2. Analyzing stress: Defini n ti t ons n a nd n C onc n e c pt p s
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1 2. Analyzing tre: Definition and Concept
2 2.1 Introduction Stre and train - The fundamental and paramount ubject in mechanic of material - Chapter 2: Stre - Chapter 3: Strain
3 2.2 Normal tre under aial loading - (Magnitude of) tre = force intenity: Force Stre = Area - Aial loading: tenion or compreion load acting along the long ai of a traight tructural member - Normal tre: tre normal to a plane (σ) F F avg =, lim A = D D A 0 DA - Sign convention of normal tre of the tetbook: tenile - poitive, compreive - negative
4 2.3 Shearing tre in connection - Shear tre: tre tangential to a plane (σ) t avg V =, t = lim A D A 0 DV DA - Punching hear: e) an action of a punch in forming rivet hole in a metal plate
5 2.4 Bearing tre - Bearing tre: compreive normal tre occurring on the urface of contact between two interacting member e) tre on the urface of contact between a bolt head and a top plate
6 2.5 Unit of tre - In SI unit (newton and meter): N/m 2 = Pa (pacal), kpa, MPa - In US unit: lb/in. 2 = pi, ki (= 1,000 pi = kip per quare inch) e) 1 lb 4.45 N, 1 inch = 2.54 cm = 1/12 feet, 1 pi 6.89 kpa Eample Problem Normal tre (ki) on mid-point between A-B, B-C, and C-D A cro-ectional area i 3.0 in. 2
7 2.5 Unit of tre Eample Problem Diameter required for each of the ection when the allowable aial tree are 125 MPa in the teel, 70MPa in the bra, and 85MPa in the aluminum.
8 2.5 Unit of tre Eample Problem Rigid: The minimum collar diameter when the bearing tre of collar i 10 ki and force on the bearing plate i 50 kip - Fleible: Calculate and plot σ ma v. collar diameter (2.5 in. d c 5.0 in.)
9 2.6 Stree on an inclined plane in an aially loaded member - Equilibrium of force: r r r P = F, F = N -V N = P co q, V = -Pinq A n = A/ coq P 2 P n = N / A n = co q = ( 1 + co 2 q ) A 2A P P t n = V / An = - inq coq = - in 2q A 2A Stre depend on force and area: tre i not a vector quantity and therefore, the law of vector addition cannot be applied to tree on different plane.
10 2.6 Stree on an inclined plane in an aially loaded member - Maimum hear tre i a half of the maimum normal tre: t - Shear tre i zero on the plane of ma. or min. normal tre: t = 0 when =, q q - Maimum hear tre i 45 apart from ma. or min. normal tre. - Magnitude of hear tre at θ i the ame a that at 90+ θ. ma min ma = ma 2
11 2.6 Stree on an inclined plane in an aially loaded member - Equality of hearing tre on orthogonal plane: å ( d dz) dy t ( ) Moment equilibrium ( Mz = 0): t y = y t = t y y dy dz d
12 2.6 Stree on an inclined plane in an aially loaded member Eample Problem Normal and hear tree on ection a-a.
13 2.6 Stree on an inclined plane in an aially loaded member Eample Problem 2-7 When A = mm, normal tre on ection a-a = 12 Mpa, - Load P - Shear tre on ection a-a - Maimum normal and hear tree in the block
14 2.7 Stre at a general point in an arbitrarily loaded member - Stre (force) at a point in an arbitrary plane can be reolved into a normal and hear component. - When an ai i et to be normal to the plane the hear component can be reolved into two hear tree coinciding with y- or z-ai. - It i cutomary to how the tree on poitive and negative urface through a point uing a mall element. (tenile/compreive i poitive/negative)
15 2.8 Two-dimenional or plane tre - A tate of plane tre occur where force can be reolved into only two component: within thin plate where z-dimenion of the body i mall and the z-component of force are zero. e) = t ( t ) = t ( t ) = 0. z z z zy yz
16 2.9 The tre tranformation equation for plane tre - Normal and hear tree on a plane can be calculated by uing the force equilibrium of a free-body diagram. å å F F + y - y = 0 : = + co 2q + t in 2q y = 0 : t = - in 2q + t co 2q 2 n n y t nt y
17 2.9 The tre tranformation equation for plane tre Eample Problem Normal and hear tree on ection a-b. - Normal and hear tree on ection c-d. - Stree on ection a-b and c-d uing a mall element.
18 2.9 The tre tranformation equation for plane tre Eample Problem Normal and hear tree on ection b-b.
19 2.10 Principal tree and maimum hearing tre plane tre - Tranformation equation for plane tre: + y - y n = + co 2q + t y in 2 q (2-12b) y t nt = - in 2q + t y co 2q 2 - Ma. and min. value of σ n : n n = p when = -( - y ) in 2q + 2t y co 2q = 0 q 2t y tan 2 q p =, t nt = 0 (2-14) - y + æ - ö ç 2 è 2 ø 2 y y 2 p1, p2 = ± + t y from (2-12b) and (2-14)
20 2.10 Principal tree and maimum hearing tre plane tre - Ma. value of τ nt : t nt t nt = t p when = -( - y ) co 2q - 2t y in 2q = 0 q - æ - ö = - = ± + 2 è ø y y 2 tan 2 qt, t p ç t y (2-17) 2t y t p = 2 t = ma p1 p2 ma - 2 from (2-15) and (2-17) min in 3D cae - The direction of the maimum hear tre mut oppoe the larger of the two principal tree.
21 2.10 Principal tree and maimum hearing tre plane tre - Invariant of tre + æ - ö = ± ç + t 2 è 2 ø y y 2 p1, p2 y + = + p1 p2 y 2 - In engineering problem, maimum will alway refer to the larget abolute value (larget magnitude).
22 2.10 Principal tree and maimum hearing tre plane tre Eample Problem Principal tree and the maimum hear tre - Locate the plane on which thee tree act and how the tree
23 2.11 Mohr circle for plane tre - Equation of Mohr circle + y - y n = + co 2q + t y in 2 q y t nt = - in 2q + t y co 2q æ + ö æ - ö ç - + t = ç + t è 2 ø è 2 ø y 2 y 2 n nt y æ - y ö R = ç + t è 2 ø 2 2 y - Horizontal coordinate: normal tre (V,F..) - Vertical coordinate: hear tre (F F, V V..) (CW - above; CCW below)
24 2.11 Mohr circle for plane tre - Line CV, CH : plane - Angle on the circle: twice the angle for an actual body (CW - above; CCW below) - Procedure for drawing Mohr circle:
25 2.11 Mohr circle for plane tre Eample Problem Principal tree and the maimum hear tre - Normal and hear tre on plane a-a
26 2.12 General tate of tre at a point - Force equilibrium on an oblique face F = S da = da l + t da m + t da n y z F = S da = t da l + da m + t da n y y y y zy F = S da = t da l + t da m + da n z z z yz z - Traction component on the oblique face S = S + S + S y z S = l + t m + t n y z S = t l + m + t n y y y yz S = t l + t m + n z z zy z - Normal/hear tre on r the oblique face = + t = g ˆ = g ( ) ( ) S, S n S, S, S l, m, n n nt n y z \ n = l + ym + zn + 2t ylm + 2t yzmn + 2t znl r r r 2 2 t = S -, t = S - nt n nt n
27 2.12 General tate of tre at a point - Principal tre when the oblique face i a principal plane S =, S = l, S = m, S = n p p y p z p p ( p ) t y t z y p t y ( y p ) t yz z p t z t zy ( z p ) ( - p ) t y t z t y ( y p ) t yz t z t zy ( z - p ) S - l = - l + m + n = 0 S - m = l + - m + n = 0 S - n = l + m + - n = 0 - = ( ) ( ) ( y z t yz yt z zt y 2t yt yzt z ) t -t -t p y z p y y z z y yz z p =
28 2.12 General tate of tre at a point - Maimum hear tre when σ, σ y, and σ z are principal tree S = l, S = m, S = n y y z z S = l + m + n y z ( l m n ) = n y z 2 ( ) 2 t = l + m + n - l + m + n nt y z y z - - t ç t 2 2 è 2 2 ø æ 1 1 ö p1 p2 ma min ma = p1 + p2 - p1 + p2 = ma =
29 2.12 General tate of tre at a point Eample Problem 2-15 = 14 ki = 12ki = 10ki y t = 8ki t = - 10ki t = 6 y yz z - Normal/hear tree on a plane whoe outward normal i oriented at angle of 61.3, 53.1, and 50.2 with -, y-, and z-ae, repectively. - Principal tree and the maimum hearing tre at the point Home work problem: 2-1, 10, 22, 35, 46, 57, 70, 81, 90, 99
30
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3.5b Stre Boundar Condition: Continued Conider now in more detail a urface between two different material Fig. 3.5.16. One a that the normal and hear tree are continuou acro the urface a illutrated. 2
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