Surface and Cntact Stress The cncept f the frce is fundamental t mechanics and many imprtant prblems can be cast in terms f frces nly, fr example the prblems cnsidered in Chapter. Hwever, mre sphisticated prblems require that the actin f frces be described in terms f stress, that is, frce divided by area. r example, if ne hangs an bject frm a rpe, it is nt the weight f the bject which determines whether the rpe will break, but the weight divided by the crss-sectinal area f the rpe, a fact nted by Galile in 68... Stress Distributins s an intrductin t the idea f stress, cnsider the situatin shwn in ig...a: a blck f mass m and crss sectinal area sits n a bench. llwing the methdlgy f Chapter, an analysis f a free-bdy f the blck shws that a frce equal t the weight mg acts upward n the blck, ig...b. llwing fr mre detail nw, this frce will actually be distributed ver the surface f the blck, as indicated in ig...c. Defining the stress t be frce divided by area, the stress acting n the blck is mg (..) The unit f stress is the Pascal (Pa): Pa is equivalent t a frce f Newtn acting ver an area f metre squared. Typical units used in engineering applicatins are the kilpascal, kpa ( Pa ), the megapascal, Pa ( 6 Pa ) and the gigapascal, GPa ( 9 Pa). mg mg (c) igure..: a blck resting n a bench; weight f the blck, reactin f the bench n the blck, (c) stress distributin acting n the blck The stress distributin f ig...c acts n the blck. By Newtn s third law, an equal and ppsite stress distributin is exerted by the blck n the bench; ne says that the weight frce f the blck is transmitted t the underlying bench. The stress distributin f ig... is unifrm, i.e. cnstant everywhere ver the surface. In mre cmplex and interesting situatins in which materials cntact, ne is mre likely t btain a nn-unifrm distributin f stress. r example, cnsider the case f a metal ball being pushed int a similarly stiff bject by a frce, as
Sectin. illustrated in ig.... gain, an equal frce acts n the underside f the ball, ig...b. s with the blck, the frce will actually be distributed ver a cntact regin. It will be shwn in Part II that the ball (and the large bject) will defrm and a circular cntact regin will arise where the ball and bject meet, and that the stress is largest at the centre f the cntact surface, dying away t zer at the edges f cntact, ig...c ( in ig...c). In this case, with stress nt cnstant, ne can nly write, ig...d, (..) d d The stress varies frm pint t pint ver the surface but the sum (r integral) f the stresses (times areas) equals the ttal frce applied t the ball. cntact regin Small regin d (c) (d) igure..: a ball being frced int a large bject, frce applied t ball, reactin f bject n ball, (c) a nn-unifrm stress distributin ver the cntacting surface, (d) the stress acting n a small (infinitesimal) area given stress distributin gives rise t a resultant frce, which is btained by integratin, Eqn.... It will als give rise t a resultant mment. This is examined in the fllwing example. Example Cnsider the surface shwn in ig..., f length m and depth m (int the page). The stress ver the surface is given by x kpa, with x measured in m frm the lefthand side f the surface. The frce acting n an element f length dx at psitin x is (see ig...b) kpa m m d d x dx the weight f the ball is neglected here the radius f which depends n the frce applied and the materials in cntact
Sectin. The resultant frce is then, frm Eqn... d xdx kpa m 4kN The mment f the stress distributin is given by d l d (..) where l is the length f the mment-arm frm the chsen axis. Taking the axis t be at x, the mment-arm is l x, ig...b, and x 6 kpa m kn m d x x dx Taking mments abut the right-hand end, x, ne has x xdx kpa m kn m d x 8 (x) m x x dx (x) igure..: a nn-unifrm stress acting ver a surface; the stress distributin, stress acting n an element f size dx.. Equivalent rces and ments Smetimes it is useful t replace a stress distributin with an equivalent frce, i.e. a frce equal t the resultant frce f the distributin and ne which als give the same mment abut any axis as the distributin. rmulae fr equivalent frces are derived in what fllws fr triangular and arbitrary linear stress distributins.
Sectin. Triangular Stress Distributin Cnsider the triangular stress distributin shwn in ig...4. The stress at the end is, the length f the distributin is and the thickness int the page is t. The equivalent frce is, frm Eqn..., x t dx t (..4) which is just the average stress times area. The pint f actin f this frce shuld be such that the mment f the frce is equivalent t the mment f the stress distributin. Taking mments abut the left hand end, fr the distributin ne has, frm.., t x ( x) dx t Placing the frce at psitin x xc, ig...4, the mment f the frce is t / x c. Equating these expressins leads t the psitin at which the equivalent frce acts:. (..5) equivalent frce (x) x igure..4: triangular stress distributin and equivalent frce Nte that the mment abut any axis is nw the same fr bth the stress distributin and the equivalent frce. rbitrary inear Stress Distributin Cnsider the linear stress distributin shwn in ig...5. The stress at the ends are and and this time the equivalent frce is t )( x / ) dx t / ( (..6) 4
Sectin. Taking mments abut the left hand end, fr the distributin ne has / 6 t x ( x) dx t The mment f the frce is t x / t Eqn...5 fllws frm..7 by setting. c. Equating these expressins leads (..7) equivalent frce (x) igure..5: a nn-unifrm stress distributin and equivalent frce The Centrid Generalising the abve cases, the line f actin f the equivalent frce fr any arbitrary stress distributin (x) is t x ( x) dx t ( x) dx xd Centrid (..8) This lcatin is knwn as the centrid f the distributin. Nte that mst f the discussin abve is fr tw-dimensinal cases, i.e. the stress is assumed cnstant int the page. Three dimensinal prblems can be tackled in the same way, nly nw ne must integrate tw-dimensinally ver a surface rather than ne-dimensinally ver a line. ls, the frces cnsidered thus far are nrmal frces, where the frce acts perpendicular t a surface, and they give rise t nrmal stresses. Nrmal stresses are als called pressures when they are cmpressive as in igs...-. 5
Sectin... Shear Stress Cnsider nw the case f shear frces, that is, frces which act tangentially t surfaces. nrmal frce acts n the blck f ig...6a. The blck des nt mve and, t maintain equilibrium, the frce is resisted by a frictin frce mg, where is the cefficient f frictin. free bdy diagram f the blck is shwn in ig...6b. ssuming a unifrm distributin f stress, the stress and resultant frce arising n the surfaces f the blck and underlying bject are as shwn. The stresses are in this case called shear stresses. igure..6: shear stress; a frce acting n a blck, shear stresses arising n the cntacting surfaces..4 Cmbined Nrmal and Shear Stress rces acting inclined t a surface are mst cnveniently described by decmpsing the frce int cmpnents nrmal and tangential t the surface. Then ne has bth nrmal stress N and shear stress S, as in ig...7. N S igure..7: a frce giving rise t nrmal and shear stress ver the cntacting surfaces The stresses cnsidered in this sectin are examples f surface stresses r cntact stresses. They arise when materials meet at a cmmn surface. Other examples wuld be sea-water pressurising a material in deep water and the stress exerted by a train wheel n a train track. 6