21 DIRECT SHEAR n n Scissors Shear stress

2/1 DIRECT SHEAR n n Scissors Shear stress still Force/Area n n but different area Strain still deflection over length n n but angular distortion at right angles =angle in radians (for small strains)

2/2 EQUALITY OF STRESSES DEFINITION OF STRAIN V Original Angle = /L Area A y z L x V Shear Force V Shear Stress = V/A Shear Strain = /L

2/3 EQUALITY OF MOMENTS n n n But that block is not in EQUILIBRIUM It will rotate There must be a matching pair of shears

2/4 SHEAR COUPLES V Original V Shear Force V Angle /2 = /2 L Area A L Shear Stress = V/A Shear Strain = /L

2/5 Sign Conventions What What is is is a a a positive positive direct stress? shear stress? moment? force? BENDING moment?

2/6 Sign for Force and Stress Y Negative X Face Area A Z -Fx Positive X Face Area A Fx x = -Fx/-A = +Fx/+A X

2/7 Sign for Shear Force & Stress Y Vxy Direction Negative X Face Area A Z Face -Vxy Positive X Face Area A Vxy = -V/-A =+V/+A X

2/8 Sign for Moments and Rotations Y Y z, Mz Z X X

2/9 SIGN CONVENTION n n n +ve faces – outward normal pointing towards +ve direction -ve faces – pointing towards origin +ve shear stress – acts on a +ve face in a +ve dirn or a –ve face in a –ve dirn +ve shear strain – when angle between two +ve or two –ve faces is reduced. +ve shear stresses give +ve shear strains

2/10 PURE SHEAR n n No normal stresses Pure shear means no change of length Can only apply in one direction Pure shear produces (implies) n n Diagonal Tension Diagonal Compression

2/11 SHEAR TESTS n n n n Stress–strain diagram in shear ( vs ) Hooke’s Law in shear = G. G = Shear Modulus of Elasticity Same units as E (Pa), steel = 75 GPa G = E/(2[1+ ]) G, E and not independent elastic props 0< <0. 5 so E/3<G<E/2

2/12 BOLT IN SINGLE SHEAR

2/13 BOLT IN SINGLE SHEAR

2/14 PIN IN DOUBLE SHEAR 2 V V ave=V/A

2/15 Shear failure of wood block in compression

2/16 OTHER CAUSES OF SHEAR n Shear also appears in torsion