ashby_data_rotated - WW 2 MOMENTS 0F AREA OF SECTIONS A...

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Unformatted text preview: WW _ ' _ ' _ 2. MOMENTS 0F AREA OF SECTIONS A beam of unifima section, loaded in simple tension by a force F. carries a stress SECTION Mmz) I (m‘) K(m‘) by,“ (m3) H(m3) _F arr—A where A is the area of the section. Its response is calculated from the appropriate constitutive equation. Here the important characteristic of the section is its area, A. For other modes of-loading, higher moments ofthe-area are involved. Those for various common sections are given on the facing page. They are defined as follows. The second moment i measures the resistance of the section to bending about a horizontal axis (shown as a broken line). It is __ I ' I= ~Im 3! law) dy where 3r is measured vertically and fly) is the rigidth of the section at y. The moment K measures the resistance ofthe section to twisting. It' is equal to the polar moment J for circular sections, where J z where r is measured radially from the centre-of the circular section. For non- circular sections K is less than J. The moment I/ym (where Jim is the normal distance from' the neutral axis of bending'to the outer surface of the beam) measures the surface stress generated by a given bending moment, M: __iln. Finally. the moment H, defined by = jsection YB 11(3)?) dy measures the resistance of the beam to fully-plastic bending. The fully plastic moment for a beam in bending is M? i Hay ' .. _ . . ., 1-213“:th um. I Autonomi- Thin or slender shapes may buckle before they yield or fracture. It is this which sets a practical limit to the thinness of tube walls and web's. WW3 When a beam is loaded by a force F or moments M, the initially straight axis is deformed into a curve. If the beam is uniform in section and properties, long in relation to its depth and nowhere stressed beyond the elastic limit, the deflection 8, and the angle of rotation, 9 , can be calculated using elastic beam theory (see Further Reading in Section 16). The basic differential equation describing the curvature of the beam at a point 1: along its length is' where y is the lateral deflection, and M is the bending moment at the point x on the beam. E is Young's modulus and I is the second moment of area (Section 2). This can be written M- 3.3. T'E[R so] where Re is the radius of curvature before applying the moment and r the radius after it is applied. Deflections 6 and rotations 0 are found by integrating these equations along the beam. Equations for the deflection, 5, and end slope, B , of beams, for various common modes of loading are shown on the facing page. The stiffness of the beam is defined by s = 1;- - It depends on Young's modulus, E, for the material of the beam, on its length, £ , and on the second moment of its section, 1. EH DEFLECTION OF BEAMS -—-———(——- C1 C2 = vousos assume (we?! DEFLECT10N [ml FORCE IN} E o . F lz-éi-ZE—‘ég 2g 24_ M = MOMENTIle I LENGTH [ml E f 2 b WIDTH [ml 192 - t ' = DEPTH (ml END SLOPE {-1 E 384 ‘ . I = SEE TABLE zlm’u y _ DISTANCE FROM N.A.lml @d% 6 - a = RADIUS 0F CURVATUREIml U - It"... - _E.. M 7 " I ' R .n a: ll NW The longimdinal (or "fibre") stress a at a point y from the neutral axis of a uniform beam loaded elastically in bending by a moment M is 0-54.: 3.-.}. y_ I E(3 3.] where I is the second-moment of area (Section 2), E is Young‘s modulus, R0 is the radius of curvature before applying the moment and R is the radius after it is applied. The tensile stress in the outer fibre of such a beam is fimlm where ym is the perpendicular distance from the neutral axis to the outer ' surface of the beam. Ifthis stress reaches the yield strength 0'5, of the material of the beam, small zones of plasticity appear at the surface (top diagram. facing page). The beam is no longer elastic, and. in this sense. has failed. If, instead, the maximum fibre stress reaches the brittle fracture strength, 0f (the "modulus of rupture", often shortened to MOE) of the material of the beam, a crack nucleates at the surface and propagates inwards (second diagram); in this case, the beam has certainly failed. A third criterion for failure is often important: that the plastic zones penetrate " through the section of the beam, linking to form a plastic hinge (third diagram). _ I The failure moments and failure loads, for each of these three types of failure, and for each of several geometries of loading, are given on the diagram. The formulas labelled "ONSET" refer to the first two failure modes; those labelled ”FULL PLASTICITY" refer to the third. Two new functions of section shape are involved. Onset of failure involves the quantity L’ym; full plasticity involves the quantity H. Both are listed in the table of Section 2, and defined in the text which accompanies it. 4‘- FAILURE OF BEAMS I—-————£_._'...g C ” Ft 1 (i...) a" looser} Em:— H03, IFULL PLASTICITYI % .. Fl C(fi-fl )5} loosen I_ 1 CH” {FULL msncrm F' F E' 2 - M, = FAILURE MOMENT {le . . Ff z FORCE AT- FAILURE [N] z = LEN'GW [ml t = DEPTH lie! I: :- WIDTH-lml ' ' 1 = SEE TABLE 2 lm‘l 8 - 3}.” see TABLE 2 {m3} @ _ H = SEE TABLE 2 (m3: a, ; YIELD STRENGTH lN/mzl c} = MODULUS OF RUPTURE lN/mzl o'.’ = 0', {PLASTICMATERIAL} F = O" {BRITTLE MATERIAL] LBLLQW If sufficiently slender, an elastic column, loaded in compression, fails by elastic buckling at a critical load, Fcrit- This load is determined by the end constraints, of which four extreme cases are illustrated on the facing page: an end may be constrained in a position and direction; it may be free to rotate but not translate (or "sway"); it may sway without rotation; and it may both sway and rotate. Pairs of these constraints applied to the ends of column lead to the five cases shown opposite. Each is characterised by a value of the constant :1 which is equal to the number of half- wavelengths of the buckled shape. The addition of the bending moment M reduces the buckling load by the amount shown in the second box. A negative value of Fem means that a tensile force is necessary to prevent buckling. An elastic foundation is one that exerts a lateral restoring pressure. p, proportional to the deflection (p = ky where k is the foundation stiflhess per unit depth and y the local lateral deflection). Its effect is to increase Fmt. by the amount shown in the third box. A thin-walled elastic tube will buckle inwards under an external pressure p', given in the last box. Here 1 refers to the second moment of area of a section of the tube wall cut parallel to the'tube axis. 5= COLUMN BUCKLING I-——-—4———-+ n F = FORCE [N] M momsm lNrnl E = muses MODULUS lN/m’} I = LENGTH {ml A - SECTION AREA (m?! I = 555 TABLE-2 {m‘l r = GYRATION RAD. (19mm! k = FOUNDATION STIFFNESS [N/n n a HALF-WAVELENGTHS IN BUCKLED SHAPE p’ = pnessuss {Iv/m?! LEQESIQEQESHAEIS 6 = TORSION 0F SHAFTS A torque, 'I‘, applied to the ends of an isotropic bar of uniform section, and acting in the plane normal to the axis of the bar, produces an angle of I 4! 1 twist it. The twist is related to the torque by the first equation on the facing " . ELASTIC DEFLECTION page, in which G is the shear modulus. For round bars and tubes of circular section, the factor K is equal to J, the polar moment of inertia of the section, defined in Section 2. For any other section shape K is less than J. Values of Kare given in Section 2. Ifthe bar ceases to deform elastically, it is said to have failed. This will happen if the maximum surface stress exceeds either the yield ETC = z—ijl-IBRITTLE FRACTLREI strength a}. of the material or the stress at which it fractures. For circular - FAILURE K = fig!- Iouss'r DFYIELDI = TORQUE leL. = ANGLE 0F TWIST sections, the shear stress at any point a distance r from the axis of rotation is T _ Tr _ Gsr 6 2 7 " f - “f“ G = SHEAR MODULUSIN/rni The maximum shear stress, tmax , and the maximum tensile stress, 1 = LEN 6TH [ml _ d = DIAMETER [ml cm“, are at the surface and have the values ,. ‘ _ “zit-Gad K = SEETABLE‘IIIT'I] 1am — 0m — 2K — 42, a, = YIELD STRENGTH INA-n21 If 1mm: exceeds cry/2 (using a Tresca yield criterion}, or if om“ exceeds 0', = MODULUS 0F RUPTURE iN/fl'lzI the MOR, the bar fails, as shown on the figure. The maximum surface stress for the solid ellipsoidal, square, rectangular and triangular sections F is at the points on the surface closest to the centroid of the section (the mid- points of the longer sides). It can be estimated approximately by inscribing the largest circle which can be contained within the section and caiculating SPRING DEFLECTIUN AND FAILURE the surface stress for a circular bar of that diametiar. More complex F = FORCE {N} section-shapes require special consideration, and, if thin, may additionally u = DEFLECTION {ml fail by buckling. Helical springs are a special case of torsional deformation. E U. R = COIL RADIUS Irn } n 2 NUMBER OF TURNS The extension of a helical spring of It turns of radius R, under a force F, and the failure force Fcn't: are given on the facing page. mm A thin disk deflects when a pressure difference Ap is applied across its two surfaces. The deflection causes stresses to appear in the disk. The first box on the facing page gives deflection and maximum stress (important in predicting failure} when the edges of the disk are simply supported. The second gives the same quantifies srhen the edges are clamped. The results for a thin horizontal disk deflecting under its own weight are found by replacing 4;: by the mass-peFMfit-area, pet. of the disk (here p is the density and g is the acceleration due to gravity}; Thick disks are more complicated; for those, see "Further Reading". Spinning disks, rings and cylinders store kinetic energy. The maxinmm rotation rate and energy are limited by the burst-strength of the disk. given by equating the maximum strength in the disk to the strength of the material. The two boxes list the kinetic energy and the maximum stress in disks and rings rotating at an angular velocity to (rsdisnsfsec). 7:5TATIC AND SPINNING DISKS DEFLECTION [ml YOUNGS MODULUSIN/rn l PRESISURE DIFF. iN/m l POISSON'S RATIO ENERGY {J1 ANGULAR VELIHAD/s} DENSITY kg/m3 i 3: CONTACT smsssss MW When surfaces are placed in contact they touch at one or a few discrete points. Ifthe surfaces are loaded, the contacts flatten elastically and the contact areas grow until failure of some sort occurs: failure by crushing (caused by the compressive stress. on), tensile fracture (caused by the tensile stress, at) or yielding {caused by the shear stress 0,). The boxes on the facing page summarise the important results for the radius, a, of the contact zone, the oentre-to-centre displacement u and the peak values of ac. at and as. The first box shows results for a sphere on a flat, when both have the same moduli and Poisson‘s ratio has the value 113. Results for the more general problem (the "Hertzian Indentation" problem) are shown in the second box: two elastic spheres (radii R1 and R2, moduli and Poisson‘s ratios E1, v1 and E2, v2) are pressed together by a force F. I If the shear stress orfl exceeds the shear yield strength a), 3'2, a plastic zone appears beneath the centre ofthe contact at a depth ofabont #2 and ' R1 R2 RAD” 0F SPHERES lrnl 5. I52 MODULH oF SPHERESlN/rnzl v v P l ’ this state is reached, the contact pressure is approximately 3 times the - l; 2 LESSTTHS RATIOS yield stress, as shown in the bottom box. a RADIUS 0F CONTACT [ml u DISPLACEMENT [ml c STRESSES tN/m’} a, YIELD srsess {ha/m?! spreads to form the fully-plastic field shown in the two lower figures. When 1"V12 1w; -1 7i E. ( 51 " E2 ) WWW Stresses and strains are concentrated at holes. slots or changes of section in elastic bodies. Plastic flow, fracture and fatigue cracking start at these places. "The local stresses at the stress concentrations can be computed numerically, but this is often unnecessary. Instead, they can be estimated using the equation shown on the facing page. The stress concentration caused by a change in section dies away at distances of the order of the characteristic dimension of the section-change (defined more fully below), an example of St Venant's principle at work. This means that the maximum local stresses in a structure can be found by determining the nominal stress distribution, neglecting local discontinuities (such as holes or grooves}, and then multiplying the nominal stress by a stress concentration factor. Elastic stress concentration factors are given approximately by the equation. In it, on“, is defined as the load divided by the minimum cross-section of the part, 9 I is the minimum radius of curvature of the stress-concentrating groove or hole, and C is the characteristic dimension: either the half-thickness of the remaining ligament, the half-length of a contained crack, the length of an edge-crack or the height of a shoulder, whichever is least. The drawings show examples of each such situation. The factor at is roughly 2 for tension, but is nearer 1f2 for torsion and bending. Though inexact, the equation is an adequate working approximation for many design problems. The maximum stress is limited by plastic flow or fracture. When plastic flow starts. the strain concentration grows‘rapidly while the stress concentration remains constant. The strain concentration becomes the more important quantity, and may not die out rapidly with distance {St Venant's principle no longer applies). 9.: STRESS CONCENTRATIONS rap- fin F. = FORCE [NI Am: MINIMUM SECTION {mil cm: F/Am-n iN/m’l p = RADIUS OFCUBVATUREiml c = CHARACTERISTIC LENGTH ImI Cl. a: 0-5 {TORSION} o'. a! 2-0 {TORSION} iii—SHAEILGBdQKS Sharp cracks (that is, stress concentrations with a tip radius of curvature 'of atomic dimensions) concentrate stress in an elastic body more acutely than rounded stress concentrations do. To a first approximation, the local stress falls off as 1km with radial distance mm r to the crack tip. A tensile stress 0, applied normal to the plane of a crack of length 29. contained in an infinite plate (as in the top figure on the facing page} gives rise to a local stress field at which is tensile in the plane containing the crack and given by Co na 0' = ‘ 2:11 where r is measured from the crack tip in the plane 0 = 0, and C is a constant. The W K1, is defined as K1 = COW Values of the constant C for various modes of loading are given on the figure. (The stress 6 for point loads and moments is given by the equations ' at the bottom.) The crack propagates when K > Kic, the fracture to ess. When the crack length is very small compared with all specimen dimensions and compared with the distance over which the applied stress varies, C is equal to 1 for a contained crack and 1.1 for an edge crack. As the crack extends in a uniformly loaded component, it interacts with the free surfaces, giving the correction factors shown opposite. If, in addition, the stress field is non-uniform (as it is in an elastically bent beam}, 0 difi'ers from 1; two examples are given on the figure. The factors, C, given here, are approximate only, good when the crack is short but not when the crack tips are very close to the boundaries of the sample. They are adequate for most design calculations. More accurate approximations, and other less common loading geometries can be found in the references listed in Further Reading. 10 = SHARP CRACKS 1-1l1 -i§l {1-0/1}? 1.111-i-f-l l1 - “413’? FAILURE WHEN K1 2 Km STRESS INTENSITY IN/m“! REMOTE STRESS {N/m’} LOAD IN] MOMENT £le :2 CRACK HALF-LENGTH 3 SURFACE CRACK LENGTH {m} HALF-WIDTH {CENTRE} ImI WIDTH [EDGE CRACK} ImI b = SAMPLE DEPTH ImI BEAM THICKNESS {m} 'l 9311ng It S II — ll POINT LOAD ON CRACK FACE: -L ”'2ab MOMENT 0N BEAM : 0' = .5”. hi 3 - POINT BENDING : _ 3F»: U ' 2bt WW Thin-walled pressure vessels are treated as membranes. The approximation is reasonable when t < W4. The stresses in the wall are given on the facing page; they do not vary significantly with radial distance, r. Those in the plane tangent to the skin, 0'9 and a, for-the cylinder and 09 and 0‘, for the sphere, are just equal to the internal pressure amplified by the ratio bit or him, depending on geometry. The radial stress or is equal to the mean of the internal and extemal stress, pit! in this case. The equations describe the stresses when an external pressure 139 is superimposed if p is replaced by (p - p,). I In thick-walled vessels. the stresses vary with radial distance r from the inner to the outer surfaces, and are greatest at the inner surface. _ The equations can be adapted for the case ofbeth internal and external pressures by noting that when the internal and external pressures are equal, the state of stress in the wall is I as = c, a -p (cylinder) or . 09 = 04, = o, = -p (sphere) allowing the term involving the external pressure to be evaluated. It is not valid tojust replace p by (p - p9.) - Pressure vessels fail by yielding when the Von Mises equivalent stress first exceed'the yield'strength, 03,. They fail by fracture if the largest tensile stress exceeds the fracture stress O'f , where C K of 3 Has and K10 is the fracture toughness, a the half-crack length and C a constant given in Section 10. 11.’ PRESSURE VESSELS THIN WALLED ' CYLINDER p = PRESSURE Iii/m2] : a WALL THICKNESSIml a ;, INNER R'ADIUS lml I: = OUTER RADIUS'lrI'Il " r s RADIAL COORDINATE [ml ."1' ' T'i" 'kJ.-.LF:L W‘ 12 1 VIBRATING BEAMS.TU BES AND DISKS Anything vibrating at its natural frequency can he reduced to the simple problem of a mass m attached to a spring of stiflhess K. The I--—-—-- f —--1 C1 BE AM'S TUBES lowest natural frequency of such a system is f=i .111. 2a m --------- ., ._ Specific cases require specific values for m and K. They can often he " .___ I 1—:‘7Afl 9'37 f = NATURAL FREQUENCY [5‘1] I I'M“ "'0' mo = pA = MASS/LENGTH l kg/ml estimated with suflicient accur to be useful ' a re 'mats modellin . m m pp 31 - g p = DENSITY [kg/m3} Higher natural frequencies are simple multiples of the lowest. -" '- 22’ L A = SECTION AREA { mg: The first box on the facing page gives the lowest natural frequencies " - .. -— ' " I = SEE T ABLE A1 of the flexural modes of uniform beams with various end-constraints. As an exam 1e, the ii at can be estimated 1) ass ' ‘ that --... ,. ---- .. .. p r . y ummg __ H, ._ _____ ,- .f , WITH A a 21m 2F! 9.87 _ 3 was i I "R t 2 where IncJ is the mass per unit length of the beam (i.e. m is half the total 3 mass ofthe beam) and that K is the bending stifihess (given by Fifi from 2 68 WW” A 3 £31. Section 3); the estimate differs from the exact value by 39%. Vibrations of a tube have a similar form, using I and mD for the tube. Circumferential vibrations can be found approximately by "unwrapping" the tube and treating it as a vibrating plate. simply supported at two ofits four edges. The second box gives the lowest natural frequencies for flat circular C2 disks with simply-supported and clamped edges. Disks with curved faces are stiffer and have higher natural frequencies. 1"“ m1 = pi = MASS/AREAlkg/mzl 23‘ t = THICKNESS {m} R = RADIUS lml v = POISSDN‘S RATIO W At temperatures above 2113 T11‘ (where Tm is the absolute melting point), mafiriels creep when loaded. It is convenient to characterise the creep of a material by its behaviour under a tensile stress or. at a temperature Tm . Under these conditions the tensile strain rate é is often found to vary as a power of the stress and exponentially with temperature: 11 - _ 3. __Q_ 8 _— 440°]- exp RT where Q is an activation energy, and R the gas constant. At constant [I . = so] 0 where go (3-1) , 60(Nl m3) and n are creep constants. temperature this becomes The behaviour of creeping components is summarised on the facing page which give the deflection rate of a beam, the displacement rate of an indenter and the change in relative density of cylindrical and spherical pressure vessels in terms of the tensile creep constants. a a STRESS Wu:2 F = FORCE-{NI Sta = DISPLACEMENT RATES [Tn/S] n.€.,o;= CREEP CO...
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