A Design Method to Induce Ductile Failure of Flexural Strengthened One-Way RC Slabs

Abstract

Strengthening existing reinforced concrete (RC) slabs using externally bonded materials is increasingly popular due to its adaptability and versatility. Nevertheless, ductility reduction of the rehabilitated flexural members with these materials can lead to brittle shear failure. Therefore, a new approach for strengthening is necessary. This paper presents a methodology to induce ductile failure of flexural strengthened one-way RC slabs. Ultimate failure loads can be considered to develop the proposed design methodology. Different failure modes corresponding to ultimate failure loads for RC slabs are addressed. Flexural and shear failure regions of RC slabs can be established by considering the failure modes. The end span of the concrete slab is shown for a case study, and numerical examples are solved to prove the essentiality of this methodology.

1. Introduction

The demand for strengthening and rehabilitation of infrastructures becomes more urgent in modern society [1,2,3]. Apart from the aging, corrosion, unexpected excessive loads, and accidental damage, the rectify of initial design and construction faults or upgrading the load capacity of reinforced concrete (RC) structures are also of interest to structural engineers and researchers [4,5,6]. Moreover, strengthening the existing structures is preferable to demolishing and building an entirely new system due to its lower costs and minimized environmental deterioration simultaneously [7].

Repairing or strengthening RC structures using reinforced concrete, ferrocement, steel, or fiber reinforced polymer (FRP) materials on the tension parts of the RC structures is one of the most common techniques [8]. In addition, the development of FRP composite materials allows improving significant loading capacity of strengthened structures [9,10,11,12,13]. Over four decades, several studies have been solved to examine the performance of the structures strengthened with FRP and proven its effectiveness due to its high tensile strength and corrosion resistance [14,15,16,17,18]. The post strengthening with FRP materials can rehabilitate the original flexural capacity of the damaged structures and even increase it significantly. There are many methods for post strengthening flexural RC slabs with FRP [19,20,21,22]. One of the typical methods is the addition of FRP on the top and bottom parts of the slabs subjected to the positive (+) and negative (–) flexural moments, respectively, as shown in Figure 1. However, the brittle failure and the ductility reduction of the rehabilitated flexural members were recognized [23,24].

Additionally, RC structures in contact with FRP composite materials have significant and unavoidable effects on their behavior [25]. The failure mode of strengthened RC structures tends to be more brittle compared with the counterpart steel-reinforced concrete structures due to the intrinsic bond conditions between FRP and concrete and the linear-elastic brittle tensile behavior of FRP as well [26]. Structural ductility should be considered an important design factor because it can prevent brittle shear modes [27,28]. A well-designed structure should warn of impending failure when it is subjected to an overload [29]. Several studies have shown the necessity of failure modes in reflecting the corresponding behavior of the structure under load [30,31,32]. Although some studies have been investigated on the failure modes of strengthened structures [33,34,35,36], there is little reported work on strengthened flexural structures avoiding sudden failure and inducing ductile failure.

This research work recommends using carbon fiber reinforced polymer (CFRP) unidirectional laminates to enhance the strength of existing continuous slabs. However, the excessive improvement of the flexural strength relative to the shear strength of the strengthened sections can lead to brittle shear failure [37]. To the best of our knowledge, there is not yet a complete design process of preventing brittle failure for strengthened RC slabs. This study proposes a new classification of failure modes that reflect the corresponding behavior of the RC slab under load. Structural evaluation through new failure modes can suggest the appropriate enhancement of the existing RC slabs. Based on that, the methodology limiting the additional strength for post-strengthened RC slabs through the failure modes to prevent brittle failure and induced ductile failure is also determined.

In this paper, different failure modes for the end span of a continuous RC slab corresponding to ultimate failure loads for RC slabs are addressed for a case study. The results would contribute to developing a design methodology for the strengthened RC slab to ensure ductile failure.

2. Failure Limits

This work focuses on determining the failure limits of the end span of a continuous slab subjected to uniformly distributed load. For flexural members in a frame, the flexural rigidities of members and supporting columns are decisive factors in the distribution of bending moments. Thus, the moment at the support and the mid-span sections of the frame members, subjected to a uniformly distributed load $w$, could be established, as $M=Cw{l}^2,$ where C is a coefficient according to flexural rigidities of respective flexural members and l is the clear span length. If infinite rigidity of columns is considered, the well-known results for a fixed-end moment of a flexural member subjected to a uniformly distributed load $w$, coefficient C will be 1/12 [38]. For practical design purposes, moment and shear coefficients for continuous RC slabs subjected to a uniformly distributed load ($w_u$) are reported by ACI 318M [39] with column support cases, as shown in Figure 2.

Considering a flexural member having unsymmetric boundary conditions, which is comparable to the end span in Figure 2, the different failure modes could be established according to the relationship between the moment limit of the negative moment M_n,N and positive moment M_n,P at the support and the mid-span sections, respectively, and the shear limit of V_n of the slab sections. In case the sections at the mid-span and the exterior face of the first interior support (N2) fail simultaneously by forming two plastic hinges, the moment limits subjected to a uniform distributed load can be written as a formula of the shear carrying capacity, V_n, as follows:

$$M_{n,N}=\frac{2C_{m,N2}}{C_{v2}}{V}_n{l}_n$$

(1)

$$M_{n,P}=\frac{2C_{m,P}}{C_{v2}}{V}_n{l}_n$$

(2)

At the same time, the moment carrying of the interior face of the exterior support (N1) can be expressed as follows,

$$M_{N1}=\frac{2C_{m,N1}}{C_{v2}}{V}_n{l}_n$$

(3)

where M_n,N and M_n,P are the moment carrying capacity of the support and mid-span sections; M_N1 is the moment carrying of the N1 section; V_n is the shear carrying capacity of the RC slab sections; C_m,N1, C_m,N2, and C_m,P are the moment coefficients for the negative moments at the N1 section, N2 section, and the positive moment at the mid-span section, respectively; C_v2 is the shear coefficient at the N2 section; and l_n is the clear span length between support columns.

Here, the signs of the moment coefficients could be neglected because they are only used to show moment directions, as shown in Figure 2. The limit of Equations (1)–(3) may be described with a given shear limit and the shear and moment coefficients (refer to Figure 3). The failure regions of RC slabs could be shown as follows.

Failure modes should be classified according to the order of plastic hinge formation at sections and the type of failure by analyzing the failure limit for each region in Figure 3. The summary of different failure modes is shown in

Table 1. Summary of different failure modes for the end span of an RC slab.

Failure Modes	First Plastic Hinge	Second Plastic Hinge	Third Plastic Hinge	Shear Failure	Failure Type
D-1	N2	N1	M	-	Ductile
D-2	N2	M	N1	-	Ductile
D-3	M	N2	N1	-	Ductile
DB-1	N2	N1	-	N2	Brittle
DB-2	N2	M	-	N2	Brittle
DB-3a	M	-	-	N2	Brittle
DB-3b	M	N2	-	N2	Brittle
B-1	N2	-	-	N2	Brittle
B-2	-	-	-	N2	Brittle

. Details of the analysis for Table 1 are described in the Appendix A of this article. Failure modes of D-1, D-2, and D-3 are ductile failures and desirable while the failure modes DB-1, DB-2, DB-3a, DB-3b, B-1, and B-2 are brittle failures and thus may not be suitable for a well-designed structure.

The limit equations of failure modes in each region are shown in Figure 4. The ultimate failure load can be determined by the superposition method considering plastic redistribution of strengthened RC slab. For failure modes of D-1, D-2, and D-3, the ultimate failure loads can be calculated as:

• Failure mode D-1

$$w_f={\varphi }_m\frac{8}{l_n^2}(M_{n,P}+M_{n,N}\frac{\left(1/8-C_{m,P}\right)}{C_{m,N2}})$$

(4)

• Failure mode D-2

$$w_f={\varphi }_m\frac{4}{l_n^2}\left(M_{n,P}+M_{n,N}\frac{\left(1/4+C_{m,N2}-C_{m,N1}-C_{m,P}\right)}{C_{m,N2}}\right)$$

(5)

• Failure mode D-3

$$w_f={\varphi }_m\frac{4}{l_n^2}\left(M_{n,P}\frac{\left(1/4-C_{m,N1}\right)}{C_{m,P}}+M_{n,N}\right)$$

(6)

For failure modes of DB-1, DB-2, DB-3a, DB-3b, B-1, and B-2, the failure loads can be calculated as:

$$w_f={\varphi }_v\frac{2V_n}{C_{v2}{l}_n}$$

(7)

where strength reduction factors of ${\Phi }_m \mathrm{and} {\Phi }_v$ are used for the flexural strength and the shear strength, respectively, as specified by ACI 318M [39].

3. Design Example

In this case, an RC slab strengthened with an externally bonded CFRP sheet has been presented. Moment and shear coefficients are considered for the end span of the column support case, where $C_{m,N1}=1/16, C_{m,N2}=1/10, C_{m,P}=1/14, C_{v1}=1,$ and $C_{v2}=1.15,$ as shown in Figure 2. The strength reduction factors of the flexural strength and shear strength are 0.90 and 0.75, respectively [39]. The reduction factor for the strength contribution of CFRP reinforcement ψ_f is 0.85 [40]. The selected CFRP along with mechanical properties (tensile strength f_fu = 717 MPa, elastic modulus E_f = 65.1 GPa) is reported by the manufacturers, as recommended by ACI 440R [41]. Additional CFRP thickness (t_F) is assumed as a design variable and installed in tensile regions of the RC slab corresponding to the width of the slab (refer to Figure 5). The clear span of the rectangular RC slab is 2.5 m long. Slab material properties and dimensions are summarized in

Table 2. The RC slab material properties and dimensions for design example.

Section	h (mm)	b (mm)	As (mm2)	d (mm)	f’c (MPa)	γc (kg/m3)	fy (MPa)	Es (GPa)
Supports	150	1000	355	120	27	2400	410	200
Mid-span

. The existing slab is computed in

Table 3. Calculation of the existing RC slab.

Calculation	Existing RC Slab
Moment and shear coefficients for the end span of the column support case [39]	$$\begin{gathered} C_{m,N1}=1/16, C_{m,N2}=1/10, C_{m,P}=1/14 \\ {C}_{v1}=1, \mathrm{and} C_{v2}=1.15 \end{gathered}$$
Designed resistance	${\varphi }_f{M}_{n,P}=15.3 \mathrm{kNm}, {\varphi }_f{M}_{n,N}=15.3 \mathrm{kNm},$ ${\varphi }_v{V}_n$ = 77.94 kNm;
Failure mode	D-2, as shown in Figure 6
Ultimate failure load: Equation (5) for D-2	$\begin{array}{l}{w}_f=\frac{4}{{2.5}^2}\left(15.3+15.3\frac{(1/4+1/10-1/16-1/14)}{1/10}\right)\\ \hspace{1em}=31 \mathrm{kN}/\mathrm{m}\end{array}$
Preparatory computations for strengthened design
Self-weight $w_D={\gamma }_{c}bh$	$w_D=(2400)(9.8\times {10}^{-3})(0.15)=3.53 \mathrm{N}/\mathrm{mm}$
Factored moment $M_D=C_m{w}_D{l}_n^2$	At N2 section $M_{D,N}=(1/10)(3.53)({2500}^2)/1000=2205 \mathrm{kNmm}$ At mid-span section $M_{D,P}=(1/14)(3.53)({2500}^2)/1000=1575 \mathrm{kNmm}$
Modulus of elasticity $E_c=4700\sqrt{f_c^{\prime }}$	$E_c=4700\sqrt{27}=24400 \mathrm{MPa}$
Cracking moment $I_{cr}$ Neutral axis depth c = kd	At both sections (N2 and mid-span sections) $I_{cr}=31.4\times 10^6{ \mathrm{mm}}^4$ $c=7.5 \mathrm{mm}$

, and the failure mode considering the relationship between factor flexural and shear resistance and slab status is extracted, as shown in Figure 6. In this analysis, a perfect bond between the strengthened materials and the RC slab is supposed up to the ultimate failure loads.

4. Results and Discussions

For the calculation of the existing slab shown in Table 3, the ultimate failure load ($w_f$) of the existing slab is determined as 31 kN/m, as indicated by Equation (5). The failure mode of the slab is named D-2, as shown in Figure 6. Notably, the existing slab can continue to be strengthened to also fail in ductile failure. The calculation procedure for 1 mm thick CFRP sheet strengthened for positive and negative moment sections of the RC slab is shown in

Table 4. Calculation of the strengthened slab.

Procedure	Strengthened Slab
1. CFRP thickness, tF	At both sections: tF,N = tF,P = 1 mm
2. Existing state of strain ${\epsilon }_{bi}=\frac{M_D\left(d_f-kd\right)}{I_{cr}{E}_c}$ kd is c in Table 3	At N2 section: ${\epsilon }_{bi,N}=\frac{(2.205\times {10}^6)\left(150.6-7.5\right)}{(31.4\times {10}^6)(24400)}=0.00041$ At mid-span section: ${\epsilon }_{bi,P}=\frac{(1.575\times {10}^6)(150.6-7.5)}{(31.4\times {10}^6)(24400)}=0.00029$
3. Design strain of CFRP ${\epsilon }_{fd}=0.41\sqrt{\frac{f_c^{\prime }}{n{E}_F{t}_F}}\le 0.9{\epsilon }_{fu}$	${\epsilon }_{fd}=0.41\sqrt{\frac{27}{(1)(65100)(1)}}=0.0083<0.9(0.011)=0.0099$
4. Assume concrete strain at failure ε Revise ε until equilibrium achieved	At both sections: ${\epsilon }_{c,N}={\epsilon }_{c,P}=0.0021$
5. Compute neutral axis depth	At both sections: $c_N=c_P=32.7 \mathrm{mm}$
6. Compute CFRP strain ${\epsilon }_{fe}={\epsilon }_c\left(\frac{h-c}{c}\right)-{\epsilon }_{bi}\le {\epsilon }_{fd}$	At N2 section: ${\epsilon }_{fe,N}=0.0021\left(\frac{150-32.7}{32.7}\right)-0.00041=0.0072<0.0099$ At mid-span section: ${\epsilon }_{fe,N}=0.0021\left(\frac{150-32.7}{32.7}\right)-0.00029=0.0073<0.0099$
7. Compute tension steel strain ${\epsilon }_s={\epsilon }_c\left(\frac{d-c}{c}\right)$	At both sections: ${\epsilon }_{s,N}={\epsilon }_{s,P}=0.0021\left(\frac{120-32.7}{32.7}\right)=0.0056>0.002 \mathrm{yield}$
8. Check for force equilibrium ${\beta }_1=\frac{4{\epsilon }_c^{\prime }-{\epsilon }_c}{6{\epsilon }_c^{\prime }-2{\epsilon }_c}$, ${\alpha }_1=\frac{3{\epsilon }_c^{\prime }{\epsilon }_c-{\epsilon }_c^2}{3{\beta }_1{{\epsilon }^{\prime }}_2^c}$ ${\epsilon }_c^{\prime }$ is strain relative to $f_c^{\prime }$ ${\epsilon }_c^{\prime }=\frac{1.7f_c^{\prime }}{E_c}$ Check the neutral axis depth $c=\frac{A_s{f}_s+t_{f}b{\epsilon }_{fe}{E}_F}{{\alpha }_1{f}_c^{\prime }{\beta }_{1}b}$	${\epsilon }_c^{\prime }=\frac{1.7(27)}{24400}=0.0019$, ${\beta }_1=\frac{4(0.0019)-0.0021}{6(0.0019)-2(0.0021)}=0.77$ ${\alpha }_1=\frac{3(0.0019)(0.0021)-{0.0021}_{}^2}{3(0.77)({0.0019}^2)}=0.92$ At N2 section: $c_N=\frac{(355)(410)+(1)(1000)(0.0072)(65100)}{(0.92)(27)(0.77)(1000)}=32.3 \mathrm{mm}$ (OK) At mid-span section: $c_P=\frac{(355)(410)+(1)(1000)(0.0073)(65100)}{(0.92)(27)(0.77)(1000)}=32.7 \mathrm{mm}$ (OK) Assumption of ε is satisfied.
9. Compute flexural strength at N2 section provided by - Steel: $M_{n,Ns}=A_{s,N}{f}_{s,N}(d-\frac{{\beta }_1{c}_N}{2})$ - CFRP: $M_{n,Nf}=t_{f}b{\epsilon }_{fe}{E}_F\left(d_f-\frac{{\beta }_1{c}_N}{2}\right)$	$M_{n,Ns}=\frac{(355)(410)}{{10}^6}\left(120-\frac{(0.77)(32.3)}{2}\right)=15.64 \mathrm{kNm}$ $\begin{array}{l}{M}_{n,Nf}=\frac{(1)(1000)(0.0072)(65100)}{{10}^6}\left(150.6-\frac{(0.77)(32.3)}{2}\right)\\ \hspace{1em}\hspace{1em} =64.25 \mathrm{kNm}\end{array}$
10. Compute flexural strength at mid-span section provided by - Steel: $M_{n,Ps}=A_{s,P}{f}_{s,P}\left(d-\frac{{\beta }_1{c}_P}{2}\right)$ - CFRP: $M_{n,Pf}=A_{f,P}{f}_{fe,P}\left(d_f-\frac{{\beta }_1{c}_P}{2}\right)$	$M_{n,Ps}=\frac{(355)(410)}{{10}^6}\left(120-\frac{(0.77)(32.7)}{2}\right)=15.64 \mathrm{kNm}$ $\begin{array}{l}{M}_{n,Pf}=\frac{(1)(1000)(0.0073)(65100)}{{10}^6}\left(150.6-\frac{(0.77)(32.7)}{2}\right)\\ \hspace{1em}\hspace{1em} =65.31 \mathrm{kNm}\end{array}$
11. The design flexural strength ${\varphi }_f{M}_n={\varphi }_f\left(M_{ns}+{\psi }_f{M}_{nF}\right)$ ${\varphi }_f=0.9; {\psi }_f=0.85$	At N2 section: ${\varphi }_f{M}_{n,N}=0.9\left(15.64+(0.85)(64.25)\right)=63.23 \mathrm{kNm}$ At mid-span section: ${\varphi }_f{M}_{n,P}=0.9\left(15.64+(0.85)(65.31)\right)=64.04 \mathrm{kNm}$
12. The design shear strength ${\varphi }_v{V}_n={\varphi }_v\left(d\sqrt{f_c^{\prime }}\right)\frac{b}{6}$ ${\varphi }_v=0.75$	${\varphi }_v{V}_n=\frac{0.75}{6}(120)\sqrt{27}=77.94 \mathrm{kN}$
13. Failure mode	B-2, as shown in Figure 6
14. Ultimate failure load Equation (7) for B-2	$w_f=\frac{2(77.94)}{(1.15)(2.5)}=54.2 \mathrm{kN}/\mathrm{m}$

, as specified by ACI 440 [40]. In step 2, the existing state of strain is determined. CFRP debonding is computed at step 3 and is used in steps 4 to 8 and shows that concrete strain is less than the failure strain of 0.003. In this case, debonding of CFRP occurs before the failure strain of concrete reaches ε = 0.0021. The coefficients of rectangular stress block α₁ and β₁ for the failure strain in the concrete at the limit state are applied. The design flexural and shear strength are calculated in steps 9 through 12, whereas the ultimate failure load ($w_f$) is computed in steps 13 through 14 with corresponding failure mode. The failure mode of the slab is named B-2, as shown in Figure 6, with the ultimate failure load $w_f$ = 54.2 kN/m using Equation (7). Although the ultimate failure load is increased 75% compared to the ultimate failure load of the existing slab, brittle failure is not a desirable result.

A similar calculation procedure is performed while the CFRP sheet thickness is assumed as a design variable and adjusted to ensure ductile flexural failure. The existing slab can be reinforced by installing 0.12 mm thick CFRP sheets for positive and negative moment sections. The ultimate failure load of the strengthened slab $w_f$, found from Equation (5), is estimated as 47.9 kN/m, which increased by 55% compared to the ultimate failure load of the existing slab. The failure mode is named D-2, as shown in Figure 7.

In another approach for ease of application [19], a 0.26 mm thick CFRP sheet could be applied at the upper side of the slab to enhance negative moment capacity. The failure mode is named D-3, as shown in Figure 7. Using Equation (6), the ultimate failure load of the strengthened slab $w_f$ is determined as 47.1 kN/m, which is increased by 52% compared to the ultimate failure load of the existing slab.

The flexural strengthened RC slab capacities are shown in

Table 5. The strengthened slab capacities for the design example.

Slabs	Failure Mode	${\mathit{w}}_{\mathit{f}}(\mathbf{kN}/\mathbf{m})$		tFN (mm)	tFP (mm)
Existing slab	D-2	31.0	[100%]	-	-
Strengthened both sections	D-2	47.9	[155%]	0.12	0.12
Strengthened negative sections	D-3	47.1	[152%]	0.26	-

. From this table, one can realize that the strengthened slab with 0.12 mm thick CFRP for positive and negative sections and 0.26 mm thick CFRP for only negative parts has the efficiency to enhance the factored design load by 155% and 152%, respectively. Especially, the failure mode of the strengthened RC slab is a desirable ductile failure to which there is little reported concern to prevent sudden failure.

5. Conclusions

Failure modes of the continuous RC slab considering the relationship between the flexural strength and shear strength to prevent brittle shear failure and induce flexural ductile failure are presented.

An efficient procedure to strengthen slabs flexurally with an externally bonded CFRP avoiding sudden failure and ensuring ductile failure is demonstrated through several examples. By using 0.12 mm thick CFRP for positive and negative sections and 0.26 mm thick CFRP for only negative parts of the slab, the factored design load can be enhanced by 155% and 152%, respectively.

A simple approach to determine the additional strength limit for flexural slabs considering failure modes is introduced. Adjusting the thickness of the CFRP sheet could achieve the desired increase in flexural strength. Additionally, CFRP discrete strips may also be used instead of CFRP sheets if their thickness is fixed by manufacturers.

The method has advantages in constructability and economic aspects. It can be applied to strengthen the existing floor slabs or bridge decks. Furthermore, the additional flexural strength with an externally bonded CFRP could be also optimized with a warning of impending failure under overload.

This study is theoretical, and it would contribute to developing a design methodology for the strengthened RC slab to ensure ductile failure, completed through finite element analysis and experimental research in further studies.