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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 ofloading, higher moments ofthearea 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 centreof 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 fullyplastic bending. The fully plastic
moment for a beam in bending is M? i Hay ' .. _ . . ., 1213“: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
lzéiZE—‘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 054.: 3..}.
y_ I E(3 3.] where I is the secondmoment 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(ﬁﬂ )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: : WIDTHlml '
' 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 thinwalled 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 TABLE2 {m‘l r = GYRATION RAD. (19mm! k = FOUNDATION STIFFNESS [N/n n a HALFWAVELENGTHS 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—ijlIBRITTLE 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 ,.
‘ _ “zitGad K = SEETABLE‘IIIT'I]
1am — 0m — 2K — 42, a, = YIELD STRENGTH INAn21 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}
sectionshapes 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 masspeFMﬁtarea, 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 burststrength 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 oentretocentre 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 fullyplastic 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 sectionchange
(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 crosssection of the part, 9
I is the minimum radius of curvature of the stressconcentrating groove or
hole, and C is the characteristic dimension: either the halfthickness of
the remaining ligament, the halflength of a contained crack, the length of
an edgecrack 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/Amn iN/m’l p = RADIUS OFCUBVATUREiml
c = CHARACTERISTIC LENGTH ImI Cl. a: 05 {TORSION}
o'. a! 20 {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 nonuniform (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 11l1 i§l {10/1}? 1.111ifl l1  “413’? FAILURE WHEN K1 2 Km STRESS INTENSITY IN/m“!
REMOTE STRESS {N/m’}
LOAD IN] MOMENT £le :2 CRACK HALFLENGTH 3 SURFACE CRACK LENGTH {m}
HALFWIDTH {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 Thinwalled 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, forthe 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 thickwalled 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 halfcrack 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 endconstraints. 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 simplysupported 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 (31) , 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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