CHAPTER 1 UNSYMMETRICAL BENDING Summary The second moments of area of a section are given by I=ydA and Iy=dA The product second moment of area of a section is defined as ln =xydA which reduces to Ixy =Ahk for a rectangle of area A and centroid distance h and k from the X and Y axes. The principal second moments of area are the maximum and minimum values for a section and they occur about the principal axes.Product second moments of area about principal axes are zero. With a knowledge of I,Iyy and Ixy for a given section,the principal values may be determined using either Mohr's or Land's circle construction. The following relationships apply between the second moments of area about different axes: Iu=(lx +Iyy)+(Ixx -Iyy)sec 20 =(Ixx+lyy)(lx-Iyy)sec20 where 6 is the angle between the U and X axes,and is given by 2Ixy tan29=1y-1x） Then Iu+Iv=Ia+lyy The second moment of area about the neutral axis is given by IN.A.=(u+)+(u-I)cos2au where o is the angle between the neutral axis(N.A.)and the U axis. Also Ix Iu Cos20+I sin20 Iyy =Iv cos20+Iu sin20 1xw=l。-lu)sin29 Ix-Iyy=(Iu-I)cos 20 1
CHAPTER 1 UNSYMMETRICAL BENDING Summary The second moments of area of a section are given by I, = 1 y2 dA and I,, = 1 x2 dA The product second moment of area of a section is defined as I,, = xydA which reduces to I,, = Ahk for a rectangle of area A and centroid distance h and k from the X and Y axes. The principal second moments of area are the maximum and minimum values for a section and they occur about the principal axes. Product second moments of area about principal axes are zero. With a knowledge of I,, I,, and I,, for a given section, the principal values may be determined using either Mohr’s or Land’s circle construction. The following relationships apply between the second moments of area about different axes: s I, = ;(I,, +I,,) + ;(I= - 1,,)sec28 I, = ;(I,, + I,,) - ;(I= - I,,)sec20 where 0 is the angle between the U and X axes, and is given by Then I, + I, = I.r, + I,, The second moment of area about the neutral axis is given by IN.^,. = ;(I, + I,) + 4 (I, - I,) COS 2a, where u, is the angle between the neutral axis (N.A.) and the U axis. Also I, = I, cos2 8 + I, sin2 8 I,, = I, cos2 8 + I, sin2 0 I,, = ;(I~ - 1,)sin20 I, - I,, = (I, - I,>) cos 28 1
2 Mechanics of Materials 2 Stress determination For skew loading and other forms of bending about principal axes a=Mi Mou where M and M,are the components of the applied moment about the U and V axes. Alternatively,with a Px+Oy Mxt PIxy +Qlx Myy=-Plyy -Qlxy Then the inclination of the N.A.to the X axis is given by P tana = 0 As a further alternative, M'n 0三 IN.A. where M'is the component of the applied moment about the N.A.,IN.A.is determined either from the momental ellipse or from the Mohr or Land constructions,and n is the perpendicular distance from the point in question to the N.A. Deflections of unsymmetrical members are found by applying standard deflection formulae to bending about either the principal axes or the N.A.taking care to use the correct component of load and the correct second moment of area value. Introduction It has been shown in Chapter 4 of Mechanics of Materials I that the simple bending theory applies when bending takes place about an axis which is perpendicular to a plane of symmetry.If such an axis is drawn through the centroid of a section,and another mutually perpendicular to it also through the centroid,then these axes are principal axes.Thus a plane of symmetry is automatically a principal axis.Second moments of area of a cross-section about its principal axes are found to be maximum and minimum values,while the product second moment of area,fxy dA,is found to be zero.All plane sections,whether they have an axis of symmetry or not,have two perpendicular axes about which the product second moment of area is zero.Principal axes are thus defined as the axes about which the product second moment of area is zero.Simple bending can then be taken as bending which takes place about a principal axis,moments being applied in a plane parallel to one such axis. In general,however,moments are applied about a convenient axis in the cross-section; the plane containing the applied moment may not then be parallel to a principal axis.Such cases are termed "unsymmetrical"or"asymmetrical"bending. The most simple type of unsymmetrical bending problem is that of"skew"loading of sections containing at least one axis of symmetry,as in Fig.1.1.This axis and the axis EJ.Hearn,Mechanics of Materials 1.Butterworth-Heinemann.1997
2 Mechanics of Materials 2 Stress determination For skew loading and other forms of bending about principal axes M,v M,u c=-+- 1, 1, where Mu and M, are the components of the applied moment about the U and V axes. Alternatively, with 0 = Px + Qy M, = PI,, + QIM Myy = -Plyy - QIxy Then the inclination of the N.A. to the X axis is given by P tana! = -- Q As a further alternative, M’n 1N.A. o=- where M’ is the component of the applied moment about the N.A., IN.A. is determined either from the momenta1 ellipse or from the Mohr or Land constructions, and n is the perpendicular distance from the point in question to the N.A. Deflections of unsymmetrical members are found by applying standard deflection formulae to bending about either the principal axes or the N.A. taking care to use the correct component of load and the correct second moment of area value. Introduction It has been shown in Chapter 4 of Mechanics of Materials 1 that the simple bending theory applies when bending takes place about an axis which is perpendicular to a plane of symmetry. If such an axis is drawn through the centroid of a section, and another mutually perpendicular to it also through the centroid, then these axes are principal axes. Thus a plane of symmetry is automatically a principal axis. Second moments of area of a cross-section about its principal axes are found to be maximum and minimum values, while the product second moment of area, JxydA, is found to be zero. All plane sections, whether they have an axis of symmetry or not, have two perpendicular axes about which the product second moment of area is zero. Principal axes are thus de$ned as the axes about which the product second moment of area is Zero. Simple bending can then be taken as bending which takes place about a principal axis, moments being applied in a plane parallel to one such axis. In general, however, moments are applied about a convenient axis in the cross-section; the plane containing the applied moment may not then be parallel to a principal axis. Such cases are termed “unsymmetrical” or “asymmetrical” bending. The most simple type of unsymmetrical bending problem is that of “skew” loading of sections containing at least one axis of symmetry, as in Fig. 1.1. This axis and the axis EJ. Hearn, Mechanics of Murerids I, Buttenvorth-Heinemann, 1997
§1.1 Unsymmetrical Bending 3 (a)Rectangular (b)I-section (c)Channei (d)T-section section section Fig.1.1.Skew loading of sections containing one axis of symmetry. perpendicular to it are then principal axes and the term skew loading implies load applied at some angle to these principal axes.The method of solution in this case is to resolve the applied moment MA about some axis A into its components about the principal axes. Bending is then assumed to take place simultaneously about the two principal axes,the total stress being given by Mv Mou 0= With at least one of the principal axes being an axis of symmetry the second moments of area about the principal axes I and I can easily be determined. With unsymmetrical sections (e.g.angle-sections,Z-sections,etc.)the principal axes are not easily recognized and the second moments of area about the principal axes are not easily found except by the use of special techniques to be introduced in $s1.3 and 1.4.In such cases an easier solution is obtained as will be shown in $1.8.Before proceeding with the various methods of solution of unsymmetrical bending problems,however,it is advisable to consider in some detail the concept of principal and product second moments of area. 1.1.Product second moment of area Consider a small element of area in a plane surface with a centroid having coordinates (x,y)relative to the X and Y axes (Fig.1.2).The second moments of area of the surface about the X and Y axes are defined as Ie=y'dA and In =xda (1.1) Similarly,the product second moment of area of the section is defined as follows: (1.2) Since the cross-section of most structural members used in bending applications consists of a combination of rectangles the value of the product second moment of area for such sections is determined by the addition of the /x value for each rectangle(Fig.1.3), ie. Iry Ahk (1.3)
$1.1 Unsymmetrical Bending 3 V V V (c) Rectangular (b) I-sectam (c) Channel (d) T-sectton section section Fig. 1 .I. Skew loading of sections containing one axis of symmetry. perpendicular to it are then principal axes and the term skew loading implies load applied at some angle to these principal axes. The method of solution in this case is to resolve the applied moment MA about some axis A into its components about the principal axes. Bending is then assumed to take place simultaneously about the two principal axes, the total stress being given by M,v M,u a=-+- 1, I, With at least one of the principal axes being an axis of symmetry the second moments of area about the principal axes I, and I, can easily be determined. With unsymmetrical sections (e.g. angle-sections, Z-sections, etc.) the principal axes are not easily recognized and the second moments of area about the principal axes are not easily found except by the use of special techniques to be introduced in $3 1.3 and 1.4. In such cases an easier solution is obtained as will be shown in 51.8. Before proceeding with the various methods of solution of unsymmetrical bending problems, however, it is advisable to consider in some detail the concept of principal and product second moments of area. 1.1. Product second moment of area Consider a small element of area in a plane surface with a centroid having coordinates (x, y) relative to the X and Y axes (Fig. 1.2). The second moments of area of the surface about the X and Y axes are defined as zXx = Jy’d~ and zYy = /x’& (1.1) Similarly, the product second moment of area of the section is defined as follows: zXy = Jxy (1.2) Since the cross-section of most structural members used in bending applications consists of a combination of rectangles the value of the product second moment of area for such sections is determined by the addition of the I,, value for each rectangle (Fig. 1.3), i.e. Zxy = Ahk (1.3)
4 Mechanics of Materials 2 12 Fig.1.2. where h and k are the distances of the centroid of each rectangle from the X and Y axes respectively(taking account of the normal sign convention for x and y)and A is the area of the rectangle. k- h+ h+ X k- k+ h- h- X Fig.1.3. 1.2.Principal second moments of area The principal axes of a section have been defined in the introduction to this chapter. Second moments of area about these axes are then termed principal values and these may be related to the standard values about the conventional X and Y axes as follows. Consider Fig.1.4 in which GX and GY are any two mutually perpendicular axes inclined at to the principal axes GV and GU.A small element of area A will then have coordinates (u,v)to the principal axes and (x,y)referred to the axes GX and GY.The area will thus have a product second moment of area about the principal axes given by uudA. .'total product second moment of area of a cross-section uvdA (x cos+y sin)(y cos-x sin)dA
4 Mechanics of Materials 2 51.2 Y t Fig. 1.2. where h and k are the distances of the centroid of each rectangle from the X and Y axes respectively (taking account of the normal sign convention for x and y) and A is the area of the rectangle. k- kt h- I hFig. 1.3. 1.2. Principal second moments of area The principal axes of a section have been defined in the introduction to this chapter. Second moments of area about these axes are then termed principal values and these may be related to the standard values about the conventional X and Y axes as follows. Consider Fig. 1.4 in which GX and GY are any two mutually perpendicular axes inclined at 8 to the principal axes GV and GU. A small element of area A will then have coordinates (u, v) to the principal axes and (x, y) referred to the axes GX and GY. The area will thus have a product second moment of area about the principal axes given by uvdA. :. total product second moment of area of a cross-section I,, = /"uvdA = S(xcosO+ysin8)(ycos8-xsine)~A
§1.2 Unsymmetrical Bending 5 (ycosysincoscossin-xysin0dA =aosg-sm2o∫yda+sn6ecos[/ah-/ran Ixy cos 20+(lx-Iyy)sin 20 dA Principal axis Fig.14. Now for principal axes the product second moment of area is zero. 0=Ixy cos20+(lxx-Iyy)sin 20 tan 20= -2= 2Ixy (Lx -Iyy)(Iyy -Ixx) (1.4) This equation,therefore,gives the direction of the principal axes. To determine the second moments of area about these axes, 1.da=(ycos0-xsina da =cos0ydA+sin20dA-2cossin xydA =Lxx cos20+lyy sin20-Ixy sin 26 (1.5) Substituting for Ixy from eqn.(1.4), 1.=1+cos201w+1-cos291n-1c0s200n-1a) =0+m2a+0-om20y[A62un-1a =(1 cos20)xx+(1-cos 20)Iyy-i sec 20(lyy -Ixx)+cos 20(lys -Ixx) =(l+Iyy)+(Ixx -Iyy)cos 20-(Iyy-Lx)sec 20+(Iyy -Ixx)cos 20
91.2 Unsymmetrical Bending 5 = /(x y cos2 8 + y2 sin 8 cos 8 - x2 cos 8 sin 8 - xy sin2 8) dA = (cos2 8 - sin2 8) /xy dA + sin 8 cos 8 [/” y2 dA - /x2 dA] Y Principal axis Fig. 1.4. Now for principal axes the product second moment of area is zero. .. o = I,, COS 28 + 4 (I, - zYy) sin 28 This equation, therefore, gives the direction of the principal axes. To determine the second moments of area about these axes, I, = v2 dA = (y cos 8 - x sin dA ss = cos2 8 y2 dA + sin2 8 /x2 dA - 2cos8 sin 8 xydA I = I, cos2 8 + I,, sin2 8 - I,,~ sin 28 Substituting for I,, from eqn. (1.4), sin228 I,= ;(1+cos28)Ixx+;(1-cos28)z,,--- 2 cos28 (I,r - 1,) (1.4) = ;(I + cos 2011, + ; (1 - cos 28)1,, - sec 213(1,, - I,) + ; cos 20(1,,. .. - I,) I - (I, + I,,) + (I, - I,,) cos 28 - (I,, - I,) sec 28 + (Ivv - I,) cos 213