Published Analytical Models

Chapter 1: Introduction

1.5 Relevant Literature

1.5.3 Published Analytical Models

The nominal shear capacity of RC deep beams (Vn) can be expressed by Equation 1.1, where Vc is the concrete contribution to the shear resistance including the dowel action and effect of steel fibers, and Vw is the contribution of the internal web shear reinforcement to the shear capacity. The concrete contribution to the shear capacity, Vcd, including the influence of dowel action is affected by the mechanical properties of the concrete, fiber volume fraction (vf), flexural steel reinforcement ratio (), and the shear span-to-effective depth ratio (a/d). The contribution of the internal web shear reinforcement, Vw, is influenced by the type of steel, area of shear reinforcement bars, and number of shear reinforcing bars intersected by the critical shear crack.

𝑉_𝑛= 𝑉_𝑐𝑑+ 𝑉_𝑤 1.1 Narayanan & Darwish (1987) proposed Equation 1.2 for the average concrete shear stress at shear failure (i.e. ultimate shear stress), vu, which is defined by Vcd/bd, where b is the beam width and d is the effective depth of the beam. The magnitude of e in Equation 1.2 depends on a/d; e = 1 when a/d

> 2.8 and e = 2.8(d/a) when a/d ≤ 2.8 whereas df is a bond factor that accounts for the bond characteristics of steel fibers and equals 0.5 for round fibers, 0.75 for crimped fibers, and 1 for indented fibers. The parameter τ is the average fiber matrix interfacial bond stress, 4.15 MPa.

𝑣_𝑢= 𝑒 [0.24𝑓_𝑠𝑝+ 80𝜌 (^𝑑

𝑎)] + 0.41𝜏(^{𝑙𝑓𝑣𝑓𝑑𝑓}

𝐷_𝑓 ) 1.2

Ashour et al. (1992) proposed Equation 1.3 to predict the ultimate shear stress of RC beams with a/d ≥ 2.5 and Equation 1.4 for beams with a/d

< 2.5.

For a/d ≥ 2.5:

𝑣_𝑢= [2.11√𝑓³ _𝑐^′+ 7 (^{𝑙𝑓𝑣𝑓𝑑𝑓}

𝐷_𝑓 )] (𝜌^𝑑

𝑎)

1⁄3

1.3 For a/d < 2.5:

𝑣𝑢= [2.11 √𝑓³ 𝑐′+ 7 (^{𝑙𝑓𝑣𝑓𝑑𝑓}

𝐷_𝑓 )] (𝜌^𝑑

𝑎)

1⁄3

[2.5 (^𝑎

𝑑)] + 0.41𝜏 (^{𝑙𝑓𝑣𝑓𝑑𝑓}

𝐷_𝑓 ) [2.5 − (^𝑎

𝑑)] 1.4 Kwak et al. (2002) proposed Equation 1.5 as a modified version of

that developed earlier by Narayanan & Darwish (1987) to predict the ultimate shear stress of RC beams. The factor k in Equation 1.5 depends on a/d; k = 1 for a/d > 3.4 and k = 3.4(d/a) for a/d ≤ 3.4.

𝑣_𝑢= 3.7𝑘(𝑓_𝑠𝑝)^2⁄3[𝜌 (^𝑑

𝑎)]

1⁄3

+ 0.8 [0.41𝜏 (^{𝑙𝑓𝑣𝑓𝑑𝑓}

𝐷_𝑓 )] 1.5 The contribution of the internal web shear reinforcement to the shear capacity, Vw, can be expressed by Equation 1.6 proposed by Kong et al.

(1977) and adopted by CIRIA guide (1984), where Cs is an empirical coefficient equal to 100 MPa for plain round bars and 225 MPa for deformed bars, h is the beam depth, and Aw is the area of an individual web steel bar, y is the depth of the intersection between the web steel bar and a potential critical shear crack measured from the compression face of the beam, and α is the angle of intersection between the web steel bar and the potential critical shear crack. The potential critical shear crack in solid RC deep beams is typically assumed to be developed in the direction of the natural load path that is the line connecting the inner points of the load and support plates.

𝑉_𝑤= 𝐶_𝑠∑ [^𝐴𝑤𝑦

ℎ ] sin²𝛼 1.6 1.5.3.2 Deep Beams with Openings

Structural idealization of reinforced concrete deep beams with openings is shown in Figure 1.2. The shear capacity of steel fiber-reinforced RC deep beams with openings, Vn, can be estimated using Equation 1.7. In this equation, Vc = concrete contribution to the shear capacity, and Vs = contribution of steel bars to the shear resistance, including web reinforcement and dowel action (Vw + Vd). The value of Vc is mainly influenced by the mechanical properties of concrete and a/d whereas that of Vs is affected by the properties and location of the steel reinforcing bars crossed by the critical shear crack.

𝑉_𝑛= 𝑉_𝑐+ 𝑉_𝑠 1.7

Figure 1.2: Structural idealization of deep beams with openings (Based on Kong & Sharp (1977), Shanmugam & Swaddiwudhipong (1988), and Ray

& Reddy (1979))

According to Kong & Sharp (1977), the shear strength of a deep beam is significantly affected only when the opening interrupts the natural load path, which is considered in this study as the line joining edges of the load and support plates. In this model, the applied load is assumed to be transmitted to the support mainly by a lower path and partly by an upper path [Figure 1.2]. Based on this structural idealization, the researchers proposed Equation 1.8 to estimate the nominal shear capacity of RC deep beams with an opening that interrupts the natural load path. In this equation, K1 and K2 = coefficients defining the position of opening, a1 and a2 = coefficients defining the opening size, X = clear shear span, fsp = cylinder-splitting tensile strength of concrete, λ = empirical coefficient, equal to 1.5 for web bars and 1 for main bars, Ab = area of an individual steel reinforcing bar, y1 = depth at which a typical bar intersects a potential critical diagonal crack (upper and lower load path in Figure 1.2), α1 = angle of inclination between a typical bar and a potential diagonal crack depending on the failure mode (upper and lower load path in Figure 8), C1 = 1.4 for normal weight concrete, C2 = 130 N/mm² for plain round bars and 300 N/mm² for deformed bars, h = total depth of the beam, and b = width of the beam.

𝑉𝑛= 𝐶1[1 − 0.35^(𝐾_(𝐾^1+𝑎1)𝑋

2−𝑎₂)ℎ] 𝑓𝑠𝑝𝑏(𝐾2− 𝑎2)ℎ +

∑ 𝜆𝐶₂𝐴_𝑏^𝑦1

ℎ sin²𝛼₁ 1.8 Shanmugam & Swaddiwudhipong (1988) proposed Equation 1.9 as a modified form of Kong and Sharp’s (1977) semi-empirical formula for nominal shear strength prediction of steel fiber-reinforced concrete deep beams. Equations 1.10 - 1.12 were utilized to calculate the required components of Equation 1.9, where, n = 1.1, f1 = reduction factor to account for the size of openings, f2 = reduction factor to account for the degree of interruption of the opening, k = the distance of the center of opening from the beam axis, r = factor depending on the location of the center of opening which is equal to 1 in case the center of opening is located in the unloading quadrants and 2 in case the center of opening is located in the loaded quadrants. The remaining notations are defined in the previous paragraph similarly to the Kong & Sharp Equation 1.8.

𝑉_𝑛= 𝐶₁𝑓₁𝑓₂[1 − 0.35^𝑋_ℎ] 𝑓𝑠𝑝𝑛𝑏ℎ + ∑ 𝜆𝐶₂𝐴_𝑏^𝑦1

ℎ sin²𝛼₁ 1.9 𝑓1= (1 − 𝑎1) (1 −^𝑎2

0.6) 1.10

𝑓₂= + 2(𝑘₂)^𝑟√ ^{(𝑘1−𝑘2)2}

[(𝑎₁𝑋)²+(𝑎₂ℎ)²]≤ 1 1.11

= 0.6 − 2𝑘 ≥ 2 1.12

Ray & Reddy (1979) proposed Equation 1.13 to predict the nominal shear strength of reinforced concrete deep beams with web openings. The researchers split Vs into two components, Vd to account for the dowel action and Vw to account for the web reinforcement (steel stirrups). In their study, the researchers postulated that the applied load may be assumed to be

transmitted to the supports through the natural load path connecting edges of the load and support plates even if the opening interferes with it. Equations 1.14 - 1.23 were utilized to calculate the required components of Equation 2.13. In these equations, ψs = 0.65, ψw = 0.5, As = area of tension steel, Aw = area of an individual web steel reinforcing bar, fy = 300 MPa for deformed bars and 130 MPa for smooth bars, f’c = cylinder compressive strength of the concrete, fsp = cylinder-splitting tensile strength of concrete, b = beam width, h = beam depth, c is a parameter dependent on the concrete compressive strength and tensile splitting strength, λ1, λ2 and λ3 are factors that account for the opening size and location, α = angle of inclination of web bar with horizontal, β = angle of inclination of the natural load path, X = clear shear span, XN = nominal shear span, ax = width of the opening, ay = height of the opening, ex and ey = eccentricities of the opening, K2 and K3 = coefficients defining the opening location. m is the ratio of path length intercepted to total path length along the natural load path.

𝑉_𝑛= 𝑉_𝑐+ (𝑉_𝑑+ 𝑉_𝑤) 1.13

𝑉_𝑐=sin 𝛽 cos 𝛽(tan 𝛽+tan 𝜑)^{𝑐𝑏ℎ(𝜆1𝜆2𝜆3)} 1.14

𝑉_𝑑= (𝜓_𝑠𝐴_𝑠𝑓_𝑦)[𝑡𝑎𝑛 𝛽 𝑡𝑎𝑛 𝜑−1

𝑡𝑎𝑛 𝛽+𝑡𝑎𝑛 𝜑] 1.15

𝑉_𝑤= 𝜓_𝑤∑ 𝐴_𝑤𝑓_𝑦[ sin 𝛼 cot 𝛽+cos 𝛼

(tan 𝛽+tan 𝜑)/(tan 𝛽 tan 𝜑)−cos 𝛼(1−tan 𝛼 tan 𝛽)

tan 𝛽+tan 𝜑 ] 1.16

𝑐 = √^{𝑓𝑐′𝑓𝑠𝑝}

2 1.17 tan 𝜑 =^{(𝑓𝑐′−𝑓𝑠𝑝)}

2√𝑓_𝑐^′𝑓_𝑠𝑝

1.18

𝜆₁= {1 −^{𝐾3𝑋𝑛}

3𝐾₂ℎ 𝑓𝑜𝑟 𝐾₃𝑋_𝑁 < 𝐾₂ℎ

2

3 𝑓𝑜𝑟 𝐾₃𝑋_𝑛𝑁 ≥ 𝐾₂ℎ 1.19 𝜆₂= 1 − 𝑚 1.20

𝝀_𝟑= {

(𝟎. 𝟖𝟓 + 𝟎. 𝟑^𝒆𝒙

𝑿𝒏𝒆𝒕) (𝟎. 𝟖𝟓 + 𝟎. 𝟑^𝒆𝒚

𝒀𝒏𝒆𝒕) ≤ 𝟏 𝒇𝒐𝒓 𝒄𝒆𝒏𝒕𝒆𝒓 𝒐𝒇 𝒐𝒑𝒆𝒏𝒊𝒏𝒈 𝒊𝒏 𝒕𝒉𝒆 𝒖𝒏𝒍𝒐𝒂𝒅𝒆𝒅 𝒒𝒖𝒂𝒅𝒓𝒂𝒏𝒕 (𝟎. 𝟖𝟓 + 𝟎. 𝟑 ^𝒆𝒙

𝑿𝒏𝒆𝒕) (𝟎. 𝟖𝟓 − 𝟎. 𝟑^𝒆𝒚

𝒀𝒏𝒆𝒕) ≤ 𝟏 𝒇𝒐𝒓 𝒄𝒆𝒏𝒕𝒆𝒓 𝒐𝒇 𝒐𝒑𝒆𝒏𝒊𝒏𝒈 𝒊𝒏 𝒕𝒉𝒆 𝒍𝒐𝒂𝒅𝒆𝒅 𝒒𝒖𝒂𝒅𝒓𝒂𝒏𝒕 1.21 𝑋𝑛𝑒𝑡 = 𝑋𝑛− 𝑎𝑥 1.22

𝑌_𝑛𝑒𝑡= 0.6ℎ − 𝑎_𝑦 1.23