【文章內容簡介】 in (b). Taking moments about an axis through the cross section where the beam is cut [ (b)] gives M=RaL/2PL/4=PL/8Mo/2Depending upon the relative magnitudes of the terms in this equation, we see that the bending moment M may be either positive or negative.To obtain the stress resultants at a cross section just to the right of the middle, we cut the beam at that section and again draw an appropriate freebody diagram [(c)]. The only difference between this diagram and the former one is that the couple Mo now acts on the part of the beam to the left of the cut section. Again summing forces in the vertical direction, and also taking moments about an axis through the cut section, we obtain V=P/4Mo/L M=PL/8+Mo/2We see from these results that the shear force does not change when the cut section is shifted from left to right of the couple Mo, but the bending moment increases algebraically by an amount equal to Mo. (Selected from:Stephen P. Timosheko and James M. Gere, Mechanics of Materials,Van NostrandReinhold Company Ltd. , 1978. )材料2梁所受剪力和彎矩現(xiàn)在我們考慮，（a）,懸臂梁的自由端受到斜向載荷P。如果我們從截面mn將梁截開，（b）所示，我們可以發(fā)現(xiàn)被移去一部分的梁（即右邊部分）在維持左邊部分的平衡上是必須的。在截面mn上的力的分布在我們學習的現(xiàn)階段不可知，但是我們知道應力的合力必須是與載荷P平衡。當帶入一個沿軸向過截面軸心的力N和一個與截面平行的剪切力V，一個作用與橫梁的彎矩。截面上的軸向力，剪切力和彎矩都是應力的結果。對于靜力平衡的橫梁，我們能夠通過靜力平衡方程知道合應力。，從第二部分圖示的自由圖解中我們可以列出3個靜力平衡方程。水平方向和垂直方向的合力，各自為： N=P cosβ V=P sinβ從通過截面mn軸心軸向的合力，我們得：M=Pxsinβ。x是自由端與截面mn的距離。因此，通過利用自由體圖示和靜力平衡方程，我們很容易的可以求出合應力。橫梁的軸向應力N單獨作用的情況我們已經(jīng)在第2單元討論過了。現(xiàn)在我們將要看到如何聯(lián)合彎矩M和剪切力V共同求得。(b)所示，應力N，v和M需假定正方向。這樣一般性假定只在討論橫梁左邊部分時才有用。如果把右邊部分加以考慮的話，我們將得到同樣大小但是方向相反的力。因此，我們必須確定數(shù)學標定不是依靠其空間的方向，例如向左或者向右，而是依靠力作用其材料上產(chǎn)生的反作用力的方向。為了說明這種情況，對于N。我們可以看到，正向軸力方向遠離截面表面表現(xiàn)為拉伸，正向剪切力沿順時針方向，正彎矩壓緊橫梁的上部分。簡單橫梁AB受到兩個載荷的作用，集中力P和力偶Mo，（a）所示。我們發(fā)現(xiàn)橫梁的剪切力和彎矩位于斷面如下：（a）距離橫梁的左端1/4個橫梁的距離。（b）距離中右端一端距離。分析橫梁的第一步是找出反作用力RA和RB。對于A，B兩端我們列出兩個平衡方程，從中我們發(fā)現(xiàn)：RA=3P/4Mo/L RB=p/4+Mo/L 然后，從中間將橫梁截斷成兩個如圖所示的自由體結構，我們以左半部分為例，（b）所示。力P和反作用力RA，還有我們先假定正向的剪切力V和彎矩M都在圖中畫出。力偶Mo沒有出現(xiàn)，因為它應用到了被分開的橫梁的左邊部分去了。垂直方向上的合力： V=RAP=P/4Mo/L 在垂直方向上的合力是負數(shù)，（b）所假設的方向相反。（b）給出的沿截面軸向的合力： M=LRA/2PL/4=PL/8Mo/2 根據(jù)方程中的大小關系，我們可以發(fā)現(xiàn)彎矩可能既不正向也不反向。為了獲得作用在右邊中間截面上合力，我們將橫梁沿截面砍斷，（c）。這份圖表和和之前的的差別在，現(xiàn)在力偶Mo作用于橫梁的左截面。我們再2假設垂直方向的力和通過截面軸心的力，我們獲得：V=P/4Mo/L M = PL/8+Mo/2 從結果分析，有以下結論：力矩Mo在梁上左右移動時，剪切力并沒有改變，但彎矩和Mo成線性比例關系。(選自Stephen P. Timosheko and James M. Gere，材料力學，范諾斯特蘭德萊茵霍爾德股份有限公司，1978)Reading Material 3Theories of Strength Principal StressesThe state of stress at a point in a structural member under a plex system of loading is described by the magnitude and direction of the principal stresses. The principal stresses are the maximum values of the normal stresses at the point。 which act on planes on which the shear stress is zero. In a twodimensional stress system,the principal stresses at any point are related to the normal stresses the x and y directions σx and σy and the shear stress, xy at the point by the following equation:Principal stresses, ()The maximum shear stress at the point is equal to half the algebraic difference between the principal stresses: Maximum shear stress , τmax=（σ1—σ2） ()Compressive stresses are conventionally taken as negative。 tensile as positive. Classification of Pressure vesselsFor the purposes of design and analysis, pressure vessels, pressure vessels are subdivided into two classes depending on the ratio of the wall thickness to vessel diameter: thinwalled vessels, with a thickness ratio of less than 1/10,and thickwalled above this ratio. The principal stresses acting at a point in the wall of a vessel, due to a pressure load, are shown in the wall is thin, the radial stressσ3 will be small and can be neglected in parison with the other stresses, and the longitudinal and circumferential stressesσ1 and σ2 can be taken as constant over the wall thickness. In a thick wall, the magnitude of the radial stress will be significant, and the circumferential stress will vary across the wall. The majority of the vessels used in the chemical and allied industries are classified as thinwalled vessels. Thickwalled vessels are used for high pressures. Twodimensional stress system Principal stress in pressurevessel wall Allowable StressIn the first two sections of this unit equation were developed for finding the normal stress and average shear in a structural member. These equations can also be used to select the size of a member if the member’s strength is known. The strength of a material can be defined in several ways, depending on the material and the environment in which it is to be used. One definition is the ultimate strength or the stress. Ultimate strength is the stress at which a material will rupture when subjected to a purely axial load. This property is determined from a tensile test of the material. This is a laboratory test of an accurately prepared specimen which usually is conducted on a universal testing machine. The load is applied slowly and is continuously monitored. The ultimate stress or strength is the maximum load divided by the original crosssectional area. The ultimate strength for most engineering materials has been accurately determined and is readily available. If a member is loaded beyond its ultimate strength it will fail—rupture. In most engineering structures it is desirable that the structure not fail. Thus design is based on some lower value called allowable stress or design stress. If, for example, a certain steel is known to have an ultimate strength of 110000psi,a lower allowable stress would be used for design, say allowable stress would allow only half the load the ultimate strength would allow. The ratio of the ultimate strength to the allowable stress is known as the factor of safety: Factor of safety=ultimate strength/allowable stress or n=Su/SA () We use S for strength or allowable stress andσfor the actual stress in a material. In a design: