# Lec.16 - 16. STRESS AT A POINT I Main Topics ...

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Unformatted text preview: 10/16/11 16. STRESS AT A POINT I Main Topics A Stress vector (trac=on) on a plane B Stress at a point C Principal stresses D Transforma=on of principal stresses to trac=ons in 2D 10/16/11 GG303 1 16. STRESS AT A POINT I Stress vector (trac=on) on a plane A τ = lim F / A A→ 0 B Trac=on vectors can be added as vectors C A trac=on vector can be resolved into normal (τn) and shear (τs) components 1 A normal trac=on (τn) acts perpendicular to a plane 2 A shear trac=on (τs) acts parallel to a plane D Local reference frame 1 The n ­axis is normal to the plane 2  The s ­axis is parallel to the plane 10/16/11 GG303 2 1 10/16/11 16. STRESS AT A POINT III Stress at a point (cont.) A Stresses refer to balanced internal "forces (per unit area)". They diﬀer from force vectors, which, if unbalanced, cause accelera=ons B "On  ­in conven=on": The stress component σij acts on the plane normal to the i ­direc=on and acts in the j ­direc=on 1 Normal stresses: i=j 2 Shear stresses: i≠j 10/16/11 GG303 3 16. STRESS AT A POINT III Stress at a point C Dimensions of stress: force/unit area D Conven=on for stresses 1  Tension is posi=ve 2  Compression is nega=ve 3  Follows from on ­in conven=on 4  Consistent with most mechanics books 5  Counter to most geology books 10/16/11 GG303 4 2 10/16/11 16. STRESS AT A POINT III Stress at a point ⎡ σ xx C ij = ⎢ σ ⎢ σ yx ⎣ D σ xy ⎤ 2 ­D ⎥ σ yy ⎥ 4 components ⎦ ⎡ σ xx σ xy σ xz ⎢ σ ij = ⎢ σ yx σ yy σ yz ⎢ ⎢ σ zx σ zy σ zz ⎣ ⎤ ⎥ 3 ­D ⎥ ⎥ 9 components ⎥ ⎦ E In nature, the state of stress can (and usually does) vary from point to point F For rota=onal equilibrium, σxy = σyx, σxz = σzx, σyz = σzy 10/16/11 GG303 5 16. STRESS AT A POINT IV Principal Stresses (these have magnitudes and orienta=ons) A Principal stresses act on planes which feel no shear stress B The principal stresses are normal stresses. C Principal stresses act on perpendicular planes D The maximum, intermediate, and minimum principal stresses are usually designated σ1, σ2, and σ3, respec=vely. E Principal stresses have a single subscript. 10/16/11 GG303 6 3 10/16/11 16. STRESS AT A POINT IV Principal Stresses (cont.) F Principal stresses represent the stress state most simply G ⎡ σ1 0 ⎤ 2 ­D ⎥ σ ij = ⎢ 0 σ 2 ⎥ 2 components ⎢ ⎣ ⎦ ⎡ σ xx σ xy σ xz ⎢ σ ij = ⎢ σ yx σ yy σ yz ⎢ σ zy σ zz ⎢σ ⎣ zx H 10/16/11 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ 3 ­D 3 components GG303 7 16. STRESS AT A POINT V Transforma=on of principal stresses to trac=ons in 2D A Descrip=on of terms 1 Three planes A, A1, and A2 form the sides of a triangular prism; these have normals in the n ­, 1 ­, and  ­2 ­direc=ons, respec=vely. 2 Plane A1 is acted on by known normal stress σ1. 3 Plane A2 is acted on by known normal stress σ2. 4 The n ­direc=on is at angle θ1 (=θ) with respect to the 1 ­ direc=on, and at angle θ2 with respect to the 2 ­direc=on. 5 The s ­direc=on is at a counter ­ clockwise 90° angle rela=ve to the n ­direc=on (like y and x). 10/16/11 GG303 8 4 10/16/11 16. STRESS AT A POINT V Transforma=on of principal stresses to trac=ons in 2D B Approach Find weigh=ng factors that determine contribu=ons of known stresses to desired trac=ons and sum contribu=ons 1 τn = wn1 σ1 + wn2 σ2 2 τs = ws1 σ1 + ws2 σ2 10/16/11 GG303 9 16. STRESS AT A POINT C Contribu=on of σ1 (on face A1 of area A1) to τn (on face A of area A) Start with the deﬁni=on of trac=on: 1 τn(1)= Fn(1))/ A Find unknowns Fn(1) and A from knowns σ2 and θ. First ﬁnd the force F1 associated with σ1 2 F1 = σ1 A1 Force = (stress)(area) Find Fn(1), the component of F1 in the n ­direc=on 3 Fn(1) = F1 cos θ1 Find A in terms of A1 A1 = A cos θ1 (see diagram at right) 4 A = A1/cos θ1 Contribu=on of σ1 to τn: 5a τn(1)= Fn(1) / A= F1 cos θ1 / (A1/cos θ1 ) 5b τn(1)= (F1 / A1) cos θ1 cos θ1=σ1 cos θ1 cos θ1 Weigh=ng factor wn1 6 wn1 = cosθ1 cosθ1 = cosθ cosθ 10/16/11 GG303 10 5 10/16/11 16. STRESS AT A POINT D Contribu=on of σ2 (on face A2 of area A2) to τn (on face A of area A) Start with the deﬁni=on of trac=on: 1 τn(2)= Fn(2))/ A Find unknowns Fn(2) and A from knowns σ2 and θ. First ﬁnd the force F2 associated with σ2 2 F2 = σ2 A2 Force = (stress)(area) Find Fn(2), the component of F2 in the n ­direc=on 3 Fn(2) = F2 cos θ2 Find A in terms of A2 A2 = A cos θ2 (see diagram at right) 4 A = A2/cos θ2 Contribu=on of σ2 to τn: 5a τn(2)= Fn(2) / A= F2 cos θ2 / (A2/cos θ2 ) 5b τn(2)= (F2 / A2) cos θ2 cos θ2=σ2 cos θ2 cos θ2 Weigh=ng factor wn2 6 wn2 = cosθ2 cosθ2 = sinθ sinθ 10/16/11 GG303 11 16. STRESS AT A POINT E Contribu=on of σ1 (on face A1 of area A1) to τs (on face A of area A) Start with the deﬁni=on of trac=on: 1 τs(1)= Fs(1))/ A Find unknowns Fs(1) and A from knowns σ1 and θ. First ﬁnd the force F1 associated with σ1 2 F1 = σ1 A1 Force = (stress)(area) Find Fs(1), the component of F1 in the s ­direc=on 3 Fs(1) =  ­F1 cos θ2 Find A in terms of A2 A1 = A cos θ1 (see diagram at right) 4 A = A1/cos θ1 Contribu=on of σ1 to τn of σ1: 5a τs(1) = Fs(1) / A=  ­F1 cos θ2 / (A1/cos θ1 ) 5b τs(1) =  ­ (F1 / A1) cos θ2 cos θ1=  ­σ1 cos θ2 cos θ1 Weigh=ng factor ws1 6 ws1 =  ­cos θ2 cos θ1 =  ­sinθ cosθ 10/16/11 GG303 12 6 10/16/11 16. STRESS AT A POINT F Contribu=on of σ2 (on face A1 of area A1) to τs (on face A of area A) Start with the deﬁni=on of trac=on: 1 τs(2)= Fs(2))/ A Find unknowns Fs2) and A from knowns σ2 and θ. First ﬁnd the force F2 associated with σ2 2 F2 = σ2 A2 Force = (stress)(area) Find Fs(2), the component of F2 in the s ­direc=on 3 Fs(2) = F2 cos θ1 Find A in terms of A1 A1 = A cos θ1 (see diagram at right) 4 A = A2/cos θ2 Contribu=on of σ1 to τn of σ1: 5a τs(2) = Fs(2) / A= F2 cos θ1 / (A2/cos θ2 ) 5b τs(2) = (F2 / A2) cos θ1 cos θ2 = σ2 cos θ1 cos θ2 Weigh=ng factor ws2 6 ws2 = cos θ1 cos θ2 = cosθ sinθ 10/16/11 GG303 13 16. STRESS AT A POINT V Transforma=on of principal stresses to trac=ons in 2D G Original equa=ons 1 τn = wn1 σ1 + wn1 σ2 2  τs = ws1 σ1 + ws1 σ2 H Original equa=ons 1 τn = cosθcosθ σ1 + sinθsinθ σ2 2  τs =  ­sinθcosθ σ1 + sinθcosθ σ2 10/16/11 Weigh=ng factors are products of two direc=on cosines GG303 14 7 ...
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