Principio de responsabilidad de la Superintendencia de Bancos y Seguros

A variety of possible approaches have been proposed for the problem of the interface between concrete and FRP. Closed form solutions have been found for highly idealised linear elastic models [101], [135] in which the relative movement between the two materials is due to elastic deformation of the bonding agent (epoxy) assumed with a finite thickness. At this level of idealization a closed form solution to the interface problem can be derived as follows.

The problem can be greatly simplified if the following assumptions are made:

• Linear elastic behaviour of all components

• Unidirectional FRP plates with their fibres aligned with the beam axis • Bond line has no axial and bending stiffnesses

• The adhesive is homogeneous and uniform along the bond line

• Linear strain distribution is assumed in the beam and strengthening plate sections

−γti = d(v0 − φ) (4.23)

where γ is the shear strain in the adhesive layer, ti is the thickness of the adhesive, d is the distance of the axis of the FRP plate from the axis of the concrete section, v0 is the rotation of the concrete section and φ is the average cross sectional rotation defined by the difference of the axial displacement of the concrete section and the plate and the distance between their axes d (see Fig. 4.10). Assuming small deformations:

φ = u2− u1

d (4.24)

According to the convention used in the figure v0 and φ are counter clock- wise while γ is a clockwise angle.

The axial strains of beams and plate are given by: u10 = C EAB , u20 = T EAp (4.25) where C is the compressive axial force in the concrete, T is the tensile axial force in the FRP and all other symbols have their usual meaning with the suffix B for beam and p for plate.

Since the deflection is assumed to be the same for all points in the section, the curvature (v00) may be directly used to define the bending moments in the beam and the FRP plate, respectively (Fig.4.11(a)).

MB = EIBv00, Mp = EIpv00 (4.26)

The interface shear stress τ is related to the shear strain γ through the shear modulus G.

τ = Gγ (4.27)

The equilibrium of an elemental segment of plate yields: τ = 1

b dT

dx (4.28)

where b denotes the width of the FRP plate.

Taking the equilibrium of forces and moments at any cross section

C = −T (4.29)

Figure 4.10: FRP strengthened beam considering interface slip: (a) beam layout; (b) cross section kinematics

Figure 4.11: Equilibrium of forces: (a) force/moment distribution in cross section; (b) interface shear stress on differential plate element

Differentiating φ with respect to x and using equation 4.29 we get: φ0 = T

dEAs

(4.31) where EAs = 1/EAb+ 1/EAp. Substituting equation 4.26 into 4.30 and

rearranging

T = M (x) − EIsv00

d (4.32)

where EIs = EIb + EIp. Differentiating equation 4.32 with respect to x

and substituting into 4.28

v000 = M 0(x) − bdτ (x) EIs

(4.33) Differentiating equation 4.23 twice with respect to x, substituting equa- tion 4.31 and rearranging

γ00 + d ti

(v000 − T 0 dEAs

) = 0 (4.34)

Substituting equation 4.28 and 4.33 into 4.34, The shear differential equa- tion is obtained: τ 00(x) − α2τ (x) = − dG tiEIs M 0(x) (4.35) where α2 _{= bG/t} i[1/EAs+ d2/EIs].

The solution of the above linear differential equation with constant coef- ficients has the well known form:

τ (x) = A cosh(αx) + B sinh(αx) + τp(x) (4.36)

where τp is a particular solution.

To calculate the shear stresses as a function of x it is necessary to find a particular solution τp(x) and impose the appropriate boundary conditions.

In the case of a simply supported partially plated beam under uniform loading the solution has the following form:

τ (x) = dG α2_t iEIs w L 2 − xp 

+ sinh(αxp) − tanh(αLp/2) cosh(αxp) α  − dG 2α2_t iEIs w(L − Lp) cosh(αxp) cosh(αLp/2) + Gw 8αtidEAs (L2− L2 p)  1 − EI EIb  [(sinh(αxp) − tanh(αLp/2)cosh(αxp) (4.37)

The results obtained by using the above approach have been compared to those obtained by using a more general FEM technique. The structural scheme is simplified in both cases (all materials are homogeneous, isotropic and linear elastic, as required by the analytical approach) and steel rein- forcement is not included. However the kinematical assumptions of the FEM model are less stringent and it can be used to test the assumptions of the analytical method.

Different shear moduli G have been used in the analyses. The values of G have been kept lower than typical values for the adhesive to reduce the effect of its axial stiffness (lower G implies lower E) in the FEM model. In fact, axial forces in the adhesive cannot be excluded in the latter. Note that the analyses have been carried out only to check a mathematical method and not to provide results on a specific structure.

The details of the modelled beam are reported in Fig 4.12 and the results are reported in Fig 4.13. The stress distribution predicted by the two meth- ods are generally in good accordance. However, local stresses at the tip of the laminate are fairly different. This is a critical point as these local stresses are often responsible for end peeling. We note that it is not demonstrated, although likely, that FEM results obtained are better than the analytical ones.

Other approaches [135] with less stringent kinematical assumptions giving also the possibility of considering more than three layers of different materials involve the use of Fourier series and are not reported for brevity.

We conclude this section observing that elastic approaches, although ap- pealing for their simplicity, have little relevance for practical applications. This is because the strongly nonlinear behaviour of cracked concrete in the range of loading conditions of practical interest and the nonlinear proper-

Figure 4.12: Elastic interface: Model structure used for solution validation

ties of the interface play a key role in the behaviour of retrofitted structures at both Serviceability and Ultimate limit states. A more appropriate ap- proach to the problem of the interface involves the use of cohesive models as described in the following section.