Shear resistance and structural hardening resistance models for hybrid NSC-UHPC elements

Abstract

Ultra-high performance concrete (UHPC) is being increasingly used as a thin overlay for deficient normal strength concrete (NSC) structural elements. Compared to reinforced NSC elements, hybrid NSC-UHPC elements have two stages' structural behaviour under high shear load before failure: monolithic action followed by a composite mechanical action. The composite mechanical action gives to hybrid elements a structural hardening capability after the occurrence of a diagonal shear crack and is not yet considered by most design standards. Two analytical models for hybrid elements under shear load are thus proposed to predict the shear resistance V_R of the monolithic action and to predict the structural hardening resistance V_Rsh of the composite mechanical action. The shear resistance model integrates the UHPC contribution in the estimation of the tensile strain at mid-height cross-section according to the general method of the Canadian code. The structural hardening behaviour is reproduced with a strut-and-tie system with plastic hinges, where the failure is controlled by the crushing of a concrete strut. Predictions of the proposed models are in very good agreement with 31 hybrid elements experimental results retrieved from the literature.

Keywords

ultra-high performance concrete (UHPC)hybrid structures shear design shear resistance structural hardening analytical model strut-and-tie model

Introduction

Ultra-high performance concretes (UHPC) feature minimum compressive strength of 120–150 MPa, minimum tensile strength of 5–7 MPa and a tensile strain hardening behaviour around 2000 µε (Naaman, 2008; AFGC, 2013; CSA, 2018; Eide and Hisdal, 2012; Haber et al., 2018). Such improved mechanical properties compared to normal strength concrete (NSC), strongly favour its utilization as thin overlay for rehabilitating or strengthening purposes, which is a growing field of application (Brühwiler, 2019; Brühwiler and Denarié, 2013; Guingot et al., 2013; Denarié et al., 2013). Through this concept, many experimental campaigns were realized to study the bending behaviour of hybrid NSC-UHPC elements (Habel, 2004; Makita, 2014; Oesterlee, 2010; Wuest, 2007; Al-Osta et al., 2017; Denarié et al., 2003; Lachance et al., 2016; Noshiravani and Brühwiler, 2013; Paschalis et al., 2018; Prem and Murthy, 2016; Safdar et al., 2016; Zingaila et al., 2017; Pharand, 2022; Yin et al., 2017). However, only a few (Pharand and Charron, 2022a; Sine et al., 2022; Noshiravani and Brühwiler, 2013; Ji and Liu, 2020; Yin et al., 2017) specifically studied the shear behaviour of hybrid NSC-UHPC elements.

Experimental campaigns dedicated to the shear behaviour showed that hybrid NSC-UHPC elements have a two stages' structural behaviour before failure. First, shear load is resisted by monolithic action alike NSC element (solid green line in Figure 1(a)). Within this first stage, the ultra-high performance concrete (UHPC) layer, reinforced or not, increases the capacity of the shear transfer mechanism in the NSC and delays the apparition of a diagonal shear crack (Noshiravani and Brühwiler, 2013; Sine et al., 2022; Pharand and Charron, 2022a; Yin et al., 2017). Second, upon the occurrence of a diagonal shear crack, shear load is resisted by a composite mechanical action (bold doted orange line in Figure 1(a)) also named double hinges mechanism by Noshiravani and Brühwiler (2014). Within this second stage, the UHPC layer, reinforced or not, enables a structural hardening behaviour by forming two plastic hinges and reduces the stress sustained by the NSC (Noshiravani and Brühwiler, 2013; Pharand and Charron, 2022a; Sine et al., 2022; Bastien-Masse and Brühwiler, 2016). Capacity gained from this second stage is such that shear failure could even be avoided on shear deficient NSC elements (Pharand, 2022; Yin et al., 2017).

Figure 1.

Behaviour of hybrid NSC-UHPC elements under high shear load, (a) load-displacement curve, (b) crack pattern and amplified deformed shape.

The composite mechanical action develops in four steps (Pharand and Charron, 2022a) (dashed yellow line in Figure 1(a)): (I) diagonal shear crack formation in NSC; (II) UHPC first hinge formation; (III) NSC-UHPC interface delamination and strut stabilization in NSC; and (IV) UHPC second hinge formation. Setting of the mechanism divides the hybrid element into three subsections along its longitudinal axe (Pharand and Charron, 2022a) (➁ in Figure 1(b)). The hybrid trapezoidal subsection S1 sustains bending stresses. The NSC trapezoidal subsection S2 has mainly compression stresses by transferring load directly from the application point to the support. The subsection S3 corresponds to the delaminated UHPC layer with a plastic hinge at each of its extremities (later called nodes). It transfers tensile stresses and a portion of shear stresses.

The main failure criteria observed for the composite mechanical action is concrete crushing of subsection S2 at the tip of the diagonal shear crack (Ji and Liu, 2020; Noshiravani and Brühwiler, 2013; Pharand and Charron, 2022a; Sine et al., 2022; Yin et al., 2017) (Figure 1(b)). Nonetheless, complete UHPC delamination beyond the loading plate (Figure 1(b)) and UHPC tensile failure prior to concrete crushing were also observed (Pharand and Charron, 2022a; Houdoin, 2022).

Presently, SIA (2016) design code is the only standard providing a resistance model specific to the second stage mechanism. The general concept of the Swiss code is to sum NSC, UHPC and stirrups’ individual contribution to the shear resistance. Although the model acknowledges the composite mechanical action in its definition, it does not account for size effect (Sine et al., 2022). Moreover, it does not allow determination of the improved resistance at the end of the first stage, prior the setting of the composite mechanical action.

Therefore, the objective of this paper is to propose new models to evaluate the shear capacity of the monolithic action ( $V_{R}$ in Figure 1(a)) and to evaluate the structural hardening capacity before failure ( $V_{R s h}$ in Figure 1(a)). Proposed models follow the Canadian design philosophy, which is based on the modified compression field theory, and are validated on experimental data from literature.

Proposed analytical resistance models

Considering the behaviour of hybrid NSC-UHPC elements under high shear load involves the succession of two bearing mechanisms, two resistance models are proposed. First, the shear resistance model is used to evaluate the resistance given by the monolithic action of the hybrid NSC-UHPC element, V_R. Then, based on the computed resistance of the monolithic action V_R, the structural hardening model is used to evaluate the resistance given by the composite mechanical action, V_Rsh.

For sake of clarity, partial safety factors were not integrated in equations detailed in next sections since it should be adjusted following the jurisdiction considered.

Shear resistance model for V_R

The proposed shear resistance model for hybrid NSC-UHPC elements is an adaptation of the Canadian general method for NSC elements shear resistance. Thus, Canadian general method for NSC elements section details the Canadian general method for NSC elements and Modified Canadian general method for NSC-UHPC elements section describes the modifications to the Canadian general method considering the UHPC contribution to the shear resistance.

Canadian general method for NSC elements

In the CSA standard, the general method (CSA, 2014) for the shear resistance of NSC elements V_R is a simplified model based on the modified compression field theory (MCFT) (Bentz and Collins, 2006). The shear load is assumed to be carried by aggregates interlock within the diagonal shear crack V_c and by yielding of stirrups that cross the diagonal shear crack V_s. The shear resistance is thus given by equation (1) where $λ$ is a factor to account for low density concrete, $β$ is a factor considering aggregates interlock in cracked concrete, $f_{c}^{'}$ is the concrete compressive strength (cylinder), $b_{w}$ is the web width, $d_{v}$ is the flexural lever arm equals to 0.9 times the static height of the centroid of longitudinal tension rebars d_s, $A_{v}$ is area of stirrups, $f_{y}$ is the yield strength of stirrups, $s$ is the stirrups spacing and $θ$ is the angle of inclination of diagonal compressive stresses to the longitudinal axis of the element.

V_{R} = V_{c} + V_{s} = λ \cdot β \cdot \sqrt{f_{c}^{'}} \cdot b_{w} \cdot d_{v} + \frac{A_{v} \cdot f_{y} \cdot d_{v}}{s \cdot \tan θ}

(1)

The concept of the Canadian general method is that aggregates interlock factor $β$ and angle of compression stresses $θ$ are controlled by the longitudinal strain at mid-depth $ε_{x}$ and by the effective crack spacing $s_{z e}$ . These two parameters, $ε_{x}$ and $s_{z e}$ , simultaneously capture geometric and loading effects such as aggregates size, element size, reinforcement ratio, applied moment, shear and normal load. The longitudinal strain at mid-depth $ε_{x}$ is conservatively approximated as half of the strain at the tensile cord. Thus, to compute the shear resistance $V_{R}$ , $β$ and $θ$ parameters must be defined through equation (2) to equation (7), where $γ$ is the a factor for size effect, $s_{z e}$ is the effective crack spacing, $s_{z}$ is the basic crack spacing and $a_{g}$ is the aggregate size. $M_{f}$ , $V_{f}$ and $N_{f}$ are respectively the applied moment, shear and normal load, $E_{s}$ is the Young’s Modulus of the longitudinal tensile rebars and $A_{s}$ is the area of the longitudinal tensile rebars.

In the Canadian general method, compression rebars are assumed to be of equal area than tension rebars and the ratio of $0.5 / \tan θ$ is assumed equal to 1 for the possible range of $θ$ . This conservative approach eliminates iterative computation between equations (6) and (7) since the term $0.5 \cdot V_{f} / \tan θ$ is simplified as $V_{f}$ in equation (7).

β = (\frac{0.4}{1 + 1500 \cdot ε_{x}}) \cdot γ

(2)

γ = (\frac{1300}{1000 + s_{z e}})

(3)

s_{z e} = \frac{35 \cdot s_{z}}{15 + a_{g}} \geq 0.85 \cdot s_{z}

(4)

s_{z} = {\begin{array}{c} d_{v}, w i t h o u t s t i r r u p s \\ 300, w i t h s t i r r u p s \end{array}

(5)

θ = 29 + 7000 \cdot ε_{x}

(6)

ε_{x} = \frac{1}{2} (\frac{(M_{f} / d_{v} + V_{f} + 0.5 N_{f})}{E_{s} \cdot A_{s}})

(7)

Modified Canadian general method for NSC-UHPC elements

In a hybrid NSC-UHPC element, the UHPC layer acts as an additional rebar layer in terms of ultimate bending resistance (Denarié et al., 2003), but also enhance its behaviour in terms of initial stiffness and crack development (Paschalis et al., 2018; Safdar et al., 2016; Habel, 2004; Oesterlee, 2010; Pharand and Charron, 2022a). This improved mechanical behaviour provides smaller crack width and greater aggregate interlock that must be accounted for the shear resistance of the monolithic action.

As suggested by the Canadian general method (CSA, 2014), strain of the tensile cord can be estimated by independent contributions of applied loads ( $M_{f}$ , $V_{f}$ , $N_{f}$ ) as depicted in Figure 2. The total strain at the tensile cord ε_t thus corresponds to the sum of each applied load contribution to the strain given by equation (8) to (10) where $ε_{t, M}$ , $ε_{t, V}$ , and $ε_{t, N}$ are the strain at the tensile cord respectively induced by applied moment, shear load and normal load, $E_{s U}$ and $A_{s U}$ are respectively the Young’s Modulus and area of longitudinal rebars located in the UHPC layer, $E_{U}$ and $A_{U}$ are respectively the Young’s Modulus and area of UHPC. As indicated by equation (11), α is the ratio the tensile rebars ( $E_{s} \cdot A_{s}$ ; $E_{s U} \cdot A_{s U}$ ) and UHPC layer axial rigidity ( $E_{U} \cdot A_{U}$ ) over the compressed rebars axial rigidity $(E_{s c} \cdot A_{s c})$ .

Figure 2.

Estimated strain at tensile cord for applied bending moment, shear load and normal load.

For a hybrid NSC-UHPC element, the tensile cord is located at the equivalent static height d_eq and is computed from the centroid $d_{i}$ of longitudinal tension rebars and of the UHPC layer respective areas $A_{i}$ (equation (12)). $f_{i}$ is either the yielding strength of rebars $f_{y}$ or the ultimate UHPC tensile strength $f_{U t u}$ , subscript i stand for each rebar layer and the UHPC layer, and the flexural lever arm d_v is 0.9 times the equivalent static depth d_eq.

ε_{t, M} = \frac{M_{f} / d_{v}}{E_{s} \cdot A_{s} + E_{s U} \cdot A_{s U} + E_{U} \cdot A_{U}}

(8)

ε_{t, V} = \frac{α \cdot V_{f} / \tan θ}{E_{s} \cdot A_{s} + E_{s U} \cdot A_{s U} + E_{U} \cdot A_{U}}

(9)

ε_{t, N} = \frac{α \cdot N_{f}}{E_{s} \cdot A_{s} + E_{s U} \cdot A_{s U} + E_{U} \cdot A_{U}}

(10)

α = \frac{E_{s} \cdot A_{s} + E_{s U} \cdot A_{s U} + E_{U} \cdot A_{U}}{E_{s c} \cdot A_{s c}}

(11)

d_{e q} = \frac{\sum_{i} d_{i} \cdot A_{i} \cdot f_{i}}{\sum_{i} A_{i} \cdot f_{i}}

(12)

Considering the tensile strain hardening behaviour of UHPC, elastic $E_{U e}$ and strain hardening $E_{U s h}$ axial rigidity contribution must also be considered in the model. Since UHPC is most likely to be in its strain hardening phase when reaching monolithic action shear resistance, a large fraction of applied stresses is used by its elastic rigidity. Hence, equation (13) becomes the modified equation for the estimation of the longitudinal strain at mid-depth ε_x of a hybrid element, where E_Ue and E_Ush are the elastic and strain hardening modulus of the UHPC, A_U is the area of the UHPC and ε_Ute is the elastic strain of the UHPC. It removes the UHPC elastic stress fraction $((E_{U e} - E_{U s h}) \cdot A_{U} \cdot ε_{U t e})$ from the applied stresses and considers UHPC strain hardening rigidity $E_{U s h}$ for the residual stresses. $ε_{x}$ computed with equation (13) must be greater than half the UHPC elastic strain $ε_{U e}$ due to the assumption that UHPC is in strain hardening phase.

ε_{x} = \frac{1}{2} (\frac{(M_{f} / d_{v} + V_{f} + 0.5 N_{f}) - ((E_{U e} - E_{U s h}) \cdot A_{U} \cdot ε_{U t e})}{E_{s} \cdot A_{s} + E_{s U} \cdot A_{s U} + E_{U s h} \cdot A_{U}})

(13)

Compared to rebars layers (terms $E_{s} \cdot A_{s}$ and $E_{s U} \cdot A_{s U}$ ), axial rigidity contribution of the UHPC layer in its strain hardening phase ( $E_{U s h} \cdot A_{U}$ ) is relatively small. If the UHPC tensile law is considered plastic rather than strain hardening, as suggested by CSA-S6 (CSA, 2019) for design purposes, the term $E_{U s h} \cdot A_{U}$ becomes equal to 0. In that condition, the value of the ratio the tensile rebars and UHPC layer axial rigidity over the compressed rebars axial rigidity α is close to 0.5, the assumed value for a NSC element. Therefore, α is considered equal to 0.5 alike a NSC element and the assumption that $0.5 / \tan θ$ is equal to 1 for the possible range of θ is kept in equation (13).

The shear resistance V_R of the monolithic action of a hybrid NSC-UHPC element is thus computed using equations (1) to equation (6) by replacing the static height of the centroid of longitudinal tension rebars d_s with the equivalent static height from equation (12) and by computing the longitudinal strain at mid-depth $ε_{x}$ with equation (13).

Structural hardening resistance model for V_Rsh

The occurrence of a diagonal shear crack within the NSC region of a hybrid element initiates a composite mechanical action which divides the element into three subsections and provides a structural hardening capability to the element, as described in Introduction section. Stresses reorganization acting in that stage takes the shape of a strut-and-tie system with plastic hinges depicted in Figure 3. Figure 3(a) and (b) present respectively principal maximum strain ε₁ and principal minimum strain ε₂ measured from digital image correlation (DIC) in Pharand and Charron (2022a) experimental program. DIC snapshot of slab UD-S were taken at V = 220 kN just before failure, which corresponds to the state ➁ in Figure 1. Strains displayed range from 0 µε in purple to 2000 µε in red for the principal maximum strain ε₁, and from −2000 µε in purple to 0 µε in red for the principal minimum strain ε₂. Those ranges were used as the 2000 µε limits corresponds to the yielding strain of Canadian reinforcement and to the assumed plastic strain of a strut-and-tie NSC node (CSA, 2014). Figure 3(c) presents the simplified proposed strut-and-tie model for the structural hardening resistance of a hybrid element.

Figure 3.

Composite mechanical action mechanism for hybrid NSC-UHPC elements, (a) principal maximum strains, (b) principal minimum strains, (c) proposed structural hardening resistance model.

In the simplified proposed model for the structural hardening resistance (Figure 3(c)), ties T₁ and T₂ carry tensile stresses from the applied load P, reduced by the plastic hinges H₁ and H₂ reactions. Both ties’ strength is defined from the combined properties of the tensile cord: NSC rebars, UHPC rebars as well as UHPC layer. Plastic moment M_U of hinges is defined only using UHPC layer section properties, which includes the UHPC and its rebars. Strut C₁ at angle θ₁ carries compression stresses from plastic hinge H₁ reaction, while strut C₂ at angle θ₂ carries compression stresses from the applied load P, reduced by the plastic hinge H₂ reaction. Both struts’ strength is defined from the NSC properties. All node locations, except for hinge H₁, are geometrically determined. Although low orientation angle (<25°) for struts is not recommended (Muttoni et al., 1996), strut C₂ (Figure 3(c)) was defined as such based on experimental data (Figure 3(b)) confirming the formation of a direct strut between loading and bearing plates. Stability of strut C₂ at plastic hinge H₂ is achieved by the fact that the plastic hinge node H₂ is a hybrid NSC-UHPC node crossed by a UHPC tie. Indeed, the UHPC tie significantly reduces the principal maximum strain ε₁ induced at the plastic hinge node H₂ (Figure 3(a)), which does not significantly reduce the effective compressive strength of the NSC within the node.

Forces and angles of inclination of the struts and ties system are given by equation (14) to equation (19) where $a$ is the distance between support and applied load and $l_{h}$ is the distance between plastic hinges. The plastic moment M_U, it is computed using Annex 8 of CSA-S6 (CSA, 2019) recommendations for full UHPC elements, as the UHPC layer between hinges (subsection S3 in Figure 1(b)) is delaminated and acts as a full UHPC element.

T_{1} = P \cdot \frac{a}{d_{v}} - 2 (\frac{M_{U}}{d_{v}})

(14)

T_{2} = P \cdot \frac{a}{d_{v}} - 2 (\frac{M_{U}}{d_{v}}) \cdot \frac{a}{l_{h}}

(15)

C_{1} = - 2 (\frac{M_{U}}{l_{h} \cdot \sin θ_{1}})

(16)

C_{2} = - (\frac{P}{\sin θ_{2}} - 2 (\frac{M_{U}}{l_{h} \cdot \sin θ_{2}}))

(17)

θ_{1} = \tan^{- 1} (\frac{d_{v}}{{a - l}_{h}})

(18)

θ_{2} = \tan^{- 1} (\frac{d_{v}}{a})

(19)

Based on the proposed model (Figure 3), four conditions can limit the structural hardening capacity of the composite mechanical action: (I) tensile strength of tie T₁ is exceeded; (II) rotation capacity of hinges H₁ or H₂ is exceeded; (III) NSC-UHPC interface strength at hinge node H₂ is exceeded; or (IV) node-to-strut interface strength of node N_c is exceeded. First two conditions are ductile failure type, while last two conditions are fragile failure type.

In this paper, the failure from concrete crushing at the node-to-strut interface of node N_c is assessed since it is the most frequent failure mode observed for elements with good NSC-UHPC interface and with good UHPC placement (Ji and Liu, 2020; Noshiravani and Brühwiler, 2013; Pharand and Charron, 2022a; Sine et al., 2022; Yin et al., 2017). Figure 4 presents longitudinal strain ε_x measured from DIC in Pharand and Charron (2022a) experimental program as well as the compression node N_c from Figure 3(c). In Figure 4(a), DIC snapshot was taken on specimen UD-S at V = 220 kN just before failure, which corresponds to the state ➁ in Figure 1. Strains displayed range from −3000 µε in purple to 3000 µε in red to show NSC compression fracture and UHPC tensile failure respectively. Thick dark lines qualitatively indicate cracks with larger opening.

Figure 4.

Reduced strut area, (a) experimental DIC results from Pharand and Charron (2022a) experiment, (b) illustration of the fictitious node.

Struts C₁ and C₂ (Figure 3) are combined into a single strut C (Figure 4(b)) by addition of their X and Y components, which eliminates the unknown variable l_h of the calculation process. The combined strut force C_c and angle of inclination $θ_{c}$ are given by equation (20) and equation (21). For the node N_c dimensions, the bearing face length l_Nc is defined by bearing plate dimension (Figure 4(b)) and the back face height h_Nc is defined by the element’s bending capacity at concrete crushing pivot point $M_{r u}$ with equation (22), where $α_{1}$ is the ratio of average stress in rectangular ultimate compression block to the specified concrete strength $f_{c}^{'}$ . The bending capacity at concrete crushing pivot point $M_{r u}$ of a hybrid element can be estimated using the simplified moment-curvature model from Pharand and Charron (2022b) or any other detailed sectional analysis method. Occurrence and widening of the diagonal shear crack near the node N_c physically impede the development of the width $w_{N c}$ (Figure 4(a)). To account for the decreasing strut efficiency due to the diagonal shear crack, a reduced width $w_{N c}^{'}$ is estimated with equation (23) by assuming a fictitious node geometry $N_{c}^{'}$ (Figure 4(b)). Its back face height $h_{N c}^{'}$ is defined by the resulting tensile cord load at V_R with equation (24) and for which the fictitious bearing face length $l_{N c}^{'}$ is unaffected and equals to l_Nc. According to the Canadian code recommendations (CSA, 2014), resistance of the node-to-strut interface R_Nc is obtained by equation (25), where $b$ is the width of the element and 0.85 is the resistance coefficient for node regions bound by struts and bearing areas (CCC zone).

C_{c} = {({(\frac{P \cdot a}{d_{v}} - \frac{{2 \cdot M}_{U}}{d_{v}})}^{2} + P^{2})}^{\frac{1}{2}}

(20)

θ_{c} = \tan^{- 1} (\frac{P \cdot d_{v}}{P \cdot a - 2 \cdot M_{U}})

(21)

h_{N c} = \frac{\frac{M_{r u}}{d_{v}}}{α_{1} \cdot f_{c}^{'} \cdot b}

(22)

w_{N c}^{'} = (l_{N c} \cdot \sin θ_{c} + h_{N c}^{'} \cdot \cos θ_{c})

(23)

h_{N c}^{'} = \frac{V_{R} \cdot \frac{a}{d_{v}}}{α_{1} \cdot f_{c}^{'} \cdot b} \leq h_{N c}

(24)

R_{N c} = w_{N c}^{'} \cdot b \cdot f_{c}^{'} \cdot 0.85

(25)

Hence, assuming the failure is controlled by concrete crushing at the node-to-strut interface of node N_c, the structural hardening resistance V_Rsh of the composite mechanical action is obtained with equation (26) by finding the load P for which the resistance of the node-to-strut interface $R_{N c}$ becomes smaller than the combined strut force $C_{c}$ .

C_{c} - R_{N c} \leq 0

(26)

Given that plastic moment of the UHPC layer is relatively small compared to the equivalent bending moment corresponding to the structural hardening resistance of the element, $θ_{c}$ can conservatively be approximated by $θ_{2}$ . The inclination $θ_{2}$ being calculated geometrically, the resistance of the node-to-strut interface does not depend anymore on the load P and is thus simplified. Therefore, by substituting equation (20) to equation (25) into equation (26), equation (26) can be transformed to a quadratic equation (equation (27)) with the load P as the independent variable is then obtained.

{({(\frac{P \cdot a}{d_{v}} - \frac{{2 \cdot M}_{U}}{d_{v}})}^{2} + P^{2})}^{\frac{1}{2}} - (l_{N c} \cdot \sin θ_{2} + \frac{V_{R} \cdot \frac{a}{d_{v}}}{α_{1} \cdot f_{c}^{\cdot} \cdot b} \cdot \cos θ_{2}) \cdot b \cdot f_{c}^{'} \cdot 0.85 \leq 0

(27)

Quadratic equation equation (27) can be rearranged as in equation (28) where A, B and C are quadratic coefficients defined by equation (29) to equation (31). Based on the context of the problem, the positive root (equation (32)) provides the load P corresponding to the structural hardening resistance $V_{R s h}$ of the element. A computed value of $V_{R s h}$ smaller than V_R indicates the composite mechanical action cannot provide structural hardening and the failure will thus occur at the end of the monolithic action given by V_R. Therefore, $V_{R s h}$ corresponds to the maximum value of P and $V_{R}$ (equation (33)).

A \cdot P^{2} + B \cdot P + C \leq 0

(28)

A = 1 + \frac{a^{2}}{{d_{v}}^{2}}

(29)

B = - 4 \cdot a \cdot M_{U} / {d_{v}}^{2}

(30)

C = 4 \cdot {M_{U}}^{2} / {d_{v}}^{2} - {R_{N c}}^{2}

(31)

P = \frac{- B + \sqrt{B^{2} - 4 \cdot A \cdot C}}{2 \cdot A}

(32)

V_{R s h} = \max ({P, V}_{R})

(33)

In the structural hardening model (Figure 3(c)), eventual stirrups located inside NSC do not directly contribute to the load-bearing capacity of the node-to-strut interface, but rather limit the incursion of the diagonal shear crack into the node N_c, which results in a larger fictitious node $N_{c}^{'}$ . This reduced interference of the shear crack to the node N_c is automatically accounted by the definition of the fictitious node $N_{c}^{'}$ through equation (24), as stirrups contribution V_s is already considered in the model of shear resistance V_R.

Experimental data

The proposed models to evaluate the shear resistance V_R and the structural hardening resistance $V_{R s h}$ of hybrid NSC-UHPC elements are validated with series of experimental data described hereafter.

A total of 31 hybrid NSC-UHPC elements that showed shear failure were retrieved from five international experimental campaigns. A schematic view of an equivalent cantilever test setup and elements’ cross-section configuration is presented in Figure 5. Configuration type (P, S, R/RS) of cross-section refers to the main function of the UHPC layer (Pharand and Charron, 2022a). In type P, the UHPC layer is unreinforced, and its purpose is to protect the NSC from environmental hazards. In type S, the UHPC layer is reinforced with additional rebars to strengthen the NSC element’s resistance. In type R, the UHPC layer is used to rehabilitate the NSC element by replacing the NSC surrounding rebars with UHPC. Type R becomes type RS if new rebars are added in the original concrete cover to replace rebars damaged by corrosion or to increase more significantly the cross-section capacity.

Figure 5.

Schematic view of equivalent cantilever test setup and section parameters of experimental data used for models validation.

Summary of cross-section and test setup parameters for each test specimen considered are presented in Table 1, where h_c is the NSC substrate thickness, h_U is the UHPC overlay thickness, l_Nc is half the bearing plate length and a is the center-to-center span length for an equivalent cantilever setup (Figure 5). Test specimens include all four configuration types (P, S, R and RS) and show a wide range of testing parameters such as width varying between 150 and 400 mm, total height varying between 100 and 690 mm, ratio of UHPC to NSC heights varying between 0.15 and 0.50 and ratio of span to equivalent static depth varying between 2.23 and 7.80.

Table 1.

Summary of cross-section and test setup parameters of the five experimental campaigns.

ID	Type	b (mm)	h_c (mm)	h_U (mm)	A_s (mm²)	d_s (mm)	A_sU (mm²)	d_sU (mm)	l_Nc (mm)	a (mm)	d_eq (mm)	a/d_eq (-)	Ref.
MN3^a	S	150	250	50	462	237	201	275	60	800	255	3.14	Noshiravani and Brühwiler (2013)
MW3^b	S	150	250	50	236	237	150	275	60	800	259	3.08
MW4^b	S	150	250	50	236	237	201	275	60	800	259	3.09
MW5^b	S	150	250	50	236	237	201	275	60	800	261	3.07
MW6^b	S	150	250	50	462	237	201	275	60	800	255	3.14
RE-20	P	300	80	20	565	74	—	—	0	600	77	7.80	Yin et al. (2017)
OV-25	P	300	100	25	565	74	—	—	0	600	82	7.28
OV-25a	S	300	100	25	565	74	393	105	0	600	90	6.66
OV-50	P	300	100	50	565	74	—	—	0	600	92	6.51
OV-50a	S	300	100	50	565	74	393	125	0	600	102	5.88
RC-U-2.7^c	P	200	500	50	2500	443	900	525	25	1200	450	2.66	Ji and Liu (2020)
RC-RU-2.4^c	S	200	500	50	2500	443	900	525	25	1050	470	2.23
RC-RU-2.7^c	S	200	500	50	2500	443	900	525	25	1200	470	2.55
RC-RU-3.1^c	S	200	500	50	2500	443	900	525	25	1400	470	2.98
VT-U3	P	150	200	30	339	180	—	—	50	600	186	3.23	Sine et al. (2022)
VT-U5	P	150	200	50	339	180	—	—	50	600	191	3.14
VT-RU1	S	150	200	50	339	180	157	225	50	600	200	3.00
VT-RU2	S	150	200	50	339	180	236	225	50	600	203	2.96
VT-RU3	S	150	200	50	339	180	339	225	50	600	206	2.92
VB0-RU	S	150	200	50	57	180	339	225	50	600	220	2.72
VB1-RU	S	150	200	50	157	180	339	225	50	600	213	2.82
VB2-RU	S	150	200	50	226	180	339	225	50	600	210	2.86
VS1-RU	S	150	200	30	339	180	101	215	50	600	192	3.13
VS2-RU	S	150	400	60	679	360	201	430	50	1200	383	3.13
VS3-RU	S	150	600	90	1018	540	302	645	50	1800	575	3.13
US-S	S	400	240	50	1500	220	300	265	60	785	236	3.32	Pharand and Charron, (2022a)
PS-S	S	400	220	70	1500	220	300	265	60	785	236	3.32
US-R	R	400	190	60	—	—	1500	220	60	785	220	3.57
PS-R	R	400	200	50	—	—	1500	220	60	785	221	3.55
US-RS	RS	400	190	60	—	—	1800	220	60	785	220	3.57
PS-RS	RS	400	200	50	—	—	1800	220	60	785	221	3.55

^aStirrups (57 mm²) with 250 mm spacing considered.

^bstirrups with 400 mm spacing not considered.

^cStirrups (100 mm²) with 250 mm spacing considered.

Main material properties measured in experiments are summarized in Table 2, where

f_{c}^{'}

and

f_{U c}^{'}

are the cylinder compressive strength of NSC and UHPC respectively,

f_{U t e}

and

f_{U t u}

are respectively the elastic and ultimate UHPC tensile strength,

ε_{U t u}

is the ultimate UHPC tensile strain reached at

f_{U t u}

f_{y, s}

and

f_{y, s U}

are respectively the NSC and UHPC rebars yielding strength. Again, test specimens include a wide range of material properties. The substrate compressive strength varies between 23 and 50 MPa, ultimate UHPC tensile strength and strain vary respectively between 7.8 and 12.5 MPa and 2500–4000 µε, and yielding strength of rebars varies between 433 and 710 MPa.

Table 2.

Summary of material properties of the five experimental campaigns.

ID	f ′_c (MPa)	f ′_Uc (MPa)	f_Ute (MPa)	f_Utu (MPa)	ε_Utu (-)	f_y,s (MPa)	f_y,sU (MPa)	Ref.
MN3	41.6	160.0	10.2	12.5	0.0030	565	710	Noshiravani and Brühwiler (2013)
MW3	41.6	160.0	10.2	12.5	0.0030	594	710
MW4	41.6	160.0	10.2	12.5	0.0030	594	516
MW5	41.6	160.0	10.2	12.5	0.0030	594	703
MW6	41.6	160.0	10.2	12.5	0.0030	565	710
RE-20	23.0	153.0	8.5	10.5	0.0030	502	—	Yin et al. (2017)
OV-25	23.0	153.0	8.5	10.5	0.0030	502	—
OV-25a	23.0	153.0	8.5	10.5	0.0030	502	475
OV-50	23.0	153.0	8.5	10.5	0.0030	502	—
OV-50a	23.0	153.0	8.5	10.5	0.0030	502	475
RC-U-2.7	46.2	150.0	8.8	12.1	0.0040	487	—	Ji and Liu (2020)
RC-RU-2.4	46.2	150.0	8.8	12.1	0.0040	487	530
RC-RU-2.7	46.2	150.0	8.8	12.1	0.0040	487	530
RC-RU-3.1	46.2	150.0	8.8	12.1	0.0040	487	530
VT-U3	35.0	135.0	7.0	7.8	0.0025	537	—	Sine et al. (2022)
VT-U5	35.0	135.0	7.0	7.8	0.0025	537	537
VT-RU1	35.0	135.0	7.0	7.8	0.0025	537	537
VT-RU2	35.0	135.0	7.0	7.8	0.0025	537	537
VT-RU3	35.0	135.0	7.0	7.8	0.0025	537	537
VB0-RU	37.0	135.0	7.0	7.8	0.0025	478	510
VB1-RU	37.0	135.0	7.0	7.8	0.0025	530	510
VB2-RU	37.0	135.0	7.0	7.8	0.0025	537	510
VS1-RU	50.0	135.0	7.0	7.8	0.0025	537	562
VS2-RU	50.0	135.0	7.0	7.8	0.0025	537	562
VS3-RU	50.0	135.0	7.0	7.8	0.0025	537	562
US-S	35.5	170.0	9.0	12.0	0.0032	433	450	Pharand and Charron, (2022a)
PS-S	35.5	170.0	9.0	12.0	0.0032	433	450
US-R	35.5	170.0	9.0	12.0	0.0032	—	433
PS-R	35.5	170.0	9.0	12.0	0.0032	—	433
US-RS	35.5	170.0	9.0	12.0	0.0032	—	436
PS-RS	35.5	170.0	9.0	12.0	0.0032	—	436

The models’ validation with such a large variation of testing parameters and materials properties for hybrid elements will provide a good estimate of their accuracy and confidence in their application.

Validation of the proposed models

Validation of the proposed models for evaluating the shear resistance

V_{R}

of the monolithic action and the structural hardening resistance

V_{R s h}

for the composite mechanical action is performed using experimental data summarized in Experimental Data section. Experimental and computed resistances are presented in Table 3, where subscript _EXP refers to experimental values, and V_bending is the corresponding shear load of the bending moment capacity calculated with the simplified moment-curvature model from Pharand and Charron (2022b). V_bending values superior to

V_{R s h}

values confirm the composite mechanical action failure happened before the bending failure of all specimens. θ_c values presented were computed using equation (21). Finally, ratios

V_{R} / V_{R EXP}

and

V_{R s h} / V_{R s h EXP}

show the accuracy of models, values inferior to 1 mean proposed models are conservative.

Table 3.

Comparison of the proposed models and the experimental results.

ID	V_{R EXP} (kN)	V_{Rsh EXP} (kN)	V_c (kN)	V_s (kN)	V_R (kN)	V_Rsh (kN)	V_bending (kN)	θ _c (°)	V_R/V_{R EXP}	V_Rsh/V_{Rsh EXP}
MN3	90.4^a	134.8	55.3	52.8	108.1	110.7	137.2	17.2	1.20	0.82
MW3	78.3	91.8	51.6	0	51.6	82.0	96.9	17.5	0.66	0.89
MW4	75.0	90.8	52.3	0	52.3	83.6	96.2	17.5	0.70	0.92
MW5	82.0	99.6	53.6	0	53.6	85.7	107.1	17.9	0.65	0.86
MW6	77.0	90.9	57.9	0	57.9	89.3	137.2	17.5	0.75	0.98
RE-20	^b	28.6	25.3	0	25.3	25.3	32.7	7.0	—	0.88
OV-25	^b	36.7	32.5	0	32.5	32.5	36.7	7.7	—	0.89
OV-25a	^b	39.0	37.5	0	37.5	42.8	53.6	9.4	—	1.10
OV-50	^b	39.0	44.1	0	44.1	44.1	47.6	10.7	—	1.13
OV-50a	^b	47.5	49.6	0	49.6	55.3	69.7	11.7	—	1.16
RC-U-2.7	^b	358.5	153.2	119.3	272.5	289.6	427.0	18.9	—	0.81
RC-RU-2.4	^b	477.5	174.8	127.4	302.2	325.6	651.1	23.1	—	0.68
RC-RU-2.7	^b	461.0	169.4	126.4	295.8	321.8	569.7	20.3	—	0.70
RC-RU-3.1	^b	380.5	163.0	125.1	288.1	316.0	488.3	17.5	—	0.83
VT-U3	44.8	53.4	34.0	0	34.0	51.1	57.5	16.0	0.76	0.96
VT-U5	52.1	60.2	39.0	0	39.3	59.5	64.6	17.1	0.75	0.99
VT-RU1	61.0	76.8	42.5	0	42.5	66.0	90.1	18.4	0.70	0.86
VT-RU2	63.0	81.3	45.4	0	45.4	71.6	102.2	19.3	0.72	0.88
VT-RU3	66.9	86.8	47.8	0	47.8	77.5	117.3	20.3	0.71	0.89
VB0-RU	56.3	60.3	45.7	0	45.7	78.2	82.1	21.4	0.81	1.30
VB1-RU	57.5	98.3	46.8	0	46.8	78.3	94.9	20.7	0.81	0.80
VB2-RU	65.8	107.1	47.6	0	47.6	78.5	103.0	20.4	0.72	0.73
VS1-RU	54.5	54.5	42.2	0	42.2	69.5	75.5	16.6	0.77	1.28
VS2-RU	94.1	94.1	77.1	0	77.1	107.3	151.0	16.8	0.82	1.14
VS3-RU	140.6	140.6	106.8	0	106.8	139.7	224.8	16.9	0.76	0.99
US-S	120.0	220.0	136.4	0	136.4	186.4	259.4	16.0	1.14	0.85
PS-S	167.0	290.0	144.1	0	144.1	199.8	279.8	16.4	0.86	0.69
US-R	146.0	228.0	125.5	0	125.5	201.0	218.7	17.7	0.86	0.88
PS-R	115.0	192.0	122.3	0	122.3	190.7	210.8	17.1	1.06	0.99
US-RS	159.0	252.0	130.6	0	130.6	212.8	248.3	18.1	0.82	0.84
PS-RS	160.0	188.0	127.6	0	127.6	201.5	240.7	17.5	0.80	1.07
Average									0.81	0.93
S.D.									0.14	0.16

^aSecond crack considered since stirrups bridged first crack.

^bUnspecified in references (Yin et al., 2017; Ji and Liu, 2020).

The predicted resistances $V_{R}$ and $V_{R s h}$ from the proposed models are plotted against experimental results $V_{R EXP}$ and $V_{R s h EXP}$ in Figure 6 with solid red markers. Solid line is the target line when predicted resistances equal the experimental results, while dashed lines are $\pm 20 %$ bounds for over and under predictions relatively to experimental results. For Figure 6(b), blue X markers are results provided by SIA (2016) model.

Figure 6.

Comparison of the proposed models' predictions and the experimental results, (a) shear resistance, (b) strain hardening resistance.

Shear resistance V_R

The proposed shear resistance model shows an average resistance prediction ratio (V_R/V_{R EXP}) of 0.81 with a SD of 0.14 (Table 3). This indicates that the proposed model provides generally conservative predictions of the monolithic action’s resistance. Analysis of results could not correlate that specific testing configuration can lead to a particular overestimation or underestimation. There are two explanations to this observation.

First, Yerzhanov et al. (2019) have demonstrated that the Canadian general method is the most accurate model compared to other international design codes model for regular NSC elements. The Canadian general method is yet conservative with a resistance prediction ratio (V_R/V_{R EXP}) of 0.74¹ based of a database of 784 beams. Since the proposed model for the shear resistance of hybrid elements integrates UHPC contribution to the general method model for NSC, it shares it conservative accuracy. Furthermore, the Canadian general method uses the effective crack spacing $s_{z e}$ to establish a simplified relationship between crack width and longitudinal strain. In addition to being a conservative simplification, it becomes even more conservative for hybrid elements as the addition of a UHPC layer on the tensile side reduces cracks width and spacing (Al-Osta et al., 2017).

Second, experimental shear resistance V_{R EXP}, which is defined by the end of the monolithic action of the hybrid cross-section, is measured differently by authors. For Noshiravani and Brühwiler (2013); Sine et al. (2022), V_{R EXP} is defined as the shear load before the sudden small drop in load-displacement curve which marks the end of stresses reorganization in Figure 1. For Pharand and Charron (2022a), V_{R EXP} is defined as the shear load marking the initiation of stresses reorganization. This initiation occurs at a lower load as shown by DIC measurements. As a result, Pharand and Charron (2022a) results are closer to the target line in Figure 6(a), while Noshiravani and Brühwiler (2013) as well as Sine et al. (2022) results are grouped around the −20% line. Overall, reasonable conservative results obtained with the shear resistance model for hybrid elements validate the proposed model which integrates UHPC contribution into the Canadian general method model.

Structural hardening resistance V_Rsh

For the proposed structural hardening resistance model, an average resistance prediction ratio (V_Rsh/V_{Rsh EXP}) of 0.93 with a SD of 0.16 are obtained (Table 3), which indicates predictions are accurate, while being slightly conservative. Indeed, for most tested specimens, prediction are grouped around the target tine (Figure 6(b)), except for Ji and Liu (2020) specimens which all predictions are grouped around the −20% line. Specimens from this experimental campaign stand out from others by two factors: these specimens contain stirrups and were tested at the lowest span to equivalent depth ratio (a/d_eq) varying from 2.23 to 2.98 (Table 1). The number of specimens within the five experimental campaigns with a diagonal shear crack intercepting stirrups is very limited. Nonetheless, MN3 falls into this category and its predictions ratio of 0.82 (Table 3) is within one SD of the average, meaning the model’s precision can be adequate with stirrups. Concerning low span to equivalent depth ratio, VB series from Sine et al. (2022) was tested with ratios varying from 2.72 to 2.86 (Table 1) and also had greater dispersion for resistance prediction ratios (0.73–1.30 in Table 3). Hence, the proposed model has a lower accuracy for elements with a/d_eq ratio smaller than 2.9.

Performance of the proposed structural hardening resistance model estimating V_Rsh can be compared to the one of the SIA design code (2016) for the same 31 hybrid NSC-UHPC elements in Figure 6(b) (blue X markers). In general, the resistance predictions from SIA (2016) model are less accurate than the proposed model: results are more conservative (average prediction ratio V_Rsh/V_{Rsh EXP} of 0.88) and show larger scatter (SD of 0.37). SIA (2016) model largely underestimate resistance of Ji and Liu (2020) specimens. Furthermore, its resistance predictions for highly reinforced specimens with a/d_eq ratios varying from 5.88 to 7.80 (Yin et al., 2017) are clearly overestimated.

As a result, very good agreement with test results of 31 hybrids beams validate the proposed structural hardening resistance model for the resistance prediction of the composite mechanical action controlled by concrete strut crushing.

Discussion

The original approach of the proposed models performs well for predicting the capacities of hybrid NSC-UHPC elements under high shear load as they integrate the key shear parameters of the two bearing mechanisms.

Bearing mechanisms

Unlike available SIA (2016) design code predicting only the ultimate shear capacity by adding NSC, UHPC and stirrups’ individual contributions, the proposed approach allows prediction of each observed bearing mechanism with distinct models. In the first stage (continuous green line in Figure 1(a)), when the hybrid element behaves monolithically, the proposed model for shear resistance corresponds to a modified version of Canadian general method. The NSC, UHPC and stirrups’ individual contributions are summed up to compute the capacity, which is adequate a monolithic structural element. In the second stage (bold doted orange line in Figure 1(a)), the hybrid element is divided into three subsections (Figure 1(b)) that enables a structural hardening behaviour. The proposed structural hardening model therefore bases the resistance prediction on a realistic strut-and-tie system observed in experimental testing (Pharand and Charron, 2022a). Hence, the proposed approach gives accurate predictions of the shear resistance $V_{R}$ and of the structural hardening resistance $V_{R s h}$ because it includes a representative model for each bearing mechanism. Moreover, the proposed approach gives designers the flexibility to consider or not the additional resistance provided by the composite mechanical action.

Shear parameters

Size effect, reinforcement ratio, span to equivalent depth ratio, compressive strength and magnitude of applied loads are important parameters influencing the shear resistance that must be considered for hybrid NSC-UHPC elements, as it is the case for NSC elements. Since the proposed model for the shear resistance $V_{R}$ of hybrid elements integrates the UHPC contribution to the Canadian general method model for NSC, it thus considers all these critical shear parameters.

In the proposed model for the structural hardening resistance $V_{R s h}$ , reinforcement ratio, span to equivalent depth ratio, as well as compressive and tensile materials strength are directly considered through the strut-and-tie system. Indeed, reinforcement ratio and tensile UHPC properties modify ties and hinges capacity, span to equivalent depth ratio impacts the inclination of the combined strut and compressive NSC strength influences the combined strut capacity. On the other hand, magnitude of applied loads and size effect are indirectly considered using a fictitious node region. As applied loads and equivalent depth increase, shear resistance V_R computed through the modified Canadian general method is reduced. Since the fictitious node region size is defined by the shear resistance V_R, the resulting combined strut capacity is thus modified accordingly by applied loads and size effect. This characteristic of the proposed model for structural hardening resistance is a significant improvement relatively to the SIA (2016) model not accounting for the size effect in hybrid elements.

Limitations

The proposed shear resistance model for V_R predicts the shear resistance of the monolithic action while the proposed structural hardening model for V_Rsh predicts the ultimate resistance of the composite mechanical action controlled by the concrete crushing of the combined strut. Both models were validated for a wide range of repair configurations (P, R, S and RS) on beams and unidirectional slabs (total height (h_c + h_U) varying between 100-690 mm, ratio of UHPC to NSC heights (h_U/h_c) varying between 0.15-0.50, ratio of span to equivalent static depth (a/d_eq) varying between 2.23-7.80) as well as various materials properties. Depending on the construction process, NSC-UHPC interface characteristics as well as fibers orientation and density may lead to other types of failure such as limited hinges rotation capacity, limited ties capacity and limited NSC-UHPC interface capacity. Further experimental and numerical research is needed to evaluate extend of the main and each secondary failure mode. Then, the model could be improved to capture all shear failure modes of hybrid NSC-UHPC elements.

Conclusion

The use of UHPC as repair material for deficient NSC structural elements is growing. However, design codes are not yet fully developed to represent both the monolithic and the composite mechanical actions showed by hybrid NSC-UHPC elements under shear load. This paper addresses the evaluation of the shear resistance V_R from the monolithic action and of the structural hardening resistance V_Rsh of the composite mechanical action for hybrid NSC-UHPC elements by proposing two models based on the Canadian design philosophy. The following point can be highlighted:

1. The shear resistance model evaluates the shear load at which the hybrid section ends its monolithic action due to the localization of a diagonal shear crack in the NSC. The model integrates UHPC contribution for the strain at mid-height estimation to the Canadian general method model.

2. The shear resistance model offers reasonably accurate and conservative results with an average predictions over experimental results ratio of 0.81 and with a SD of 0.14.

3. The structural hardening model evaluates the ultimate load the composite mechanical action can sustain. It is modelled using a strut-and-tie system with plastic hinges. Struts are defined by NSC properties, ties are defined by the total tensile cord properties, while plastic hinges are defined only by the UHPC layer and its embedded rebars properties.

4. The concept of the structural hardening model differs from conventional models by evaluating the resistance of a NSC node-to-strut interface rather than by summing NSC, UHPC and stirrups individual shear resistance contributions.

5. The structural hardening resistance model offers accurate and mostly conservative results with an average predictions over experimental results ratio of 0.93 and with a SD of 0.16.

The proposed models are an initial attempt to offer simple and intuitive design tools for the shear resistance of hybrid NSC-UHPC elements. Even though predictions are in very good agreement with experimental data, additional modules should be developed to evaluate capacity of other possible types of failure within the mechanical composite action.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This project was financially supported by the Discovery Grant of the Nature Science and Engineering Research Council of Canada (NSERC) granted to Prof. J.-P. Charron; and by a PhD Scholarship of the Fonds de recherche du Québec – Nature et technologies (FRQNT) awarded to M. Pharand.

Credit authors statement

Pharand, M.: Conceptualization, Methodology, Software, Investigation, Formal analysis, Writing – Original draft, Review and Editing, Visualization.

Charron, J.-P.: Conceptualization, Methodology, Formal Analysis, Resources, Funding acquisition, Writing – Review and Editing, Supervision.

Data availability statement

All data and models used during the study appear in the submitted article or are available from cited references.

ORCID iD

Martin Pharand

Note

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