§7.2 Shear Stress Distribution 157 (d +M)ndy ndy 1 dM 起了bdy Therefore total out-of-balance force from all sections above height y h [am For equilibrium,this force is resisted by a shear force set up on the section of length dx and breadth b,as shown in Fig.7.4. h aM ybdy Fig.7.4. Thus if the shear stress is t,then dM ybdy (7.10 h But ybdy first moment of area of shaded portion of Fig.7.3b about the N.A. =A5 where A is the area of shaded portion and y the distance of its centroid from the N.A. dM Also rate of change of the B.M. =S.F.O at the section t=24y Ib (7.2) or,alternatively, ydA where dA =bdy (7.3) 7.2.Application to rectangular sections Consider now the rectangular-sectioned beam of Fig.7.5 subjected at a given transverse cross-section to a S.F.O.$7.2 Shear Stress Distribution 157 Therefore total out-of-balance force from all sections above height y = jgybdy I Y For equilibrium, this force is resisted by a shear force set up on the section of length dx and breadth b, as shown in Fig. 7.4. h C D [ ybdy Y Fig. 7.4. Thus if the shear stress is T, then h But j ybdy = first moment of area of shaded iortion of Fig. 7.3b about the N.A. Y = Aj where A is the area of shaded portion and j the distance of its centroid from the N.A. dM ~ = rate of change of the B.M. dx Also = S.F. Q at the section .. QAY z=- lb or, alternatively, z = 2 ydA where dA = bdy lb Y 7.2. Application to rectangular sections Consider now the rectangular-sectioned beam of Fig. 7.5 subjected at a given transverse cross-section to a S.F. Q