Flexural strengthening

Reinforced concrete elements, such as beams, slabs and columns, may be strengthened in flexure through the use of FRP composites epoxy-bonded to their tension zones, with the direction of fibres parallel to that of high tensile stresses (member axis). The calculations described address both the Ultimate Limit State (ULS) and the Serviceability Limit State (SLS).

Input:

The calculations are based on the assumption that one of the following two desirable failure modes govern the behaviour:

The first step in the calculations is to find the initial strain, ₀, that develops in the extreme fibre of the cross section when the strengthening operations take place. This strain is the result of a moment M₀ (service moment) acting at the critical cross section during strengthening (e.g. due to the self-weight of the structure), and may be calculated based on equilibrium of internal forces and moments.

Serviceability Limit State

For the SLS (Serviceability Limit State), the analysis of the critical cross section is performed, according to EC2, for the two possible load combinations:

• Rare load
• Quasi-permanent load.

For the case of Rare Load the calculations are performed as in the case of the ULS, with the following modifications:

(a)	0.85fcd is replaced by fck;
(b)	Mrd is replaced by the acting moment (under the rare load combination) Mser,r
(c)	fyd (the tension steel stress) is replaced by fs1;
(d)	the stress limitations are fs1 <= 0.8fyk (for steel) and c <= 0.6fck, where the stress in the concrete is given by the following stress-strain relationship of concrete (for c less than 0.002):

For the case of Quasi-permanent Load the calculations are performed as in the case of the ULS, with the following modifications:

(a)	0.85fcd is replaced by fck;
(b)	Mrd is replaced by the acting moment (under the quasi-permanent load combination) Mser,q-p;
(c)	fyd (the tension steel stress) is replaced by fs1;
(d)	c is replaced by c/(1+), where is the creep coefficient;
(e)	the stress limitations are fs1 <= 0.8fyk (for steel) and c <= 0.45fck, where the stress in the concrete is calculated with c replaced by c/(1+).

Bond check

For user-defined dimensions of the FRP cross section geometry (n strips of width b_f and thickness t_f placed in m layers, n/m should be an integer if m > 1) the program calculates the maximum force, N_bd,max, that can be carried by the total number of strips, and the associated bond length, l_bd,max,before debonding of the external reinforcement initiates at the ends (anchorage zone).

At each cross section (say A), equilibrium and strain compatibility equations yield the tensile force N_fd,A carried by each strip. If this force does not exceed N_bd,max, then the bond check is verified, that is failure of the anchorage is not expected, provided that the appropriate bond length l_bd will be available. The bond length corresponding to N_fd,A is calculated.

It was mentioned above that N_fd,A is the tensile force carried by the FRP. This is calculated by multiplying the cross sectional area A_f by the product of elastic modulus times strain, E_ff, where _f results through cross section equilibrium and compatibility. The equations in this case are identical to those used in the ULS, with the provision that the tensile steel reinforcement may not be yielding. Hence the same formulas used for the ULS apply, with:

(a)	Mrd replaced by the design value of the bending moment acting at section A, Msd,A
(b)	fyd replaced by fsd1;
(c)	o taken approximately equal to that corresponding to Mo, times the reduction factor (Msd,A/Msd). This implies the assumption that the bending moment during strengthening at cross section A, Mo,A , is equal to Mo (acting at the critical section) reduced by the factor Msd,A/Msd (note that Msd is acting at the critical section).