Analysis on Flexural Stiffness of GFRP-concrete Hybrid Beams

2.1. GFRP Pultruded Profile

For the convenience of adjustment to the cross section size of beam, an L-shaped GFRP pultruded profile was first fabricated, then two L-shaped profiles were connected with GFRP bar to form a U-shaped profile. To ensure an efficient section, composite action between the GFRP permanent formwork and the concrete is essential. So the mechanical bond is provided with some T-up-stands on the GFRP internal surface which was covered with a curing epoxy mortar, as shown in Fig. (1).

2.2. Beam Specimens

Five rectangular beams and one T-beam were fabricated. All the beams were 2000mm long. The rectangular beams were 250mm in height and 150mm in width. The flange width and thickness of the T-beam were 450mm and 80mm, respectively. The height of the T-section was 310mm and the thickness of web was 150mm, as shown in Fig. (1).

The different configurations of tested beams are shown in Table 1, where beams FB1, FB2 and FB3 were RC beams combined with U-shaped GFRP profile, PB was control RC beam, RB1 was RC beam combined with GFRP only at the bottom of beam and TB was T-beam combined with U-shaped GFRP only around the web of beam.

3.1. The Moment-curvature Curves

For the limitation of space, the moment-curvature curve of the tested beam TB is only shown in Fig. (2).

3.2. The Change Characteristics of Flexural Stiffness

From the moment-curvature curve shown in Fig. (2), it can be seen that both moment-curvature relation and flexural stiffness of hybrid beam change along with the increase of bending moment, and it can be divided into three stages:

(1) Stage I: It was stage I from the initial loading to the cracking of concrete in which the beam behaves elastically while the relationship between moment and curvature was nearly linearity.

(2) Stage II: It was stage II from the cracking of concrete to the yield of tensile steel. The moment-curvature curve appeared a turning. The curvature increased fast while the flexural stiffness decreased with the increase of bending moment.

(3) Stage III: It was stage III after the yield of tensile steel. The moment-curvature curve appeared a second turning. Flexural stiffness continued to decrease while the curvature increased further.

3.3. The Calculation of Flexural Stiffness

Based on tests and for the sake of simplicity the following basic assumptions are made in the derivation of short-term flexural stiffness for hybrid beams:

1. Plane sections before bending remain plane after bending.
2. The GFRP profile web's contribution to the flexural stiffness is ignored.
3. The lower flange area of GFRP profile is converted to the tensile steels area according to the strength equivalent principle.

Based on the above assumptions, the derivation of short-term flexural stiffness for hybrid beam is shown as follows [10]:

1. Geometric relationship According to the plane section assumption, the relationship between mean curvature and mean strain is:

   (1)

2. Physical relationship The tensile steels in beams have not yet yielded in serviceability state, so the stress-strain relationship of steel is still nearly linearity. The elastic-plastic properties should be considered in the stress-strain relationship of concrete’s compression area. The physical relations can be expressed as:

   (2)

3. Equilibrium relationship For the cracked rectangular cross section, as shown in Fig. (3), the mean stress of equivalent rectangular stress block in the compressive area of concrete is ωσ_ck, the compressive depth of section is ξh ₀, and the internal lever arm from the resultant internal compressive force of concrete to tensile force of steel is ηh ₀.

The lower flange area of GFRP profile is converted to the steel area according to the strength equivalent principle.

(3)

By equilibrium of forces at the cracked section we have:

(4)

From the above equations we have:

(5)

By the relationship between mean strain and cracked section strain, namely and , combined with Eq.(2) and (5), the mean strain in tensile steel and extreme compressive fiber of concrete can be obtained by

(6)

(7)

Where, is a compound coefficient of mean strain in the extreme compressive fiber of concrete.

For T-section with compressive flange shown in Fig. (4), by equilibrium of forces we have:

(8)

Where, is the strengthen coefficient for compressive flange.

By using Eq.(1) and substituting Eq. (6) and Eq.(7) into the following Equation, we have:

(9)

By introducing α_E = E_s/E_c, , the expression for flexural stiffness Bs under bending moment M_k is:

(10)

According to the expression of short-term flexural stiffness for reinforced concrete beam provided by Code for Design of Concrete Structures (GB50010-2010), the expression of flexural stiffness for hybrid beams can be taken as:

(11)

(12)

Where: ψ — uneven coefficient of longitudinal tensile reinforcement strain between cracks: If ψ<0.2, take ψ=0.2; if ψ>0.2, take ψ=1.0.

f_tk—characteristic value of tensile strength of concrete.

ρ_te—effective ratio of reinforcement.

σ_sk— tensile stress in steel under the short-term bending moment, which is calculated by Eq.(5).

η— internal lever arm factor of cracked section. Take η=0.87.

3.4. The Calculation of Midspan Deflections

All the hybrid beams were tested under three points loading, so the midspan deflections of which under short-term moment can be taken as follows:

(13)

Where, l is the span of tested beam. The expression of Bs is shown in Eq. (11).

The theoretical values of midspan deflections for hybrid beam can be calculated by using the above formula. The comparison between tested values and computed values is shown ​in Table 2. During serviceability state, the short-term moment M_k of hybrid beams was generally in stage II of moment-curvature curve, so M_k in Table 2 should be taken as the moment M_y when the steel yielded.

It can be seen from the Table 2 that the tested​ values agree quite well with the computed values.