# Analysis of Stresses and Strains

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1 Chapter 7 Analysis of Stresses and Strains 7.1 Introduction axial load = P / A torsional load in circular shaft = T / I p bending moment and shear force in beam = M y / I = V Q / I b in this chapter, we want to find the normal and shear stresses acting on any inclined section for uniaxial load and pure shear, this relation are shown in chapters and 3, now we want to derive the transformation relationships that give the stress components for any orientation this is referred as stress transformation when an element is rotated from one orientation to another, the stresses acting on the faces of the element are different but they still represent the same state of stress, namely, the stress at the point under consideration 7. Plane Stress consider the infinitesimal element with its edges parallel to x, y, and z axes if only the x and y faces of the element are subjected to stresses, it is called plane stress, it can be shown as a two dimension stress element 1

2 equal normal stresses act on opposite faces, shear stress has two subscripts, the first denotes the face on which the stress acts, and the second gives the direction of that face xy : acts on yx : acts on x face directed to y axis y face directed to x axis sign convention : acts on! face and directed to! axis as + as discussed in chapter 1, xy = yx consider an element located at the same point and whose faces are perpendicular to x 1, y 1 and z 1 axes, in which z 1 axis coincides with the z axis, and x 1 and y 1 axes are rotated counterclockwise through an angle w.r.t. x and y axes, the normal and shear stresses acting on this new element are denoted x1, y1, x1y1 and y1x1 also x1y1 = y1x1 the stresses acting on the rotated x 1 y 1 element can be expressed in terms of stress on the xy element by using equation of static equilibrium choose a wedge-shaped element force equilibrium in x 1 -direction x A 0 sec - x A 0 cos - xy A 0 sin 1 - y A 0 tan sin - yx A 0 tan con = 0 force equilibrium in y 1 -direction x1y1 A 0 sec + x A 0 sin - xy A 0 cos

3 - y A 0 tan cos + yx A 0 tan sin = 0 with xy = yx it is obtained x1 = x cos + y sin + xy sin cos x1y1 = - ( ) sin cos + xy (cos - sin ) for = 01 x1 = x x1y1 = xy for = 901 x1 = y x1y1 = - xy from trigonometric identities 1 1 cos = C (1 + cos ) sin = C (1 - cos ) 1 sin cos = C sin the above equations can be expressed in a more convenient form x + y x1 = CCC + CCC cos + xy sin x1y1 = - CCC sin + xy cos this is the transformation equations for plane stress substituting for in x1 equation x + y y1 = CCC - CCC cos - xy sin also we obtain the following equation for plane stress 3

4 x1 + y1 = x + y i.e. the sum of the normal stresses actin on perpendicular faces for aplane stress element is constant, independent of x1 and x1y1 versus the angle of rotation can be plotted as for uniaxial stress case, y = 0, xy = 0 x1 = x cos = x (1 + cos ) / x1y1 = - x sin cos = - x sin / for pure shear stress case, x = y = 0 x1 = xy sin cos = xy sin x1y1 = xy (cos - sin ) = xy cos same as derived in previous chapters for biaxial stress case, xy = 0 x + y x1 = CCC + CCC cos x1y1 = - CCC sin Example 7-1 x = 110 MPa y = 40 MPa xy = yx = 8 MPa determine the stresses for = 451 4

5 x + y CCC = 75 MPa CCC = 35 MPa sin = sin 901 = 1 cos = cos 901 = 0 x + y x1 = CCC + CCC cos + xy sin = x x 1 = 103 MPa x1y1 = - CCC sin + xy cos = - 35 x x 0 = - 35 MPa x1 + y1 = x + y y1 = x + y - x1 = = 47 MPa Example 7- x = - 46 MPa y = 1 MPa xy = yx = - 19 MPa determine the stresses for = x + y CCC = - 17 MPa CCC = - 9 MPa sin = sin (- 301) = cos = cos (- 301) = x + y x1 = CCC + CCC cos + xy sin = (- 9) (- 19) (-0.5) = MPa 5

6 x1y1 = - CCC sin + xy cos = - (- 9) (-0.5) + (- 19) = - 31 MPa x1 + y1 = x + y y1 = x1 + y - x = (- 3.6) = MPa 7.3 Principal Stresses and Maximum Shear Stresses x1 and x1y1 vary continuously as the element is rotated through the angle for design purpose, the largest positive and negative stresses are usually needed, the maximum and minimum normal stresses are called the principal stresses consider the stress transformation equation x + y x1 = CCC + CCC cos + xy sin to find the maximum normal stress, we may set d x1 / d = 0 d x1 CC = -( ) sin + xy cos = 0 d we get xy tan p = CCC p defines the orientation of the principal plane, two values of p from 0 ~ 3601 and differ by

7 p has two values differ by 901, we conclude that the principal stresses occur on mutually perpendicular plane also ( ) / cos p = CCCCC R xy sin p = C R where R = [(CCC) + xy ] substitute cos p and sin p into the expression of x1 x + y 1 = ( x1 ) max = CCC + [(CCC) + xy ] and the smaller principal stress denoted by is obtained 1 + = x + y x + y = ( x1 ) min = CCC - [(CCC) + xy ] the principal stresses can be written as x + y 1, = CCC! [ (CCC) + xy ] + sign gives the larger principal stress - sign gives the smaller principal stress p1 and p can be determined, but we cannot tell from the equation which angle is p1 and which is p an important characteristic concerning the principal plane : the shear is 7

8 zero on the principal plane x1y1 = - CCC sin + xy cos substitute p into this equation, we can get x1y1 = 0 for uniaxial and biaxial stress states, xy = 0 tan p = 0 p = 01 and 901 for pure shear stress, x = y = 0 tan p = p = 451 and 1351 for the three-dimensional stress element, z = 0 is also a principal stress, note that there are no shear stresses on the principal plane Maximum Shear Stress x1y1 = - CCC sin + xy cos d x1y1 CC = -( ) cos - xy sin = 0 d tan s = - CCC xy 8

9 s has two values between 0 and 3601 s has two values between 0 and 1801, and differ by 901 comparing the angle s and p, it is shown that plane 1 tan s = - CCC = - cot p tan p i.e. s z p s = p! 901 or s = p! 451 the plane of maximum shear stress occur at 451 to the principal similarly we have xy cos s = CC R sin s = - CCC R and the corresponding maximum shear stress is max = [ (CCC) + xy ] the algebraically minimum shear stress min has the same magnitude a usually expression for the maximum shear stress can be obtained by 1 and 1 - max = CCC the max is equal to one half the difference of the principal stress normal stresses also act on the planes of maximum, substituting s1 in the formula of x1, it is obtained 9

10 x + y x1 = CCC = ave = y1 ave acts on both the plane of maximum and minimum planes if we make a three-dimensional analysis, we can establish that there are possible positions of element for maximum shear stress 1 ( max ) x1 =! C rotate the element 451 about x 1 axis ( max ) y1 =! C rotate the element 451 about y 1 axis 1 - ( max ) z1 =! CCC rotate the element 451 about z 1 axis if 1 > > 0, then max = 1 / = ( max ) y1 for the element Example 7-3 x = 84 MPa y = - 30 MPa xy = - 3 MPa determine the principal stresses and maximum shear stress and their directions 10

11 the principal angles p can be obtained xy (- 3) tan p = CCC = CCCCC = (- 30) p = or p = or x + y CCC = (84-30) / = 7 MPa CCC = ( ) / = 57 MPa x + y x1 = CCC + CCC cos + xy sin for p = ( p = ) for p = ( p = 75.31) 1 = 9.4 MPa = MPa check 1 + = x + y (O. K.) alternative method for principal stresses x + y 1, = CCC! [ (CCC) + xy ] = 7! [ (57) + (- 3) ] = 7! 65.4 thus 1 = 9.4 MPa = MPa p1 = p = the maximum shear stresses are given by 1 - max = CCC = 65.4 MPa s1 = p1-451 =

12 and s = = and the normal stress acting on the planes of maximum shear stress are x + y ave = CCC = 7 MPa 7.4 Mohr's Circle for Plane Stress the transformation of plane stress can be represented in graphical form, known as Mohr's circle the equation of Mohr's circle can be derived from the transformation equations for plane stress x1 - x + y CCC = CCC cos + xy sin x1y1 = - CCC sin + xy cos to eliminate the parameter, we square both sides of each equation and then add together, it can be obtained x + y ( x1 - CCC) + x1y1 = (CCC) + xy x + y let ave = CCC R = (CCC) + xy then the above equation can be written ( x1 - ave ) + x1y1 = R 1

13 this is a equation of circle with x1 and x1y1 as coordinates, the radius is R and center at x1 = ave, x1y1 = 0 positive shear stress is plotted downward and a positive angle is plotted counterclockwise positive shear stress is plotted upward and a positive angle is plotted clockwise Construction of Mohr's Circle (1) locate the center C at x1 = ave, x1y1 = 0 () locate point A which is at = 0, x1 = x, x1y1 = xy (3) locate point B which is at = 901, x1 = y1, x1y1 = - xy [Note that the line AB must passes through point C] (4) draw the circle through points A and B with center at C this circle is the Mohr's circle with radius R R = [ (CCC) + xy ] the stress state on an inclined element with an angle is represented at point D on the Mohr's circle, which is measured an angle counter- clockwise from point A 13

14 to show the coordinate at D x1 = ave + R cos ( p - ) = ave + R (cos p cos + sin p sin ) R cos p = R CCC = CCC R xy R sin p = R C = xy R x1 = ave + CCC cos + xy sin xy = R sin ( p - ) = R (sin p cos - cos p sin ) = - CCC sin + xy cos same results as the transformation equations point D' represents the stress state on the face of the face 901 from the face represented by point D, i.e. y 1 face 14

15 A ( x, xy ) B ( y, - xy ) C ( ave, 0) D ( x1, x1y1 ) D' ( y1, - x1y1 ) at point P 1 on the circle, x1 = max = 1 hence, P 1 represents the stress state at principal plane the other principal plane ( min = ) is represented by P x + y 1 = OC + CP 1 = CCC + R x + y = OC - CP = CCC - R the principal angle p1 can be obtained by xy cos p1 = CCC or sin p1 = CC R R and p = p comparing the Mohr's circle and the stress element, it is observed Mohr's Circle stress element A P 1 ( p1 ) x x 1 ( p1 ) A P ( p ) x x 1 ( p ) or ( p1 ) or (901 - p1 ) P 1 S (901 ) x 1 max (451 ) points S and S' representing the points of maximum and minimum shear stresses, are located on the circle at 901 from points P 1 and P 15

16 i.e. the planes of maximum and minimum shear stress are at 451 to the principal planes, and max = R = [ (CCC) + xy ] if either x of y is negative, part of the circle will be located to the left of the origin Mohr's circle makes it possible to visualize the relationships between stresses acting planes at various angles, and it also serves as a simple memory device for obtaining the stress transformation equation Example 7-4 x = 90 MPa y = 0 MPa xy = 0 = 301 x + y ave = CCC = CCCC = 55 MPa A ( = 0) x1 = 90 x1y1 = 0 B ( = 901) x1 = 0 x1y1 = 0 R = [ (CCC) + xy ] = 35 MPa = 301 = 601 (point D) x1 = ave + R cos 601 = cos 601 = 7.5 MPa x1y1 = - R sin 601 = - 35 sin 601 = MPa 16

17 = 101 = 401 (point D') x1 = ave - R cos 601 = cos 601 = 37.5 MPa x1y1 = R sin 601 = 35 sin 601 = 30.3 MPa Example 7-5 x = 100 MPa y = 34 MPa xy = 8 MPa determine the stresses on the face of = 401 determine 1, and max x + y ave = CCC = CCCC = 67 MPa A ( = 0) x1 = 100 x1y1 = 8 B ( = 901) x1 = 34 x1y1 = - 8 R = [ (CCC) + xy ] = 43 MPa = 401 = 801 (point D) tan ACP 1 = 8 / 33 = ACP 1 = p1 = DCP 1 = ACP 1 = x1 = ave + R cos = 100 MPa 17

18 x1y1 = - R sin = MPa principal stresses are represented by P 1 and P 1 = ave + R = 110 MPa p1 = p1 = = ave - R = 4 MPa p = p = maximum shear stress max = R = 43 MPa s = p1-451 = x1 = ave = 67 MPa Example 7-6 x = - 50 MPa y = 10 MPa xy = - 40 MPa = 301 determine x1, x1y1 on = 451 determine 1, and max x + y ave = CCC = CCCC = - 0 MPa A ( = 0) x1 = - 50 x1y1 = - 40 B ( = 901) x1 = 10 x1y1 = 40 R = [ (CCC) + xy ] = 50 MPa 18

19 = 451 = 901 (point D) tan ACP = 40 / 30 = ACP = p = DCP = ACP = x1 = cos = - 60 MPa x1y1 = 50 sin = 30 MPa at D' x1 = (- 60) = 0 MPa x1y1 = - 30 MPa principal stresses are represented by P 1 and P 1 = ave + R = 30 MPa p1 = p = p1 = = ave - R = - 70 MPa p = 6.61 maximum shear stress max = R = 50 MPa s = p1-451 = x1 = ave = - 0 MPa 19

20 7.5 Hook's Law for Plane Stress for the plane stress with normal stresses x and y, the normal strains are x = ( x y ) / E y = ( y x ) / E z = - ( x + y ) / E the first two equations can be solved for the stresses in terms of strains x = E ( x + y ) / (1 ) y = E ( y + x ) / (1 ) for the pure shear stress xy, the shear strain xy is xy = xy / G or xy = G xy the three material parameters with the relation G = E / [ (1 + )] volume change V 0 = a b c V 1 = a (1 + x ) b (1 + y ) c (1 + z ) = V 0 (1 + x ) (1 + y ) (1 + z ) = V 0 (1 + x + y + z + x y + y z + x z + x y z ) j V 0 (1 + x + y + z ) and the volume change is 0

21 V = V 1 - V 0 = V 0 ( x + y + z ) the unit volume change or dilatation e is defined e = V / V 0 = x + y + z for uniaxial stress x only e = V / V 0 = x (1 ) / E for plane stress x and y e = V / V 0 = ( x + y )(1 ) / E Strain-energy density in plane stress u = ( x x + y y + xy xy ) = ( x + y x y ) / E + xy / G = E ( x + y + x y ) / [(1 )] + G xy / 7.6 Triaxial Stress a stress element subjected to normal stress x, y and z is said to be in a triaxial stress state on the inclined plane parallel to the z axis, only and on this plane, the maximum shear stress occurs on the plane by a 451 rotation about z axis is similarly ( max ) z = ± CCC 1

22 y - z ( max ) x = ± CCC x - z ( max ) y = ± CCC the absolute maximum shear stress is the difference between algebraically largest and smallest of the three principle stresses the Mohr s circles for a 3-D element is shown Hooke s law for triaxial stress, the normal strains are x = x / E ( y + z ) / E y = y / E ( x + z ) / E z = z / E ( x + y ) / E stresses in terms of strains are x = [E (1 ) x + ( y + z )] / (1 + ) (1 ) y = [E (1 ) y + ( x + z )] / (1 + ) (1 ) z = [E (1 ) z + ( x + y )] / (1 + ) (1 ) in the special case of biaxial stress, z = 0, the result are the same as in section 7.5 the unit volume change is also obtained e = V / V 0 = x + y + z = ( x + y + z )(1 ) / E and the strain energy density is u = ( x x + y y + z z ) = ( x + y + z ) / E ( x y + x z + y z ) / E

23 E [(1 )( x + y + z ) + ( x y + x z + y z )] = CCCCCCCCCCCCCCCCCCC (1 + )(1 ) for spherical stress x = y = z = 0 then the normal strains are 0 = 0 (1 ) / E and the unit volume change is e = 3 0 = 3 0 (1 ) / E define the bulk modulus of elasticity as K = E / 3 (1 ) then e may expressed as e = 0 / K and the bulk modulus is K = 0 / e for an object submerged in water, the stress is spherical state, it is often called hydrostatic stress 7.7 Plane Strain the normal and shear strains at a point in body vary with direction, for plane strain, the strain components are x, y, xy and z = xz = yz = 0 plane stress : z = xz = yz = 0 but z g 0 plane strain : z = xz = yz = 0 but z g 0 3

24 plane stress and plane strain do not occur simultaneously in general special cases for plane stress ==> plane strain 1. x = - y z = - ( x + y ) / E = 0. = 0 z = 0 for z = 0 we will derive the strain transformation equations for the case of plane strain, the equations actually are valid even when z exists assume x, y, xy associated with x and y axes are known, to determine x1, y1, x1y1 associated with x 1 and y 1 axes where are rotated counterclockwise through an angle form x and y axes consider first the strain x in x direction x x 1 = x dx = x dx cos similarly for y in y direction y x 1 = y dy = y dy sin consider shear strain xy in xy plane x 1 = xy dy cos then the total increase <d in x 1 direction is <d = x dx cos + y dy sin + xy dy cos 4

25 and the strain in x 1 direction is <d dx dy dy x1 = CC = x C cos + y C sin + xy C cos ds ds ds ds but dx / ds = cos and dy / ds = sin thus x1 = x cos + y sin + xy sin cos substituting for, the y1 is obtained y1 = x sin + y cos - xy sin cos hence x1 + y1 = x + y to obtain the shear strain x1y1, this strain is equal to the decrease in angle between lines that where initially along x 1 and y 1 axes similarly x1y1 = + = dx dy dy = - x C sin + y C cos - xy C sin ds ds ds = - ( ) sin cos - xy sin = - ( ) sin cos + xy cos then the shear strain x1y1 is x1y1 = - ( ) sin cos + xy (cos - sin ) use some trigonometric identities, the transformation equations for plane strain are 5

26 x + y xy x1 = CCC + CCC cos + C sin x1y1 xy CC = - CCC sin + C cos the equations are the counterparts of plane stress stresses strains x xy y x y xy / x1 y1 x1y1 x1 y1 x1y1 / principal strains exist on perpendicular planes with angles p xy tan p = CC the principal strains can be calculated x + y xy 1, = CCC! [ (CCC) + (C) ] the maximum shear strains exists at 451 to the direction of the principal strains max xy CC = [ (CCC) + (C) ] and the normal strains in the directions of maximum shear strains are x + y max = CCC 6

27 the principal strains and principal stresses occur in the same directions Mohr's Circle for Plane Strain plane strains at a point can be measured by strain rosette, then the stresses at this point can be calculated, also the principal strains and principal stresses can be obtained Example 7-7 x = 340 x 10-6 y = 110 x 10-6 xy = 180 x 10-6 determine the strains for = 301, principal strains, maximum shear strain x + y ( )10-6 CCC = CCCCCCC = 5 x

28 ( )10-6 CCC = CCCCCCC = 115 x 10-6 xy / = 90 x 10-6 = 301 = 601 x + y xy x1 = CCC + CCC cos + C sin = 5 x (115 x 10-6 ) cos (90 x 10-6 ) sin 601 = 360 x 10-6 x1y1 xy CC = - CCC sin + C cos = - 55 x 10-6 x1y1 = x 10-6 y1 = x + y - x1 = 90 x 10-6 x + y xy 1, = CCC! [ (CCC) + (C) ] = 5 x 10-6! [(115 x 10-6 ) + (90 x 10-6 ) ] = (5! 146) x = 370 x 10-6 = 80 x 10-6 the angles of principal directions are xy 180 tan p = CC = CCCC = p = 381 and 181 8

29 p = 191 and = 370 x 10-6 p = 191 = 80 x 10-6 p = 1091 note that 1 + = x + y the maximum shear strain is max xy CC = [ (CCC) + (C) ] = 146 x 10-6 max = 9 x 10-6 s1 = p1-451 = - 61 s = s = 641 the normal strains at this direction is x + y ave = CCC = 5 x 10-6 all of this results can be obtained from Mohr's circle 9

30 Example 7-8 the plane strains measured by a 451 strain rosette are a, b and c determine x, y and xy x = a y = b x + y xy x1 = CCC + CCC cos + C sin for = 451, x1 = b a + c a - c xy b = CCC + CCC cos C sin 901 solve for xy, we get xy = b - a - c the strains x, y and xy can be determined from the strain-gage reading also the strains x1, y1 and x1y1 can be calculate at any angle 30

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