CHAPTER 4 BENDING Summary The simple theory of elastic bending states that M G E T-y-R where M is the applied bending moment(B.M.)at a transverse section,I is the second moment of area of the beam cross-section about the neutral axis(N.A.),o is the stress at distance y from the N.A.of the beam cross-section,E is the Young's modulus of elasticity for the beam material,and R is the radius of curvature of the N.A.at the section. Certain assumptions and conditions must obtain before this theory can strictly be applied: see page 64. In some applications the following relationship is useful: M=Zo max where Z=I/ymaxand is termed the section modulus;omaxis then the stress at the maximum distance from the N.A. The most useful standard values of the second moment of area for certain sections are as follows(Fig.4.1片 bd3 rectangle about axis through centroid 2=14 b rectangle about axis through side 3=1xx circle about axis throun 64 =INA NA NA X Fig.4.1. 62
CHAPTER 4 BENDING Summary The simple theory of elastic bending states that MaE _- _- _- IYR where M is the applied bending moment (B.M.) at a transverse section, I is the second moment of area of the beam cross-section about the neutral axis (N.A.), 0 is the stress at distance y from the N.A. of the beam cross-section, E is the Young’s modulus of elasticity for the beam material, and R is the radius of curvature of the N.A. at the section. Certain assumptions and conditions must obtain before this theory can strictly be applied: see page 64. In some applications the following relationship is useful: M = Zomax where Z = Z/y,,,and is termed the section modulus; amaxis then the stress at the maximum distance from the N.A. The most useful standard values of the second moment of area I for certain sections are as follows (Fig. 4.1): bd3 12 rectangle about axis through centroid = ~ = ZN,A, bd3 3 nD4 64 rectangle about axis through side = __ = I,, circle about axis through centroid = - = ZN,A, Fig. 4.1. 62
Bending 63 The centroid is the centre of area of the section through which the N.A.,or axis of zero stress, is always found to pass. In some cases it is convenient to determine the second moment of area about an axis other than the N.A.and then to use the parallel axis theorem. INA=IG+Ah2 For composite beams one material is replaced by an equivalent width of the other material given by =El where E/E'is termed the modular ratio.The relationship between the stress in the material and its equivalent area is then given by E a-Era' For skew loading of symmetrical sections the stress at any point (x,y)is given by Mxx y土 the angle of the N.A.being given by Myy Ixx tanf=±MxIy For eccentric loading on one axis, P Pey 0= the N.A.being positioned at a distance yN=士AE from the axis about which the eccentricity is measured. For eccentric loading on two axes, P Ph Pk 。=±，x y For concrete or masonry rectangular or circular section columns,the load must be retained within the middle third or middle quarter areas respectively. Introduction If a piece of rubber,most conveniently of rectangular cross-section,is bent between one's fingers it is readily apparent that one surface of the rubber is stretched,i.e.put into tension, and the opposite surface is compressed.The effect is clarified if,before bending,a regular set of lines is drawn or scribed on each surface at a uniform spacing and perpendicular to the axis
Bending 63 The centroid is the centre of area of the section through which the N.A., or axis of zero stress, is always found to pass. In some cases it is convenient to determine the second moment of area about an axis other than the N.A. and then to use the parallel axis theorem. IN,*. = I,+AhZ For composite beams one material is replaced by an equivalent width of the other material given by where EIE is termed the modular ratio. The relationship between the stress in the material and its equivalent area is then given by E E 0 =yo‘ For skew loading of symmetrical sections the stress at any point (x, y) is given by o=Iy+lX Mxx Myy xx YY the angle of the N.A. being given by Myy Ixx Mxx I,, tan6 = f- - For eccentric loading on one axis, P Pey (J=-+- A- I the N.A. being positioned at a distance I Ae y,= ffrom the axis about which the eccentricity is measured. For eccentric loading on two axes, P Ph Pk a=-+-Xf-y A -I,, Ixx For concrete or masonry rectangular or circular section columns, the load must be retained within the middle third or middle quarter areas respectively. Introduction If a piece of rubber, most conveniently of rectangular cross-section, is bent between one’s fingers it is readily apparent that one surface of the rubber is stretched, i.e. put into tension, and the opposite surface is compressed. The effect is clarified if, before bending, a regular set of lines is drawn or scribed on each surface at a uniform spacing and perpendicular to the axis
64 Mechanics of Materials §4.1 of the rubber which is held between the fingers.After bending,the spacing between the set of lines on one surface is clearly seen to increase and on the other surface to reduce.The thinner the rubber,i.e.the closer the two marked faces,the smaller is the effect for the same applied moment.The change in spacing of the lines on each surface is a measure of the strain and hence the stress to which the surface is subjected and it is convenient to obtain a formula relating the stress in the surface to the value of the B.M.applied and the amount of curvature produced.In order for this to be achieved it is necessary to make certain simplifying assumptions,and for this reason the theory introduced below is often termed the simple theory of bending.The assumptions are as follows: (1)The beam is initially straight and unstressed. (2)The material of the beam is perfectly homogeneous and isotropic,i.e.of the same density and elastic properties throughout. (3)The elastic limit is nowhere exceeded. (4)Young's modulus for the material is the same in tension and compression. (5)Plane cross-sections remain plane before and after bending. (6)Every cross-section of the beam is symmetrical about the plane of bending,i.e.about an axis perpendicular to the N.A. (7)There is no resultant force perpendicular to any cross-section. 4.1.Simple bending theory If we now consider a beam initially unstressed and subjected to a constant B.M.along its length,i.e.pure bending,as would be obtained by applying equal couples at each end,it will bend to a radius R as shown in Fig.4.2b.As a result of this bending the top fibres of the beam will be subjected to tension and the bottom to compression.It is reasonable to suppose, therefore,that somewhere between the two there are points at which the stress is zero.The locus of all such points is termed the neutral axis.The radius of curvature R is then measured to this axis.For symmetrical sections the N.A.is the axis of symmetry,but whatever the section the N.A.will always pass through the centre of area or centroid. (0】 (b) Fig.4.2.Beam subjected to pure bending (a)before,and (b)after,the moment M has been applied. Consider now two cross-sections of a beam,HE and GF,originally parallel (Fig.4.2a). When the beam is bent (Fig.4.2b)it is assumed that these sections remain plane;i.e.H'E'and G'F',the final positions of the sections,are still straight lines.They will then subtend some angle 0
64 Mechanics of Materials $4. I of the rubber which is held between the fingers. After bending, the spacing between the set of lines on one surface is clearly seen to increase and on the other surface to reduce. The thinner the rubber, i.e. the closer the two marked faces, the smaller is the effect for the same applied moment. The change in spacing of the lines on each surface is a measure of the strain and hence the stress to which the surface is subjected and it is convenient to obtain a formula relating the stress in the surface to the value of the B.M. applied and the amount of curvature produced. In order for this to be achieved it is necessary to make certain simplifying assumptions, and for this reason the theory introduced below is often termed the simple theory of bending. The assumptions are as follows: (1) The beam is initially straight and unstressed. (2) The material of the beam is perfectly homogeneous and isotropic, i.e. of the same density and elastic properties throughout. (3) The elastic limit is nowhere exceeded. (4) Young's modulus for the material is the same in tension and compression. (5) Plane cross-sections remain plane before and after bending. (6) Every cross-section of the beam is symmetrical about the plane of bending, i.e. about an (7) There is no resultant force perpendicular to any cross-section. axis perpendicular to the N.A. 4.1. Simple bending theory If we now consider a beam initially unstressed and subjected to a constant B.M. along its length, i.e. pure bending, as would be obtained by applying equal couples at each end, it will bend to a radius R as shown in Fig. 4.2b. As a result of this bending the top fibres of the beam will be subjected to tension and the bottom to compression. It is reasonable to suppose, therefore, that somewhere between the two there are points at which the stress is zero. The locus of all such points is termed the neutral axis. The radius of curvature R is then measured to this axis. For symmetrical sections the N.A. is the axis of symmetry, but whatever the section the N.A. will always pass through the centre of area or centroid. Fig. 4.2. Beam subjected to pure bending (a) before, and (b) after, the moment M has been applied. Consider now two cross-sections of a beam, HE and GF, originally parallel (Fig. 423). When the beam is bent (Fig. 4.2b) it is assumed that these sections remain plane; i.e. HE and GF', the final positions of the sections, are still straight lines. They will then subtend some angle 0
§4.1 Bending 65 Consider now some fibre AB in the material,distance y from the N.A.When the beam is bent this will stretch to A'B'. Strain in fibre AB=_ extension A'B'-AB original length AB But AB=CD,and,since the N.A.is unstressed,CD C'D'. A'B'-C'D'(R+y)0-R0 y strain = CD' Re R stress But Young's modulus E strain strain=E Equating the two equations for strain, G y E-R G E or y R (4.1) Consider now a cross-section of the beam (Fig.4.3).From eqn.(4.1)the stress on a fibre at distance y from the N.A.is E 0= NA Fig.4.3.Beam cross-section. If the strip is of area A the force on the strip is E F=08A-RY6A This has a moment about the N.A.of E Fy= 6A
54.1 Bending 65 Consider now some fibre AB in the material, distance y from the N.A. When the beam is bent this will stretch to A’B’. A’B’ - AB - extension . original length AB Strain in fibre AB = But AB = CD, and, since the N.A. is unstressed, CD = C‘D’. A‘B‘ - C‘D‘ (R + y)8 - RB y C’D‘ RB R strain = - -- But stress strain -- - Young’s modulus E 0 strain = - E .. Equating the two equations for strain, or Consider now a cross-section of the beam (Fig. 4.3). From eqn. (4.1) the stress on a fibre at E R distance y from the N.A. is 0=-y Fig. 4.3. Beam cross-section. If the strip is of area 6A the force on the strip is E R F = 06A = -y6A This has a moment about the N.A. of
66 Mechanics of Materials §4.2 The total moment for the whole cross-section is therefore M=发A E EEyOA since E and R are assumed constant. The term Ey26A is called the second moment of area of the cross-section and given the symbol I. E ME -RI and T-R M= (4.2) Combining eqns.(4.1)and(4.2)we have the bending theory equation M G E T-)-R (4.3) From eqn.(4.2)it will be seen that if the beam is of uniform section,the material of the beam is homogeneous and the applied moment is constant,the values of I,E and M remain constant and hence the radius of curvature of the bent beam will also be constant.Thus for pure bending of uniform sections,beams will deflect into circular arcs and for this reason the term circular bending is often used.From eqn.(4.2)the radius of curvature to which any beam is bent by an applied moment M is given by: EI R= M and is thus directly related to the value of the quantity EI.Since the radius of curvature is a direct indication of the degree of flexibility of the beam(the larger the value of R,the smaller the deflection and the greater the rigidity)the quantity El is often termed the flexural rigidity or flexural stiffness of the beam.The relative stiffnesses of beam sections can then easily be compared by their EI values. It should be observed here that the above proof has involved the assumption of pure bending without any shear being present.From the work of the previous chapter it is clear that in most practical beam loading cases shear and bending occur together at most points. Inspection of the S.F.and B.M.diagrams,however,shows that when the B.M.is a maximum the S.F.is,in fact,always zero.It will be shown later that bending produces by far the greatest magnitude of stress in all but a small minority of special loading cases so that beams designed on the basis of the maximum B.M.using the simple bending theory are generally more than adequate in strength at other points. 4.2.Neutral axis As stated above,it is clear that if,in bending,one surface of the beam is subjected to tension and the opposite surface to compression there must be a region within the beam cross-section at which the stress changes sign,i.e.where the stress is zero,and this is termed the neutral axis
66 Mechanics of Materials $4.2 The total moment for the whole cross-section is therefore E R = - Z y26A since E and R are assumed constant. symbol I. The term Cy26A is called the second moment of area of the cross-section and given the (4.2) Combining eqns. (4.1) and (4.2) we have the bending theory equation MaE IYR ---=- - (4.3) From eqn. (4.2) it will be seen that if the beam is of uniform section, the material of the beam is homogeneous and the applied moment is constant, the values of I, E and M remain constant and hence the radius of curvature of the bent beam will also be constant. Thus for pure bending of uniform sections, beams will deflect into circular arcs and for this reason the term circular bending is often used. From eqn. (4.2) the radius of curvature to which any beam is bent by an applied moment M is given by: and is thus directly related to the value of the quantity El. Since the radius of curvature is a direct indication of the degree of flexibility of the beam (the larger the value of R, the smaller the deflection and the greater the rigidity) the quantity El is often termed the jexural rigidity or flexural stiflness of the beam. The relative stiffnesses of beam sections can then easily be compared by their El values. It should be observed here that the above proof has involved the assumption of pure bending without any shear being present. From the work of the previous chapter it is clear that in most practical beam loading cases shear and bending occur together at most points. Inspection of the S.F. and B.M. diagrams, however, shows that when the B.M. is a maximum the S.F. is, in fact, always zero. It will be shown later that bending produces by far the greatest magnitude of stress in all but a small minority of special loading cases so that beams designed on the basis of the maximum B.M. using the simple bending theory are generally more than adequate in strength at other points. 4.2. Neutral axis As stated above, it is clear that if, in bending, one surface of the beam is subjected to tension and the opposite surface to compression there must be a region within the beamcross-section at which the stress changes sign, i.e. where the stress is zero, and this is termed the neutral axis