Influence of Shear Span-to-Effective Depth Ratio on Behavior of High-Strength Reinforced Concrete Beams

Arowojolu, Olaniyi; Ibrahim, Ahmed; Almakrab, Abdullah; Saras, Nicholas; Nielsen, Richard

doi:10.1186/s40069-020-00444-7

International Journal of Concrete Structures and Materials

Table 6 Shear prediction models for beams with and without shear reinforcement.

From: Influence of Shear Span-to-Effective Depth Ratio on Behavior of High-Strength Reinforced Concrete Beams

Author	Shear prediction model (SI unit)
ACI 318–14	$V_{u} = \left( {\left( {0.16\sqrt {f_{c}^{^{\prime}} } + 17.6\rho_{w} \frac{{V_{u} d}}{{M_{u} }}} \right)b_{w} d} \right) + \frac{{A_{v} f_{y} d}}{s}$
BS8110	$V_{u} = \left( {\left( {\left( {\frac{0.79}{{\gamma_{m} }}} \right)\left( {\frac{{100A_{s} }}{{b_{w} d}}} \right)^{1/3} \left( \frac{400}{d} \right)^{1/4 } \left( {\frac{{f_{c}^{^{\prime}} }}{25}} \right)^{1/3} } \right)b_{w} d } \right) + \frac{{A_{v} f_{y} d}}{s} $ for a/d $\ge 2.0$; $V_{u} = \left( {\left( {\left( {\frac{0.79}{{\gamma_{m} }}} \right)\left( {\frac{{100A_{s} }}{{b_{w} d}}} \right)^{1/3} \left( \frac{400}{d} \right)^{1/4 } \left( {\frac{{f_{c}^{^{\prime}} }}{25}} \right)^{1/3} \left( {\frac{2.0d}{a}} \right)} \right)b_{w} d } \right) + \frac{{A_{v} f_{y} d}}{s} $ for a/d $< 2.0$
Bazant et. al	$V_{u} = 0.54\sqrt[3]{\rho }\left[ {\sqrt {f_{c}^{^{\prime}} } + 249\sqrt {\frac{\rho }{{\left( {{\raise0.7ex\hbox{$a$} \!\mathord{\left/ {\vphantom {a d}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$d$}}} \right)^{5} }}} } \right]\left[ {\frac{{1 + \sqrt {{\raise0.7ex\hbox{${5.08}$} \!\mathord{\left/ {\vphantom {{5.08} {d_{a} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${d_{a} }$}}} }}{{\sqrt {\left( {1 + \left( {{\raise0.7ex\hbox{$d$} \!\mathord{\left/ {\vphantom {d {25d_{a} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${25d_{a} }$}}} \right)} \right)} }}} \right]b_{w} d$
Zsutty	$V_{u} = \left( {2.1746\left( {f_{c}^{^{\prime}} \rho {\raise0.7ex\hbox{$d$} \!\mathord{\left/ {\vphantom {d a}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$a$}}} \right)^{1/3} b_{w} d} \right) + \frac{{A_{v} f_{y} d}}{s} $ for a/d $\ge 2.5$; $V_{u} = \left( {2.1746\left( {f_{c}^{^{\prime}} \rho {\raise0.7ex\hbox{$d$} \!\mathord{\left/ {\vphantom {d a}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$a$}}} \right)^{1/3} \left( {2.5\frac{d}{a}} \right)b_{w} d} \right) + \frac{{A_{v} f_{y} d}}{s} $ for a/d $< 2.5$
Campione et al	$V_{u} = \left( {1.07\sqrt \rho \sqrt {f_{c}^{^{\prime}} } + \rho_{w} f_{yw} } \right)b_{w} d$
Huber et al	$V_{c} = k\left( {100\rho_{l} \frac{{f_{c}^{^{\prime}} }}{{a_{cs} }}d_{dg} } \right)^{1/3} b_{w} d$ $V_{s} = \sum \sigma_{sw,i} \frac{{\emptyset_{w}^{2} \pi }}{4}$

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Author	Shear prediction model (SI unit)
ACI 318–14	\(V_{u} = \left( {\left( {0.16\sqrt {f_{c}^{^{\prime}} } + 17.6\rho_{w} \frac{{V_{u} d}}{{M_{u} }}} \right)b_{w} d} \right) + \frac{{A_{v} f_{y} d}}{s}\)
BS8110	\(V_{u} = \left( {\left( {\left( {\frac{0.79}{{\gamma_{m} }}} \right)\left( {\frac{{100A_{s} }}{{b_{w} d}}} \right)^{1/3} \left( \frac{400}{d} \right)^{1/4 } \left( {\frac{{f_{c}^{^{\prime}} }}{25}} \right)^{1/3} } \right)b_{w} d } \right) + \frac{{A_{v} f_{y} d}}{s} \) for a/d \(\ge 2.0\); \(V_{u} = \left( {\left( {\left( {\frac{0.79}{{\gamma_{m} }}} \right)\left( {\frac{{100A_{s} }}{{b_{w} d}}} \right)^{1/3} \left( \frac{400}{d} \right)^{1/4 } \left( {\frac{{f_{c}^{^{\prime}} }}{25}} \right)^{1/3} \left( {\frac{2.0d}{a}} \right)} \right)b_{w} d } \right) + \frac{{A_{v} f_{y} d}}{s} \) for a/d \(< 2.0\)
Bazant et. al	\(V_{u} = 0.54\sqrt[3]{\rho }\left[ {\sqrt {f_{c}^{^{\prime}} } + 249\sqrt {\frac{\rho }{{\left( {{\raise0.7ex\hbox{$a$} \!\mathord{\left/ {\vphantom {a d}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$d$}}} \right)^{5} }}} } \right]\left[ {\frac{{1 + \sqrt {{\raise0.7ex\hbox{${5.08}$} \!\mathord{\left/ {\vphantom {{5.08} {d_{a} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${d_{a} }$}}} }}{{\sqrt {\left( {1 + \left( {{\raise0.7ex\hbox{$d$} \!\mathord{\left/ {\vphantom {d {25d_{a} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${25d_{a} }$}}} \right)} \right)} }}} \right]b_{w} d\)
Zsutty	\(V_{u} = \left( {2.1746\left( {f_{c}^{^{\prime}} \rho {\raise0.7ex\hbox{$d$} \!\mathord{\left/ {\vphantom {d a}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$a$}}} \right)^{1/3} b_{w} d} \right) + \frac{{A_{v} f_{y} d}}{s} \) for a/d \(\ge 2.5\); \(V_{u} = \left( {2.1746\left( {f_{c}^{^{\prime}} \rho {\raise0.7ex\hbox{$d$} \!\mathord{\left/ {\vphantom {d a}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$a$}}} \right)^{1/3} \left( {2.5\frac{d}{a}} \right)b_{w} d} \right) + \frac{{A_{v} f_{y} d}}{s} \) for a/d \(< 2.5\)
Campione et al	\(V_{u} = \left( {1.07\sqrt \rho \sqrt {f_{c}^{^{\prime}} } + \rho_{w} f_{yw} } \right)b_{w} d\)
Huber et al	\(V_{c} = k\left( {100\rho_{l} \frac{{f_{c}^{^{\prime}} }}{{a_{cs} }}d_{dg} } \right)^{1/3} b_{w} d\) \(V_{s} = \sum \sigma_{sw,i} \frac{{\emptyset_{w}^{2} \pi }}{4}\)