MECH 320 Problems with Solutions, Fall 2010

PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com 7 - 5 A single horizontal force P of 150 lb magnitude is applied to end D of lever ABD. Determine (a) the normal and shearing stresses on an element at point H having sides parallel to the x and y axes, (b) the principal planes and principal stresses at the point H. SOLUTION: • Determine an equivalent force-couple system at the center of the transverse section passing through H. • Evaluate the normal and shearing stresses at H. • Determine the principal planes and calculate the principal stresses. PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com 7 - 6 SOLUTION: • Determine an equivalent force-couple system at the center of the transverse section passing through H. ( )( ) ( )( ) kip in 5.1 lb 10in 150 kip in 7.2 lb 18in 150 lb 150 ⋅ = = ⋅ = = = x M T P • Evaluate the normal and shearing stresses at H. ( )( ) ( ) ( )( ) ( )4 2 1 4 4 1 in 6.0 in 6.0 kip in 7.2 in 6.0 in 6.0 kip in 5.1 π τ π σ ⋅ = + + = ⋅ = + + = J Tc I Mc xy y .7 96ksi .8 84ksi 0 = + = + = y y x τ σ σ PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com 7 - 7 • Determine the principal planes and calculate the principal stresses. ( ) ° ° − = = − − = − = ,119 61 0. 2 8.1 .8 84 0 .7 96 2 2 2 tan p y x xy p θ σ σ τ θ ° ° = − , 59 5. 30 5. p θ ( )2 2 2 2 min max, .7 96 2 .8 84 0 2 .8 84 0 2 2 +     − ± + =  +      − ± + = xy y x y x τ σ σ σ σ σ 68ksi .4 .52ksi 13 min max − = + = σ σ PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com 7 - 8 PB 1 • Cylindrical vessel with principal stresses σ1 = hoop stress σ2 = longitudinal stress ( ) ( ) t pr r x p t x Fz = ∆ − ∆ = = ∑ 1 1 2 2 0 σ σ • Hoop stress: ( ) ( ) 2 1 2 2 2 2 2 2 0 σ σ σ π π σ = = − = = ∑ t pr r p rt Fx • Longitudinal stress: PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com 7 - 9 • Points A and B correspond to hoop stress, σ1, and longitudinal stress, σ2 • Maximum in-plane shearing stress: t pr 4 2 1 2 max(in plane) = = − σ τ • Maximum out-of-plane shearing stress corresponds to a 45o rotation of the plane stress element around a longitudinal axis t pr 2 2 max = = σ τ PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com 7 - 10 • Spherical pressure vessel: t pr 2 2 1 = = σ σ • Mohr’s circle for in-plane transformations reduces to a point 0 constant -plane) max(in 2 1 = = = = τ σ σ σ • Maximum out-of-plane shearing stress t pr 4 1 2 1 max = = σ τ PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com