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Assume a beam with stirrups inclined at 60o angle with respect to the longitudinal steel resisting cracks developing at 45o angles. Derive analytically the equation that allows you to calculate the shear resistance provided by the stirrups as a function of the stirrup and crack angle.

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Answer:

Shear resistance of a inclined stirrup is given by  

 Vᵇ = Asb(0.95*Fy)(cos α - sin α*cotβ) ((d-d')/sᵇ)

Explanation:

The shear resistance is checked in accordance with BS8110: Part 1, section 3.5.5. Shear resistance of a inclined stirrup is given by  

 Vᵇ = Asb(0.95*Fy)(cos α - sin α*cotβ) ((d-d')/sᵇ)

where,

 Vᵇ=design shear resistance of inclined bars

Asb=cros-sectional area of the inclined bars

Fy=characteristics strength of the stirrups

α=angle between the inclined bars and the axis of the beam

β=angle between the compression strut of inclined bars and the axis of the beam

sᵇ=spacing of the inclined bars

d= effective depth   and d'=effective depth minus cover to reinforcement

The shear stress is given by    

v=V/bd  

 where V is the shear force due to ultimate loads. If v is shear stress

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