Elasticity Chapter 3 two dimensional Problems in Rectangular Coordinates
1 Elasticity
弹单性生力学
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Chapter 3 Two-dimensional Problem in Rectangular Coordinates 83-1 Solution by Polynomials 8 3-2 Determination of Displacements 83-3 Bending of a Simply Supported Beam by Uniform Load 83-4 Loading of a Wedge by Gravity and Hydraulic Pressure 83-5 Solution by Series 8 3-6 Bending of a Simply Supported Beam by arbitrary Lateral Load [D Exercises
3 Chapter 3 Two-dimensional Problem in Rectangular Coordinates §3-1 Solution by Polynomials §3-2 Determination of Displacements §3-3 Bending of a Simply Supported Beam by Uniform Load §3-4 Loading of a Wedge by Gravity and Hydraulic Pressure §3-6 Bending of a Simply Supported Beam by Arbitrary Lateral Load §3-5 Solution by Series Exercises
平冒的立么解答 第三章平面问题的直角坐标解答 §3-1多项式解答 §3-2位移分量的求出 「§3-3简支梁受均布载荷 □§3-4楔形体受重力和液体压力 「§3-5级数式解答 §3-6简支梁受任意横向载荷 习题课
4 第三章 平面问题的直角坐标解答 §3-1 多项式解答 §3-2 位移分量的求出 §3-3 简支梁受均布载荷 §3-4 楔形体受重力和液体压力 §3-5 级数式解答 §3-6 简支梁受任意横向载荷 习题课
s3-1 Solution by Polynomials IThe Stress Function in the form of a polynomial of the First degree o=a+bx+cy The Stress Components o=0,0,=0,rn=m=0 The Stress Boundary Condition X=Y=0 Conclusion: (1 The linear stress function is corresponding to the state of no surface force and no stress.( 2) There's no effect to the stress to add a linear function to any stress function of two-dimensional problem 2. The Stress function in the form of a polynomial of the second degree o=ax+bxy+cy 1. Corresponding too=ax, the stress components 0,S=2a,t=t=0 x y y yx 5
5 1.The Stress Function in the form of a Polynomial of the First Degree §3-1 Solution by Polynomials = a + bx + cy The Stress Components: x = 0, y = 0, xy = yx = 0 The Stress Boundary Condition： X = Y = 0 Conclusion:（1）The linear stress function is corresponding to the state of no surface force and no stress.（2）There’s no effect to the stress to add a linear function to any stress function of two-dimensional problem. 2.The Stress Function in the form of a Polynomial of the Second Degree 2 2 = ax +bxy+ cy 。 1.Corresponding to ,the stress components 2 = ax = 0, = 2 , = = 0 x y xy yx s s a t t
平动血解答 §3-1多项式解答 一、应力函数取一次多项式 p=a+bx+cy 应力分量: 0.o.=0.r 应力边界条件: Ⅹ=Y=0 结论:(1)线性应力函数对应于无面力、无应力的状态 (2)把任何平面问题的应力函数加上一个线性函 数,并不影响应力 二、应力函数取二次多项式 =ax +bxy+cy 1.对应于9=mX,应力分量x=0,,=2a,Txy=rx=0
6 一、应力函数取一次多项式 §3-1 多项式解答 = a + bx + cy 应力分量： x = 0, y = 0, xy = yx = 0 应力边界条件： X = Y = 0 结论：（1）线性应力函数对应于无面力、无应力的状态。 （2）把任何平面问题的应力函数加上一个线性函 数，并不影响应力。 二、应力函数取二次多项式 2 2 = ax +bxy+ cy 1.对应于 ，应力分量 。 2 = ax x = 0, y = 2a, xy = yx = 0
Conclusion: 1. The stress function= ax 2 can be used to solute the problem of uniformly distributed tensile(if a>0)or compressive(if a<O) stresses on y-axis of rectangular plate. Fig3-la) 2aliltil x 2a b 2c 2c y a) (b) (c) Fig3-1 2. Correspondingto =bxy,stress components 0.6.=0.x= Conclusion: The stress function =bxy can used to solute the problemof uniformly distributed shearing stresses rectangular plate (Fig3-1b) 7
7 2.Corresponding to ,stress components = bxy x = 0, y = 0, xy = yx = −b Conclusion: The stress function can used to solute the problem of uniformly distributed shearing stresses rectangular plate.(Fig3-1b) = bxy Fig.3-1 x y o 2a 2a （a） x y o b b b b （b） x y o 2c 2c （c） Conclusion：1.The stress function can be used to solute the problem of uniformly distributed tensile(if ) or compressive(if ) stresses on y-axis of rectangular plate.(Fig3-1a) a 0 a 0 j = ax 2
平冒的立么解答 结 论:应力函数q=ax2能解决矩形板在y方向受均布拉力 (设a>0)或均布压力(设a<0)的问题。如图3-1a)。 2 b O 2a b 2c 2c y (a) (b) (c) 图3-1 2对应于=bxy,应力分量ax=0.,=0,xy=x=-b 结论:应力函数=bxy能解决矩形板受均布剪力问题。如 图3-1(b) 8
8 2 结论：应力函数 = ax 能解决矩形板在 方向受均布拉力 （设 ）或均布压力（设 ）的问题。如图3-1（a）。 y a 0 a 0 x y o b b b b x y o 2a 2a x y o 2c 2c 2.对应于 = bxy ，应力分量 x = 0, y = 0, xy = yx = −b 。 结论：应力函数 能解决矩形板受均布剪力问题。如 图3-1（b）。 = bxy 图3-1 （a） （b） （c）
3. The stress function =cy- can be used to solute the problem of uniformly distributed tensile(if c0)stresses on -axis of rectangular plate. Fig3-Ic) 3. The Stress Function in the form of a polynomial of the Third degree 9=a13边 The Corresponding Stress Components o=6a], 0 =0,T=T=0 (a) Conclusion: The stress function (a)can be used to solute the problem of pure bending of rectangular beam. The rectangular beam is shown in Fig3-2 h x图 y Fig 3-2 9
9 3.The Stress Function in the form of a Polynomial of the Third Degree 3 = ay The Corresponding Stress Components Conclusion：The stress function（a) can be used to solute the problem of pure bending of rectangular beam. The rectangular beam is shown in Fig3-2. x = 6ay, y = 0, xy = yx = 0 （a） − + M M h l 2 h 2 y h x x 图 x y Fig.3-2 1 3.The stress function can be used to solute the problem of uniformly distributed tensile(if ) or compressive(if ) stresses on -axis of rectangular plate.(Fig3-1c) 2 = cy c 0 c 0
平冒的立么解答 3.应力函数￠=qy能解决矩形板在x方向受均布拉力 (设c>0)或均布压力(设c<0)的问题。如图3-1(c)。 三、应力函数取三次式 p=ay 对应的应力分量:x=6ay,n=0,xm=m=0(a) 结论:应力函数=qy3能解决矩形梁受纯弯曲的问题。如图 3-2所示的矩形梁。 h x图 图3-2 10
10 3.应力函数 能解决矩形板在 x 方向受均布拉力 （设 ）或均布压力（设 ）的问题。如图3-1（c）。 2 = cy c 0 c 0 三、应力函数取三次式 3 = ay 对应的应力分量： x = 6ay, y = 0, xy = yx = 0 （a） − + M M h l 2 h 2 y h x x 图 x y 图3-2 1 结论：应力函数 能解决矩形梁受纯弯曲的问题。如图 3-2所示的矩形梁。 3 = ay