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'- - - ---:- t, ,, - - t --- ,,_.., _.,,-.. -;--he 1_ R prog Tr o-cbje-ts- L--Tilts--,0-- _fzt,tr- Ile- r ----' --- i\j a-t-Lo kJ. es a , _ - --_-; _,_...._ r- a 1 .d - -Lt -_,--,. ___. ---,--,---0-1t---e-(4 ____..__ ,...., _, - ..e._ - . _ -- - --.._ , 1..‘ -t , ,-,--, ....-,, ., a' br.comehe -r)/"7-re -_i a--i in_ -° c3d d ----27 _ .a -11:r6 y 1 1; r -.4 :;_y- a.11*- --111:8:-Y -?----1....mi.o_ _ --tloha 1.4.u----b 1_'------_ _-_,'-- ....! -, : _ ----- - - --:.‘ ----L.-- 4--- '---,----._-' -,---. -,7-8.51 -Et. _-_ -__-_,, , _-- - - -- --- - 1,, ___ 2 _ :,,,_ - --= i_ - • --' - - _- _ , -__. , _.,. ... ,- ivr-iTrien . N n . .1 7,1 7 __1 - _ ' - - - _ --- --' - - Stress Strain tension) and shear stresses acting upon this plane can be described in terms of two principal stresses and the angle between the direction of stress normal to the plane and the direction 2 of one of the principal stresses.L1. The stresses are described by the following well known equations: f - f cos 2 0 u f e s e f . :extension) of this line and its rota:tion with respect to a principal axis :of strain, both caused by the appli:cation of the stress, can be described :in terms of two principal strains and :the angle between the line and the direc2 :tion of one af the principal strains.— The strains are described by the :following well known equations: . 2 + e s= so') e cos u fe (f ) sinOcose u e = angle of normal of plane to s,J) = - (e 3 u e (1) v ) sinCnosD angle of line to principal axis (u) of strain measured positive counter clockwise. principal axis (u) of stress measured positive counter clockwise,. e = component of stress normal to = longitudinal strain of line. plane. fs e = component of stress parallel to :eso = angle of rotation of line with plane (shear stress). respect to a principal axis of strain. fu = principal stress. : e u = principal strain. f v = principal stress. :e v = principal strain. Both sets of equations are purely geometrical, and apply no matter what kind of material is considered and no matter what the elastic constants of the material are. The rotational strain (e s q) is half of the conventional shear strain. It has positive or negative values, depending upon the direction of rotation. It is used, rather than the conventional shear strain, to make the two sets of equations identical in form and to facilitate the use of the strain circles. 2 --The descriptions are a little more general than the case of plane stress, but are applied only to the case of plane stress in this report. *"Strength of Materials," Timoshenko. Ft.1,p.50, also Zivilingenieur,1882,p.113. Note that 6 is an angle related to stress and that a) is an angle related to strain. In some cases both angles Till appear upon a single diagram so that the distinction is necessary. Mimeo. No. 1317 -2- Mohr represented the above relations graphically, as is shown in figure 1. Such graphs .are called Mohr's stress circles or Mohr's strain circles, depending upon whether the particular diagram represents stress or strain. • -,.n••••n•••n••n• Strain Stress The unit extension of the line, e The stress normal to the plane, lotted hori-:previously described, is plotted horireviously described, is p :zontally positive to the right and the ontally positi w to the right, and :rotation of the line is plotted posi the shear stress upan the plane is :tive downward if a counter-clockwise lotted vertically. Positive shear :rotation is considered positive, stresses are plotted downward if a counter-clockwise shear is considered :Points upon the circumferen ce of the ositive, Points upon the circumfer- ;circle (fig. 1, B) give the relations :between the strains indicated by the ence of the circle( fig. 1, A), give the relations between the stresses :above equations. A simple geometric indicated by the above equations. A :analysis will demonstrate this factb simple geometric analysis will demonstrate this fact. z p p There are a number of useful geometrical relationship s in the strress and strain circles which apply to either circle. The most important of these have to do with the strains (or stresses) in two mutually perpendicular directions (1 and 2) and to the strains (or stresses) in a direction (3) at 45 degrees. These relationships involve the radius (r) of the strain (or stress) circle and the distance (c) of its center from the origin. Both distances be obtained by the use of the similar triangles shown in figure 2, which can is a strain circle. The relationshi p s are: Stress C = 1/2 (f 1f2) f sl = Strain + 1/2 (( e l +02' (2) (2) !e si = e 3c -C o C) 2 il dmeo. No. 1317 2 /(e 1 + (e sl ) 2 Stress and Strain Circles for Wood In an isotropic material. (such as steel or aluminum) the principal axes of strain are always parallel to the principal axes of stress, so that fl) is always equal to e. These angles are not shown as equal in figure 1, and they are not necessarily equal - for an orthotrOpid material like wood. Again, in an isotropic material the principal strains are related to the principal stresses in a way which is independent of the direction of the principal stresses. In an orthotropic material this is not true; however, simple relationships exist between the stresses and strains if the stresses act in the directions of the natural axes of the material, such as the grain or longitudinal (L) direction, tangential (T) direction, and radial (R) direction of wood. (See fip--) . 1, C.) Since this report is concerned only with the two dimensional case, the results will be given in terms of the (L) anct the (T) directions, but can he used for any other two directions by making the necessary substitutions. If the stresses are resolved into the (L) and (T) directions, the relations between th3 stresses and the strains in those directions are given by the follo*ing equations: f f e L = , 7 'TL z f em = J_ E f e T fL T L (3 ) L m? SL SL Vhere EL = modulus of elasticity in the (L) direction. ET = modulus of elasticity in the (T) direction. modulus of rigidity in the (LT) plane. ' LT = Poisson's ratio of the contraction in the (L) direction, due to tension in the (T) direction, to the extension in the (T) direction. 1/LT J, , Jmeo. ,T2 Poisson's ratio of the contraction in the (T) direction, due to tension in the (L)) direction, to the extension in the (L) direction. No. 1317 -4- Solving equations (3) for the stresses, we have: 'L . L f E T f eTPTL SL e T + e L LI T 1 - LT TL (4) 2G e LT SL (The modulus 'of rigidity is multiplied by 2 because the rotational-strain ) is only one-half the conventional shear .strain.) (e SL Determination of Strains from Stresses lathematical equations have been developed for orthotropic materials, but these equations are complicated. The use of the stress and strain circles provides a geometric method of calculation in place of these equations and gives a better picture of the resulcs obtained. The following is an example of the method. Figure 3 shows a block of wood laid out at an angle to the grain direction. The block is so cut that its face, which lies in the plane of the flat page, is parallel to the L, T plane of the wood; that is, i is part of a flat-sawn board. The complete block shown in figure 3 is located at a distance from the edges of the board, and the ends and edges of the board are loaded in a manner which will cause the stresses at the edges of the block which are indicated in figure j. These stresses are: fi = 80 pounds per square inch tension. f 260 pounds per square inch tension. f Sl = 40 pounds per square inch shear. f 82- -7-- 40 . pounds per square inch shear. Of course, fS2 will always equal -f 51 because of equilibrium conditions. The stress circle can bc drawn from these values: 1/2 (12 1 + f2 ) = 70 0, R - \/(fl Yimeo. Foe 1317 ) + (f ) S1 2 _5_ 41.25 The values of the r,iven stresses allow the determination of the direction in , Ihich those stresses act with respect to the stress circle (fic,. 4, A). The circle is drawn to the proper radius (R). The origin is located the distanc e (J) to the left (right if C is negative) of the center of the circle. The stress (fs1) is positive and therefore down and because it causes counter-clockwise rotation. A point (1) upon the circle is found which satisfies the conditions f l = 80 and f 31 = 40. The line (H,1) is drawn in the direction of the stress (f ). (See fig. 1, A,) The grain 1 direction makes an angle of (a) to line (H,1) and therefore the grain direction can be located upon the stress circle. This direction is given 'by the line (H,L), This stress circle completely determines the stress distribution, and the stresses in any direction can be found by ,1 -;he use of simple geometry and trigonometry. The stresses in the directions of the natural axes of the -rood can be found as follows or by any other geometric method: f Tr? = S1 ; _TTT 1 0 + R -7 ' e = 142° '1 = tr-) -L 180° 2t + a 168° 36 - ' = 11° 2 4 ' The stress circle is redrawn in figure 4, B omitting some of the lines which are no longer needed. From similar triangles HLB and HLF (fig. 4, B): , L - 2R sin (180° —00 coo U_60 f, = 2k cos l. ( f T = 2C - • - = 15.98 + C = 108,05 I (from equations(2)) 31.95 and f 01 = f - 3L = -15'98 Thus, all the stresses upon the inner block of figure 3 Nhich are parallel to th natural axes of the wood have been determined. They can be substituted in equations (3) to determine the strains parallel to the natural axes. From these strains the strain circle can be drawn. It may be noted that the stress &rc2e need not be drawn accurately to scale,as it serves only to determine the method of calculaton. The same is true of the strain circle. Ydmeo. No. 1:317 -6- If the wood is spruce, we have E = 0,036 E L G = 0.037 E LT L = 0.0194 O.539 'LT If these values are substituted in equations (3) together with the stresses obtained above, the following, values of the strains result, 91.2 eL L = 828.8 T e SL = 216 E L These strains are given in terms of the modulus of elasticity in the (E ) and therefore this constant need not be substidirection of the tuted in the equations until the problem is solved, thus avoiding the use of exceedingly small numbers in the calculations. From equations (2) we have: c = 1/2 (e L + em) = r = 2 (e Lc) 460 2 427.5 (eSL) = The strain circle can now be drawn with the radius (r) as shown in _A- An axis is drawn through the circle and the distance (c) laid fir,-ure negative) of the center of the circle. The off to the left (right if (c) value of (eSL) is positive, so that ib is laid off downward. A point (L) is 216 0 1.2 found upon the circle -which satisfies the conditions e Land en = -°- E . L Line A, L is dra-wn which gives the direction of the grain vdth respect to the strain circle. acferrinE , to figure 5, A: • e SL LB = = -368 tan %L = c+e AB r L :)T = 107 12 The strain circle completely determines the strain distribution, and the strains in any direction oan be determined by the use of simple geometry and trigonometry. It is evident from a comparison of figure 4, A, with figure 5, A, that the principal axes of strain are not parallel to the principal axes of stress. The major principal axis of stress can be drawn in figure 5, A, by drawing a line through point (A) at angle ( BL ) to the grain direction, as shown. To illustrate the method, the strains in the directions of the original stresses shown in figure 3 will be determined. The direction of stress (f 1 ) acts at angle (a) to the grain direction; thus, the direction of this stress can be shown by line A,1 in figure 5, B (which is a reproduction of figure A, -with some lines omitted). m 78 0 38 D From similar triangles AlF and An es 1 = -2r cos (15 1 vin C= 165.2 L r e l = 2r cos 2 17 1 + c 2c - e e1 65.73 E L = 0,9,27 1 E (from equations (2)) L and e S2 = e S1 1 )542 EL The strain 'can also be deterMined in any other direction. In particular, the strain (e 3 ) in a direction at 45° to (I) and (2) ad in the order (1,3,2) counter clockwise can be found by U4e of equations (2) thus: e 3 = 0 31 + c 2:4,4 E L Determination of Stresses from Strains In testing procedure, it is often desired to measure strains and to compute the stresses from the measurements. Linear strain 8 are more con veniently measured than rotational strains. With records of such strains in three different directions, the strain circle can be drawn and the stress circle derived. The work is greatly simplified if two of the directions are mutually perpendicular and the third is at angles of 45° to the others. The work for wood is further simplified - if the mutually perpendicular directions are those parallel and at right angles to the grain; the more general case will, however, be illustrated. Mimeo. No. 1317 _8_ Fio.ure 6 shows the directions in which the strains are measured relative to the grain direction upon a face of a plane sawed spruce board. The plane of the paper of figure 6 is therefore parallel to the L,T plane of the spruce. The strains obtained in the previous example will be used here. We have: e 1 e2 = 65.73 = 8514.27 E L EL From equations (2): 4Co 1/2 (e l + 6 2) = e r 1 6 c-,..2 =) = e, - S1 427.5 (eS1)2 V(e lc)2 EL The strain circle can now- be drawn as shown in figure 5, B. Line A, 1 gives the direction of (e ) relative to the strain circle. 1 -S1 tan C = Di = i AD C 1 + r =4,975 - 7 0 ° 3' The direction of the grain can be drawn upon the strain circle at • angle (a) to A, 1 as shown, and (±) 2 = (1) + a = 105' 12' From similar triangles ALP and BLF (fig. 5, A): - 3L = c 2r cos 2I e = L e, 2c - -L = 828'8 L Mimeo. No. 1317 216 E L -2r cos n sin L -9- r =91.2 E L (from equations (2)) These strains are parallel to the natural axes of the wood and can be substituted in equations (4) to obtain the corresponding stresses. The followinc7, valuer, are obtained using the elastic constants for spruce previously given. "1f L = 1.08.05 31-95. f T f SL = Usin,7 equations (2) we have: C 1/2 + fT) = \j“' 70 0) + f 1 S1 = 41.25 The stress circle can now be drawn (fig. 11-, B), The grain direction is given by the line H,L. f FL HE" tan OL SL fL C R = -0. 2017 = 1b8° 56' L The stress in any direction can now be determined. For example, the stress ,-.;s in the directions of the original strains shown in figure 6 will be determined .(fig. 4, A). The line 11,1 can 1-)e drawn at angle (a) to che line H,L. This line gives the direction of the strain (e, ) relative to the stress circle. 1 - From similar triangles Elf and 111K eL - a = 142' 2' we have: 21,-), cos (180- Si = 2R cos (180 -• f 2 2 1" ' )sin (180- 1 ) + -1 = /1-0 R = 80 f = 60 1 Lit will he noted that the valu-e-, .of E cancels out. If the original strains L had not been expressed in terms of E L ; this constant would appear in the values " of stres . t as a multir)lier. Mimeo. No. 1317 -10- and f = -f. = LO S1 These are the stresses shown in figure 3. Stress and Strain Circles for Plywood Plywood can be considered as an orthotropic material provided the grain direction of each ply is perpendicular to that of the adjacent plies and the surfaces of the plywood are not bent or warped by the application of the loads. The assumption is slightly in error, as will be pointed out later, but the error is exceedingly small. The strains in plywood can be obtained from the stresses, or the stresses from the strains, in exactly the salrie manner as described for wood if the apparent:1 elastic properties of the plywood . are used instead of the elastic properties of wood and the direction of the face grain of the plywood is used in p lace of the grain direction of the wood. It remains only to derive the apparent elastic properties of plywood in terms of the elastic properties of wood. Figure 7 shows a small unit square of plywood located at a distance from the edge of the plywood panel. The grain directions of the various plies are parallel both to the edges of the small square and those of the panel. A compressive force is applied to the upper edges of the panel, causing the small square to become deformed, as shown by the dotted lines in figure 7. The top edge of the small square will exert an average compressive stress such as to oppose the load, but the compressive stress in any one ply need not equal the compressive stress in another ply. . The compressive stresses in the various plies are indicated in figure 7. The plies are all glued together, so that the compressive strain is the same for all plies as shown. The small square will expand sidewise due to the apparent Poisson's ratio of the plywood. Since the plies are glued together, this expansion will be the same for all plies. Each ply, however, would expand according to its own Poisson's ratio if it were allowed to do so. As it is restrained by the glue, the stresses sho-yn at the left of figure 7 are induced by shear stresses across the glue lines near the edges of the panels. These shear stresses, however, are limited to parts of the panel that are close to its edges, and hence do not appear in the small square. The relations between the deformations and the stresses for a single ply (the ith ply) are: -"The word "apparent" is applied to a summation average value of each property. Its meaning is made clear in the subsequent discussion. Mimeo. No. 1317 -11- ai f. i. 11,- • rY),_ bi f. ai f. 'ja f e = b E. abi ai Ebi where: . - stress in the ith ply in direction (a). = stress in the ith ply in direction (b) al ' bi E a = modulus of elasticity of the ith ply in direction (a) direction (b) ci = modulus of elasticity of the ith ply in = Poisson's ratio of the ith ply ratio of expansion in the (b) direction, due to a compression in the (a) direction, to the contraction in the (a) direction. = Poisson's ratio of the ply: ratio of expansion in the (a) direction, due to a compression in the (b) direc lAon, to the contraction in the (h) direction. These equations are identical with the first two of equations (3). They can be solved for the stresses, and the following, equations are obtained: f f ai bi = E . e a + e, o Da a.. bi The force upon the top eage of any one ply is obtained by multiplying the stress by the area of the ply, and, since the small square is a unit square, the area of the bop edge of a sil]Kle ply is equal to its thickness. The sum of these forces is equal to the total force, or the average stress multiplied by the total thickness. The following equation results: e pt = r a 1 - abl bat. The apparent modulus of elasticity of the plywood is the average stress divided by the deformation, or b ur 1 + E Mimeo No. 1317 a = 7.1 tf• E bai • al-71--- abilbai -12- but eb = 1-Lab a = the apparent Poisson's ratio of the plywood. ij' ab Therefore a (5) 1 - t'abir-bai the apparent modulus of elasticity of the plywood in the (a) direction t = the thickness of the plywood = the apparent Poisson's ratio of the plywood: ratio of expansion in the (b) direction, due to a compression in the (a) direction, to the contraction in the (a) direction. Similarly: =1 t E 1 - i bi Eb abi it is evident that each of the numerators and denominators in the is only slightly less than unity, and by very similar and E equation for L b a amounts. The following equations ar e , therefore, good approximations: E 1 t 5 E = b T approx. t•E . a_ al (6) a-oprox. t E i bi The summations of the forces upon the left edge of the sma l l square must be equal to zero, since no load was applied to the plywood in this direction. It follows that: b t -v., • -- 1) 1 1 - ea abi 4 = 0 ai Separating the terms in the numerator: e. t. E 1 b i n•• nn•••n • ab i lo a i Mimeo. No. 1317 t E bi k a bi e a — 1 _ -13- Dividing both sides of the equation by e a and remembering that eb Pab th -Piab 1 - tE bi + . abi birabi 0 Sabi ^bai t.Ebi pomi 1 bai (7) tihbi 1 - i4bi Similarly: t i Eal. tbal. 1- t'abi t.E ai Pba 1- Pbai The denominators in both the numerator and denominator are all slightly less than Unity, and all by about the same amounts. Therefore, the following equation is a good approximtion: t i obi ba abi 7 t. E . bi d ab • or = tL b and iLba 11D i Ic] f" a b i from equations (6)) i -ai bai The accurate value of (La ) can be obtained by substituting the accurate value of 4±r ,00 ) in the equation for (E a ), thus: Mimeo. No. 1317 (8) i bi P'abi 1 1 - ha" 1 a t Ebi • 2" -t-T;' 1 ab • - (9) abi The appro x imate equ a tions are, however, sufficiently accurate for practical use in connection with plywood. In setting up these equations it has been assumed that the small square of plywood shown in figure 7 is located at a distance from the edge of the larger sheet. If the small square is located at the left edge of the sheet, the stresses shown acting upon the left edge of the square are each equal to zero and the argument used in the derivation of the equations breaks down. A complicated distribution of stress and strain exists in this case. Very near to the edge, the value of (L a ) approaches more nearly the value which would be obtained if the various plies were not glued together. This is t approximate value given by equation (6), which has been justified by test._ It remains to determine the apparent modulus of rigidity of plywood. Imagine a small square of plywood similar to that shown in figure 7, except that the direct stresses are replaced by shear stresses. The shear strain is constant throughout the small square because the various plies are glued together. The shear stress in the lth ply is: f si = 2e Sabi S1 It is evident at once that the average shear stress is 2e 3i L. s Gabi and that the apparent modulus of rigidity of the plywood is fs G ab 2es, t b. G . a.bi (Twice the rotational strain is used in these equations because . the rotational strain is half of the conventional shear strain.) Forest Products Laboratory Restricted Mimeograph No. 1315, "Tentative method of calculating the strength and modulus of elasticity of plywood in compression." himeo. No. 1317 -15- Examples It is quite possible to solve typical problems by means of the stress circles, obtaining, a general formula for a specific case. Two examples are given for plywood, one with tension at 45' to the direction of the face grain and the other with shear at 49' to the direction of the face grain. Plywood in Tension at 45' to the Direction of the Face Grain FiFure shows the specific case considered. From equations (2) 1 f1 for f 2 = 0 2 = 1 f 2 1 R= for f s1 =0 The stress circle can nolif ha drawn as shown in figure 9. The circle passes through the origin, since C = R. It is evident that: fa 1 '1 fb f sa 1 =— f 2 1 It may be recalled here that f a and f sb act parallel to the face grain and f sa acts at riht angles to these two, The strains in these directions tions (3): f f, a 0a I-43a 1;0 ob = e fb can be obtained by the use of equa- - - \ ."13 La 1 f a , ba) (1 1 — 2 (1 f l - Ea' f sa 1 1 f 9r, sa 1 "ab 2S ab2 = The strain circle can nova be drawn, and is shown in figure 10. _Numerical values for drawing it are not available because a general case is being dealt with. The radius of the circle is chosen at random. 31imeo, No. 1317 -16- L'_ax-,ell's theorem shov,is: hpa = Pab E a b a It follows from equations (11) that if E>. E b , e a <e b , and figure 10 was drawn with this fact in mind. It should be noted that the principal axis of strain is not parallel to the direction of the tonsil() stress. The first of equations (2) gives: ( 1 11 , .11 = ea + 2 1 + - 'b Lb P' ab a The second of equations (2) gives: a ksalo + 1 ) 7---Tab a 'b 111-- 1 4,( 1 '1 E esa el = '13 This relation could have been obtained directly from the geometry of figure 10. The modulus of elasticity at Li-° to the direction of the face grain is defined as the ratio of a tensile (or compressive) stress in this direction to the strain in this direction which the stress produces, hence f L 450 1 1 1 Ea ab 1 ba Gab "n'b However, there will be associated with the tensile strain a shear strain (e s1 ). It is found from the gemoetry of figure 10 that: e sl e _ c 4 1 E b E a '.-2J a The use of :!.ia.xwell's relation reduces this equation to: e sl l, ( 1 1 IF - 1 EtEa The special case in -Jvhich E t = E a is interesting. In this case - 0, and the principal axis of strain is parallel to the direction of l s the stress. Then; e Mimeo. No. 1317 -17- a + 20ao Plywood in Shear at 45' to the Direction of the Face Grain Figure 11 shows the specific case considered. From equations ( w.e have: C = 0, because f and f 2 - 0. .1 It follows that the center of the stress circle is at the origin, and R = f si , because C and fl0, The stress circle can now he drawn and is shown in figure 12, The figure shows that the principal axis of stress is parallel to the grain direction. It also shows that; sl fa f = f sl b fs a=0 The resulting strains can be found by substituting these values in equations (3): f, a 1 77--) b E a E b Pl a = sl kb ra Eh E a ab e f sl ( 1 E, + /lab ha sa , 0 ab As :In the previous specific cse e a ( (3 13 (numerically) if Ea .> Et . The strain circle can be drawn, as before, usinE, an arbitrary value for ra(lius. Again using; equations (2): 1 — lame°. No. 1317 a + -b - 2 sl -18- la b a 1 7 - T- a Using Yaxwell's relation, this reduces to: 1 = 1 ( 1 f The value of 0 The resulting strain circle is shown in figure 13. is negative (fsl is negative), therefore the center of the circle lies to the left of the origin. Also the direction of e a , and therefore the direction of the face grain, is parallel to the direction of the principal axis because esa = 0. It is evident from the geometry of the strain circle that: C esl Caa Both e 51 and C are negative and e a is positive. f sl + 7 f sl -b / 1 1 f sl e sl = -b T7-- 'a 1 aB Ea - 7a /1 1 'ba\ (1 0 51 .6b 1 E aEb The modulus of rigidity at 45: to the grain direction (G4 5 0) is defined as the shear divided By the shear deformation, hence: f .) Le • sl 1 = (-4 45° • and = B ID Pjab E a + lib a 1 a E b Associcted with the shear strain is a direct strain e l . The strain circle shows: 1 /1 1 e lC=—f 2 s l EtLa Again it is of interest to examine the special case in which E a = Et. In 'this case e l = 0 and the center of the strain circle coincides with the origin. The formula for the modulus of rigidity at 45° becomes (using Naxwell's relation): E a 1 G 2 1 45° It is interesting to note that this is exactly the relation obtained for isotropic materials. Mimeo. No. 1317 -19-