Equilibrium of Stress

Transcription

1 Equilibrium f Stress Cnsider tw perpendicular planes passing thrugh a pint p. The stress cmpnents acting n these planes are as shwn in ig a. These stresses are usuall shwn tgether acting n a small material element f finite size, ig b. It has been seen that the stress ma var frm pint t pint in a material but, if the element is ver small, the stresses n ne side can be taken t be (mre r less) equal t the stresses acting n the ther side. B cnventin, in analses f the tpe which will fllw, all stress cmpnents shwn are psitive. p ( a) igure 3.4.1: stress cmpnents acting n tw perpendicular planes thrugh a pint; (a) tw perpendicular surfaces at a pint, small material element at the pint The fur stresses can cnvenientl be written in the matri frm: ij (3.4.1) It will be shwn belw that the stress cmpnents acting n an ther plane thrugh p can be evaluated frm a knwledge f nl these stress cmpnents Smmetr f the Shear Stress Cnsider the material element shwn in ig b, reprduced in ig. 3.4.a belw. The element has dimensins is and is subjected t unifrm stresses ver its sides. The resultant frces f the stresses acting n each side f the element act thrugh the sidecentres, and are shwn in ig. 3.4.b. The stresses shwn are psitive, but nte hw 5 Kell

2 psitive stresses can lead t negative frces, depending n the definitin f the aes used. The resultant frce n the cmplete element is seen t be zer. ( a) igure 3.4.: stress cmpnents acting n a material element; (a) stresses, resultant frces n each side B taking mments abut an pint in the blck, ne finds that { Prblem 1} (3.4.) Thus the shear stresses acting n the element are all equal, and fr this reasn the stresses are usuall labelled, ig a, r simpl labelled, ig b. ( a) igure 3.4.3: shear stress acting n a material element 3.4. Three Dimensinal Stress The three-dimensinal cunterpart t the tw-dimensinal element f ig is shwn in ig Again, all stresses shwn are psitive. 53 Kell

3 z z z z z zz igure 3.4.4: a three dimensinal material element Mment equilibrium in this case requires that,, (3.4.3) z z z z The nine stress cmpnents, si f which are independent, can nw be written in the matri frm z ij z (3.4.4) z z zz A vectr has ne directin assciated with it and is characterised b three cmpnents (,, z ). The stress is a quantit which has tw directins assciated with it (the directin f a frce and the nrmal t the plane n which the frce acts) and is characterised b the nine cmpnents f Eqn Such a mathematical bject is called a tensr. Just as the three cmpnents f a vectr change with a change f crdinate aes (fr eample, as in ig...1), s the nine cmpnents f the stress tensr change with a change f aes. This is discussed in the net sectin fr the tw-dimensinal case. (The cncept f a tensr will be eamined mre clsel in Bks II and especiall IV.) Stress Transfrmatin Equatins Cnsider the case where the nine stress cmpnents acting n three perpendicular planes thrugh a material particle are knwn. These cmpnents are,, etc. when using, z, aes, and can be represented b the cube shwn in ig a. Rtate nw the planes abut the three aes these new planes can be represented b the rtated cube shwn in ig b; the aes nrmal t the planes are nw labelled,, z and the crrespnding stress cmpnents with respect t these new aes are,, etc. 54 Kell

4 zz z z z z z z z zz z z z (a) igure 3.4.5: a three dimensinal material element; (a) riginal element, rtated element There is a relatinship between the stress cmpnents,, etc. and the stress cmpnents,, etc. The relatinship can be derived using Newtn s Laws. The equatins describing the relatinship in the full three-dimensinal case are ver length the will be discussed in Bks II and IV. Here, the relatinship fr the twdimensinal case will be derived this D relatinship will prve ver useful in analsing man practical situatins. Tw-dimensinal Stress Transfrmatin Equatins Assume that the stress cmpnents f ig a are knwn. It is required t find the stresses arising n ther planes thrugh p. Cnsider the perpendicular planes shwn in ig b, btained b rtating the riginal element thrugh a psitive (cunterclckwise) angle. The new surfaces are defined b the aes. 55 Kell

5 ( a) igure 3.4.6: stress cmpnents acting n tw different sets f perpendicular surfaces, i.e. in tw different crdinate sstems; (a) riginal sstem, rtated sstem T evaluate these new stress cmpnents, cnsider a triangular element f material at the pint, ig Carring ut frce equilibrium in the directin, ne has (with unit depth int the page) : AB OB cs OAsin OB sin OAcs 0 (3.4.5) Since OB AB cs, OA AB sin, and dividing thrugh b AB, cs sin sin (3.4.6) B O A igure 3.4.7: a free bd diagram f a triangular element f material 56 Kell

6 The frces can als be reslved in the directin and ne btains the relatin ( )sin cs cs (3.4.7) inall, cnsideratin f the element in ig ields tw further relatins, ne f which is the same as Eqn igure 3.4.8: a free bd diagram f a triangular element f material In summar, ne btains the stress transfrmatin equatins: cs sin sin cs sin cs ( sin sin ) cs D Stress Transfrmatin Equatins (3.4.8) These equatins have man uses, as will be seen in the net sectin. In matri frm, cs sin sin cs cs sin sin cs (3.4.9) Bd rce, Acceleratin and Nn-Unifrm Stress Here, it will be shwn that the Stress Transfrmatin Equatins are valid als when (i) there are bd frces, (ii) the bd is accelerating and (iii) the stress and ther quantities are nt unifrm. 57 Kell

7 Suppse that a bd frce i j acts n the material and that the material b b b is accelerating with an acceleratin a ai aj. The cmpnents f bd frce and acceleratin are shwn in ig (a reprductin f ig ). The bd frce will var depending n the size f the material under cnsideratin, e.g. the frce f gravit b mg will be larger fr larger materials; therefre cnsider a quantit which is independent f the amunt f material: the bd frce per unit mass, b / m. Then, Eqn nw reads : AB OB cs OAsin OB sin OAcs / m m cs / m msin ma cs ma sin 0 b b (3.4.10) where m is the mass f the triangular prtin f material. The vlume f the triangle is AB /sin s that, this time, when is divided thrugh b AB, ne has cs sin sin AB / m / sin / m / cs a / sin a / cs b b (3.4.11) where is the densit. Nw, as the element is shrunk in size dwn t the verte O, AB 0, and Eqn is recvered. Thus the Stress Transfrmatin Equatins are valid prvided the material under cnsideratin is ver small; in the limit, the are valid at the pint O. B a a b b O A igure 3.4.9: a free bd diagram f a triangular element f material, including a bd frce and acceleratin inall, cnsider the case where the stress is nt unifrm ver the faces f the triangular prtin f material. Intuitivel, it can be seen that, if ne again shrinks the prtin f material dwn in size t the verte O, the Stress Transfrmatin Equatins will again be 58 Kell

8 valid, with the quantities,, etc. being the values at the verte. T be mre precise, cnsider the stress acting ver the face OB in ig N matter hw the stress varies in the material, if the distance OB is small, the stress can be apprimated b a linear stress distributin, ig b. This linear distributin can itself be decmpsed int tw cmpnents, a unifrm stress f magnitude (the value f at the verte) and a triangular distributin with maimum value. The resultant frce n the face is then OB /. This time, as the element is shrunk in size, 0 and Eqn is again recvered. The same argument can be used t shw that the Stress Transfrmatin Equatins are valid fr an varing stress, bd frce r acceleratin. B B O O igure : stress varing ver a face; (a) stress is linear ver OB if OB is small, linear distributin f stress as a unifrm stress and a triangular stress Three Dimensins Re-visited As the planes were rtated in the tw-dimensinal analsis, n cnsideratin was given t the stresses acting in the third dimensin. Cnsidering again a three dimensinal blck, ig , there is nl ne tractin vectr acting n the plane at the material particle, t. This tractin vectr can be described in terms f the, z, aes as t i j k, ig a. Alternativel, it can be described in terms f the z z zz,, z aes as t zi zj zzk, ig b. 59 Kell

9 z z z t z t z z zz zz (a) igure : a three dimensinal material element; (a) riginal element, rtated element (rtatin abut the z ais) With the rtatin nl happening in the plane, abut the z ais, ne has, zz zz k k. One can thus eamine the tw dimensinal plane shwn in ig , with i j i j. (3.4.1) z z z z Using sme trignmetr, ne can see that z z cs z sin. (3.4.13) sin cs z z z z t z z j i z igure 3.4.1: the tractin vectr represented using tw different crdinate sstems 60 Kell