Selection of material and shape
Shape factors
The loads on a component can be decomposed into those that are axial,
those that exert bending moments and those that exert torques. One of
these usuallydominates to such an extent that structural elements are
specially designed to carry it, and these have common names. Thus ties
carry tensile loads; beams carry bending moments; shafts carry torques;
columns carry compressive axial loads. Figure 11.3 shows these modes of
loading applied to shapes that resist them well. The point it makes is that
the best material-and-shape combination depends on the mode of loading.
In what follows, we separate the modes, dealing with each separately.
In axial tension, the area of the cross-section is important but its shape is
not: all sections with the same area will carry the same load. Not so in
bending: beams with hollow-box or I-sections are better than solid
sections of the same cross-sectional area. Torsion too, has its efficient
shapes: circular tubes, for instance, are more efficient than either solid
sections or I-sections. To characterize this we need a metric—a way of
measuring the structural efficiency of a section shape, independent of the
material of which it is made. An obvious one is that given by the ratio of
_ the stiffness or strength of the shaped sectionsection with the same
cross-sectional area, and thus the same mass per unit length, as the shaped
section.1 to that of a ‘neutral’ reference shape, which we take to be that of
a solid square
Elastic bending of beams and twisting of shafts (Figure 11.3b and c)
The bending stiffness S of a beam is proportional to the product EI
Here E is Young’s modulus and I is the second moment of area of the
beam about the axis of bending (the x axis):
where y is measured normal to the bending axis and dA is the differential
element of area at y (Figure 11.3). Values of the moment I and of the area
A for common sections are listed in the first two columns of Table 11.2.
Those for the more complex shapes are approximate, but completely
adequate for present needs. The second moment of area, Io, for a
reference beam of square section with edge-length bo and section-area
is simply
(Here and elsewhere the subscript ‘o’ refers to the solid square section.) The bending stiffness of the
shaped section differs from that of a square one with the same area A by the factor
We call
where
the shape factor for elastic bending. Note that it is
dimensionless— I has dimensions of (length)4 and so does A2. It depends
only on shape, not on scale: big and small beams have the same value of
if their section shapes are the same.2 This is shown in Figure 11.4.
The three members of each group differ in scale but have the same shape
factor—each member is a magnified or shrunken version of its neighbors.
Shape-efficiency factors
for common shapes, calculated from the
expressions for A and I in Table 11.2, are listed in the first column of
Table 11.3. Solid equiaxed sections (circles, squares, hexagons, octagons)
all have values very close to 1—for practical purposes they can be set
equal to 1. But if the section is elongated, or hollow, or of I-section,
things change; a thin-walled tube or a slender I-beam can have a value of
of 50 or more. A beam with
beam of the same weight.
=50 is 50 times stiffer than a solid
Shapes that resist bending well may not be so good when twisted. The
stiffness of a shaft—the torque T divided by the angle of twist, (Figure
11.3c)—is proportional to GK, where G is its shear modulus and K its
torsional moment of area. For circular sections K is identical with the
polar moment of area, J:
where dA is the differential element of area at the radial distance r,
measured from the center of the section. For non-circular sections, K is
less than J; it is defined such that the angle of twist is related to the
torque T by
where L is length of the shaft and G the shear modulus of the material of
which it is made. Approximate expressions for K are listed in Table 11.2.
The shape factor for elastic twisting is defined, as before, by the ratio of
the torsional stiffness of the shaped section, ST, to that, STo, of a solid
square shaft of the same length L and cross-section A, which, using
equation (11.5), is:
The torsional constant Ko for a solid square section (Table 11.1, top row
with b=h) is
It, too, has the value 1 for a solid circular square section, and values near
1 for any solid, equiaxed section; but for thin-walled shapes, particularly
tubes, it can be large. As before, sections with the same value of
differ in size but not shape. Values derived from the expressions for K
and A in Table 11.2 are listed in Table 11.3.
Onset of failure in bending and twisting
Plasticity starts when the stress, somewhere, first reaches the yield
strength,
; fracture occurs when this stress first exceeds the fracture
strength,
; fatigue failure if it exceeds the endurance limit we use
the symbol
for the failure stress, meaning ‘‘the local stress that will
first cause yielding or fracture or fatigue failure’’. In bending, the stress
is largest at the point ym in the surface of the beam that lies furthest
from the neutral axis; it is:
where M is the bending moment. Failure occurs when this stress first
exceeds
. Thus, in problems of failure of beams, shape enters through
the sectionmodulus, Z=I/ym. The strength-efficiency of the shaped beam,
, is measured by the ratio Z/Zo, where Zo is the section modulus of a
reference beam of square section with the same cross-sectional area, A:
Like the other shape-efficiency factor, it is dimensionless and therefore
independent of scale, and its value for a beam with a solid square section
is unity. Table 11.3 gives expressions for other shapes derived from the
values of the section modulus Z, which can be found in Table 11.2. A
beam with an failure shape-efficiency factor of 10 is 10 times stronger in
bending than a solid square section of the same weight.
In torsion the problem is more complicated. For circular rods or tubes
subjected to a torque T (as in Figure 11.3c) the shear stress
is a
maximum at the outer surface, at the radial distance rm from the axis of
bending:
The quantity in J/rm twisting has the same character as I/ym in bending.
For non-circular sections with ends that are free to warp, the maximum
surface stress is given instead by
where Q, with units of m3 now plays the role of J/rm or Z. This allows
the definition of a shape factor,
, for failure in torsion, following the
same pattern as before:
Values of Q and
are listed in Tables 11.2 and 11.3. Shafts with a
solid equiaxed sections all have values of
close to 1.
Axial loading and column buckling
A column of length L, loaded in compression, buckles elastically when
the load exceeds the Euler load
where n is a constant that depends on the end-constraints. The resistance
to buckling, then, depends on the smallest second moment of area, I min,
and the appropriate shape factor
is the same as that for elastic bending
(equation (11.4)) with I replaced by Imin.
Elastic bending of beams and twisting of shafts. Consider the selection
of a material for a beam of specified bending stiffness SB and length L
(the constraints), to have minimum mass, m (the objective). The mass m
of a beam of length L and section area A is given, as before, by
Its bending stiffness is
where C1 is a constant that depends only on the way the loads are
distributed on the beam. Replacing I by
using equation (11.3) gives
Using this to eliminate A in equation (11.24) gives the mass of the beam.
For beams with the same shape (for which
is constant) the best choice
is the material with the greatest value o
_—the result derived in
Chapter 5. But if we want the lightest material-shape combination, it is
the one with the greatest value of the index
The procedure for elastic twisting of shafts is similar. A shaft of section A
and length L is subjected to a torque T. It twists through an angle
required that the torsional stiffness,
. It is
, meet a specified target, ST, at
minimum mass. Its torsional stiffness is
where G is the shear modulus. Replacing K by
gives
Using this to eliminate A in equation (11.24) gives
using equation (11.7)
The best material-and-shape combination is that with the greatest value of
1=2. The shear modulus, G, is closely related to Young’s modulus E. For
the practical purposes we approximate G by 3/8E; when the index
becomes
Failure of beams and shafts. The procedure is the same. A beam of
length L, loaded in bending, must support a specified load F without
failing and be as light as possible. When section-shape is a variable the
best choice is found as follows. Failure occurs if the load exceeds the
moment
where Z is the section modulus and
is the stress at which failure
occurs. Replacing Z by the shape-factor
B of equation (11.10) gives
Substituting this into equation (11.24) for the mass of the beam gives
The best material-and-shape combination is that with the greatest value of
the index
A similar analysis for torsional failure gives:
Co-selecting material and shape
The indices allow comparison and selection of material-shape
combinations. shape combinations.
Co-selection by calculation. Consider as an example the selection of
a material for a stiff shaped beam of minimum mass. Four materials are
available, listed in Table 11.5, each with their properties and typical
(modest) values for shapes, characterized by
. We seek the
combination with the largest value of the index M1 of equation (11.28)
which, repeated, is
It identifies the material-shape combinations with the lowest mass for a
given stiffness. The second last column of the table shows the simple
‘‘fixed shape’’ index
: wood has the greatest value, more than twice
as great as steel. But when each material is shaped efficiently (last
column) wood has the lowest value ofM1—even steel is better; the
aluminum alloy wins, surpassing steel and GFRP. Selection for strength
follows a similar routine, using the index M3 of equation (11.35).
Graphical co-selection. The material index for elastic bending (equation
(11.28)) can be rewritten as
The equation says: a material with modulus E and density
, when
structured, can be thought of as a new material with a modulus and
density of
The
chart is shown schematically in Figure 11.16. The ‘‘new’’
material properties
and can be plotted onto it. Introducing shape
moves the materialMto the lower left along a line of slope
1, from the position E, to the position E/10, /10 as shown in the
figure. it is shown, for one value of
as a broken line. The
introduction of shape has moved the material from a position below this
line to one above; its performance has improved. Elastic twisting of shafts
is treated in the same way.
Materials selection based on strength (rather than stiffness) at a minimum
weight uses a similar procedure. The material index for failure in bending
(equation (11.35)), can be rewritten as follows
The material with strength
and density
, when shaped, behaves in
bending like a new material of strength and density
Introducing shape
moves material M along a line of
slope 1, taking it, in the schematic, from a position , below the
material index line (the broken line) to the position /10, /10 that lies
above it. The performance has again improved. Torsional failure is
analyzed by using
T in place of
Forks for a racing bicycle
The first consideration in bicycle design (Figure 12.4) is strength (Table
12.6). Stiffness matters, of course, but the initial design criterion is that
the frame and forks should not yield or fracture in normal use. The
loading on the forks is predominantly bending. If the bicycle is for racing,
then the mass is a primary consideration: the forks should be as light as
possible. What is the best choice of material and shape?
The model and the selection. We model the forks as beams of length L
that must carry a maximum load P (both fixed by the design) without
plastic collapse or fracture. The forks are tubular, of radius r and fixed
wall-thickness, t. The mass is to be minimized. The fork is a light, strong
beam. Further details of load and geometry are unnecessary: the best
material and shape, read from Table 12.1, is that with the greatest value
of
Table 12.7 lists seven candidate materials with their properties. If the
forks were solid, meaning that
, spruce wins (see the second last
column of the table). Bamboo is special because it grows as a hollow tube
with a macroscopic shape factor
of about 2.2, giving it a bending
strength that is much higher than solid spruce (last column). When shape
is added to the other materials, however, the ranking changes. The shape
factors listed in the table are achievable using normal production
methods. Steel is good; CFRP is better; Titanium 6-4 is better still. In
strength-limited applications magnesium is poor despite its low density.
Postscript. Bicycles have been made of all seven of the materials listed in
the table—you can still buy bicycles made of six of them. Early bicycles
were made of wood; present-day racing bicycles of steel, aluminum or
CFRP, sometimes interleaving the carbon fibers with layers of glass or
Kevlar to improve the fracture-resistance. Mountain bicycles, for which
strength and impact resistance are particularly important, have steel or
titanium forks.
Ultra-efficient springs
Springs, we deduced in Section 6.7, store energy. They are best made of a
material with a high value of
volume, then of
, or, if mass is more important than
. Springs can be made more efficient still by
shaping their section. We take as a measure of performance the energy
stored per unit volume of solid of which the spring is made; we wish to
maximize this energy. Energy per unit weight and per unit cost are
maximized by similar procedures (Table 12.4).
The model and the selection. Consider a leaf spring first (Figure 12.3a).
A leaf spring is an elastically bent beam. The energy stored in a bent
beam, loaded by a force F, is
where SB, the bending stiffness of the spring, is given by equation
(11.25), or, after replacing I by
, by equation (11.26), which, repeated,
is:
The force F in equation (12.2) is limited by the onset of yield; its
maximum value is
(The constants C1 and C2 are tabulated in Appendix A, Sections A.3 and
A.4). Assembling these gives the maximum energy the spring can store:
where V=AL is the volume of solid in the spring. The best material and
shapefor the spring—the one that uses the least material—is that with the
greatestvalue of the quantity
For a fixed section shape, the ratio involving the two
is a constant:
then the best choice of material is that with the greatest value
—the
same result as before. When shape is a variable, the most efficient shapes
are those with large
Values for these ratios are tabulated for
common section shapes in Table 12.5; hollow box and elliptical sections
are up to three times more efficient than solid shapes.
Torsion bars and helical springs are loaded in torsion (Figure 12.3b). A
similar calculation gives
The most efficient material and shape for a torsional spring is that with
the largest value of
(where G has been replaced by 3E/8). The criteria are the same: when
shape is not a variable, the best torsion-bar materials are those with high
values of
with a ratio of
. Table 12.5 shows that the best shapes are hollow tubes,
that is twice that of a solid cylinder; all other
shapes are less efficient. Springs that store the maximum energy per unit
weight (instead of unit volume) are selected with indices given by
replacing E by
in equations (12.6) and (12.8). For maximum energy
per unit cost, replace E by
where Cm is the material cost per kg.