulation the theory of limit analysis refers to perfectly plastic structures, made of standard materials, and is based on two fundamental theorems, namely: the upper and the lower bound theorem [14,27]. However, the field of applicability investigated in this paper lies outside the realm of standard materials, as in fact, concrete is a nonstandard pressure dependent material as well as FRP is a nonstandard material and this implies the use of a nonstandard limit analysis approach, that is, the search of two bounds, an upper and a lower bound, to the real collapse load value of the whole RC element. To this purpose two distinct limit analysis numerical methods are considered in the following. The two methods are known in the relevant literature as Linear Matching Method (LMM) and Elastic Compensation Method (ECM), respectively. They were conceived in [6,25,15] with reference to von Mises materials and then it has been proved that their applicability is also effective with reference to a more general class of materials [20–24, 26] the only requisite being the strict convexity of the involved yield criterion. Both methods make use of linear elastic

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analyses to mimic the behavior of the structure at collapse and are performed iteratively. 3.1. The Linear Matching Method The LMM is a numerical method based on the kinematic approach of limit analysis which is able to ‘‘build’’ the collapse mechanism of the analyzed structure and to compute an upper bound to the peak load multiplier. Looking at the geometrical interpretation of the method sketched in Fig. 1, at the current iteration, say at the ðk  1Þth FE-analysis, a fictitious structure (i.e. the structure under study with its real geometry, boundary and loading conditions but made of fictitious material) is analyzed under loads i , with Pðk1Þ load multiplier and p i assigned reference loads. P ðk1Þ p The fictitious linear solutions computed at each Gauss Point of the FE mesh, can be represented, at the generic Gauss Point (GP), ðk1Þ

by a point PL lying on the complementary dissipation rate equipotential surface referred to the fictitious viscous material, say ‘ ðk1Þ ðk1Þ  jðk1Þ Þ ¼ W ðk1Þ , whose geometrical dimen; DI ;r W ðk1Þ ðrj ðk1Þ

sions and center position depend on the fictitious values DI  ðk1Þ j

r

and

fixed at the current GP (I ranging over the elastic constants

entering the considered material; j ranging over the needed stress ðk1Þ

components). The point PL

with its coordinates, say

r‘j ðk1Þ in

the chosen principal stress space, shown in the sketch of Fig. 1, represents the fictitious solution in terms of stresses while the outward ðk1Þ ‘ ðk1Þ , say the normal of components e_ j , represents the normal at PL fictitious solution in terms of linear viscous strain rates. The fictiðk1Þ PL

is tious moduli and initial stresses are then modified so that brought onto the yield surface of the real constitutive material the

Fig. 2. Geometrical sketch, in the principal stress space, of the ECM at current iteration ðk  1Þ of the current sequence s. Stress points representing the elastic ðk1Þ solution at elements #1; #2; . . . ; #e; . . . ; #n; with PR ‘‘maximum stress’’ among all the elements.

analyzed structure is made with. The latter surface is here presented ðk1Þ

by the ellipsoidal shaded surface of Fig. 1. Namely PL

is brought to

ðk1Þ PM ,

ðk1Þ

having the same outward normal as PL identify with point but lying on the real material yield surface. The described

Fig. 1. Geometrical sketch, in the principal stress space, of the matching procedure, from iteration ðk  1Þ to ðkÞ at the current GP within the current element.

380

D. De Domenico et al. / Composite Structures 118 (2014) 377–384 ðk1Þ

‘ ðk1Þ c ðk1Þ u_ j  u_ j , define a collapse mechanism. The related stresses

ðk1Þ

j modification of DI and r implies that the ‘‘modified’’   ‘ ðkÞ ðkÞ ðkÞ ðkÞ rj ; DI ; r j ¼ W ðk1Þ matches the yield surface at point W

rjY ðk1Þ are the pertinent stresses at yield. If the expounded rationale is repeated at all GPs of the mesh, a c ðk1Þ c ðk1Þ ; u_ i ) with the related stresses at

ðk1Þ

PM , this step is the so-called ‘‘matching procedure’’, see again Fig. 1. The fictitious solution in terms of strain rates, namely e_ ‘j ðk1Þ  e_ jc ðk1Þ , where the apex c stands for ‘‘at collapse’’, as well as ðk1Þ

the stress coordinates of PM

Y ðk1Þ

, say the stresses at yield rj

collapse mechanism, (e_ j yield,

ðkÞ

bound value to the collapse load multiplier, say P UB , can be evaluated at current ðk  1Þth FE elastic analysis. However, the above stress at yield, computed through the matching, do not meet the i and the equilibrium conditions with the acting loads Pðk1Þ p

, give

all the information pertaining to a state of incipient collapse built at the current GP. In particular, the fictitious strain rates e_ ‘j ðk1Þ  e_ jc ðk1Þ , with the associated displacement rates

slab opening (if any)

(a)

Lo

rjY ðk1Þ can be defined for the whole structure and an upper

Pp

Pp

z

t

do

x

do

y

L1

b

L2 L

L1

(b)

600

600 z

bary 3@300 25

tf

y

z

t = 125

barx 5# 4

bary 3@300

tf

y

barx 5# 4

FRP sheet Num of layers variable (1-2) along the slab span

FRP strips G1-2; G1- 3

G1-1

t = 125 25

strip width N. layers variable (1-3;1-6) along the slab span; N. 3 strips - width 150.

G2 -1; G2 -1;G2 - 3

N. 3; 4; 5 strips -width 100.

Lo

960 d0 z

tf

bary Φ10@250 y

t t = 150

barx 11Φ10 FRP strips beside the opening

do

y x

S8 - S13 Fig. 3. Mechanical model of the analyzed slabs (with and without openings): (a) geometry, loading and boundary conditions; (b) cross-section geometry (dimensions in mm) with reinforcement arrangement (after [3,4]).

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D. De Domenico et al. / Composite Structures 118 (2014) 377–384

procedure, as said, is carried on iteratively until the difference between two subsequent P UB values is less than a fixed tolerance. A flow-chart scheme of the LMM is reported below with different sets of stress coordinates according to the stress reference system employed for the three materials yield criteria (see Eqs. (1), (4), (5)). Flow-chart of the Linear Matching Method Initialization  Generate FE-mesh: 3D-solid concrete elements; 2D-solid plane stress FRP elements; 1D truss steel elements; 0 0  Set strength parameters for each material: (f c ; f t ; e) for concrete; (X t ; X c ; Y t ; Y c ; S) for FRP; f y for steel; ð0Þ  ð0Þ  Assign fictitious parameters DI and r to FEs; precisely: j ð0Þ ð0Þ ð0Þ  ð0Þ  ð0Þ  (K ; G ) and (n ; qx ; qy ) to concrete elements; ð0Þ ð0Þ  ð0Þ (Ej ; m12 ) and r (with j ¼ 1; 2; 6) to FRP elements; Eð0Þ j to steel re-bars elements; ð0Þ i (p i = reference loads; P ð0Þ  Assign loads P UB p UB load multiplier); ðk1Þ ð0Þ  To start iterations (FE analyses), set: k ¼ 1; PUB ¼ P UB ¼ 1. Step #1: FE-code ðk1Þ i  Perform a linear elastic FE-analysis under loads PUB p with fictitious parameters ðÞðk1Þ . Output at each GP of the FE mesh:  linear viscous fictitious strain rates and associated linear fictitious stresses; the former being the components of the outward normal e_ ‘ ðk1Þ of Fig. 1, the latter locating ðk1Þ point PL , i.e.:     ‘ ðk1Þ ‘ ðk1Þ _e‘Cðk1Þ e_ ‘vðk1Þ ; e_ ‘dðk1Þ ; e_ d‘ ðk1Þ ; PðCÞ n‘ ðk1Þ ; qx ; qy L x y in concrete elements.   



e_ ‘F ðk1Þ e_ ‘j ðk1Þ ; PðFÞ r‘j ðk1Þ (with j ¼ 1; 2; 6) L in FRP elements.  e_ ‘ ðk1Þ ; PðSÞ r  ‘ ðk1Þ in steel elements. L S ‘ ðk1Þ  displacement rates u_ i Step #2: Fortran-code  Matching procedure in concrete, FRP and steel elements: ðCÞ

ðCÞ

Y ðk1Þ

(with j ¼ 1; 2; 6). ðSÞ ðSÞ  Y  f y ); compute EðkÞ . PL ) PM (r Step #3: Compute the upper bound multiplier ‘ ðk1Þ c ðk1Þ  e_ C ;  Assume as collapse mechanism: e_ C ‘ ðk1Þ c ðk1Þ ‘ ðk1Þ c ðk1Þ ‘ ðk1Þ c ðk1Þ e_ F  e_ F ; e_ S  e_ S ; ui  ui  Compute the upper bound multiplier integrating on the whole mesh: ðkÞ

PUB ¼

V

Specimen label

L (mm)

L1 (mm)

L2 (mm)

Lo (mm)

do (mm)

tf (mm)

G1-1 G1-2 G1-3 G2-1 G2-2 G2-3 S8 S9 S10 S11 S12 S13

2440 2440 2440 2440 2440 2440 3000 3000 3000 3000 3000 3000

870 870 870 870 870 870 5486 5486 5486 3660 4570 772

600 600 600 600 600 600 1000 1000 1000 1000 1000 1000

– – – – – – 1250 250 1300 300 1350 350

– – – – – – 500 500 400 400 300 300

0.165 0.165 0.165 1.20 1.20 1.20 0.12 0.12 0.12 0.12 0.12 0.12

Also the ECM can easily be explained by means of a geometrical sketch as the one given in Fig. 2 with reference to a generic yield sur j Þ ¼ 0. The ECM starts with a first sequence, say s ¼ 1, face Fðrj ; DI ; r of FE analyses, carried on the structure endowed with the proper (real) material elastic parameters and suffering applied initial loads ðsÞ i ¼ P ð1Þ  PD p D pi , and by the initial real values of the elastic parameters. At the current iteration, say at the ðk  1Þth FE analysis, the elastic stress solution is computed at the GPs of the mesh. Such values, averaged within the current element #e, allow to define a solution ‘‘at element level’’, which, as shown in the sketch of Fig. 2, locates in e ðk1Þ

 Y e_ cS nY e_ cv þ qYx e_ cdx þ qYy e_ cdy þ rYj e_ cj þ r R c ðk1Þ i u_ i dS p S

ðk1Þ

dV

Y ðk1Þ

the principal stress space a stress point, say P#e . P#e denotes the corresponding stress point at yield (i.e. lying on the yield surface) ! ! e e . In the figure are reported measured on the direction OP#e = OP#e other stress points, representing the average stress elastic solution within elements #1; #2; . . . ; #e; . . . ; #n. If the elastic solution at ! e ðk1Þ ! Y ðk1Þ OP > , then the elethe #eth element is such that OP#e #e ment’s Young modulus is reduced according to the formula:

Y ðk1Þ

PL ) PM ðnY ðk1Þ ; qx ;q Þ; compute   y ðkÞ ðkÞ ðkÞ  ðkÞ  ðkÞ  (K ; G Þ; n ; qx ; qy       ðFÞ ðFÞ Y ðk1Þ ðkÞ ðkÞ  jðkÞ P L ) P M rj ; compute Ej ; m12 , r

R 

Table 1 Geometrical data of the analyzed slabs.

ðkÞ E#e

¼

2 ! 32 Y ðk1Þ ðk1Þ 4jOP#e j 5 ; E#e ! e ðk1Þ jOP#e j

ð7Þ

where the square of the updating ratio, within the square brackets, is used to increase the convergence rate. After the above moduli variation, the maximum stress value has to be detected in the whole FE mesh, namely the value corresponding to the stress point farthest away from the yield surface, say ! ! ðk1Þ ðk1Þ PR in the sketch of Fig. 2. If jOPR jðk1Þ is greater than OPYR (as drawn Fig. 2) a new FE analysis is performed within the current sequence trying to re-distribute the stresses within the structure; and this by keeping fixed the applied loads but with the updated ðkÞ

a new FE-analysis with the adjusted fictitious parameters

E#e values given by Eq. (7). The iterations are carried on, inside the given sequence, until all the stress points just reach or are below their corresponding yield values, which means that an admissible stress field has been built. Increased values of loads are then considered in the subsequent sequences of analyses, each

ðÞðkÞ . Set k ¼ k  1 and go to Step #1

one with an increased value of P D , till further load increase does

t

Step #4: Check for convergence ðkÞ ðk1Þ  If PUB  P UB is less than a fixed tolerance exit, else perform

ðsÞ

ðk1Þ

3.2. The Elastic Compensation Method The ECM is a numerical method based on the static approach of limit analysis which is oriented to ‘‘build’’ a statically and plastically admissible stress field (corresponding to a given load) so giving a lower bound to the peak load multiplier.

to be brought below yield by the not allow the stress point PR re-distribution procedure. A P LB load multiplier can then be evaluated at last admissible stress field attained for a maximum acting ðsÞ i , say at s ¼ S, and at last FE analysis, say at k ¼ K, as: load PD p

! ðKÞ PLB ¼ OPYR

ðSÞ

PD !

jOPR jðKÞ

:

ð8Þ

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D. De Domenico et al. / Composite Structures 118 (2014) 377–384

The flow-chart scheme of the ECM is reported below. Flow-chart of the Elastic Compensation Method Initialization  Generate FE-mesh: 3D-solid concrete elements; 2D-solid plane stress FRP elements; 1D truss steel elements; 0 0  Set strength parameters for each material: (f c ; f t ; e) for concrete; (X t ; X c ; Y t ; Y c ; S) for FRP; f y for steel; ðsÞ i (p i = reference loads; PðsÞ  Assign loads PD p D load multiplier);  To start the sequence of FE analyses, set s ¼ 1. Step #1: Start iterations within the current sequence  To start the first FE analysis of the sequence, set k ¼ 1;  Assign the (real) material parameters to FEs: (Eðk1Þ ; m) to   ðk1Þ concrete elements; Ej ; m12 (with j ¼ 1; 2; 6) to FRP elements; Eðk1Þ to steel re-bars elements; Step #2: FE-code ðsÞ i with  Perform a linear elastic FE-analysis under loads P D p ðk1Þ material parameters ðÞ . Output the average solution at current element #e:   e ðk1Þ e ðk1Þ e ðk1Þ e ðk1Þ  P#e in concrete elements n#e ; q#e ; h#e   eðk1Þ eðk1Þ  r#e (with j ¼ 1; 2; 6) in FRP elements r#e j eðk1Þ  #e  r in steel elements Step #3: Fortran-code Y ðk1Þ  Find the corresponding stress point at yield: P#e in conY ðk1Þ  Y  f y in steel FEs; crete FEs; r#e in FRP FEs; r  Update elastic moduli within ‘‘critical’’ elements #e (where the computed stress exceed the corresponding yield stress): ðk1Þ  ! ðk1Þ 2 ðkÞ ðk1Þ ! Y e  E#e ¼ E#e OP #e in concrete OP #e elements ðk1Þ  ðk1Þ 2 e ðkÞ ðk1Þ (with j ¼ 1; 2; 6) in  E#ej ¼ E#ej rY#e r#e

FRP elements h i2 ðkÞ ðk1Þ eðk1Þ  #e f y =r in steel elements

 E#e ¼ E#e

 Locate the ‘‘maximum stress’’ in each element-type group ðk1Þ ðk1Þ within the FE-mesh: PR in concrete FEs; rR in FRP  ðk1Þ FEs; r in steel FEs R  Compute a unique value of the ‘‘rescaling factor’’ as the minimum among the three values evaluated within the three element-type groups:  

! ðk1Þ ! vðk1Þ :¼ min OP RY =jOP R jðk1Þ ; rYR ðk1Þ =jrR jðk1Þ ; f y =r ðk1Þ R

Step #4: Check for redistribution ðsÞ  If vðk1Þ < vðk2Þ the redistribution failed for the current PD . ðs1Þ

Set PLB  PLB and exit.  else if vðk1Þ P 1  A statically admissible stress field is constructed for the acting load of current sequence. Compute the lower ðsÞ

experimental findings provided by the literature. In particular, according to the aims of the paper, attention is focused on FRP strengthened structural slabs, with and without openings, that can be used in hospital rooms to overcome structural problems that can arise from increment of service loads due to installation of specific (heavy) equipments, or from the realization of openings for situating cables, pipes, fire systems etc. To demonstrate the effectiveness of the promoted method, experimental tests on 12 large scale slabs have been numerically reproduced and the obtained results have been compared with those given by the laboratory tests. Among the analyzed specimens, six, without openings, refer to the experimental work of Al Rousan et al. [3] which considers slabs strengthened with different layers and configurations of carbon FRP (CFRP) sheets. In the following these slabs are labeled as: G1-1, G1-2, G1-3, G2-1, G2-2, G2-3. For the other six slabs reference is made to the experimental work of Anil et al. [4] which investigates the flexural behavior of CFRP strengthened RC slabs with openings having different size and different location; these slabs in the following are indicated as: S8, S9, S10, S11, S12, S13. All the analyzed slabs are simply supported and  (with p  refersubjected to two line loads each having intensity P p ence line load), and symmetrically located about mid-span. For all the examined examples, the resultant of the reference line load is equal to 100kN. Fig. 3(a) gives a schematic representation of the mechanical model, constrains and loading conditions of the slabs, while Fig. 3(b) provides information about internal steel bars and stirrups reinforcements as well as information about the strengthening CFRP arrangements of the different prototypes. Supplementary geometrical data are given in Table 1. For a detailed description of the tested prototypes the reader could refer to the two above quoted papers. Tables 2 and 3 gives the properties of Table 2 Material properties and geometrical data of reinforcement bars of the analyzed slabs. Reinforcement bar

d (mm)

A (mm2)

f y (MPa)

Es (GPa)

#3 #4 U10

9.5 12.7 10

71 129 78.5

410 410 480

200 200 200

Table 3 Mechanical parameters of the CFRP lamina utilized in the analyzed slabs. Specimen label

CFRP lamina moduli

CFRP lamina strengths

E1 (GPa)

From G1-1 228 to G1-3 From G2-1 165 to G2-3 From S8 to S13 65.4

E2 (GPa)

G12 (GPa)

Xt Xc Yt Yc S (MPa) (MPa) (MPa) (MPa) (MPa)

10.3

7.2

4275

1440

57

228

71

10.8

7.6

3030

1875

49

246

83

5.9

2.9

894

779

27

196

63

m12 is equal to 0.27 for all the specimens.

ðsÞ

bound multiplier: P LB ¼ PD vðk1Þ  Increase the load multiplier and perform a new ðsþ1Þ ðsÞ sequence of elastic analyses: set P D > P LB ; s ¼ s þ 1; and go to Step #1  else if vðk1Þ < 1, try to redistribute within the current sequence s, performing a new FE analysis with the current load but with the updated moduli ðÞðkÞ : set k ¼ k  1 and go to Step #2

4. Validation by comparison with experimental findings The expounded design methodology is hereafter tested and validated by comparing the obtained numerical results with

Table 4 Mechanical parameters of concrete of the analyzed slabs. Specimen label

Concrete properties 0

From G1-1 to G2-3 S8 S9 S10 S11 S12 S13

0

f c (MPa)

f t (MPa)

Ec (GPa)

55.00 20.10 20.80 19.90 20.20 19.70 20.70

2.45 1.48 1.51 1.47 1.48 1.46 1.50

36.69 27.13 27.41 27.04 27.17 26.96 27.37

383

G2-3

0.965 1.019 1.168

1.723 1.678 1.921

1.561 1.428 1.778

S10

G2-2

0.981 1.051 1.138

S9

1.508 1.629 1.678

S8

G2-1

1.071 0.985 1.258

1.994 2.034 2.152 G1-3

0.906 0.986 1.011

1.053 1.095 1.390 G1-2

0.881 0.912 1.135

1.503 1.664 1.745 G1-1

0.872 0.938 1.017

peak load multipliers

peak load multipliers

D. De Domenico et al. / Composite Structures 118 (2014) 377–384

S11

S12

S13

Fig. 4. Comparison between numerical and experimental peak load multipliers for the analyzed slabs.

Fig. 5. Prediction of the failure mechanism for slab S8. Band plots of principal strain rates, in the deformed configuration, at last converged solution of the LMM. (a) RC elements; (b) FRP laminate elements; (c) photograph of slab S8 at failure (after [4]).

384

D. De Domenico et al. / Composite Structures 118 (2014) 377–384

the re-bars and CFRP reinforcement respectively, while Table 4 reports the concrete properties. In all the examined cases, the elastic analyses performed within the LMM and ECM have been carried out using FE commercial code ADINA [2] with meshes of 3D-solid 8-nodes elements with 2  2  2 GPs per element for modeling concrete, 2D-solid 4-nodes membrane elements, under plane stress hypothesis and with 2  2 GPs per element for modeling CFRP sheets and 2-nodes, 1-GP, truss elements for modeling steel bars and stirrups. Each node is endowed with the three translational degrees of freedom, while a perfect bond between concrete and steel re-bars as well as between concrete and FRP sheets is postulated in the FE-model [10]. Finally, a Fortran main program has been utilized to control the ‘‘adjusting’’ of the elastic parameters at each GP of each element to accomplish the matching, when performing the LMM, and to realize the stress redistribution, within the ECM. The obtained numerical results in terms of upper (PUB ) and lower (PLB ) bound to the peak load multiplier are given in Fig. 4 together with the experimentally detected ones (P EXP ) for all the examined specimens. The numerical findings highlight the very good prediction obtained with the proposed three-yield-criteria limit analysis approach. The LMM gives also some hints on the type of failure mechanisms. Such a prediction in fact is given by the possibility to point out the plastic zone (portions of the FE-mesh where the collapse mechanism has been eventually located) at ‘‘last converged solution’’ of the LMM. Just to show one of the analyzed cases, in Fig. 5(a and b) are reported respectively the plot of the principal (compressive) strain rates e_ cxx of concrete FEs in the deformed configuration and the one of the strain rates at collapse of FRP FEs in the fiber direction, i.e. e_ c1 , for slab S8, at convergence. By inspection of Fig. 5(a) it is possible to observe plasticized zones, corresponding to plastic hinges arising at the opening edges. These zones appear reasonably narrow and located where the damaged zones have been actually experimentally detected, as confirmed by the photograph given in Fig. 5(c) concerning slab S8 at failure. In Fig. 5(b) the most critical FRP zones are highlighted and, obviously, these zones are those where the FRP sheets, to a greater extent, bear the load and act compositely with concrete in the global collapse mechanism. Moreover, though not reported here for brevity, the stresses numerically obtained in the steel FEs at the mid-span (where a plastic hinge develops) are just yielded as observed in the experimental outcomes. Although the band plots of Fig. 5(a and b) provides only qualitative information of the failure mechanism, such potentiality of the procedure can be very useful to localize critical zones or weaker members when dealing with larger structural systems. 5. Concluding remarks  The paper has proposed a limit analysis procedure for a preliminary design of RC elements strengthened by externally bonded FRP sheets.  The procedure, based on a multi-yield-criteria limit analysis approach, has led to a reliable prediction of peak loads and failure modes of the analyzed elements (slabs) by simultaneously considering the limit state of the constituent materials, so resulting very useful in many applications of engineering interest.  The attention has been focused on hospital applications in which increment of service loads or realization of openings can weaken some structural elements that have been strengthened by FRP sheets.

 The promoted approach has been tested by comparing the numerical predictions with some experimental findings on large-scale prototypes available in literature and the goodness of the obtained results seems to be promising for the applicability of the whole approach.

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