【文章內(nèi)容簡介】 ns i nc o s.,).(.2222,??????????????????? dldl dldldlNdldA dldA dldA xyxyxxyxyyxxafeaefyxyx????????????????????????????????????逆時針為正是負的。是正的順時針為正左上右下為正以拉為正Analysis of biaxial stress statesAnalytical method Normal stress is positive as it in tensile stress and shear stress is positive as its direction is up on left side of unit cube, or is down on right side of unit cube τx is positive, butτy is negative, ? is positive as N rotates counterclockwise from x axis. )(c o ss i n 472220?????? xyx????????討論： 角度為二倍。 )(c o sc o s)(s in)s in (s in??????????????222222 反映二個平行平面上的應(yīng)力完全相等。若斜面的夾角超過 90可用小于 90的處理。 Angle in trigonometry is twice the rotation angle of normal line, it reflects that stresses on two parallel planes are pletely identical. If the angle of oblique exceeds to 90 degree, we can select an angle less than 90 degree. 正應(yīng)力和剪應(yīng)力為角度的函數(shù)。 )()c o ss i n(c o s)s i n(572202222222220??????????????? tg ddyxxxyxxyx?????????????????（ 75）式有二個角，它們相差 90。一個為最大主應(yīng)力 , 另一個為最小主應(yīng)力。 在第二章中告訴我們，脆性材料破壞時由拉應(yīng)力產(chǎn)生的。最大主應(yīng)力方位決定了破壞的方向。 ???????? ?2222s i nc o s xyxyx?????Normal stress and shear stress are functions of rotation angle. Equation 75 can lead two angles, their difference is 90 degree, one represents the maximum stress Another one represents the minimum stress. In Ch. 2, we have found that failure of brittle materials results from tension stress, the orientation of maximum principle stress determines the direction of failure. )()()()(co s)(s i nm a x672222222222222?????????????? xyxyxxyxyxxyxx?????????????????)()()()(co s)(s i nm i n672222222222222?????????????? xyxyxxyxyxxyxx?????????????????x??2yx ?? ?222 xyx ??? ?? )(?20101022m i nm ax2222145122)87()2()2(2co s)2(2/)(2s i n90)77(22022s i n22co s2相差和可見。令差剪應(yīng)力最大的二個面相??????????????????????????????? tgtg tgdd s t r e s s s h ea r of va l u esex t r e m e t h e D e t er m i n g xyxxyxxxyxyxxyxxyx???????????????????????????（ 3）相互垂直面的正應(yīng)力之和為常數(shù)。 ?????? ? 2c o s2s i n2 xyx ???Sum of normal stresses on mutually perpendicular planes is a constant. Difference between ?0 and ?1 is 45 degree. The orientations of the maximum shear stresses is in 90 degree difference. m i nm a x222s i n2c os22)2(2s i n)2(2c os222s i n2c os222c os)2(2c i n)2(2s i n,2,?????????????????????????????????????????????????????????????????????????????yxxyxyxxyxyxxyxyx h av e w e pl ane s or t h ogonal t w o F orThe first stress invariant 例 72 分析鑄鐵試件受扭時破壞面的 方位。 T T T 分析：已知 D， T。由剪應(yīng)力公式 可知剪應(yīng)力的大小。方向以左上 右下，因內(nèi)力圖為負的。其方向如 圖示。 010m i n22m a x4590)2(2)2(2????????????????????????????? tgyxxxxxyxyx?破壞面如圖示。方向垂直最大拉應(yīng)力。 Figure out the orientation of failure plane as cast iron rod is subjected to torsion. Known diameter D and torque T, the shear stress can be calculated by shear stress equation. Failure plane is shown in the figure, it is perpendicular to the maximum tension stress. 167。 74 二向應(yīng)力狀態(tài)分析 圖解法 222222)2()0,2(,)2()2(2c os2s i n22s i n2c os222s i n2c os22xyxyxxyxxyxyxxyxxyxyxxyxyx ?????????????????????????????????????????????????????????????????????半徑圓心程是未知的。上式是圓方而已知Analysis of biaxial stress statemethod of geometric solution center radius Equation of circle x?x?x?y?y??x?x?d D E F A C B A1 B1 o ????????????????????？v坐標(biāo)為點的橫坐標(biāo)為點交于為起點，逆時針、以為半徑作圓。點為圓心，以點。二點，交橫坐標(biāo)于、連、點、建立橫坐標(biāo)和縱坐標(biāo)求已知E ECDCDC CDddD yyxxxyx24321),(),.(,.,?0??2 2 Build a Cartesian coordinate system using transverse and longitudinal coordinates σandτ Find the positions of points D and d. Connect D and d, line Dd intersects transverse axis at point c. Use point c as center of cycle, line cd as radius to plot a cycle Start from cd and rotate an angle of 2? in counterclockwise, then get point c from circumference. y?d D E F A C B A1 B1 o 02??2???????????????22222222222000c oss i ns i nc oss i nc osc oss i n)s i n (xyx CAAD CDCD CEEF??????????????????????????????22222222222000s i nc o ss i ns i nc o sc o s)c o s (,xyxyxxyxyx CDCDoC CEoC CFoCoFBdAD oC ,Bo ,oA????????????????????證明： 討論： 應(yīng)力圓圓周上一點 D。代表單元體 X面上的 應(yīng)力情況。 應(yīng)力圓直徑上的二點分別代表單元體兩個正 交面上的應(yīng)力情況，它們轉(zhuǎn)向相同，圓心角 是 2倍。 A1點是最大正應(yīng)力。它的大小為 oC+ CD，方向是順時針。和 X面上的剪應(yīng)力方向 相同。且當(dāng) σxσy時它對應(yīng)的圓心角 900 D is a point on circumference of stress circle and represents stress state on x plane of unit cube 2222 xyxyx ????? ???? )(Central angle is double the rotation angle ?. Point A1 represents the maximum normal str