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CHAPTER 14 COMPLEX STRAIN AND THE ELASTIC CONSTANTS Summary The relationships between the elastic constants are E = 2G(1 +v) and E = 3K(1-2v) Poisson's ratio v being defined as the ratio of lateral strain to longitudinal strain and bulk modulus K as the ratio of volumetric stress to volumetric strain. The strain in the x direction in a material subjected to three mutually perpendicular stresses in the x, y and z directions is given by Qx 0 0, 1 & =--v-L,-v-=-((a -VQ - V Q ) " E E E E " " Similar equations apply for E, and E,. since Thus the principal strain in a given direction can be found in terms of the principal stresses, For a two-dimensional stress system (i.e. u3 = 0), principal stresses can be found from known principal strains, since E (-52 + VEl) (1 - v2) E and o2 = (81 + YE21 (1 - v2) 61 = When the linear strains in two perpendicular directions are known, together with the associated shear strain, or when three linear strains are known, the principal strains are easily determined by the use of Mohr's strain circle. 14.1. Linear strain for tri-axial stress state Consider an element subjected to three mutually perpendicular tensile stresses Q,, Q, and Q, If Q, and Q, were not present the strain in the x direction would, from the basic definition of as shown in Fig. 14.1. Young's modulus E, be QX E, = - E 36 1 362 Mechankr o j Materials g14.2 t Fig. 14.1 The effects of Q, and cZ in the x direction are given by the definition of Poisson's ratio v to be Q 0 - v 2 and - v P respectively E E the negative sign indicating that if Q, and Q, are positive, i.e. tensile, then they tend to reduce the strain in the x direction. Thus the total linear strain in the x direction is given by E, = - ( ( 6 , - V f J y - V ~ , ) 1 E Le. (14.1) Similarly the strains in the y and z directions would be 1 E E, = - (Qy - vox - va,) E, = -(a, 1 - vox - YOy) E The three equations being known as the "generalised Hookes Law" from which the simple uniaxial form of $1.5 is obtained (when two of the three stresses are reduced to zero). 14.2. Principal strains in terms of stresses In the absence of shear stresses on the faces of the element shown in Fig. 14.1 the stresses ox, CJ) and Q, are in fact principal stresses. Thus the principal strain in a given direction is obtained from the principal stresses as 1 E E 1 = - (ul - vu2 - vu3) 514.3 or or Complex Strain and the Elastic Constants 1 E - -(az-val - V 6 3 ) z- E E3 = - ( 6 3 - v u 1 - v a z ) 1 E 363 (14.2) 14.3. Principal stresses in terms of strains - two-dimensional stress system For a two-dimensional stress system, Le. a3 = 0, the above equations reduce to 1 E 1 E 1 E E1 = - (crl - va2) E2 = - (az - Vbl) E3 = - (- Val- YO2) and with .. = a1 - va2 EEZ = 0 2 - V O ~ Solving these equations simultaneously yields the following values for the principal stresses: and (14.3) 14.4. Bulk modulus K It has been shown previously that Young's modulus E and the shear modulus G are defined as the ratio of stress to strain under direct load and shear respectively. Bulk modulus is similarly defined as a ratio of stress to strain under uniform pressure conditions. Thus if a material is subjected to a uniform pressure (or volumetric stress) 0 in all directions then volumetric stress volumetric strain bulk modulus = a i.e. K = - E, the volumetric strain being defined below. 14.5. Volumetric strain ( 14.4) Consider a rectangular block of sides x, y and z subjected to a system of equal direct stresses a on each face. Let the sides be changed in length by Sx, 6y and 6z respectively under stress (Fig. 14.2). 364 Mechanics of Materials 914.6 Fig. 14.2. Rectangular element subjected to uniform compressive stress on all faces producing decrease in size shown. The volumetric strain is defined as follows: change in volume 6 V SV volumetric strain = =- =- original volume V xyz The change in volume can best be found by calculating the volume of the strips to be cut off Then the original size of block to reduce it to the dotted block shown in Fig. 14.2. sv = xysz + y(z - 6Z)hX + (x - 6x) (z - 6z)sy strip at strip at side strip on top back and neglecting the products of small quantities .. 6 v = xysz + yzdx + xzsy volumetric strain = = E, (xydz + yzsx + xzhy) XYZ . . (14.5) i.e. volumetric strain = sum of tbe three mutually perpendicular linear strains 14.6. Volumetric strain for unequal stresses It has been shown above that the volumetric strain is the sum of the three perpendicular linear strains E , = E, + Eg + E, Substituting for the strains in terms of stresses as given by eqn. (14.1), 1 1 E E 1 E E, = - (a, - VbY - va,) +-(ay - va, - vaJ + - (a, - Vb, - vay) 1 EV = ~(u*+U,+u,)(l-2") (14.6) $14.7 Complex Strain and the Elastic Constants 365 It will be shown later that the following relationship applies between the elastic constants E, v and K , Thus the volumetric strain may be written in terms of the bulk modulus as follows: E=3K(1-2~) (14.7) This equation applies to solid bodies only and cannot be used for the determination of internal volume (or capacity) changes of hollow vessels. It may be used, however, for changes in cylinder wall volume. 14.7. Change in volume of circular bar A simple application of eqn. (14.6) is to the determination of volume changes of circular Consider, therefore, a circular bar subjected to a direct stress 6 applied axially as shown in bars under direct load. Fig. 14.3. EY * % c Direct stress U Fig. 14.3. Circular bar subjected to direct axial stress u. cy= 6, cx = 0 and 0, = 0 Here Therefore from eqn. (14.6) 6V E, = -(1-2v) = - E V 6 U V E .. This formula could have been obtained from eqn. (14.5) with change of volume = 6 V = - (1 - 2v) 6 d E, = - and E, = E, = E,, = - v - E E 6V V then E , = E , + E , + E , = - (14.8) 366 Mechanics of Materials $14.8 14.8. Effect of lateral restraint (a) Restraint in one direction only Consider a body subjected to a two-dimensional stress system with a rigid lateral restraint provided in the y direction as shown in Fig. 14.4. Whilst the material is free to contract laterally in the x direction the "Poisson's ratio" extension along they axis is totally prevented. UY I I Rigid restraint Fig. 14.4. Material subjected to lateral restraint in the y direction. Therefore strain in the y direction with a, and oy both compressive, i.e. negative, 1 ' E = E = -- (ay-va,) = 0 .. a, = va, Thus strain in the x direction 1 E = E, = --(a, - va,) 1 E = - - (a, - v%,) (14.9) Thus the introduction of a lateral restraint affects the stiffness and hence the load-carrying capacity of the material by producing an effective change of Young's modulus from E to E/(1 -v*) (b) Restraint in two directions Consider now a material subjected to a three-dimensional stress system a,, a, and a, with restraint provided in both the y and z directions. In this case, ( 1 ) (2) 1 ' E 1 ' E E = --(ay-va,-va,) = o E = --(a,-vo,-vay)=0 and $14.9 Complex Strain and the Elastic Constants From (l), .. 6, = vox + voz 6, = (0, - vox) - 1 V Substituting in (2), 1 - (oy -vox) - vox - Yoy = 0 V .. o, - vox - v 2 0 x - v 2 0 y = O o,(l-v2)= o,(v+v2) v ( l +v) 0, = 6,- (1 - v2) fJXV =- (1 - v) and from (3), .. cz=- '[ -- vox v o x ] = o x [ (1-v) ] ( 1 - v ) v (1 -v) 1 - (1 - v) V O X =- ox o o strain in x direction = - - + v--li + v? E E E E ( 1 - V ) ( 1 - V ) 367 (3) (14.10) Again Young's modulus E is effectively changed, this time to 14.9. Relationship between the elastic constants E, G, K and v (a) E, G and v Consider a cube of material subjected to the action of the shear and complementary shear Assuming that the strains are small the angle ACB may be taken as 45". Therefore strain on diagonal OA forces shown in Fig. 14.5 producing the strained shape indicated. BC ACcos45" AC 1 AC O A - ad2 a ~ 2 ~ ~ 2 = 2 a =-A =- 368 Mechanics of Materials $14.9 Fig. 14.5. Element subjected to shear and associated complementary shear. But AC = ay, where y = angle of distortion or shear strain. . . Now .. . . From Q aY Y strain on diagonal = - = - 2a 2 shear stress z shear strain y = G z y = - G t strain on diagonal = - 2G 3.2 the shear stress system can be replaced by a system of direct stresses at 5", as shown in Fig. 14.6. One set will be compressive, the other tensile, and both will be equal in value to the applied shear stresses. u2=-r /_I ~ - 4x=T _- r- Fig. 14.6. Direct stresses due to shear Thus, from the direct stress system which applies along the diagonals: 0 1 0 2 strain on diagonal = -- v- E E (-4 V- =-- E E T = -(1 +v) E Combining (1) and (2), t z -=-(1+v) 2G E E = 2G(l+v) (14.11) $14.9 (b) E, K and v Complex Strain and the Elastic Constants 369 Consider a cube subjected to three equal stresses 6 as in Fig. 14.7 (Le. volumetric stress = 6). Fig. 14.7. Cubical element subjected to uniform stress u on all faces ("volumetric" or "hydrostatic" stress). 6 0 6 Total strain along one edge = - - v- - v- E E E 6 = -(1 -2v) E But volumetric strain = 3 x linear strain (see eqn. 14.5) 30 E = -(I -2v) By definition: volumetric stress volumetric strain bulk modulus K = Equating (3) and (4), . . 0 volumetric strain = - K 0 30 K E - = -(1-2v) (3) (4) (14.1 2) (c) G, K and v Equations (14.1 1) and (14.12) can now be combined to give the final relationship as follows: From eqn. (14.1 l), E 370 and from eqn. (14.12), Mechanics of Materials Therefore, equating, .. i.e. 9KG (3K + G) E = 414.10 (14.13) 14.10. Strains on an oblique plane (a) Linear strain Consider a rectangular block of material OLMN as shown in the xy plane (Fig. 14.8). The strains along Ox and Oy are E, and cy, and yxy is the shearing strain. C, a cos 8+ xv a sin 8 / Q +.-acosR-+ + a cos 8 E , Fig. 14.8. Strains on an inclined plane. Let the diagonal OM be of length a; then ON = a cos 8 and OL = a sin 8, and the increases in length of these sides under strain are &,a cos 8 and E ~ U sin 8 (i.e. strain x original length). If M moves to M', the movement of M parallel to the x axis is &,a cos 8 + ?,,a sin 8 Eya sin 8 and the movement parallel to the y axis is $14.10 Complex Strain and the Elastic Constants 37 1 Thus the movement of M parallel to OM, which since the strains are small is practically coincident with MM', is (&,a cos 8 + y,,a sin 0)cos 8 + (&,a sin 6) sin 8 Then extension original length strain along OM = = (E, cos 0 + yxy sin e) cos 8 + (ey sin e) sin 8 .. E, = E, cosz e + E, sinZ e + yxy sin e cos e .. Eo = 3 (E, + E,,) + 3 (E, - E,,) cos 28 + 3 yxp sin 28 (14.14) This is identical in form with the equation defining the direct stress on any inclined plane 8 with E, and cy replacing ox and oy and &J, replacing T , ~ , i.e. the shear stress is replaced by HALF the shear strain. (b) Shear strain To determine the shear strain in the direction OM consider the displacement of point P at the foot of the perpendicular from N to OM (Fig. 14.9). cX acos B + X , asin 9 w Fig. 14.9. Enlarged view of part of Fig. 14.8. In the strained condition this point moves to P' Since But .. strain along OM = E, extension of OM = OM extension of OP = O P ~ E , extension of OP = a cosz eEo OP = (a COS e) COS e 372 Mechanics of Materials 614.1 1 During straining the line PN rotates counterclockwise through a small angle u. (&,a cos e) cos e - a cos2 8Ee a cos 8 sin 8 U = = (E, -Ee) cot 8 The line OM also rotates, but clockwise, through a small angle (&,a cos 8 + yxya sin 8) sin 8 - (&,a sin 8) cos 8 a B = Thus the required shear strain Ye in the direction OM, i.e. the amount by which the angle OPN changes, is given by ye = u + B = (E, - Ee)cot 8 + (E,COS 8 + y,,sin 8)sin 8 - &,sin 8 cos 8 Ye = 2 (E, - E,) cos e sin 8 - yxy (cos2 e - sin2 e) Substituting for from eqn. (14.14) gives .. which again is similar in form to the expression for the shear stress 't on any inclined plane e. For consistency of sign convention, however (see 6 14.1 1 below), because OM' moves clockwise with respect to OM it is considered to be a negative shear strain, Le. 370 = 3 (E, - E,) sin 28 -$y,, cos 28 +ye = - [ 3 ( ~ , - & ~ ) ~ i 0 2 e - + y ~ , ~ 0 ~ 2 8 ] (14.H) 14.11. Principal strain- Mohr's strain circle Since the equations for stress and strain on oblique planes are identical in form, as noted above, it is evident that Mohr's stress circle construction can be used equally well to represent strain conditions using the horizontal axis for linear strains and the vertical axis for halfthe shear strain. It should be noted, however, that angles given by Mohr's stress circle refer to the directions of the planes on which the stresses act and not to the direction of the stresses themselves. The directions of the stresses and hence the associated strains are therefore normal @e. at 90") to the directions of the planes. Since angles are doubled in Mohr's circle construction it follows therefore that for true similarity of working a relative rotation of the axes of 2 x 90 = 180" must be introduced. This is achieved by plotting positive shear strains vertically downwards on the strain circle construction as shown in Fig. 14.10. 0 + +Y Fig. 14.10. Mohr's strain circle. $14.1 1 Complex Strain and the Elastic Constants 373 The sign convention adopted for strains is as follows: Linear strains: extension positive Shear strains: The convention for shear strains is a little more difficult. The first subscript in the symbol y,,, usually denotes the shear strain associated with that direction, i.e. with Ox. Similarly, y,,, is the shear strain associated with Oy. If, under strain, the line associated with the first subscript moves counterclockwise with respect to the other line, the shearing strain is said to be positive, and if it moves clockwise it is said to be negative. It will then be seen that positive shear strains are associated with planes carrying positive shear stresses and negative shear strains with planes carrying negative shear stresses. compression negative. Thus, Yxy = -Yyx Mohr's circle for strains E,, E,, and shear strain y,,, (positive referred to x direction) is therefore constructed as for the stress circle with iy,, replacing 7,,, and the axis of shear reversed, as shown in Fig. 14.10. The maximum principal strain is then E, at an angle 8, to E, in the same angular direction as that in Mohr's circle (Fig. 14.11). Again, angles are doubled on Mohr's circle. YXY Fig. 14.1 1. Strain system at a point, including the principal strains and their inclination. Strain conditions at any angle a to E, are found as in the stress circle by marking off an angle 2u from the point representing the x direction, i.e. x'. The coordinates of the point on the circle thus obtained are the strains required. Alternatively, the principal strains may be determined analytically from eqn. (14.14), i.e. E, = 3 (E, + E,,) + +(E, - E,,) cos 28 + iy,, sin 28 As for the derivation of the principal stress equations on page 331, the principal strains, i.e. the maximum and minimum values of strain, occur at values of 8 obtained by equating ds,/d8 to zero. The procedure is identical to that of page 331 for the stress case and will not be repeated here. The values obtained are E1 or E2 = +(E, + cy) k 3 JCt% - q2 + Yxy21 i.e. once again identical in form to the principal stress equation with E replacing 0 and +y replacing T. (14.16) Similarly, (14.17) 374 Mechanics of Materials $14.12 14.12. Mohr's strain circle-alternative derivation from the general stress equations The direct stress on any plane within a material inclined at an angle 0 to the xy axes is given by eqn. (13.8) as: a g = $ (a, + a,) + f (a, - a,) cos 28 + zxy sin 28 .. oe+9o = 3 (0, +a,) +$ (a, - a,)cos (28 + 1800) + z,, sin (28 + 1800) = f (a, + a,) - b; (a, - a,) cos 28 - z,, sin 28 Also, from eqn. (1 3.9), zg = + (a, - 0,) sin 28 - rXy cos 28 Fig. 14.12. Now for the two-dimensional stress system shown in Fig. 14.12, 1 E - -(%-V%+90) e - E 1 E = - { [+(a, + 0,) + + (a, - a,) cos 28 + z,, sin 281 - v[$(a, + 0,) -$(a, - a,) cos 28 - z,,, sin 281 ] 1 E = - [f(1 - v) (a, +ay) +3(1+ v) (a, - a,)cos 28 + (1 + v)z,, sin 281 1 ' E 1 E E =-(a,-va,) E, = - (a, - va,) a, = ____ [E, + Y E X I (1 - v2) a* = ~ [E, + YE, 3 (1 - VZ) But and E from which E and E(l+v) .. 3 (a, + by) = 2(1 - v2) (CY + E x ) 514.13 Complex Strain and the Elastic Constants 375 2(1 - v2) .. 1 E(1 + v, (l- v, + 2(1 -v2) - cos 28 + (1 + v) z,, sin 28 = 3 (E, + E,) + 3 (E, - E,) COS 28 + o, - (' + sin 20 E 5 Now - = G .. ' T = G ~ and E=2G(l+v) Y .. Similarly, substituting for $(ox - cy) and T,, in (l), Eg = f (E, +ey) + 3 (E, - E~)COS 28 + f yxy sin28 But (1 4.14) Yxy COS 28 E E ( E X - EJ sin 28 E .. 2(1+v)Ye= 2(1+v) 2(1+ v) ye = (E, - E,,) sin 28 - y,, cos 28 $ye = 3 (E, - E,) sin 28 - 3 y,, cos 28 .. Again, for consistency of sign convention, since OM will move clockwise under strain, the above shear strain must be considered negative, i.e. &ye = -[3(~,--~)sin28-fy,,cos28] (14.15) Equations (14.14) and (14.15) are similar in form to eqns. (13.8) and (13.9) which are the basis of Mohr's circle solution for stresses provided that iy,, is used in place of T,, and linear stresses c are replaced by linear strains E. These equations will therefore provide a graphical solution known as Mohr's strain circle if axes of E and 3 7 are used. 14.13. Relationship between Mohr's stress and strain circles Consider now a material subjected to the two-dimensional principal stress system shown For Mohr's stress circle (Fig. 14.13b), in Fig. 14.13a. The stress and strain circles are then as shown in Fig. 14.13(b) and (c). (61 + cz) 2 OA x stress scale = 3 76 Mechanics of Materials 514.13 .. and (b) Stress circle (c) Strain circle Fig. 14.13. (01 + 0 2 ) 2 x stress scale OA = radius of stress circle x stress scale = +(a, - a,) For Mohr's strain circle (Fig. 14.13c), (E1 + 62) OA' x strain scale = ___ 2 But and .. 1 E 1 E1 + E2 = - [(a, + a,) - v(o1+ a,)] = E (a1 +a,) (1 - v) (61 + (72) (1 - v) .. OA' = 2E x strain scale (3) Thus, in order that the circles shall be concentric (Fig. 14.14), OA = OA' $14.13 Complex Strain and the Elastic Constants 377 4 * Stress circle Strain circle Fig. 14.14. Combined stress and strain circles. Therefore from (1) and (3) - (61 + 0 2 ) (1 - v) - (01 + 0 2 ) 2 x stress scale 2E x strain scale x strain scale .. stress scale = - Now radius of strain circle x strain scale E (1 - V I = $(E1 - -52) = -[(a, - va2) - (a2 - V a l ) ] =-(a1 -aJ(l + v ) +(a1 - 0 2 ) - (01 - 0 2 ) (1 + v) 1 2E 1 2E radius of stress circle x stress scale - .. radius of strain circle x strain scale 1 2E E =- (1 + v) radius of stress circle E strain scale radius of strain circle (1 + v) stress scale =- X 1.e. R, E (1 - v ) - =-x- RE (1 + v ) E - (1 - v ) (1 + V I -- In other words, provided suitable scales are chosen so that (14.18) (4) (1 4.19) x strain scale stress scale = - (1 - V I E 378 Mechanics of Materials 414.14 the stress and strain circles will have the same centre. If the radius of one circle is known the radius of the other circle can then be determined from the relationship radius of stress circle = - (' - ") x radius of strain circle (1 + V I Other relationships for the stress and strain circles are shown in Fig. 14.15. '0, 6 5-t Fig. 14.15. Other relationships for Mohr's stress and strain circles. 14.14. Construction of strain circle from three known strains (McClintock method)-rosette analysis In order to measure principal strains on the surface of engineering components the normal experimental technique involves the bonding of a strain gauge rosette at the point under consideration. This gives the values of strain in three known directions and enables Mohr's strain circle to be constructed as follows. Consider the three-strain system shown in Fig. 14.16, the known directions of strain being at angles a,, ab and a, to a principal strain direction (this being one of the primary requirements of such readings). The construction sequence is then: Fig. 14.16. System of three known strains. 514.14 Complex Strain and the Elastic Constants 379 (1) On a horizontal line da mark off the known strains E., &b and E, to the same scale to give (2) From a, b and c draw perpendiculars to the line da. (3) From a convenient point X on the perpendicular through b mark off lines corresponding to the known strain directions of E, and E, to intersect (projecting back if necessary) perpendiculars through c and a at C and A. Note that these directions must be identical relative to X b as they are relative to cb in Fig. 14.16, i.e. XC is a, - a b counterclockwise from X b and X A is ab - a, clockwise from X b is then the centre of Mohr's strain circle (Fig. 14.18). points a, b and c (see Fig. 14.18). (4) Construct perpendicular bisectors of the lines X A and XC to meet at the point X which Fig. 14.17. Useful relationship for development of Mohr's strain circle (see Fig. 14.18). (5) With centre Yand radius YA or YC draw the strain circle to cut X b in the point B. (6) The vertical shear strain axis can now be drawn through the zero of the strain scale da; the horizontal linear strain axis passes through Y. (7) Join points A, B and C to I: These radii must then be in the same angular order as the original strain directions. As in Mohr's stress circle, however, angles between them will be double in value, as shown in Fig. 14.18. Fig. 14.18. Construction of strain circle from three known strains - McClintock construction. (Strain gauge rosette analysis.) 380 Mechanics of Materials 914.14 The principal strains are then and c2 as indicated. Principal stresses can now be determined either from the relationships E E 6 1 =- [ E ~ +vs2] and o2 = ~ [E2 + V E l l (1 - v2) (1 - v2) or by superimposing the stress circle using the relationships established in 414.13. The above construction applies whatever the values of strain and whatever the angles between the individual gauges of the rosette. The process is simplified, however, if the rosette axes are arranged: (a) in sequence, in order of ascending or descending strain magnitude, (b) so that the included angle between axes of maximum and minimum strain is less than 180". For example, consider three possible results of readings from the rosette of Fig. 14.16 as shown in Fig. 14.19(i), (ii) and (iii). ( 1 1 ) (11)) Fig. 14.19. Three possible orders of results from any given strain gauge rosette. These may be rearranged as suggested above by projecting axes where necessary as shown in Fig. 14.20(i), (ii) and (iii). ( 1 ) ( 1 1 ) (111) Fig. 14.20. Suitable rearrangement of Fig. 14.19 to facilitate the McClintock construction. In all the above cases, the most convenient construction still commences with the starting point X on the vertical through the intermediate strain value, and will appear similar in form to the construction of Fig. 14.18. Mohr's strain circle solution of rosette readings is strongly recommended because of its simplicity, speed and the ease with which principal stresses may be obtained by superimpos- ing Mohr's stress circle. In addition, when one becomes familiar with the construction procedure, there is little opportunity for arithmetical error. As stated in the previous chapter, the advent of cheap but powerful calculators and microcomputers may reduce the effectiveness of Mohr's circle as a quantitative tool. It remains, however, a very powerful $14.15 Complex Strain and the Elastic Constants 381 medium for the teaching and understanding of complex stress and strain systems and a valuable "aide-memoire" for some of the complex formulae which may be required for solution by other means. For example Fig. 14.21 shows the use of a free-hand sketch of the Mohr circle given by rectangular strain gauge rosette readings to obtain, from simple geometry, the corresponding principal strain equations. * t. Fig. 14.21. Free-hand sketch of Mohr's strain circle. 14.15. Analytical determination of principaI strains from rosette readings The values of the principal strains associated with the three strain readings taken from a strain gauge rosette may be found by calculation using eqn. (14.14), i.e. E~ = f (E, + E?) + f (E, - E?) COS 28 + f yxr sin 28 This equation can be applied three times for the three values of 0 of the rosette gauges. Thus with three known values of E* for three known values of 0, three simultaneous equations will give the unknown strains ex, and yxy. The principal strains can then be determined from eqn. (14.16). E1 or E2 = +(&+E?)--+ + J c ( ~ x - - ? ) * + Y X , ~ l The direction of the principal strain axes are then given by the equivalent strain expression to that derived for stresses [eqn. (13.10)], i.e. (14.20) angles being given relative to the X axis. 0 = 0", 45" and 90" or delta rosettes with 0 = 0", 60" and 120" (Fig. 14.22). The majority of rosette gauges in common use today are either rectangular rosettes with 382 Mechanics of Materials 514.15 Rectangular A Delta Fig. 14.22. Typical strain gauge rosette configurations. In each case the calculations are simplified if the X axis is chosen to coincide with 8 = 0. Then, for both types of rosette, eqn. (14.14) reduces (for 0 = 0) to Eo = +(E, + E y ) ++(E, - E y ) = E, and E, is obtained directly from the E~ strain gauge reading. Similarly, for the rectangular rosette is obtained directly from the reading. If a large number of rosette gauge results have to be analysed, the calculation process may be computerised. In this context the relationship between the rosette readings and resulting principal stresses shown in Table 14.1 for three standard types of strain gauge rosette is recommended. TABLE 14.1. Principal strains and stresses from strain gauge rosettes* (Gauge readings = u,, u2 and a,; Principal stresses = up and uq.) I I Delta (equiangular) Rosett--Arbitiardy oriented with respect to principal axes. Tee Rorans--Gage elements must be aligned with principal axes. U P I Rosette Gage-Numbering Considerations The equations at the left for calculating Princi- pal strains and stresses from rosette main meas urements a w m e that the gage elements are numbered in a particular manner Improper numbering of the gage elements will lead to ambiguity in the interpretation of 0, g , and, in the case of the rectangular rosette. c& also cause errors in the calculated principal strains and stresses. Treating the latter situation first. It 13 always necessary ~n a rectanguiar rosette that gage numberr 1 and 3 be assigned to the two mutual ly perpendicular gager. Any other numbering arrangement will produce inwrren principal strains and stresses. Ambiguities in the interpretation of $p,Q for both rectangular and delta roOttes can be eliminated by numbering the gage elements as follows In a rectangular rosette, Gage 2 must be 45' away from Gage 1 ' and Gage 3 must be 90" away. in the same direction . SLmllarlv. ~n a delta rosette Gager 2 and 3 must be 60' and 120" away respectively, in the same direnian from Gage 1. By definition, $p,q IS the angle from the axis of Gage 1 to the nearest principa axs. When $p,sir positive. the direction is the same ar that of thegage numbering. and. when negative, the oppoiite. * Reproduced with permission from Vishay Measurements Ltd wall chart. 514.16 Complex Strain and the Elastic Constants 383 14.16. Alternative representations of strain distributions at a point Alternative forms of representation for the distribution of stress at a point were presented The values of the direct strain 60 and shear strain ye for any inclined plane 8 are given by in 413.7; the directly equivalent representations for strain are given below. equations (14.14) and (14.15) as 60 = $(E, + E,) ++(E, - cy) cos 28 + f y,, sin 28 f ye = - [$(E, - E ~ ) sin 28 - + y,, cos 281 Plotting these values for the uniaxial stress state on Cartesian axes yields the curves of Fig. 14.23 which can then be compared directly to the equivalent stress distributions of Fig. 13.12. Again the shear curves are "shifted by 45" from the normal strain curves. i" I I I I I I I 1 1 - 8 0" 4 5 O 90' 135" 180' 225O 270" 315' 360" ( X ) ( Y ) ( X ) ( Y ) ( X I (P) ( q ) ( P ) (q) ( P I Fig. 14.23. Cartesian plot of strain distribution at a point under uniaxial applied stress. Comparison with Fig. 13.12 shows that the normal stress and shear stress curves are each in phase with their respective normal strain and shear strain curves. Other relationships between the shear strain and normal strain curves are identical to those listed on page 335 for the normal stress and shear stress distributions. The alternative polar strain representation for the uniaxial stress system is shown in Fig. 14.24 whilst the Cartesian and polar diagrams for the same biaxial stress systems used for Figs. 13.14 and 13.15 are shown in Figs. 14.25 and 14.26. 384 Mechanics of Materials 414.16 X __-- / $ x shear strain=L 2 Fig. 14.24. Polar plot of strain distribution at a point under uniaxial applied stress. Ye ( P ) ( q ) ( P ) ( q ) Fig. 14.25. Cartesian plot of strain distribution at a point under a typical biaxial applied stress system. 414.17 Complex Strain and the Elastic Constants Y 4 385 Fig. 14.26. Polar plot of strain distribution at a point under a typical biaxial applied stress system. 14.17. Strain energy of three-dimensional stress system (a) Total strain energy Any three-dimensional stress system may be reduced to three principal stresses cr,, g2 and o3 acting on a unit cube, the faces of which are principal planes and, therefore, by definition, subjected to zero shear stress. If the corresponding principal strains are E,, e2 and z3, then the total strain energy U, per unit volume is equal to the total work done by the system and given by the equation since the stresses are applied gradually from zero (see page 258). ut = X+Q& u, = ~ U , & , ++az&, ++a,&, .. Substituting for the principal strains using eqn. (14.2), 1 2E u, = -[Ca,(a, - vu2 - V 6 3 ) + oz(a2 - VQ3 - V b , ) + U3(Q3 - va2 -Val)] 1 2E v, = -[ a: + uf + ai - 2v(o,a2 + ~ 2 ~ 3 + a3al)] per unit volume (14.21) .. (b) Shear (or "distortion") strain energy As above, consider the three-dimensional stress system reduced to principal stresses ol, o2 and o3 acting on a unit cube as in Fig. 14.27. For convenience the principal stresses may be 386 Mechanics of Materials 614.17 General stress state Hydrostcrtic stresses Deviatoric stresses Fig. 14.27. Resolution of general three-dimensional principal stress state into "hydrostatic" and "deviatoric" components. written in terms of a mean stress 5 = +(a, + b2 + u3) and additional shear stress terms, i.e. 6, = f(6, + 6 2 + 6 3 ) + j(6, - 6 2 ) ++(a, - 6 3 ) 6 2 = +(a, + 6 2 + 6 3 ) +$(a, - a,) +$(a2 - 6 3 ) 6 3 = $(6, + 6 2 + 63) + 3 0 3 - 6 1 ) ++(a3 - 62) The mean stress term may be considered as a hydrostatic tensile stress, equal in all directions, the strains associated with this giving rise to no distortion, i.e. the unit cube under the action of the hydrostatic stress alone would be strained into a cube. The hydrostatic stresses are sometimes referred to as the spherical or dilatational stresses. The strain energy associated with the hydrostatic stress is termed the volumetric strain energy and is found by substituting into eqn. (14.21), i.e. 6 1 = 02 = 03 = +(a,+ 02 + 6 3 ) volumetric strain energy = - 2E [("1+y+03)i](l-2v) (1 - 2v) .. U" = ___ [ (al + u2 + a3)'] per unit volume (14.22) 6E The remaining terms in the modified principal stress equations are shear stress terms (i.e. functions of principal stress differences in the various planes) and these are the only stresses which give rise to distortion of the stressed element. They are therefore termed distortional or deviatoric stresses. Now total strain energy per unit volume = shear strain energy per unit volume + volumetric strain i.e. Therefore shear strain energy per unit volume is given by: energy per unit volume u, = us + U" us = U,-U, 1 (1 - 2v) + 0; + 03 - 2v(n102 + ~ 2 0 3 + 03al)] - ~ [I (61 + 6 2 + O S ) ~ I i.e. Us = -[a2 6E 2E 414.17 Complex Strain and the Elastic Constants 387 This simplifies to us = - (l+ [(a, - a2)z + (a2 - a3)2 + ( 6 3 - fJ1)2] 6E and, since E = 2G (1 + v), 1 12G 1 6G us = - [a, - 0 2 )2 + (a2 - a3 l2 + (a3 - 0, 121 (14.23a) or, alternatively, Us = - [d + 0: + u: - (ala2 + 6263 + 0301)] It is interesting to note here that even a uniaxial stress condition may be divided into hydrostatic (dilatational) and deviatoric (distortional) terms as shown in Fig. 14.28. (14.23b) Z=Q1 Q I ~-g+$$+~ A f- 5 Principal (uniaxial) stress Hydrostotic stresses Deviatoric stresses Fig. 14.28. Resolution of uniaxial stress into hydrostatic and deviatoric components. Examples Example 14.1 When a bar of 25 mm diameter is subjected to an axial pull of 61 kN the extension on a 50 mm gauge length is 0.1 mm and there is a decrease in diameter of 0.013 mm. Calculate the values of E, v, G, and K. Solution load 61 x lo3 area $n(0.025)* Longitudinal stress = - = = 124.2 MN/m2 0.1 x 103 = 2 10-3 Longitudinal strain = extension - original length lo3 x 50 stress 124.2 x lo6 strain 2 x io+ Young's modulus E = - = = 62.1 GN/m2 change in diameter 0.013 x lo3 = 0.52 x 10-3 - - Lateral strain original diameter lo3 x 25 0.52 x 10 - 3 = 0.26 - lateral strain Poisson's ratio (v) = longitudinal strain 2 x Mechanics of Materials 388 Now Also E E=2G(l+v) .*. G=- 2(1+ v) 62.1 x 109 G = = 24.6 GN/m2 2( 1 + 0.26) E 3(1 - 2 ~ ) E = 3K(l-2~) .'. K = 62.1 x 109 K = = 43.1 GN/m2 3 x 0.48 Example 14.2 A bar of mild steel 25 mm diameter twists 2 degrees in a length of 250 mm under a torque of 430 N m. The same bar deflects 0.8 mm when simply supported at each end horizontally over a span of 500 mm and loaded at the centre of the span with a vertical load of 1.2 kN. Calculate the values of E, G, K and Poisson's ratio v for the material. Solution a a J = -D4 = - (0.025)4 = 0.0383 x m4 32 32 a Angle of twist 8 = 2 x - = 0.0349 radian 180 T ce . TL J L J8 From the simple torsion theory - = - .. G=- 430 x 250 x lo6 0.0349 x lo3 x 0.0383 G = = 80.3 x lo9 N/m2 = 80.3 GN/m2 For a simply supported beam the deflection at mid-span with central load Wis WL3 6 = - 48EI Then WL3 E=--- and 4861 a x I = -D4 = - (0.025)4 = 0.0192 x m4 6 4 6 4 1.2 x 103 x (0.5)3 x 106 x 103 E = = 203 x lo9 N/mZ 48 x 0.0192 x 0.8 = 203 GN/mZ Complex Strain and the Elastic Constants 389 Now .. Also E 2G E=2G(l+v) .'. v = - - l 1 = 0.268 203 2 x 80.3 v=-- E 3(1 - 2 ~ ) E = 3K(1 - 2 ~ ) .'. K=- 203 x 109 K = = 146 x lo9 N/mZ 3(1-0.536) = 146 GN/mZ Example 14.3 A rectangular bar of metal 50 mm x 25 mm cross-section and 125 mm long carries a tensile load of 100 kN along its length, a compressive load of 1 MN on its 50 x 125 mm faces and a tensile load of 400 kN on its 25 x 125 mm faces. If E = 208 GN/mZ and v = 0.3, find (a) the change in volume of the bar; (b) the increase required in the 1 MN load to produce no change in volume. Solution load area 50 x 25 100 x lo3 x lo6 fJ,=-- - = 80MN/m2 fJy = = 128MN/mZ 6, = = - 160 MN/m2 (Fig. 14.29) 400 x 103 x io6 125 x 25 - 1 x lo6 x 106 125 x 50 Fig. 14.29. From $14.6 V change in volume = - (6, + Cy + 6,) (1 - 2v) E - (125 50 25) 10-9[80 + 128 + (- 160)]106 x 0.4 208 x 109 125 x 50 x 25 x 48 x 0.4 208 x 1OI2 m3 = 14.4mm3 - i.e. the bar increases in volume by 14.4 mm3. 390 Mechanics of Materials (b) If the 1 MN load is to be changed, then 6, will be changed; therefore the equation for the change in volume becomes (125 x 50 x 25) change in volume = 0 = 10-9(80 + 128 + a,)106 x 0.4 208 x 109 Then 0 = 80+128+a, 6, = - 208 MN/m' Now .. load = stress x area new load required = - 208 x lo6 x 125 x 50 x = - 1.3 MN Therefore the compressive load of 1 MN must be increased by 0.3 MN for no change in volume to occur. Example 14.4 A steel bar ABC is of circular cross-section and transmits an axial tensile force such that the total change in length is 0.6 mm. The total length of the bar is 1.25 m, AB being 750 mm and 20 mm diameter and BC being 500 mm long and 13'mm diameter (Fig. 14.30). Determine for the parts AB and BC the changes in (a) length, and (b) diameter. Assume Poisson's ratio v for the steel to be 0.3 and Young's modulus E to be 200 GN/m'. Fig. 14.30. Solution (a) Let the tensile force be P newtons. Then =- MNIm' P - load stress in AB = - area )n(0.02)2 loon =- MN/m2 P stress in BC = tn(0.013)' 42n Complex Strain and the Elastic Constants 39 1 Then and .. .. Then and P = - x 10-6 stress P x 106 strain in AB = -- - E i o o n x 2 ~ x 109 2on = P x 10-6 P x lo6 42n x 200 x lo9 strain in BC = 8.471 x 750 x lo-' = 11.95P x P x change in length of AB = 20n change in length of BC = x 500 x 10-3 = i8.95p x 10-9 8 . 4 ~ total change in length = (11.95P+ 18.95P)10-9 = 0.6 x lo-' P(11.95 + 18.95)10-9 = 0.6 x = 19.4 kN 0.6 x 109 103 x 30.9 P = change in length of AB = 19.4 x lo3 x 11.95 x = 0.232 x m = 0.232 mm change in length of BC = 19.4 x lo3 x 18.95 x = 0.368 x = 0.368 mm (b) The lateral (in this case "diametral") strain can be found from the definition of Poisson's ratio v. lateral strain longitudinal strain V = lateral strain = strain on the diameter (= diametral strain) = v x longitudinal strain VP x 0.3 x 19.4 x lo3 - Lateral strain on AB = 2011 20n x lo6 = 92.7 x = 92.7 p) compressive VP x 0.3 x 19.4 x lo3 8.4n x lo6 - Lateral strain on BC = 8.4 x n = 220.5 x (= 220.5 pe) compressive Then, change in diameter of AB = 92.7 x x 20 x lo-' = 1.854 x low6 = 0.00185 mm and change in diameter of BC = 220.5 x = 2.865 x x 13 x = 0.00286 mm Both these changes are decreases. 392 Example 14.5 Mechanics of Materials At a certain point a material is subjected to the following strains: E, = 400 x E, = 200 x y,, = 350 x radian Determine the magnitudes of the principal strains, the directions of the principal strain axes and the strain on an axis inclined at 30" clockwise to the x axis. Solution Mohr's strain circle is as shown in Fig. 14.31. Fig. 14.31. By measurement: &I = 500 x e, = - = 300 E 2 = 100 x e, = 900 + 300 = 1200 60" 2 = 200 x the angles being measured counterclockwise from the direction of E,. Example 14.6 A material is subjected to two mutually perpendicular strains, E, = 350 x and together with an unknown shear strain y,,. If the principal strain in the E, = 50 x material is 420 x determine: Complex Strain and the Elastic Constants 393 (a) the magnitude of the shear strain; (b) the other principal strain; (c) the direction of the principal strain axes; (d) the magnitudes of the principal stresses if E = 200 GN/m' and v = 0.3. Solution 3Y Fig. 14.32 Mohr's strain circle is as shown in Fig. 14.32. The centre has been positioned half-way between E, and E,, and the radius is such that the circle passes through the E axis at 420 x Then, by measurement: Shear strain yxy = 2 x 162 x Other principal strain = - 20 x Direction of principal strain Direction of principal strain E' = 90" + 23'30 = 113"30. The principal stresses may then be determined from the equations = 324 x 47" 2 radian. (compressive). = - = 23"30. [420 + 0.3( - 20)] x 200 x lo9 61 = 1 - (0.3)' 414 x 200 x 103 - = 91 MN/m2 tensile 0.91 Mechanics of Materials 394 and (- 20 + 0.3 x 42O)lO-j x 200 x lo9 a' = 1 - (0.3)' io6 x 200 x 103 = 23.3 MN/mZ tensile - 0.9 1 Thus the principal stresses are 91 MN/m2 and 23.3 MN/m2, both tensile. Example 14.7 The following strain readings were recorded at the angles stated relative to a given horizontal axis: E, = -2.9 x E~ = E, = -0.5 x lo-' at 140" as shown in Fig. 14.33. Determine the magnitude and direction of the principal stresses. E = 200 GN/m'; v = 0.3. at 20" 3.1 x lo-' at 80" Fig. 14.33. Solution Consider now the construction shown in Fig. 14.34 giving the strain circle for the strain For a strain scale of 1 cm = 1 x lo-' strain, in order to superimpose a stress circle values in the question and illustrated in Fig. 14.33. concentric with the strain circle, the necessary scale is x 10-5 200 x 109 x 1 x 10-5 = E l a = - (1 - v) 0.7 Also .. = 2.86 MN/mZ radius of strain circle = 3.5 cm (1 - V ) 3.5 x 0.7 radius of stress circle = 3.5 x - - (1 + v ) - 1.3 = 1.886cm Complex Strain and the Elastic Constants 395 it 5=-3.55 + e , = 3 , 4 5 4 Fig. 14.34. Superimposing the stress circle of radius 1.886 cm concentric with the strain circle, the principal stresses to a scale 1 cm = 2.86 MN/mZ are found to be o1 = 1.8 x 2.86 x lo6 = 5.15 M N / d o2 = - 2.0 x 2.86 x lo6 = - 5.72 M N / d 144 The principal strains will be at an angle - = 72" and 162" counterclockwise from the 2 direction of E,, i.e. 92" and 182" counterclockwise from the given horizontal axis. The principal stresses will therefore also be in these directions. Example 14.8 A rectangular rosette of strain gauges on the surface of a material under stress recorded the following readings of strain: gauge A +450 x gauge B, at 45" to A +200 x lop6 gauge C, at 90" to A -200 x the angles being counterclockwise from A. Determine: (a) the magnitudes of the principal strains, (b) the directions of the principal strain axes, both by calculation and by Mohr's strain circle. 396 Mechanics of Materials Solution If E~ and c2 are the principal strains and EO is the strain in a direction at 8 to the direction of = +(cl + E,) + +(e1 - E~)COS 28 E ~ , then eqn. (14.14) may be rewritten as since yxy = 0 on principal strain axes. Thus if gauge A is at an angle 8 to E ~ : 450 x io-6 = +(E1 +E2)++(E1 -EZ)COS2e 2 0 0 ~ 10-6 =+(E1+&2)+3(E1-EZ)COS(900+2e) - &,)COS (180" + 28) - 200 x 10-6 = +(E1 + E 2 ) = + E Z ) - E2)COS 28 Adding (1) and (3), 250 x = + E , Substituting (4) in (2), 200 x 10-6 = 125 x 10-6 -EZ)COS (900 + 28) .. 75 x = -&,)sin28 Substituting (4) in (l), 450 x 10-6 = 125 x i0-6++(~, -E2)COS2e .. 325 x 10-6 = - E2)COS 28 Dividing (5) by (6), 75 x - - + ( E ~ -c2)sin28 - 325 x 10-6 - E2)COS 28 .. tan 28 = - 0.23 1 .. 28 = - 13" or 180" - 13" = 167" .. e = 83030' Thus gauge A is 83'30 counterclockwise from the direction of cl. Therefore from (6), 325 x = - E~)COS 167" = -667 x 2 x 325 x .. E 1 - E , = - 0.9744 (3) (4) But from eqn. (4), Therefore adding, and subtracting, el + E , = 250 x 2c1 = -417 x .*. = - 208.5 x 2~~ = 917 x .'. E Z = 458.5 x Complex Strain and the Elastic Constants 397 Thus the principal strains are - 208 x the former being on an axis 83"30' clockwise from gauge A. Alternatively, these results may be obtained using Mohr's strain circle as shown in Fig. 14.35. The circle has been drawn using the construction procedure of $14.14 and gives principal strains of = 458.5 x lo6, tensile, at 6'30 counterclockwise from gauge A E~ = 208.5 x lo6, compressive, at 8Y30 clockwise from gauge A and 458.5 x Fig. 14.35. Problems 14.1 (A). A bar of 40 mm diameter carries a tensile load of 100 kN. Determine the longitudinal extension of a Young's modulus E = 210 GN/m2 and Poisson's ratio v = 0.3. c0.019, 0.0045 mm.] 14.2 (A). Establish the relationship between Young's modulus E, the modulus of rigidity G and the bulk modulus 50 mm gauge length and the contraction of the diameter. K in the form 9KG E = - 3K+G 14.3 (A). The extension of a 100 mm gauge length of 14.33 mm diameter bar was found to be 0.15 mm when a tensile load of 50 kN was applied. A torsion specimen of the same specification was made with a 19 mm diameter and a 200 mm gauge length. On test it twisted 0.502 degree under the action of a torque of 45 N m. Calculate E, G, K and v. c206.7, 80.9, 155 GN/m2; 0.278.1 14.4 (A). A rectangular steel bar of 25 mm x 12 mm cross-section deflects 6 mm when simply supported on its 25 mm face over a span of 1.2 m and loaded at the centre with a concentrated load of 126 N. If Poisson's ratio for the material is 0.28 determine the values of (a) the rigidity modulus, and (b) the bulk modulus. C82, 159 GN/mZ.] 14.5 (A). Calculate the changes in dimensions of a 37 mm x 25 mm rectangular bar when loaded with a tensile load of 600 kN. Take E = 210 GN/mZ and v = 0.3. C0.034, 0.023 mm.] 398 Mechanics of Materials 14.6 (A). A rectangular block of material 125 mm x 100 mm x 75 mm carries loads normal to its faces as follows: 1 MN tensile on the 125 x 100 mm faces; 0.48 MN tensile on the 100 x 75 mm faces; zero load on the 125 x 75 mm faces. If Poisson's ratio = 0.3 and E = 200 GN/m2, determine the changes in dimensions of the block under load. What is then the change in volume? c0.025, 0.0228, - 0.022 mm; 270 mm'.] 14.7 (A). A rectangular bar consists of two sections, AB 25 mm square and 250 mm long and BC 12 mm square and 250 mm long. For a tensile load of 20 kN determine: (a) the change in length of the complete bar; (b) the changes in dimensions of each portion. Take E = 80 GN/mz and Poisson's ratio v = 0.3. c0.534 mm; 0.434, -0.003, 0.1, -0.0063 mm.] 14.8 (A). A cylindrical brass bar is 50 mm diameter and 250 mm long. Find thechange in volume of the bar when an axial compressive load of 150 kN is applied. Take E = 100 GN/m2 and v = 0.27. [ 172.5 mm'.] 14.9 (A). A certain alloy bar of 32 mm diameter has a gauge length of 100 mm. A tensile load of 25 kN produces an extension of 0.014 mm on the gauge length and a torque of 2.5 kN m produces an angle of twist of 1.63 degrees. Calculate E, G, K and v. C222, 85.4, 185 GN/m2, 0.3.1 14.10 (A/B). Derive the relationships which exist between the elastic constants (a) E, G and v, and (b) E, K and v. Find the change in volume of a steel cube of 150 mm side immersed to a depth of 3 km in sea water. Take E for steel = 210 GN/m2, v = 0.3 and the density of sea water = 1025 kg/m'. [580 mm'.] 14.1 1 (B). Two steel bars have the same length and the same cross-sectional area, one being circular in section and the other square. Prove that when axial loads are applied the changes in volume of the bars are equal. 14.12 (B). Determine the percentage change in volume of a bar 50 mm square and 1 m long when subjected to an axial compressive load of 10 kN. Find also the restraining pressure on the sides of the bar required to prevent all lateral expansion. For the bar material, E = 210 GN/m2 and v = 0.27. C0.876 x lo-' %, 1.48 MN/mZ.] 14.13 (B). Derive the formula for longitudinal strain due to axial stress ux when all lateral strain is prevented. A piece of material 100 mm long by 25 mm square is in compression under a load of 60 kN. Determine the change in length of the material if all lateral strain is prevented by the application of a uniform external lateral pressure of a suitable intensity. For the material, E = 70 GN/m2 and Poisson's ratio v = 0.25. c0.114 mm.] 14.14 (B). Describe briefly an experiment to find Poisson's ratio for a material. A steel bar of rectangular cross-section 40 mm wide and 25 mm thick is subjected to an axial tensile load of 100 kN. Determine the changes in dimensions of the sides and hence the percentage decrease in cross-sectional area if E = 200 GN/rn2 and Poisson's ratio = 0.3. [ - 6 x lo-', - 3.75 x lo+, -0.03 %.I 14.15 (B). A material is subjected to the following strain system: Determine: (a) the principal strains; (b) the directions of the principal strain axes; (c) the linear strain on an axis inclined at 50" counterclockwise to the direction of E,. 14.16 (B). A material is subjected to two mutually perpendicular linear strains together witha shear strain. Given that this system produces principal strains of 0.0001 compressive and 0.0003 tensile and that one of the linear strains is 0.00025 tensile, determine the magnitudes of the other linear strain and the shear strain. E, = 200 x cy = -56 x y,, = 230 x radian [244, - 100 x 21"; 163 x [- 50 x 265 x 14.17 (C). A 50 mm diameter cylinder is subjected to an axial compressive load of 80 kN. The cylinder is partially enclosed by a well-fitted casing covering almost the whole length, which reduces the lateral expansion by half. Determine the ratio between the axial strain when the casing is fitted and that when it is free to expand in diameter. Take v = 0.3. c0.871.1 14.18 (C). A thin cylindrical shell has hemispherical ends and is subjected to an internal pressure. If the radial change of the cylindrical part is to be equal to that of the hemispherical ends, determine the ratio between the thickness necessary in the two parts. Take v = 0.3. C2.43 : 1 .] 14.19 (B). Determine the values of the principal stresses present in the material of Problem 14.16. Describe an experimental technique by which the directions and magnitudes of these stressescould be determined in practice. For the material, take E = 208 GN/m2 and v = 0.3. C61.6, 2.28 MN/rn2.] Complex Strain and the Elastic Constants 399 14.20 (B). A rectangular prism of steel is subjected to purely normal stresses on all six faoes (i.e. the Stresses are principal stresses). One stress is 60 MN/m2 tensile, and the other two are denoted by u, and uy and may be either tensile or compressive, their magnitudes being such that there is no strain in the direction of uy and that the maximum shearing stress in the material does not exceed 75 MN/m2 on any plane. Determine the range of values within which u, may lie and the corresponding values of uy. Make sketches to show the two limiting states of stress, and calculate the strain energy Der cubic metre of material in the two limiting conditions. Assume that the stresses are not sufficient to cause elastG-failure. For the prism material E = 208 GN/m2; v = 0.286. [U.L.] [ - 90 to 210; - 8.6, 77.2 MN/mZ.] For the following problems on the application of strain gauges additional information may be obtained in $21.2 (Vol. 2). 14.21 (A/B). The following strains are recorded by two strain gauges, their axes being at right angles: E, = 0.00039; ey = -0.00012 (Le. one tensile and one compressive). Find the values of the stresses ux and uy acting along these axes if the relevant elastic constants are E = 208 GN/m2 and v = 0.3. C80.9, -0.69 MN/mZ.] 14.22 (B). Explain how strain gauges can be used to measure shear strain and hence shear stresses in a material. Find the value of the shear stress present in a shaft subjected to pure torsion if two strain gauges mounted at 45" to the axis of the shaft record the following values of strain: 0.00029; - 0.00029. If the shaft is of steel, 75 mm diameter, G = 80 GN/m2 and v = 0.3, determine the value of the applied torque. c46.4 MN/mZ, 3.84 kNm.] 14.23 (B). The following strains were recorded on a rectangular strain rosette: E, = 450 x Determine: (a) the principal strains and the directions of the principal strain axes; (b) the principal stresses if E = 200 GN/mZ and v = 0.3. at 91" clockwise from A; 98, 29.5 MN/m2.] 14.24 (B). The values of strain given in Problem 14.23 were recorded on a 60" rosette gauge. What are now the [484 x -27 x 104 MN/mZ, 25.7 MN/mZ.] 14.25 (B). Describe briefly how you would proceed, with the aid of strain gauges, to find the principal stresses Find, by calculation, the principal stresses present in a material subjected to a complex stress system given that and + 40 x For the material take E = 210 GN/mZ and v = 0.3. [59, 25 MN/m'.] 14.26 (B). Check the calculation of Problem 14.25 by means of Mohr's strain circle. 14.27 (B). Aclosed-ended steel pressure vessel of diameter 2.5 mand plate thickness 18 mm has electric resistance strain gauges bonded on the outer surface in the circumferential and axial directions. These gauges have a resistance of 200 ohms and a gauge factor of 2.49. When the pressure is raised to 9 MN/mZ the change of resistance is 1.065 ohms for the circumferential gauge and 0.265 ohm for the axial gauge. Working from fust principles calculate the value of Young's modulus and Poisson's ratio. [I.Mech.E.] c0.287, 210 GN/m2.] 14.28 (B). Briefly describe the mode of operation of electric resistance strain gauges, and a simple circuit for the measurement of a static change in strain. The torque on a steel shaft of 50 mm diameter which is subjected to pure torsion is measured by a strain gauge bonded on its outer surface at an angle of45" to the longitudinal axis of the shaft. If the change of the gauge resistance is 0.35 ohm in 200 ohms and the strain gauge factor is 2, determine the torque carried by the shaft. For the shaft material E = 210 GN/mZ and v = 0.3. CI.Mech.E.1 C3.47 kN .] 14.29 (A/B). A steel test bar of diameter 11.3 mm and gauge length 56 mm was found to extend 0.08 mm under a load of 30 kN and to have a contraction on the diameter of 0.00452 mm. A shaft of 80 mm diameter, made of the same quality steel, rotates at 420 rev/min. An electrical resistance strain gauge bonded to the outer surface of the shaft at an angle of 45" to the longitudinal axis gave a recorded resistance change of 0.189 R. If the gauge resistance is l00R and the gauge factor is 2.1 determine the maximum power transmitted. [650 kW.] 14.30 (B). A certain equiangular strain gauge rosette is made up of three separate gauges. After it has been installed it is found that one of the gauges has, in error, been taken from an odd batch; its gauge factor is 2.0, that of the other two being 2.2. As the three gauges appear identical it is impossible to say which is the rogue and it is decided to proceed with the test. The following strain readings are obtained using a gauge factor setting on the strain gauge equipment of 2.2: Gauge direction 0" 60" 120' E& = 230 x E, = 0. [451 x at 1" clockwise from A, - 1 x values of the principal strains and the principal stresses? present on a material under the action of a complex stress system. strain readings in directions at On, 45" and 90" to a given axis are + 240 x respectively. + 170 x Strain x low6 + 1 -250 +200 400 Mechanics of Materials Taking into account the various gauge factor values evaluate the greatest possible shear stress value these readings can represent. [City U.] [44 MN/m2.] For the specimen material E = 207 GN/m2 and v = 0.3. 14.31 (B). A solid cylindrical shaft is 250 mm long and 50 mm diameter and is made of aluminium alloy. The periphery of the shaft is constrained in such a way as to prevent lateral strain. Calculate the axial force that will compress the shaft by 0.5 mm. Determine the change in length of the shaft when the lateral constraint is removed but the axial force remains unaltered. Calculate the required reduction in axial force for the nonconstrained shaft if the axial strain is not to exceed 0.2 % Assume the following values of material constants, E = 70 GN/m2; v = 0.3. [C.E.I.] [370 k N 0.673 mm; 95.1 kN.] 14.32 (C). An electric resistance strain gauge rosette is bonded to the surface of a square plate, as shown in Fig. 14.36. The orientation of the rosette is defined by the angle gauge A makes with the X direction. The angle between gauges A and B is 120" and between A and C is 120". The rosette is supposed to be orientated at 45" to the X direction. To check this orientation the plate is loaded with a uniform tension in the X direction only (Le. uy = 0), unloaded and then loaded with a uniform tension stress of the same magnitude, in the Ydirection only (i.e. a, = 0), readings being taken from the strain gauges in both loading cases. Y t B,y:. C Fig. 14.36. _. - x Denoting the greater principal strain in both loading cases by el, show t h t if the rosette is correctly orientated, (a) the strain shown by gauge A should be then for both load cases, and (b) that shown by gauge B should be Eg = __ (' -') E 1 +- (' - fq for the u, case 2 2 or E g = ~ (' - E l -- (' + - i3 for the uy case 2 2 Hence obtain the corresponding expressions for E,. [As for B, but reversed.]
