Evaluation of the increased load bearing capacity of steel beams strengthened with pre-stressed FRP laminates

We analyse the problem of a simply supported steel beam subjected to uniformly distributed load, strengthened with a pre-stressed fibre-reinforced polymer (FRP) laminate. We assume that the laminate is first put into tension, then bonded to the beam bottom surface, and finally fixed at both its ends by suitable connections. The beam and laminate are modelled according to classical beam theory. The adhesive is modelled as a cohesive interface with a piecewise linear constitutive law defined over three intervals (elastic response, softening response, debonding). The model is described by a set of differential equations with suitable boundary conditions. An analytical solution to the problem is determined, including explicit expressions for the internal forces and interfacial stresses. As an application, we consider the standard IPE series for the steel beam and the Sika® CarboDur® system for the adhesive and laminate. For each considered cross section, we first carry out a preliminary design of the unstrengthened steel beam. Then, we imagine to apply the FRP strengthening and calculate the loads corresponding to the elastic limit states in the steel beam, adhesive, and laminate. Lastly, we take into account the ultimate limit state corresponding to the plasticisation of the mid-span steel cross section and evaluate the increased load bearing capacity of the strengthened beam.


INTRODUCTION
ibre-reinforced polymers (FRP) are increasingly used in civil engineering for the strengthening of existing constructions.In such applications, the existing structural elements (made of traditional materials such as, for instance, masonry, wood, concrete, steel, etc.) are strengthened by adhesively bonding FRP laminates onto their external surfaces.The type of fibre, shape, thickness, and other characteristics of the laminate vary according to the F element to be strengthened and the desired level of structural performance [1].For the strengthening of steel structures, carbon fibre reinforced polymers (CFRP) are preferred because of their superior mechanical properties [2,3].Furthermore, CFRP laminates can be pre-stressed, which enables more effective use of the composite material, contribution of the strengthening in carrying out the dead load, closure of cracks in concrete [4,5], and increased fatigue life in steel [6].The existing structure and FRP laminate behave as a composite structure with a key role played by the adhesive layer, which transfers the stresses between the bonded elements.As a matter of fact, debonding of the FRP laminate due to high interfacial stresses is a relevant failure mode for this type of interventions.Therefore, a wide number of theoretical and experimental studies have been conducted to achieve reliable and accurate evaluation of such interfacial stresses.Smith and Teng [7] presented a review of the theoretical models for predicting the interfacial stresses and also developed a solution for strengthened beams in bending.Al-Emrani and Kliger [8] determined the interfacial shear stresses in beams strengthened with pre-stressed laminates subjected to mid-span concentrated loads.Benachour et al. [9] extended the previous solutions to distributed loads and multidirectional laminates used as strengthening.All the aforementioned models consider the adhesive layer as an elastic interface to obtain simple closed-form solutions.A more realistic modelling of the adhesive can be achieved by introducing a non-linear (or piecewise linear) cohesive law for the interface [10][11][12].Bennati et al. [13] used a cohesive-zone model to determine the overall non-linear response of an FRP-strengthened beam in pure bending.In a preliminary version of the present paper [14], such model has been extended to account for the prestressing of the laminate.The beam is considered simply supported and subjected to uniformly distributed load.According to the assumed application technology, the laminate is first put into tension, then bonded to the beam bottom surface, and finally fixed at both its ends by suitable connections.The beam and laminate are modelled according to classical beam theory.The adhesive is modelled as a cohesive interface with a piecewise linear constitutive law defined over three intervals (elastic response, softening response, debonding).The model is described by a set of differential equations with suitable boundary conditions.Here, we determine an analytical solution to the stated problem, including explicit expressions for the internal forces and interfacial stresses.As an application, we consider the standard IPE series [15] for the steel beam and the Sika ® CarboDur ® system [16] for the FRP strengthening.The latter consists of an epoxy resin adhesive (Sikadur-30 ® ) and pultruded carbon fibre-reinforced polymer laminates (CarboDur ® S).For each considered steel cross section, we first carry out a preliminary design to determine the length and permanent load of the "existing" unstrengthened beam.Then, we imagine to apply the FRP strengthening and calculate the loads corresponding to the elastic limit states in the steel beam, adhesive, and laminate.Lastly, we take into account the ultimate limit state corresponding to the plasticisation of the mid-span steel cross section and evaluate the increased load bearing capacity of the strengthened beam [17,18].Calculations are carried out according to the Eurocodes [19][20][21] and Italian regulations on FRP strengthening [22,23].

MECHANICAL MODEL
et us consider an I-section steel beam AB of length 2L, simply supported at its ends and subjected to a uniformly distributed load per unit length, p (Fig. 1).As better specified in the following, the load p will actually be a combination of the beam self-weight, g 1 , a permanent load due to non-structural elements, g 2 , and an imposed load, q.The beam is strengthened by an FRP laminate of length 2l adhesively bonded to its bottom surface.As concerns the application technique, we assume that the laminate is first pre-stressed by a suitable axial force, P, then adhesively L bonded to the beam, and finally fixed at both its end sections, C and D. We denote with a = L -l the distance of the anchor points from the end sections of the beam.We denote with b b and h b respectively the width and height of the steel cross section and with h f the thickness of the flange (Fig. 2).Furthermore, we indicate with A b , I b , W b , and Z b respectively the area, moment of inertia, elastic modulus, and plastic modulus of the cross section of the beam.Besides, we denote with bf and tf respectively the width and thickness of the laminate and with t a the thickness of the adhesive layer.Lastly, we indicate with A f = b f t f the area of the cross section of the laminate.Thanks to symmetry, the model can be limited to the left-hand half system (Fig. 3).The generic cross section of the beam is identified by a curvilinear abscissa, s, measured from the anchor point of the laminate, C. Similarly, the generic cross section of the laminate is identified by a curvilinear abscissa, s*, also measured from point C. We denote with w b (s) the axial displacements of points at the beam bottom surface and with wf(s*) the axial displacements of the laminate cross sections.In the proposed mechanical model, the steel beam is considered as a flexible beam, while the FRP laminate is modelled as an extensible strip.An elastic-perfectly plastic behaviour is assumed for steel with Young's modulus E s and design yield stress f yd (Fig. 4a).The behaviour of FRP is assumed elastic-brittle with Young's modulus E f and design tensile strength f fd (Fig. 4b).The adhesive layer is represented by a zero-thickness cohesive interface, which transfers shear stresses, , and no normal stresses.The interfacial stresses depend on the relative displacements, w = wf -wb, between the laminate and the bottom surface of the beam.The interface behaviour is assumed linearly elastic for shear stresses up to a limit value,  ; then, a linear softening stage, corresponding to progressive damage, follows; lastly, debonding occurs.For w ≥ 0, the cohesive interface law is given by the following piecewise linear relationship (Fig. 4c where k and ks are the elastic constants for the elastic and softening responses, respectively; w0 and wu are the relative displacements at the elastic limit and start of debonding, respectively.The elastic constant for the elastic response can be taken as k = Ga / ta, where Ga is the shear modulus of the adhesive.

STRUCTURAL RESPONSE
o determine the structural response of the FRP-strengthened beam, it is necessary to distinguish between different stages of behaviour.In what follows, stage 0 refers to the unstrengthened beam, subjected to its self-weight and permanent loads.In stage 1, the laminate is pre-stressed and fixed to the beam.At this point, there is yet no composite action between the beam and laminate, which however are both stressed and deformed because of the dead load and pre-stressing.In stage 2, the beam and laminate behave as an elastic composite structure under the imposed loads.This stage ends when one of the composing elements -beam, laminate, or adhesive -reaches its elastic limit.In stage 3, non-linear response is expected due to plasticity of the steel beam and/or softening of the adhesive layer.However -as the numerical examples will show -the steel beam turns out to be always the weakest element.Therefore, the load bearing capacity of the system is governed by the plasticisation of the steel beam.

Stage 0 -Unstrengthened beam
The unstrengthened beam is subjected to its self-weight, g 1 , and permanent loads, g 2 , both assumed here as uniformly distributed.Such loads will cause both stress and deformation, however within the linearly elastic behaviour regime.At this stage, the axial force, shear force, and bending moment in the beam respectively are with a s l    .When evaluating ultimate limit states, loads in Eq. ( 2) should be suitably factored [19,21].In real applications, cambering of the beam is often introduced to compensate for the deflection due to dead loads.For simplicity, here we do not consider any cambering and assume the deformed configuration of the unstrengthened beam as the reference configuration for strains and displacements.

Stage 1 -Pre-stressing and fixing of the laminate
During the pre-stressing stage, the laminate is put into tension by an axial force, N f,P = P, applied through the anchor points on the beam bottom surface.Simultaneously, the beam is compressed by the same axial force, which produces also bending because of the eccentricity of the load application point with respect to the beam centreline.The internal forces produced by pre-stressing in the strengthened part of the beam ( The internal forces given by Eqs.(3) are to be added to those given by Eqs. ( 2) to obtain the total internal forces in the beam at the end of the pre-stressing stage.All loads should be suitable factored when evaluating ultimate limit states.
T Due to pre-stressing, points belonging to the beam bottom surface will move towards the mid-span cross section; conversely, points on the laminate will move towards the anchor point C. From the classic assumption of beam theory, that plane sections remain plane, and the constitutive laws for the beam and laminate, we determine the following (positive) displacement for a point S on the beam bottom surface at the abscissa s: and the following (negative) displacement for a point S* of the laminate at the abscissa s*: On curing of the adhesive, the beam and laminate behave as a composite structure.To determine the interfacial stresses through Eq. ( 1), the relative displacements at the interface should be evaluated with respect to the deformed configuration at the end of the pre-stressing stage.To this aim, we consider that points S and S*, initially not aligned, be placed on the same cross section at the end of the pre-stressing operation (Fig. 5).4) and ( 5) into (6), we determine the following relationship between s and s*:

Stage 2 -Application of imposed loads -Linear response
When imposed loads are applied to the strengthened beam, the relative displacement at the interface (with respect to stage 1) turns out to be where w b,Q (s) and w f,Q (s*) respectively are the axial displacements of the beam bottom surface and laminate produced by the imposed load, q.In Eq. ( 8), the abscissa s* should be calculated through Eq. (7).Forw ≤ w0, the interface behaves elastically, so that Eq. ( 1) yields the interface shear stress Fig. 6 shows a free-body diagram of an elementary segment of the strengthened beam included between the cross sections at s and s + ds.From static equilibrium, the following equations are deduced: where Nb,Q, Vb,Q, and Mb,Q respectively are the axial force, shear force, and bending moment in the beam; Nf,Q is the axial force in the laminate due to the imposed load.By solving the differential problem defined by Eqs. ( 9) and ( 10) -as explained in the Appendix -the following expressions are obtained for the interfacial shear stress, and internal forces in the beam, ) where and  are constant parameters defined by Eq. (A10) in the Appendix.

Stage 3 -Application of imposed loads -Non-linear response and failure of the system
In stage 3, non-linear response is expected due to either plasticity of the steel beam or softening of the adhesive.However, as the numerical examples below will demonstrate, for current material properties and geometry, the steel beam turns out to be the weakest element of the system, while the adhesive behaves elastically up to very high values of the imposed load.
A detailed description of the structural response in stage 3 would require considering the progressive plasticisation of the beam at the mid-span and neighbouring cross sections.At the same time, large deformations and displacements are expected to occur, bringing further non-linearity into the problem.For the sake of simplicity, here we do not explicitly analyse the above sketched non-linear response and limit ourselves to consider the ultimate limit state corresponding to the complete plasticisation of the mid-span cross section of the beam.

APPLICATION
or illustration purposes, we apply the model to standard IPE steel beams [15].For each cross section of the series from IPE 120 to IPE 600, we first carry out a preliminary design to determine the span and permanent load of the "existing" unstrengthened beam.Then, we imagine to apply the Sika ® CarboDur ® FRP strengthening system [16] and calculate the loads corresponding to the elastic limit states in the beam, adhesive, and laminate.Lastly, we take into account the ultimate limit state corresponding to the plasticisation of the mid-span steel cross section and evaluate the increased load bearing capacity of the strengthened beam.

Material properties and geometry of structural elements
The material properties, factored according to the Eurocodes [20,21] and Italian regulations on FRP strengthening [22,23]

F Pre-sizing of steel beams
The span of the "existing" beam, 2L, and permanent load due to non-structural elements, g 2 , are fixed by assuming that, under the action of dead loads, the maximum stress in the mid-span cross section of the beam is less than 33% of the yield stress and the mid-span deflection is less than 1/800 of the span: (2 ) 384 800 By taking the equal sign in inequalities (13)

Pre-stressing of FRP laminates
The pre-stressing tensile axial force in the FRP laminates is assumed to correspond to 50% of the design tensile strength: Depending on the steel cross section, one or two laminates CarboDur ® type S613 or S626 are used, as illustrated in Tab.
3. The Table also shows the normal stresses at the upper and lower surfaces of the mid-span cross section of the beam,  1  and  1  , respectively, produced by the dead loads and pre-stressing.By recalling Eqs. ( 2) and ( 3), such stresses can be computed as follows: It can be verified that all the computed values are in magnitude less than the design yield stress, f yd .

Elastic limit state of the system
Now, we turn to evaluate the imposed load for which the strengthened beam abandons the range of linearly elastic behaviour.This load will be the minimum between the loads corresponding to the elastic limit states in the beam, adhesive, and laminate.We start from the elastic limit state of the adhesive, which occurs when the maximum shear stress in the interface reaches the design strength, .The abscissa, s , where the shear stress is maximum is first determined by differentiating Eq. ( 11) and setting the derivative to zero: Hence, the maximum shear stress value turns out to be By putting max 0    , the corresponding imposed load is obtained, where Q = 1.5 is the partial factor for variable actions [19,21].
Moving on to consider the elastic limit state of the laminate, we observe that the maximum axial force, Nf max, occurs at the mid-span cross section of the beam.By recalling Eqs. ( 3), (10), and ( 12), we obtain where  P = 1.0 is the partial factor for pre-stressing actions [19,21].By putting N f max = f fd A f , the corresponding imposed load is obtained, Lastly, we consider the elastic limit state of the beam.To this aim, the normal stresses at the upper and lower surfaces of the mid-span cross section of the beam, 2  and2  , respectively, are computed.By recalling Eqs. ( 2), (3), and ( 12), we find where G (equal to G1 = 1.3 for the self-weight, g1, and G1 = 1.5 for the other dead loads, g2) represents the partial factor for permanent actions.Hence, by putting  2  = f yd and 2  = f yd , respectively, and taking the minimum between the two resulting values of the imposed load, we obtain the imposed load corresponding to the elastic limit state of the beam, Tab. 4 summarises the results obtained by applying Eqs. ( 18), (20), and (22).The weakest element turns out to be always the steel beam, whose yielding stress determines the load corresponding to the elastic limit state of the system.

Evaluation of increased bearing capacity
To evaluate the increase in the load bearing capacity given by the strengthening system, we first calculate the load bearing capacity of the unstrengthened steel beam.The imposed load, q u , corresponding to the complete plasticisation of the midspan cross section of the beam is calculated as follows: where M p = f yd Z b is the plastic moment of the steel cross section.

Cross section
Limit load for adhesive q a (kN/m)  Moving on to the strengthened beam, the results of the previous section suggest that the ultimate limit state of the system corresponds to plasticisation of the steel beam and not to debonding of the adhesive or rupture of the laminate.In the strengthened beam, the mid-span cross section is subjected to both axial force and bending moment.According to the Eurocodes [20], Eq. ( 23) can still be used, provided that Mp is replaced by MpN, which takes into account a possible reduction of the plastic moment due to the axial force: , for 1 0.5 where, in our notation, By substituting Eqs. ( 2), (3), and ( 12) into ( 23)-( 25), and solving for q, the load bearing capacity of the strengthened beam, q s , is determined.Tab. 5 summarises the results obtained for the analysed cases, showing the increase in the load bearing capacity of the system, q = qs -qu.The percent increase ranges between 30% and 50%.

CONCLUSIONS
e have presented the mechanical model of a simply supported steel beam subjected to uniformly distributed load, strengthened with a pre-stressed FRP laminate.An analytical solution has been determined for the differential problem that describes the structural response of the strengthened beam.The model has been applied to study IPE steel beams strengthened by using the Sika ® CarboDur ® FRP strengthening system.Analysis of the elastic limit states in the steel beam, adhesive, and laminate has shown that the steel beam is always the weakest element of the system.The softening response of the adhesive is not reached in practice being preceded by the plasticisation of steel.For the sake of simplicity, however, we have not analysed the non-linear structural response resulting from the progressive plasticisation of the beam at the mid-span and neighbouring cross sections.This behaviour could indeed be the subject of future studies, which should also take into account the expected large deformations and displacements, as well as the possible lateral torsional buckling.Here, the complete plasticisation of the mid-span cross section of the steel beam has been assumed as the ultimate limit state of the system.Correspondingly, the increased load bearing capacity of the strengthened beam has been evaluated.
Then, we differentiate also Eq. ( 7) with respect to s and obtain the following expression, whose numerical values turn out to be approximately equal to 1 for current geometric and material properties: By substituting Eqs. ( 4), (5), and (A1) into (A3), and considering (A4), we obtain By further differentiating Eq. (A5) and recalling (A2), we have Lastly, by substituting Eqs. ( 10) into (A6), after simplification, we obtain the following differential equation for the interfacial shear stress: where the shear force is b Q V s q l s , ( ) ( ).   (A8)

Solution of the differential problem
The differential problem stated in the previous section can be solved analytically.In fact, the general solution to Eq. (A7) for the interfacial shear stress is where C 1 and C 2 are integration constants and the constant parameters where the approximations are justified by the values of the exponential functions in practical problems.Lastly, we obtain the following final expression for the interfacial shear stress: which is equivalent to Eq. ( 11) in the main text above.By substituting Eqs.(A8) and (A13) into (10) and integrating, we obtain where C 3 and C 4 are integration constants.These are determined by imposing the continuity of the axial force and bending moment at the cross section of the anchor point:

Figure 2 :
Figure 2: Cross section of the strengthened beam.

Figure 3 :
Figure 3: Mechanical model of the strengthened beam.

Figure 5 :
Figure 5: Displacements of beam and laminate at the end of the pre-stressing stage.Alignment of points S and S* after pre-stressing requires that

Figure 6 :
Figure 6: Free-body diagram of an elementary beam segment.
, are the following:  steel (grade S235): Young's modulus, E s = 210 GPa; characteristic yield stress, f yk = 235 MPa; partial factor for material,  s = 1.05; design yield stress, f yd = f yk /  s = 223.81MPa;  adhesive (Sikadur ® -30): shear modulus, Ga = 4.923 GPa; characteristic strength, k = 15 MPa; environmental conversion factor,  a = 0.85; partial factor for material,  a = 1.2; design strength,  0 =  a  k /  a = 10.63 MPa;  laminate (CarboDur ® S): longitudinal Young's modulus, Ef = 165 GPa; characteristic tensile strength, ffk = 3100 MPa; partial factor for material,  f = 1.1; design tensile strength, f fd =  a f fk /  f = 2395.45MPa.Concerning the geometry of structural elements, the geometric properties of the steel cross sections are listed in Tab.1.The thickness of the adhesive layer is assumed t a = 1 mm.The laminates have width b f = 60 mm and thickness t f = 1.3 (type S613) or 2.6 mm (type S626), depending on the steel cross section.The distance between the beam supports and laminate anchor points is a = 500 mm for all cross sections.
for the internal forces in the beam turn out to be ):

Table 1 :
Geometric properties of steel cross sections.

Table 2 :
, the maximum theoretical values of 2L and g 2 are first determined.Then, by rounding down such values to the nearest integer multiples of 500 mm and 0.50 kN/m, respectively, the final values assumed in subsequent calculations are determined (see Tab. 2).Pre-sizing of steel beams.

Table 4 :
Imposed loads at elastic limit states.

Table 5 :
Load bearing capacity of the system.