【正文】 平面應(yīng)變問題，軸對(duì)稱應(yīng)力 ?r=A/r2+2C ??=A/r2+2C ?r?=0 ?r )r=a =q ur )r=b=0 彈性力學(xué) 第四章 92 4125 A wedge of infinite length is subjected to uniform shearing forces of intensity q on its edges. Find stress ponents by using Eq. () 彈性力學(xué) 第四章 93 ? ?=r2(A cos2?+B sin2?+C?+D) ? ?r=??/(r?r)+?2?/(r2??2) =2A cos2?2B sin2? +2C?+2D ??= ?2?/?r2 = 2A cos2?+2B sin2? +2C?+2D ?r?= (?/?r)[??/(r??)] = 2A sin2?2B cos2? C ??)?=??/2 = 0 ?? r) ?=??/2 = ?q ABCD 。 彈性力學(xué) 第四章 86 Settlementvertical displacement, positive downward. 沉陷 鉛直向位移，向下為正 ? u?=2Psin?lnr/(?E)+(1+?)Psin?/(?E)(1?) P?cos?/(?E)Isin? ? M(r, ?=?/2)arbitrary point on the surface M點(diǎn)沉陷 =u?)M=2Plnr/(?E)(1+?)P/(?E)+I ? M(r, ?=?/2)arbitrary point on the surface M點(diǎn)沉陷 = u?)M=2Plnr/(?E)(1+?)P/(?E)+I ? B(s, ?=??/2)base point B點(diǎn)沉陷 =2Plns/(?E)(1+?)P/(?E)+I ? M點(diǎn)相對(duì)沉陷 =M點(diǎn)沉陷 B點(diǎn)沉陷 = u?)M u?)B = 2P/(?E) ln(s/r) 彈性力學(xué) 第四章 87 P y x 中文書 410 B. Uniform normal loads on a straight boundary 直線邊界上作用法向分布荷載 彈性力學(xué) 第四章 88 ? 習(xí)題作業(yè)（英文書） 彈性力學(xué) 第四章 89 ? 4121 Derive the following equations for coordinate transformation of displacement ponents: ? 推導(dǎo)位移分量的坐標(biāo)變換式： ur=u cos?+v sin? u?=u sin?+v cos? u= ur cos? u? sin? v= ur sin?+ u? cos? 彈性力學(xué) 第四章 90 ? 4122 A hollow cylinder with inner radius a and outer radius b is subjected to an internal pressure of intensity q. Find the change of the inner radius, the outer radius and the thickness. ? 41 設(shè)有內(nèi)半徑為 a外半徑為 b的圓筒受內(nèi)壓 q，求內(nèi)半徑 外半徑和圓筒厚度的改變。 ? 置坐標(biāo)原點(diǎn)于洞中心， x 軸在 ?1 方向， y 軸在 ?2 方向，將 q1= ?1 和 q2=?2 代入式 ()得欲求的解答。 ? q=q1 in Eqs. ()用 q=q1 代入式 () ? q=q2 and replace ? by ?+?/2 in Eqs. ()式 () 中 q用 q2代 , ? 用 ?+?/2 代 . ? 3. Adding the results together, we obtain Eqs.()上述結(jié)果相加得本問題解答 () 彈性力學(xué) 第四章 74 C. A plate of any shape in plane stress or strain condition with a small circle hole located at some distance away from the boundary is subjected to any external forces. ? Assume that there is no hole, find the stress ponents and then the magnitudes and directions of the principal stresses, ?1 and ?2, at the point corresponding to the center of the hole. ? Place the origin of coordinates at the center of the hole, with x and y axes along ?1 and ?2 respectively, and apply Eqs.() with q1= ?1 and q2=?2 . 彈性力學(xué) 第四章 75 C. 任意平面問題的板中有一離邊界較遠(yuǎn)的半徑為 a的小圓孔，受到任意荷載 作用 。 彈性力學(xué) 第四章 65 A. a rectangular plate with a small circular hole of radius a and subjected to uniform tensile force of intensity q in the x direction. 具有半徑為 a的小圓孔的矩形板在 x方向受均布荷載 q ? p71(E) Fig. 彈性力學(xué) 第四章 66 The stress at any point on the circle r=b ??a will be the same as if there were no hole at all. 半徑為 ba的圓周上的應(yīng)力同無孔時(shí)的應(yīng)力 ? ?x)r=b=q ?y)r=b=0 ?yx )r=b=0 p61(E) () to () ?r=+ cos2? ??= cos2? ?r?= sin2? 彈性力學(xué) 第四章 67 彈性力學(xué) 第四章 68 ?r=+ cos2? ?r?= sin2? ?r= ?r?= 0 part 1 ?r= + cos2? part2 ?r?= sin2? 彈性力學(xué) 第四章 69 Part 1 r=b: ?r= ?r?=0 ? P66 Eqs. () with qb= and a/b=0 bee Eqs. () p72 彈性力學(xué) 第四章 70 Part 2 r=b:?r= cos2? ?r?= sin2? ? Since ?r=??/(r?r)+?2?/(r2??2) ??= ?2?/?r2 ?r?= (?/?r)[??/(r??)] we assume ? = f(r) cos2? ? ?4?=[?/(r?r)+?2/(r2??2)+?2/?r2 ]2? =0 () ? cos2?[f(4) (r) +2/r f???(r)9/r2f??+9/r3f?]=0 ? f(4) (r) +2/r f???(r)9/r2f??+9/r3f?=0 ? f=Ar4+Br2+C+D/r2 ? ? = cos2?[Ar4+Br2+C+D/r2 ] ? p73 Eqs. () ?r ?? ?r? 彈性力學(xué) 第四章 71 Part 2boundary condition ?r)r=a=0 ?r? )r=a =0 ?r)r=b= cos2? ?r? )r=b = sin2? A B C D ? Solution=part1+part 2 Eqs. () 彈性力學(xué) 第四章 72 ?? )r=a =q(12cos2?) max?? )r=a =3q independent of a 兩端受拉，產(chǎn)生壓應(yīng)力。 彈性力學(xué) 第四章 63 Effect of circular holes on stress distribution 圓孔對(duì)應(yīng)力分布的影響 ? A phenomenon of stress concentration in the neighborhood of a hole. an elastic body without hole is subjected to a certain stress distribution. If a small hole is made inside the body, large additional stress in the neighborhood of the hole will take place. However the stress distribution is almost the same at distances which are large in parison with the dimension of the hole. It is called the phenomenon of stress concentration. It is of a localized character. 彈性力學(xué) 第四章 64 Effect of circular holes on stress distribution 圓孔對(duì)應(yīng)力分布的影響 ? 孔邊應(yīng)力集中現(xiàn)象 : 無孔彈性體中有某種應(yīng)力分布，在該彈性體中有一小孔后，孔邊產(chǎn)生很大的附加應(yīng)力，而離孔較遠(yuǎn)處應(yīng)力基本無變化。central angle is ?.內(nèi)半徑 a,外半徑 b,中心角 ?. ? The moment of each couple per unit width of the beam at the ends is M. 彈性力學(xué) 第四章 56 Axisymmetrical stresses of the curved beam 曲梁的軸對(duì)稱應(yīng)力 ? Since the bending moment is constant along the beam, the stress distribution is the same on all cross sections and the solution of the problem can therefore be obtained by using Eqs. () to