材料力學(xué)06_梁的剪應(yīng)力.ppt

1、Shearing Stress in Beams,06,06 - 2,Introduction,For most beams, both the bending moment and shear force will present.,If V = 0 (pure bending), txy = txz =0,Now V 0, both txy and txz may not be zero.,txz may not be zero, but the average is zero,06 - 3,Calculation of Shearing Stress: Preliminary,The d

2、istribution of txy can not be determined from statics alone. Analysis of deformation may not help too much.,However, we know that shearing stress comes in pair, arrow to arrow and tail to tail, as shown. If we can determine tyx, then from txy = tyx, we can obtain txy.,06 - 4,Shear on the Horizontal

3、Face of a Beam Element,06 - 5,Shear on the Horizontal Face: First Moment of Area,Shear Flow,First Moment of Area,I is the second moment of area of the entire section about the centroidal axis of the entire section.,A* is area of the isolated area, from where the shear flow is calculated to a free su

4、rface.,06 - 6,Example 1,Problem,A beam is made of three planks, nailed together. Knowing that the spacing between nails is 25 mm and that the vertical shear in the beam is V = 500 N, determine the shear force in each nail.,06 - 7,Example 1: Solution Strategy,If the beam were one piece, there would b

5、e shear stress (shear flow) between the cap and the web.,Solution Strategy,The nails will carry the force resulted from the shear flow.,06 - 8,Example 1: Solution,06 - 9,Shearing Stress in a Beam,On the upper and lower surfaces of the beam, tyx= 0. It follows that txy= 0 on the upper and lower edges

6、 of the transverse sections.,If the width of the beam is comparable or large relative to its depth, the shearing stresses at D1 and D2 are significantly higher than at D.,06 - 10,Shear Stress in a Narrow Rectangular Beam,Free Body Diagram,Shear Stress,parabolic,06 - 11,I Beam: Shear Stress txy,Shear

7、 Stress,For Wide Beams,06 - 12,Example 2,Problem,A timber beam is to support the three concentrated loads shown. Knowing that for the grade of timber used, the allowable stresses are sall = 1800 psi and tall = 120 psi, determine the minimum required depth d of the beam based on the maximum normal st

8、ress and shear stress.,Solution Strategy,Develop shear and bending moment diagrams. Identify the maximums.,Determine the beam depth based on allowable normal stress.,Determine the beam depth based on allowable shear stress.,Required beam depth is equal to the larger of the two depths found.,06 - 13,

9、Example 2: Solution,06 - 14,Longitudinal Shear on a Beam Element of Arbitrary Shape,06 - 15,Example 3,Problem,A square box beam is constructed from four planks as shown. Knowing that the spacing between nails is s =1.5 in. and the beam is subjected to a vertical shear of magnitude V = 600 lb, determ

10、ine the shearing force in each nail.,Solution Strategy,Obtain the shear flow at the connection.,The shearing force in each nail equals qs, where q is the shear flow carried by one row of nails.,06 - 16,Example 3: Solution,Note that q is carried by two rows of nails. The shear flow carried by one row

11、 of nails equals q/2.,06 - 17,Shearing Stresses in Thin-Walled Members,Consider a segment of a wide-flange beam subjected to the vertical shear V.,The longitudinal shear force on the element is,The corresponding shear stress is,Previously found a similar expression for the shearing stress in the web

12、,06 - 18,Shearing Stresses in Thin-Walled Members,The variation of shear flow across the section depends only on the variation of the first moment.,For a box beam, q grows smoothly from zero at A to a maximum at C and C and then decreases back to zero at E.,The sense of q in the horizontal portions of the section may be deduced from the sense in the vertical portions or the sense of the shear V.,06 - 19,Shearing Stresses in Thin-Walled Members,For a wide-flange beam, the shear flow increases symmetrically