Use of FRP on Concrete Specimen, RC Elements and Components for Higher Load Resistance

Abstract

Most of the RC structures exposed to corrosion environment are getting distressed and there is a loss of capacity of structural elements and components. In addition, existing RC structures are generally weak at joint levels and causing failures due to external events like earthquakes. Present chapter is made explaining the detailed calculations to improve the concrete properties by FRP confinement. A detailed procedure is provided to evaluate the improved strength of RC elements strengthened with FRP plate. Also procedure explained to evaluate improved strength of RC structural component (joint). All these procedures are validated with published related experimental data.

Appendices

Annexure A1 Element Stiffness Matrices

The element stiffness matrices of frame structural elements such as column and beam considering bending deformations are given as follows.

Euler theory

Timoshenko theory

$$\left[ {\begin{array}{*{20}c} {12\,{\text{C}}_{1} } & {6\,{\text{L}}_{1} {\text{C}}_{1} } & { - 12\,{\text{C}}_{1} } & {6{\text{L}}_{{1}} {\text{C}}_{1} } \\ {} & {(4 + \phi_{y1} ){\text{L}}_{1}^{2} C_{1} } & { - 6{\text{L}}_{1} {\text{C}}_{1} } & {(2 - \phi_{y1} {\text{)L}}_{1}^{2} {\text{C}}_{1} } \\ {} & {} & {12\,{\text{C}}_{1} } & { - 6\,{\text{L}}_{1} {\text{C}}_{1} } \\ {{\text{Symmetric}}} & {} & {} & {{(4 + }\phi_{{{\text{y}}1}} {\text{)L}}_{1}^{2} {\text{C}}_{1} } \\ \end{array} } \right]$$

where $\phi_{yi} = \frac{{12\,EI_{yi} }}{{GA_{syi} L_{i}^{2} }}\quad C_{i} = \frac{{EI_{yi} }}{{L_{i}^{3} (1 + \phi_{yi} )}}$

where

L_i:

Length of the ith beam element

Iyi:

Moment of inertia of ith beam element about z-axis

Asyi:

Shear area of the ith beam element along y-axis

E:

Modulus of elasticity

G:

Shear Modulus

In the case to consider axial and torsional deformations in beam or column the following stiffness matrices can be added suitably.

Annexure A2 Concrete Stress Strain Constitutive Relations and Equivalent Compressive Stress Block Parameters

Stress–Strain Models for Confined Concrete

In practice, the concrete in structures is always confined by transverse reinforcement commonly in the form of closely spaced steel spirals or rectangular hoops. In this case, at low levels of stress in concrete, the transverse reinforcement is hardly stressed; hence the concrete is unconfined. The concrete becomes confined when at stresses approaching the uniaxial strength, the transverse strains become very high because of progressive internal cracking and the concrete bears out against the transverse reinforcement, which then applies a confining reaction to the concrete. Thus the transverse reinforcement provides passive confinement [14].

The confinement by transverse reinforcement has little effect on the stress–strain curve until the concrete reaches its maximum stress. The shape of the stress–strain curve at high strains is a function of many variables, the major ones being the following:

• (i) The ratio of the volume of transverse steel to the volume of concrete core, because a high transverse steel content will mean a high transverse confining pressure.

• (ii) The yield strength of the confining steel, because this gives an upper limit to the confining pressure.

• (iii) The ratio of the spacing of the transverse steel to the dimensions of the concrete core, because a smaller spacing leads to more effective confinement.

• (iv) The ratio of the diameter of the transverse bars to the unsupported length of the transverse bar, because a large bar diameter leads to more effective confinement. If the flexural stiffness of the hoop bar is small (small diameter compared to unsupported length), the hoops bow outward rather than effectively confining the concrete.

• (v) The content and size of longitudinal reinforcement, because this steel will also confine the concrete.

• (vi) The strength of the concrete, because low-strength concrete is more ductile than high-strength concrete.

Kent and Park Model

In 1971, Kent and Park [16] proposed a stress–strain curve for concrete confined by rectangular hoops as shown in Fig. A2.1 The suggested relationship combines many of the features of previously proposed curves. A second-degree parabola represents the ascending part of curve and assumes that the confining steel has no effect on the shape of this part of curve or the strain at maximum stress. This essentially means that the ascending curve is exactly the same for both confined and unconfined concrete. It is also assumed that the maximum stress reached by confined concrete is equal to the cylinder strength $f_{c}^{\prime}$ that is reached at a strain of 0.002.

Fig. A2.1

Kent and Park [16] model for Stress–strain curve for concrete confined by rectangular hoops

The relationship for the ascending parabola is given as,

Region AB, $\varepsilon_{c} \le 0.002$

$$f_{c} = f_{c}^{\prime } \left[ {\frac{{2\varepsilon_{c} }}{0.002} - \left( {\frac{{\varepsilon_{c} }}{0.002}} \right)^{2} } \right]$$

(A2.1)

The descending part of the confined concrete is modelled as per following formulations.

Region BC, $0.002 \le \varepsilon_{c} \le \varepsilon_{20c}$

$$f_{c} = f_{c}^{\prime } \left[ {1 - Z(\varepsilon_{c} - 0.002)} \right]$$

(A2.2)

where,

$$Z = \frac{0.5}{{\varepsilon_{50u} - \varepsilon_{50h} - 0.002}}$$

(A2.3)

$$\varepsilon_{50u} = \frac{{3 + 0.002f_{c}^{\prime } }}{{f_{c}^{\prime } - 1000}}$$

(A2.4)

$$\varepsilon_{50h} = \frac{3}{4}\rho_{s} \sqrt {\frac{{b^{\prime \prime } }}{{s_{h} }}}$$

(A2.5)

$f_{c}^{\prime }$:

Concrete cylinder strength in psi

$\rho_{s}$:

Ratio of volume of transverse reinforcement to volume of concrete core measured to outside of hoops.

$$\rho_{s} = \frac{{2(b^{\prime \prime } + d^{\prime \prime } )A_{s} }}{{b^{\prime \prime } d^{\prime \prime } s_{h} }}$$

(A2.6)

$A_{s}$:

Cross-sectional area of the stirrup reinforcement

$b^{\prime \prime }$:

Width of confined core measured to outside of hoops

$d^{\prime \prime }$:

Depth of confined core measured to outside of hoops

$s_{h}$:

Spacing of hoops

Figure A2.2 shows the various parameters and symbols. The parameter Z defines the slope of the assumed linear falling branch. $\varepsilon_{50u}$ is the value of the strain when the stress has fallen to $0.5f_{c}^{\prime }$ (50% of the strength is lost) for the case of unconfined concrete. The corresponding value of strain for confined concrete is $\varepsilon_{50c}$. $\varepsilon_{50h}$ is the additional ductility due to transverse reinforcement ($\varepsilon_{50c} = \varepsilon_{50u} + \varepsilon_{50h}$). It is assumed that the cover concrete has spalled off by the time the stress had fallen to $0.5f_{c}^{\prime }$.

Fig. A2.2

Transverse confining in RC members

Region CD, $\varepsilon_{c} \, \ge \,\varepsilon_{20c}$

$$f_{c} = 0.2f_{c}^{\prime }$$

(A2.7)

This equation accounts for the ability of concrete to sustain some stresses at very large strains.

Putting, $f_{c} = 0.2f_{c}^{\prime }$ and $\varepsilon_{c} \, = \,\varepsilon_{20c}$ in Eq. (A2.2), we get

$$\varepsilon_{20c} = \frac{0.8}{Z} + 0.002$$

(A2.8)

This concludes the Kent and park stress–strain model.

Modified Kent and Park Model.

The modified form of Kent and Park model was proposed by Park et al. [15]. This model makes an allowance for the enhancement in the concrete strength due to confinement. Figure A2.3 shows the modified Kent and Park model.

Fig. A2.3

Modified Kent and Park model for concrete confined by rectangular hoops

The maximum stress reached (at point B) is assumed to be $Kf_{c}^{\prime }$ at a strain of $\varepsilon_{0} = 0.002\,{\text{K}}$, in which,

$$K = 1 + \frac{{\rho_{{_{s} f_{yh} }} }}{{f_{c}^{\prime } }}$$

(A2.9)

$f_{yh}$:

yield strength of steel hoops.

The modified Kent and Park stress–strain curve can be defined as,

Region AB, $\varepsilon_{c} \le 0.002\,{\text{K}}$.

$$f_{c} = Kf_{c}^{\prime } \left[ {\frac{{2\varepsilon_{c} }}{0.002 K} - \left( {\frac{{\varepsilon_{c} }}{0.002 K}} \right)^{2} } \right]$$

(A2.10)

Region BC, $0.002K \le \varepsilon_{c} \le \varepsilon_{20m,c}\,\,0.002K \le \varepsilon_{c} \le \varepsilon_{20m,c}$

$$f_{c} = Kf_{c}^{\prime } \left[ {1 - Z_{m} (\varepsilon_{c} - 0.002K)} \right] \ge 0.2Kf_{c}^{\prime }$$

(A2.11)

where,

$$Z = \frac{0.5}{{\frac{{3 + 0.29f_{c}^{\prime } }}{{145f_{c}^{\prime } - 1000}} + \frac{3}{4}\rho_{s} \sqrt {\frac{{b^{\prime \prime } }}{{s_{h} }}} - 0.002\,K}}$$

(A2.12)

$f_{c}^{\prime }$:

Concrete cylinder strength in mega Pascal (N/mm²)

Region CD,

$$f_{c} = 0.2Kf_{c}^{\prime }$$

(A2.13)

This equation accounts for the ability of concrete to sustain some stresses at very large strains.

Putting $f_{c} = 0.2Kf_{c}^{\prime }$ and $\varepsilon_{c} = \varepsilon_{20m,c}$ in Eq. (A2.2), we get

$$\varepsilon_{20m,c} = \frac{0.8}{{Z_{m} }} + 0.002\,K$$

(A2.14)

This concludes the Modified Kent and park stress–strain model.

The equivalent stress block parameters are calculated using Eqs. A2.15 and A2.16 for different values of ε_cm depending on whether ε_cm lies in zone AB, BC or CD of Kent and Park model or modified Kent and Park model.

• 1. Stress block parameters for Kent and Park model

Region ‘AB’: ε_cm ≤ 0.002.

The mean stress factor, α and the centroid factor, γ for any strain ε_cm at the extreme compression fiber can be determined for rectangular sections from the stress–strain relationship as follows

$${\text{Area}}\,{\text{under}}\,{\text{stress}} - {\text{strain}}\,{\text{curve}} = \int\limits_{0}^{{\varepsilon_{cm} }} {f_{c} d\varepsilon_{c} } = \alpha f_{c}^{\prime } \varepsilon_{cm}$$

$$\therefore \alpha = \frac{{\int\limits_{0}^{{\varepsilon_{cm} }} {f_{c} d\varepsilon_{c} } }}{{f_{c}^{\prime } \varepsilon_{cm} }}$$

(A2.15)

First moment of area about origin of area under stress–strain curve

$$= \int\limits_{0}^{{\varepsilon_{cm} }} {f_{c} \varepsilon_{c} d\varepsilon_{c} } = (1 - \gamma )\varepsilon_{cm} \int\limits_{0}^{{\varepsilon_{cm} }} {f_{c} d\varepsilon_{c} }$$

$$\therefore \gamma = 1 - \frac{{\int\limits_{0}^{{\varepsilon_{cm} }} {\varepsilon_{c} f_{c} d\varepsilon_{c} } }}{{\varepsilon_{cm} \int\limits_{0}^{{\varepsilon_{cm} }} {f_{c} d\varepsilon_{c} } }}$$

(A2.16)

Using Eq. (A2.1) in Eqs. (A2.15) and (A2.16) we get

$$\, f_{c} = f_{c}^{\prime } \left[ {\frac{{2\varepsilon_{c} }}{0.002} - \left( {\frac{{\varepsilon_{c} }}{0.002}} \right)^{2} } \right]$$

$$\alpha = \frac{{\varepsilon_{cm} }}{0.002}\left[ {1 - \frac{{\varepsilon_{cm} }}{0.006}} \right]$$

(A2.17)

$$\gamma = 1 - \frac{{\left[ {\frac{2}{3} - \left( {\frac{{\varepsilon_{cm} }}{0.008}} \right)} \right]}}{{\left[ {1 - \left( {\frac{{\varepsilon_{cm} }}{0.006}} \right)} \right]}}$$

(A2.18)

Region ‘BC’: 0.002 ≤ ε_cm ≤ ε_20,c

Using Eqs. (A2.10) and (A2.11) in Eq. (A2.15) and (A2.16) we get

$$\alpha = \frac{1}{{\varepsilon_{cm} }}\left[ {\frac{0.004}{3} + (\varepsilon_{cm} - 0.002) - \frac{Z}{2}\left( {\varepsilon_{cm} - 0.002} \right)^{2} } \right]$$

(A2.19)

$$\gamma = 1 - \frac{1}{{\varepsilon_{cm} }}\left[ {\frac{{\left[ {\left( {\frac{{\varepsilon_{cm}^{2} }}{2} - \frac{{\left( {0.002} \right)^{2} }}{12}} \right) - Z\left( {\frac{{\varepsilon_{cm}^{3} }}{3} - 0.001\varepsilon_{cm}^{2} + \frac{{(0.002)^{3} }}{6}} \right)} \right]}}{{\left[ {\left( {\varepsilon_{cm} - \frac{0.002}{3}} \right) - Z\left( {\frac{{\varepsilon_{cm}^{2} }}{2} - 0.002\varepsilon_{cm} + \frac{{(0.002)^{2} }}{2}} \right)} \right]}}} \right]$$

(A2.20)

Region ‘CD’: ε_cm > ε_20,c

Using Eqs. (A2.10), (A2.11) and (A2.13) in Eqs. (A2.15) and (A2.16) we get

$$\alpha = \frac{1}{{\varepsilon_{cm} }}\left[ {\frac{0.004}{3} + \frac{0.32}{Z} + 0.2\varepsilon_{cm} - 0.0004} \right]$$

(A2.21)

$$\gamma = 1 - \frac{1}{{\varepsilon_{cm} }}\left[ {\frac{{1.2667 \times 10^{ - 6} + \frac{0.00064}{Z} + \frac{{0.8^{3} }}{{6Z^{2} }} + 0.1\varepsilon_{cm}^{2} }}{{\frac{0.004}{3} - \frac{0.32}{Z} + 0.2\varepsilon_{cm} - 0.0004}}} \right]$$

(A2.22)

• 2. Stress block parameters for Modified Kent and Park model

Region ‘AB’: ε_cm ≤ 0.002 K.

Using Eq. (A2.10) in Eqs. (A2.16) and (A2.17) we get

$$\alpha = \frac{{\varepsilon_{cm} }}{0.002K}\left[ {1 - \frac{{\varepsilon_{cm} }}{0.006K}} \right]$$

(A2.23)

$$\gamma = 1 - \frac{{\left[ {\frac{2}{3} - \left( {\frac{{\varepsilon_{cm} }}{0.008K}} \right)} \right]}}{{\left[ {1 - \left( {\frac{{\varepsilon_{cm} }}{0.006K}} \right)} \right]}}$$

(A2.24)

Region ‘BC’: 0.002 K ≤ ε_cm ≤ ε_20m,c

Using equations (A2.10) and (A2.11) in Eqs. (A2.15) and (A2.16) we get

$$\alpha = \frac{1}{{\varepsilon_{cm} }}\left[ {\frac{0.004K}{3} + (\varepsilon_{cm} - 0.002K) - \frac{{Z_{m} }}{2}\left( {\varepsilon_{cm} - 0.002K} \right)^{2} } \right]$$

(A2.25)

$$\gamma = 1 - \frac{1}{{\varepsilon_{cm} }}\left[ {\frac{{\left[ {\left( {\frac{{\varepsilon_{cm}^{2} }}{2} - \frac{{\left( {0.002K} \right)^{2} }}{12}} \right) - Z_{m} \left( {\frac{{\varepsilon_{cm}^{3} }}{3} - 0.001K\varepsilon_{cm}^{2} + \frac{{(0.002K)^{3} }}{6}} \right)} \right]}}{{\left[ {\left( {\varepsilon_{cm} - \frac{0.002K}{3}} \right) - Z_{m} \left( {\frac{{\varepsilon_{cm}^{2} }}{2} - 0.002K\varepsilon_{cm} + \frac{{(0.002K)^{2} }}{2}} \right)} \right]}}} \right]$$

(A2.26)

Region ‘CD’: ε_cm > ε_20m,c

Using Eqs. (A2.10), (A2.11) and (A2.15) in Eqs. (A2.16) we get

$$\alpha = \frac{1}{{\varepsilon_{cm} }}\left[ {\frac{0.004K}{3} + \frac{0.32}{{Z_{m} }} + 0.2K\varepsilon_{cm} - 0.0004K} \right]$$

(A2.27)

$$\gamma = 1 - \frac{1}{{\varepsilon_{cm} }}\left[ {\frac{{1.2667 \times 10^{ - 6} K^{2} + \frac{0.00064K}{{Z_{m} }} + \frac{{0.8^{3} }}{{6Z_{m}^{2} }} + 0.1\varepsilon_{cm}^{2} }}{{\frac{0.004K}{3} - \frac{0.32}{{Z_{m} }} + 0.2K\varepsilon_{cm} - 0.0004K}}} \right]$$

Annexure A3 N–Raphson Technique

Consider a cantilever is subjected to monotonic loading with increase in amplitude. Maximum moment will be at the fixed end and when the member reinforcement starts yielding hinge forms and mathematical model is modified as shown in Fig.A3.1. The hinge is replaced with translational and rotational springs and to evaluate rotation induced translation within the hinge length a rigid link is introduced.

Fig. A3.1

Cantilever and its inelastic model for nonlinear static analysis

Equilibrium equation is written as:

$$\left[ K \right]\left\{ \delta \right\} = \left\{ P \right\}$$

Stiffness matrix will have single element stiffness till yielding and will get modified with hinge springs. The stiffness matrices are given in Appendix 1. If hinge moment rotation characteristics are known, at every load step, the stiffness will get modified with the iterations [17] as explained in Fig. A3.2.

$$\theta_{i + 1}^{(0)} = \theta_{i} \quad {\text{M}}_{{\text{S}}}^{(0)} = (M)_{i} \quad \Delta {\text{R}}^{(1)} = \Delta M_{e} \quad {\text{K}}_{{\text{T}}} = K_{i} \quad j = 1,2,2$$

Fig. A3.2

Inelastic moment-rotation characteristics of beam/column hinge

• 1. $K_{T} \times \Delta \theta^{(1)} = \Delta R^{(1)}$

• 2. $\theta_{i + 1}^{(j)} = \theta_{i + 1}^{(j - 1)} + \Delta \theta^{(1)}$

• 3. $\Delta M_{p}^{j} = {\text{M}}_{{\text{s}}}^{{({\text{j}})}} - {\text{M}}_{{\text{s}}}^{{({\text{j}} - 1)}}$

• 4. $\Delta R^{(j + 1)} = \Delta R^{(j)} - \Delta M_{p}^{j} .$