Deflection behavior of a prestressed concrete beam reinforced with carbon fibers at elevated temperatures

Abstract

Fiber reinforced polymer(FRP) have unique advantages like high strength to weight ratio, excellent corrosion resistance, improving deformability and cost effectiveness. These advantages have gained wide acceptance in civil engineering applications. FRP tendons for prestressing applications are emerging as one of the most promising technologies in the civil engineering industry. However, the behavior of such members under the influence of elevated temperatures is still unknown. The knowledge and application of this could lead to a cost effective and practical considerations in fire safety design. Therefore, this study examines the deflection behavior of the carbon fiber reinforced polymer(CFRP) prestressed concrete beam at elevated temperatures. In this article, an analytical model is developed which integrates the temperature dependent changes of effective modulus of FRP in predicting the deflection behavior of CFRP prestressed concrete beams within the range of practical temperatures. This model is compared with a finite element mode (FEM) of a simply supported concrete beam prestressed with CFRP subjected to practical elevated temperatures. In addition, comparison is also made with an indirect reference to the real behavior of the material. The results of the model correlated reasonably with the finite element model and the real behavior. Finally, a practical application is provided.

Keywords

FRP / CFRP / concrete / elevated temperatures / deflections / prestress

Introduction

FRP has outstanding characteristics as a non-corrosive reinforcement. This increases the service life of structures [¹]. Further advantages of FRP reinforcement in the field of structural applications are its high strength-to-weight ratio, good fatigue behavior, low relaxation, electromagnetic neutrality, and easy handling and installation [²]. FRP, generally have a linear elastic response in tension up to failure and fails in brittle manner [³].

Prestressing the fiber reinforced polymer tendons provide stiffer behavior, crack formation in the shear span is delayed, and the cracks, when they appear, are smaller [⁴]. Thus, as a result of reduced cracking, serviceability and durability is enhanced when the fiber reinforced polymer tendons are prestressed [⁵]. In a different study, it was found that the deflection behavior of prestressed concrete beams reinforced with FRP tendons depends on the deformation of material, prestressing degree and bond strength [⁶]. Their method predicted improved results over the American Concrete Institute’s method. Concrete members prestressed with basalt fiber reinforced polymers were experimentally studied for deflection, cracking growth and flexural stiffness [⁷]. Basalt polymers were found to be useful in prestressed applications. Numerical investigation has been carried out to reveal the flexural behavior of continuous concrete beams prestressed with bonded FRP and steel tendons at room temperature [⁸]. It was found that FRP tendons show improved behavior with respect to crack pattern, deformation characteristics and neutral axis depth compared to steel tendons [⁸].

As a result of the crack formation in the concrete, the steel reinforcement is exposed to the environment. Exposure to oxygen and moisture for a longer period of time leads to the corrosion of the steel rebars affecting the serviceability of the structure [⁹]. When the structure is reinforced with carbon fiber reinforced polymer, the corrosion effect can be significantly reduced.

Concrete gets de-moisturized rapidly resulting in the formation of cracks when it is subjected to elevated temperatures [³]. This causes the FRP to burn and eventual de-bonding of the polymer. The fire also causes FRP delaminat ion, cracking and deformation [¹⁰]. Polymer resin or matrix is the key component contributing to the thermo-mechanical behavior of FRP rebars. The polymer resin softens and loses its stiffness at higher temperatures. Glass transition temperature (T_g) is the midpoint of the range of temperatures over which the FRP polymer matrix undergoes a change from hard and brittle to viscous and rubbery [¹⁰]. When the temperature exceeds the glass transition temperature, the elastic modulus of the polymer decreases significantly. The glass transition region is the most significant practical region of FRP for design purposes. This is because the system undergoes significant plastic deformations beyond this region, resulting in structural collapse [¹]. Due to the softening and the reduced stiffness at this temperature (T_g), the resin is no longer able to transfer stress from concrete to fiber. This leads to increased crack widths and deflections [¹].

In structural applications, CFRP composites are effective in increasing the shear capacity of reinforced concrete beams. These composites have been available over the past two decades [¹¹] to strengthen the shear capacity of reinforced concrete beams. Common approaches include externally bonded CFRPs and wrapping prestressed CFRP sheets around concrete beams. CFRP fabric is available in continuous unidirectional sheets supplied on rolls that can be easily tailored to fit any geometry and can be wrapped around almost any profile.

Additionally, CFRP tendons are also proving to be promising as the reinforcements for presstressing in concrete structures as well. This is due to its high modulus of elasticity, high resistance to corrosion and excellent long-term and fatigue characteristics. While several studies [¹²–¹⁴] have been conducted on CFRP prestressed concrete beams which were carried out at room temperatures, very little attention has been given to their behavior at elevated temperatures. Therefore, this basic work examines the effect of higher temperatu-res on the deflection of prestressed carbon fiber reinforced polymer beams.

Proposed approach

Formulation of E-modulus

The assumptions made during the formulation of effective modulus of the CFRP prestressed beam at elevated temperatures are as follows:

• The fiber composite was assumed to be unidirectional and the fibers to be continuous and parallel.

• Good bond exists between fibers and matrix.

• Strains experienced by the fibers, matrix and concrete are approximately equal [¹⁵].

The effective modulus of prestressed FRP system can be approximated as:

(1)E_{p,frp}=\frac{f_{ps}}{ϵ}

Where f_{ps} is the stress in the prestressing system. Substituting the values of strain (ϵ) and coefficient of thermal expansion({\alpha }_L)into equation (1) provides the effective modulus with respect to temperature (\mathrm{d}T),

(2)E_{p,frp@T°\mathrm{C}}=\frac{f_{ps}}{{\alpha }_L\left(\mathrm{d}T\right)}

Development of a deflection model

At elevated temperatures, the concrete starts to demoisturize and shrinks. This results in cracks. Thus, the deflection ({\mathrm{\Delta }}_T) at elevated tempe-ratures depends on the shrinkage curvature ({\phi }_{et}), length (L) and shrinkage support conditions (K_s) [¹]. The shrinkage based support conditions of the concrete members have been provided in ACI 435R [¹⁶]. Fig. 1 shows the eccentricity (e) and the prestress force (P) acting on a beam.

(3){\mathrm{\Delta }}_T=K_s{\phi }_{et}L

Assuming that the plain sections remain plain and the creep strain is proportional to stress, the shrinkage curvature due to elevated temperatures can be expressed as [¹].

{\phi }_{et}=\frac{M}{EI}=\frac{Pe}{EI}=\frac{Pe}{E_c{I}_e}

{\phi }_{et}=\frac{Pe}{E_c{I}_e}

Where E_{c\prime } and I_e are respectively moment, modulus of elasticity of concrete, and effective moment of inertia.

P=A_{p,frp}{ϵ}_h{E}_{p,frp@T°\mathrm{C}}

A_{p,frp} is the area of prestressed tension reinforcement and is strain in concrete due to shrinkage.

Substituting the value of P into the equation of shrinkage curvature, we get

(4){\phi }_{et}=\frac{A_{p,frp}{E}_{p,frp@T°\mathrm{C}}{eϵ}_h}{E_c{I}_e}

Using the above equation in equation (3) and simplifying provides:

{\mathrm{\Delta }}_T=\frac{K_s{eLA}_{p,frp}}{I_e}\left[\frac{E_{p,frp@T°\mathrm{C}}{ϵ}_h}{E_c}\right]

It is well known that the modulus of elasticity of concrete is dependent on the water-cement ratio in the mixture, the age of concrete, the method of conditioning, and the amount and nature of the aggregates. The modulus decreases rapidly with the rise in temperature [¹⁷]. This degradation in modulus of elasticity can be attributed to excessive thermal stresses and physical and chemical changes in concrete microstructure [¹⁷]. Thus, the modulus of elasticity of concrete can be expressed as E_{c}r where r is the strength reduction factor due to elevated temperatures. This reduction of elastic modulus with varying temperatures has been reported by [¹⁷–¹⁹].

Assume \varphi =K_s{eLA}_{p,frp/I_e}

where \varphi is the deflection constant.

Therefore,

{\mathrm{\Delta }}_T=\varphi {ϵ}_h\left[\frac{E_{p,frp@T°\mathrm{C}}}{E_{c}r}\right]

{ϵ}_h can be expressed as {\alpha }_c{T}_{frp}, where {\alpha }_c is the CFRP expansion per degree temperature of variation and T_{frp} is the temperature. Therefore, temperature dependent deflection of the prestressed FRP system is:

(5){\mathrm{\Delta }}_T=\varphi {\alpha }_c{T}_{frp}\left[\frac{E_{p,frp@T°\mathrm{C}}}{E_{c}r}\right]

FEM and description of the model

The failure of reinforced concrete structures, such as prestressed beams, under increasingly monotonic static loading conditions are of interest to us. These failures have been studied using discrete crack models by [²⁰–²³], among many. Most of these approaches have disadvantage of being computationally expens-ive and in one of the approaches [²⁴], the failure is dependent on the geometry and the topology of the mesh. Remeshing and refinement can overcome this challenge; however, these computations are also expensive. Mesh free methods have been proposed by [²⁵–²⁹]. However, mesh free methods also tend to be computationally more expensive than FEM. Therefore, in view of our financial constraints, a very simple first approach was under taken here. This can later be used by other researchers to advance the knowledge. In addition, the following basic guidelines were used for the FEM beam models: a) define constants, b) material properties, c) establish the FE mesh, element numbers and nodes, d) define real loading, boundary conditions, and constraints, and e) identify the unknown quantities.

The modeling of the beam was carried out using Solidworks and the analysis was performed using ANSYS Workbench. A commonly used cross section of 375×550 mm in the prestressing applications was preferred as the simply supported beam to be modeled and analyzed using ANSYS. The span of the beam was 4500 mm. Based on the cross section and span of the beam, a general adapted loading of symmetric nature was applied to the beam. Two vertical loads, each of 200 kN were acting at a distance of 1.5 m and 3.0 m respectively from the right support. A uniform load of 5.67 kN/m inclusive of the self-weight of the beam was considered to be superimposed live load based on the minimum load specified by ASCE 7-05 [³⁰]. The beam was considered to be prestressed with CFRP. Eight #6 CFRP rebars distributed along two layers were used as the tensile reinforcement to achieve the under reinforced section due to which the reinforcement yields prior to concrete. To analyze the prestressed beam reinforced with CFRP, a steady-state thermal analysis was carried out using ANSYS and the results of the analysis were imported into the static structural analysis of the software.

Modeling using Solidworks

Modeling of a concrete beam is discussed in this section. Initially, a rectangular beam of 375 × 550 mm cross section was modeled using Solidworks. Then, taking the overall concrete cover as 75 mm, four circular profiles of 19.05 mm diameter were cut to procure space for the bottom layer of tension reinforcement. Similarly, the spaces for the upper layer of tension reinforcement of 19.05 mm diameter were extruded with a spacing of 75 mm center to center. The distance between the upper and the bottom layer of reinforcement was 50 mm center to center. Finally, the rectangular cross section was extruded along its length to develop a span of 4500 mm. Fig. 2 shows the front and isometric view of the beam geometry modeled in Solidworks.

Thereafter, the upper and bottom layers of tension reinforcements were individually modeled in Solidworks. Eight circular cross sectional bars of 19.05 mm diameter were extruded to a length of 4500 mm to form the tension reinforcement, four each for the upper and bottom layers of reinforcement respectively. The horizontal distance between the reinforcements was 75 mm and the vertical distance between the upper and bottom reinforcement was 50 mm. Fig. 3 shows the setup of the tension reinforcement modeled with the above procedure.

Using the assembly feature of Solidworks, the components of plain concrete beam and the reinforcements were assembled. Fig. 4 displays the front and isometric views of the assembled beam. This means the reinforcement has been placed in the beam.

Analysis using ANSYS

The geometry of the beam modeled in Solidworks was imported in ANSYS which is shown in Fig. 5. The properties of the materials defined are shown in Table 1.

Firstly, the defined material properties of CFRP and concrete were assigned to the parts of the beams. Next, in order to carry out the finite element analysis of the model beam, a fine size mesh was created on the beam. A fine size meshing of the model divides the beam in 44179 elements and 87347 nodes. Fig. 6 shows the fine sized meshing generated on the model beam.

At elevated temperatures, the polymer transitions from a hard or brittle state to viscous or rubbery state. This is known as glass transition range. The special carbon material used in this analysis has this approximate range from 200–300°C. The glass transition region is the most significant practical region of FRP for design purposes. This is because the system undergoes significant plastic deformations within this range and results in structural collapse [³¹]. A steady state thermal analysis module in finite element method was used to determine the deflection effect of temperature on the model.

In order to evaluate worst practical scenario, a temperature range of 200–350°C was applied to the body of the beam including the CFRP reinforcements. Fig. 7 displays the steady-state analysis.

The solution obtained through the steady-state analysis was imported in the setup of the static structural analysis. This includes all the material properties, geometry, meshing and thermal loads. A simply supported beam was defined by fixing bottom edge of one face and applying a displacement at the other edge of the opposite face. As can be seen in Fig 8, the simple supports are defined with label A and label E respectively. A uniformly distributed live load of 1.92 N/mm² was applied at the top surface of the beam, as defined by label G in Fig 8. Two concentrated loads of 200 kN were applied at distances of 1.5 m and 3.0 m respectively from the right support. From Fig 8, Labels C and D define the concentrated loads acting on the beam. The prestress force of 750 kN acting at the two faces of the beam can be seen in Fig. 8, which are labeled as B and F respectively.

With all the defined boundary conditions, live load, concentrated load and prestress force, the modeled was solved for directional deformation in y-axis. The solution of static structural analysis for the concrete beam prestressed with CFRP reinforcement at a temperature of 300°C yielded a deformation of 44.24 mm. Fig. 9 shows the deformation obtained through the simulation of model beam in ANSYS.

Comparison of the models

The results of the analytical model were compared with the finite element model at various temperatures ranging from 200°C to 350°C. The results of this comparison are presented in Table 2. It is clear from the table that the results of the analytical model correlated well with the results of the finite element model.

The overall percentage difference between the trendlines of analytical deflections and finite element deflections was within the range of±11%. Since the polymer matrix undergoes significant practical changes when the temperature exceeds upper limit of glass transition temperature, a reversal in the trend of the deflections was noticed at temperatures above 300°C.

Application and comparison of results

FRP due to its outstanding characteristics has been increasingly gaining applications in the field of civil engineering. Fiber reinforced polymers have been widely used as a replacement to conventional steel reinforcement. However, the performance of prestressed CFRP structures in fire conditions is relatively unknown. The lack of design code for the design of prestressed CFRP at elevated temperatures has restricted the applications of CFRP. The preceding predictive modeling and analysis of prestressed CFRP beam at elevated temperatures using ANSYS has led to a cost effective, efficient and practical considerations in fire safety design.

At elevated temperatures, fiber reinforced polymers tend to deflect more when compared to steel reinforcement as the matrix in the fiber composite loses its stiffness when the temperature exceeds glass transition temperature. An example with the explanations to calculate deflection of the prestressed CFRP beam is presented in the following discussion.

A simply supported rectangular beam reinforced with CFRP of cross section 375 mm × 550 mm is used. The span of the beam is 4500 mm as shown in Fig. 10. Two vertical loads of 200 kN are spaced at 1500 mm; f_c^{\prime }=40\mathrm{N}/{\mathrm{m}\mathrm{m}}^2; Eight #6 CFRP rebars were used as the tension reinforcement. Overall concrete cover=75 mm. Density of Concrete= 2400kg/m³; E_f=550\mathrm{k}\mathrm{N}/{\mathrm{m}\mathrm{m}}^2; E_m=3.5\mathrm{k}\mathrm{N}/{\mathrm{m}\mathrm{m}}^2; V_f=0.7; V_m=0.3; {\alpha }_f=0.02\times {10}^{-6}/°\mathrm{C}; {\alpha }_m=10\times {10}^{-6}/°\mathrm{C}; k_s=0.49; Initial Prestress force acting on the beam is 750 kN. .

In order to calculate the deflection, we need the effective moment of inertia. But, effective moment of inertia is a function of gross mome-nt of inertia (I_g), cracking moment (M_{cr}), applied moment (M_a) & cracking moment of inertia (I_{cr}). The calculations to determine these quantities are as follows:

Since the loading on the simply supported beam is symmetrical, reactions at the supports of this simply supported beam are calculated as: R_A= R_B = [{200+200+(5.67×4.5)}]/2. Therefore, the reactions (R_A & R_B) at the supports are approximately 212.75 kN.

The applied moment or the maximum moment occurs at the mid-span of the beam which can be calculated as:

M_a=\left\{R_A(L/2)\right\}-\left\{w\left(L\right)(L/2)\right\}-\{200\times 0.75\}, where w is the superimposed live load inclusive of self-weight. Therefore,

{\mathrm{M}}_{\mathrm{a}}=(212.75\times 2.25)-(5.67\times 2.25\times (2.25/2))-(200\times 0.75)=156\mathrm{k}\mathrm{N}\cdot \mathrm{m}

.

The gross moment of inertia of a rectangular cross section is provided by I_g={\mathrm{b}\mathrm{d}}^3/12, where bis the width of the beam and dis the total depth of the beam. Hence, I_g=(375\times {550}^3)/12=52\times {10}^8{\mathrm{m}\mathrm{m}}^4.

The cracking moment (M_{cr}) is defined as the moment when exceeded causes concrete to begin cracking. The cracking moment depends on the modulus of rupture and the gross moment of inertia which can be expressed as M_{cr}=(f_r{I}_g/y_t). Modulus of rupture (f_r) is calculated using the following equation: f_r=0.7\sqrt{f_c^{\prime }}. Where f_c^{\prime } is the compressive strength of the concrete. The distance from the center of gravity of the beam to the extreme tension fiber is referred as y_t, which can be expressed as y_t=h/2. Therefore,

f_r=0.7\sqrt{40}=4.42\mathrm{N}/{\mathrm{m}\mathrm{m}}^2; y_t=550/2=275\mathrm{m}\mathrm{m} and M_{cr}=\frac{f_r{I}_g}{y_t}=\frac{4.42\times 52\times {10}^8}{275}=8.37\times {10}^7\mathrm{N}\cdot \mathrm{m}\mathrm{m}.

Modular ratio, h is the ratio of Elastic modulus’ of FRP and concrete. Elastic modulus of concrete can be calculated as E_c=4700\sqrt{f_c^{\prime }}=29.725kN/mm². From rule of mixtures, elastic modulus of FRP can be calculated as:

E_{frp}=E_f{V}_f+E_m{V}_m=(550\times 0.7)+(3.5\times 0.3)=386.05\mathrm{k}\mathrm{N}/{\mathrm{m}\mathrm{m}}^2

. Therefore,

\eta =E_{frp}/E_c=386.05/29.725=12.98

.

Where E_f, V_f, E_m, V_m and are respectively elastic modulus of fibers, volume fraction of fibers, elastic modulus of matrix, and volume fraction of matrix.

Cracks start to develop as the tensile stresses in concrete are exceeded which leads to the reduction in the stiffness of concrete. The cracked moment of inertia (I_{cr}) is calculated using the transformed area diagram as shown in Fig. 11. In order to transform the FRP to an equivalent concrete area, location of neutral axis is required. Distance of neutral axis from the top fiber (×) can be calculated using ({bx}^2/2)={\eta A}_{p,frp}(d_e-x), where bis the width of the beam, d_e is the effective depth of the beam, \eta is the modular ratio and as defined earlier is the area of the tensile reinforcement. {\eta }_{p,frp}=8*285.06=2280.48{\mathrm{m}\mathrm{m}}^2. Substituting the above quantities into the expression of neutral axis and solution of the quadratic equation yields neutral axis at a distance of 199.04 mm from top fiber.

Using the values of the modular ratio and the neutral axis, the cracking moment of inertia is calculated using I_{cr}={bx}^3/3+{\eta A}_{p,frp}{(d_e-x)}^2. Therefore,

I_{cr}=\left[(375\times {199.04}^3)/3\right]+[12.98\times 2280.48\times (450-{199.04}^2)]=28.5\times {10}^8{\mathrm{m}\mathrm{m}}^4

.

When the applied moment exceeds the cracking moment, the moment of inertia of a beam reduces from gross moment of inertia to cracking moment of inertia. This decrease in moment of inertia is considered using the effective moment of inertia (I_e) which can be calculated using the Branson’s equation for effective moment of inertia provided in ACI 318-05 [³²].

I_e=\left[{(M_{cr}/M_a)}^3{I}_g\right]+\left[\{1-{(M_{cr}/M_a)}^3\}{I}_{cr}\right]. Substituting the above determined values of M_{cr}, M_a, I_g and I_{cr} in the expression gives I_e = [(8.37×10⁷/156×10⁶)³× 52×10⁸] + [{1-(8.37×10⁷/156×10⁶)³} × 28.5×10⁸] = 28.93 × 10⁸ mm⁴. Therefore, as specified by ACI 318-05 (ACI 318R-05), I_e<I_g.

Moment of inertia can now be used to calculate the deflection constant which is \varphi =\left(K_s{eLA}_{p,frp}\right)/I_e, where the eccentricity is the distance from the centroid of the beam to the center of the tensile reinforcement which is calculated as, e= 550–75–(50/2) = 175 mm. Therefore, the deflection constant

\varphi =(0.49\times 175\times 4500\times 2280.48)/(28.5\times {10}^8)=0.30

.

The initial prestress force acting on the beam produces stresses at top and bottom surfaces of concrete. To determine these stresses, we need the overall concrete area (A_c), section modulus (S₁ & S₂) and radius of gyration (r²). Reinforcements are neglected in the calculation of overall concrete area which is provided by A_c = 375×550= 206.25 × 10³ mm². Section modulus (S₁ & S₂) at top and bottom surfaces is given by I_g/c₁ & I_g/c₂ respectively, where c₁ and c₂ corresponds to the distances of the centroid from top and bottom surfaces respectively and I_g is the gross moment of inertia. Therefore, S₁= S₂ = (52×10⁸)/275= 189 × 10⁵ mm³. Radius of gyration (r²) is given by r² = I_g/A_c. Substituting the gross moment of inertia and the concrete area in the above equation yields r² = 25.2 × 10³ mm².

Stresses due to intial prestress

The top and bottom stresses due to initial prestress force only are given by f₁ = -P_i/A_c[1-(ec₁/r²)] and f₂= -P_i/A_c[1+(ec₂/r²)] respectively.

Thus,

f_1=\left[\frac{-750\times {10}^3}{206.25\times {10}^3}\right]\times [1-\left(\frac{175\times 275}{25.2\times {10}^3}\right)]=+3.308\mathrm{N}/{\mathrm{m}\mathrm{m}}^2

and

f_2=\left[\frac{-750\times {10}^3}{206.25\times {10}^3}\right]\times [1+\left(\frac{175\times 275}{25.2\times {10}^3}\right)]=-10.58\mathrm{N}/{\mathrm{m}\mathrm{m}}^2

Stresses due to dead load

The dead load acting on the beam is given by, w_o = Area of concrete × unit weight of concrete= 0.20625×24= 4.95 kN/m. This dead load produces a mid-span moment which is calculated as M_o = w_oL²/8= 4.95×4.5²/8= 12.53 kN·m. This moment causes stresses at top and bottom surfaces which is computed as follows.

f_1=-\frac{M_o}{S_1}=-\frac{12.53\times {10}^6}{189\times {10}^5}=-0.66\mathrm{N}/{\mathrm{m}\mathrm{m}}^2

f_2=+\frac{M_o}{S_2}=+\frac{12.53\times {10}^6}{189\times {10}^5}=+0.66\mathrm{N}/{\mathrm{m}\mathrm{m}}^2

Stresses due to intial prestress and dead load

The total stresses due to initial prestress and the dead load acting on the beam is given by

f_1=-\frac{P_i}{A_c}(1-\frac{{ec}_1}{r^2})-\frac{M_o}{S_1}=+3.308-0.66=2.648\mathrm{N}/{\mathrm{m}\mathrm{m}}^2

f_2=-\frac{P_i}{A_c}(1+\frac{{ec}_2}{r^2})+\frac{M_o}{S_2}=-10.58+0.66=-9.92\mathrm{N}/{\mathrm{m}\mathrm{m}}^2

Stresses due to effective prestress

At full service loads, the initial prestress incurs the losses due to shrinkage, creep and steel relaxation. The total losses due to shrinkage, creep and steel relaxation accounts approximately for 15% of the initial prestress force. Thus, the effective prestress force can be evaluated as P_e = 0.85×P_i = 0.85×750= 637.5 kN.

Accordingly, the concrete stresses due to effective prestress is given by f₁ = -P_e/A_c[1-(ec₁/r²)] and f₂ = -P_e/A_c[1+(ec₂/r²)]. Thus, the concrete stress in the top and bottom surfaces are:

f_1=+3.308\times 0.85=+2.812\mathrm{N}/{\mathrm{m}\mathrm{m}}^2

and

f_2=-10.58\times 0.85=-9.0\mathrm{N}/{\mathrm{m}\mathrm{m}}^2

Stresses due to superimposed live loads

The live load (w_L) acting on the beam is 5.67-4.95=0.72 kN/m. Thus, this live load produces a mid-span moment evaluated by M_L = w_L²/8= (0.72×4.5²)/8= 1.825 kN·m. Stress produced due to this moment is given by,

f_1=-\frac{M_L}{S_1}=-\frac{1.825\times {10}^6}{189\times {10}^5}=-0.096\mathrm{N}/{\mathrm{m}\mathrm{m}}^2

f_2=+\frac{M_L}{S_2}=+\frac{1.825\times {10}^6}{189\times {10}^5}=+0.096\mathrm{N}/{\mathrm{m}\mathrm{m}}^2

Stresses due to effective prestress, dead load, and live loads

Final concrete stresses produced in the top and bottom faces due to the effective prestress, selfweight and superimposed live load is

f_1=-\frac{P_e}{A_c}(1-\frac{{ec}_1}{r^2})-\frac{M_o}{S_1}-\frac{M_L}{S_1}=+2.812-0.66-0.096=+2.056\mathrm{N}/{\mathrm{m}\mathrm{m}}^2

f_2=-\frac{P_e}{A_c}(1+\frac{{ec}_2}{r^2})+\frac{M_o}{S_2}+\frac{M_L}{S_2}=-9.0+0.66+0.096=-8.244\mathrm{N}/{\mathrm{m}\mathrm{m}}^2

The concrete stresses in the top surface controls the design of the member. Therefore, the final stress at the top surface can be used to evaluate the effective modulus using Eq. (2) at a temperature of 300 °C. The bond between CFRP and concrete is influenced by elevated temperatures. This is accounted for by calculating the thermal expansion of the material [¹⁵]. is the thermal expansion of the CFRP in longitudinal direction. This expansion can be calculated using:

{\alpha }_L=(E_f{V}_f{\alpha }_f+E_m{V}_m{\alpha }_m)/E_L where E_L is the young’s modulus of FRP composite calculated earlier. Using the calculated value of young’s modulus and the other provided parameters, {\alpha }_L can be computed as:

{\alpha }_L=\left[(550\times 0.7\times 0.02\times {10}^{-6})+(3.5\times 0.3\times 10\times {10}^{-6})\right]/386.50=0.0471\times {10}^{-6}/°\mathrm{C}

.

Hence,

E_{p,frp@T°\mathrm{C}}=\frac{2.056}{0.0471\times {10}^{-6}\times 300}=145.5\mathrm{k}\mathrm{N}/{\mathrm{m}\mathrm{m}}^2

CFRP expansion per degree variation of temperature ( can be calculated using the expression provided by [³³]. Hence, {\alpha }_c={\alpha }_L\times T_{frp}\times L\times \mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}. Substituting the coefficient of thermal expansion in longitudinal direction of CFRP ({\alpha }_L) as 0.0471×10^–6, temperature as 300 °C and the length of the beam as 4500 mm yields an expansion of CFRP of 0.063 mm per length of the beam. Finally, the deflection of prestressed CFRP beam subjected to elevated temperatures is calculated using equation (5). Therefore, with

the above derived parameters the deflection is computed as:

{\mathrm{\Delta }}_T=(0.3\times 0.063\times 300\times 145.5)/(29.75\times 0.65)=42.69\mathrm{m}\mathrm{m}

. The percent error in the results obtained from the ANSYS is {(44.24–42.69)/44.24}×100 ≈ 3.5%. In order to make an indirect reference to the real behavior of the material, researchers [³⁴] validated their work against the fire test data on CFRP strengthened reinforced concrete beams. Their beams had similar cross sectional and tension reinforcement areas as ours. It was found that their deflection was approximately 39 mm compared to 42.69 mm of ours. This 9.4% difference can be attributed to higher superimposed loads on our beam.

Conclusions and future studies

In this article, a basic analytical model is developed which integrates the temperature dependent changes of effective modulus of FRP in predicting the deflection behavior of CFRP prestressed concrete beams within the range of practical temperatures. This model is compared with a finite element model of a simply supported concrete beam prestressed with CFRP subjected to practical elevated temperatures. In addition, comparison is also made with an indirect reference to the real behavior of the material. It was found that the results of model correlated reasonably with the finite element model and the real behavior..

It may be noted that this work is a basic first step towards cost effective, efficient and practical consideration in fire safety design of FRPs. There are various prospects of extending the scope of this work. Some of the future areas of study may be to develop models with different fibers, perform experimental work, and carry out comparative analyses. These may further progress considerations in fire safety design.

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