6_shearing_stresses

6_shearing_stresses - 6 Shearing Stresses in Beams and...

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1 SCHOOL OF ENGINEERING, SCHOOL OF ENGINEERING, DEPARTMENT OF MECHANICS DEPARTMENT OF MECHANICS ° C 6 Shearing Stresses in Beams and Thin- Walled Members
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2 SCHOOL OF ENGINEERING, SCHOOL OF ENGINEERING, DEPARTMENT OF MECHANICS DEPARTMENT OF MECHANICS ° C Contents Introduction Shear on the Horizontal Face of a Beam Element Example 6.01 Determination of the Shearing Stress in a Beam Shearing Stresses τ ξψ ιν Χομμον Τψπεσ οφ Βεαμσ Further Discussion of the Distribution of Stresses in a ... Sample Problem 6.2 Longitudinal Shear on a Beam Element of Arbitrary Shape Example 6.04 Example 6.04 Shearing Stresses in Thin-Walled Members Plastic Deformations Sample Problem 6.3 Unsymmetric Loading of Thin-Walled Members Example 6.05 Example 6.06
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3 SCHOOL OF ENGINEERING, SCHOOL OF ENGINEERING, DEPARTMENT OF MECHANICS DEPARTMENT OF MECHANICS ° C Introduction ( 29 ( 29 M y M dA F dA z M V dA F dA z y M dA F x z xz z x y xy y xy xz x x x = - = = = = = - = = = - = = = σ τ σ τ τ τ σ 0 0 0 0 Distribution of normal and shearing stresses satisfies Transverse loading applied to a beam results in normal and shearing stresses in transverse sections. Longitudinal shearing stresses must exist in any member subjected to transverse loading. When shearing stresses are exerted on the vertical faces of an element, equal stresses must be exerted on the horizontal faces
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4 SCHOOL OF ENGINEERING, SCHOOL OF ENGINEERING, DEPARTMENT OF MECHANICS DEPARTMENT OF MECHANICS ° C Example:
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5 SCHOOL OF ENGINEERING, SCHOOL OF ENGINEERING, DEPARTMENT OF MECHANICS DEPARTMENT OF MECHANICS ° C Shear on the Horizontal Face of a Beam Element Consider prismatic beam For equilibrium of beam element ( 29 - = - + = = A C D A C D x dA y I M M H dA H F σ σ 0 x V x dx dM M M dA y Q C D A = = - = Note, flow shear I VQ x H q x I VQ H = = = = Substituting,
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6 SCHOOL OF ENGINEERING, SCHOOL OF ENGINEERING, DEPARTMENT OF MECHANICS DEPARTMENT OF MECHANICS ° C Shear on the Horizontal Face of a Beam Element where section cross full of moment second above area of moment first ' 2 1 = = = = + A A A dA y I y dA y Q Same result found for lower area H H Q Q q I Q V x H q - = = = + - = = = axis neutral to respect h moment wit first 0 flow shear I VQ x H q = = = Shear flow,
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7 SCHOOL OF ENGINEERING, SCHOOL OF ENGINEERING, DEPARTMENT OF MECHANICS DEPARTMENT OF MECHANICS ° C Example 6.01 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. SOLUTION: Determine the horizontal force per unit length or shear flow q on the lower surface of the upper plank. Calculate the corresponding shear force in each nail.
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8 SCHOOL OF ENGINEERING, SCHOOL OF ENGINEERING, DEPARTMENT OF MECHANICS DEPARTMENT OF MECHANICS ° C Example 6.01 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 4 6 2 3 12 1 3 12 1 3 6 m 10 20 . 16 ] m 060 . 0 m 100 . 0 m 020 . 0
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