Material Limits for Shape Efficiency

Progress in Marerials Science Vol. 4 I, pp. 6 1 to 128. I997 T 1998 Elsevier Science Ltd. All rights reserved

Printed in Great Britain 0079-6425/98/$19.00

PII: s0079-6425(97)ooo34-0

MATERIAL LIMITS FOR SHAPE EFFICIENCY

Cambridge

I. INTRODUCTION

P. M. Weaver and M. F. Ashby University Engineering Department, Trumpington Street,

Cambridge, CB2 lPZ, U.K.

CONTENTS

I I, Material and shape 1.2. The perspective of fhis study

2. THE LIMITING EFFICIENCY OF STANDARD SECTIONS 2.1. Shape.fuctors

2. I I. Elastic e.utension 2. I .2. Elastic bending and twisting 2.1.3. Failure in bending and twisting 2.1.4. Axial loading and column buckling

2.2. Practical limits for the esfciencies of standard sections 3. MATERIAL LIMITS FOR SHAPE FACTORS

3.1. The approach 3.2. Failure modes and their interaction

4. SHAPE LIMITS FOR TUBES 4. I Axially-loaded tubes 4.2. Tubes loaded in bending 4.3. Torsion of tubes

5. BOX SECTIONS 5.1. Axial loading of bor secttons 5.2. BO.Y sections in bending 5.3. Box sections in torsion

6. I-SECTIONS 6. I. A.vial compression of l-beams 6.2. I-beams in bending

6.2. I. Relationships between dimensions for strength-limited I-sections 62.2. Relationships between dimensions,for srtffiness-limited I-sections 62.3. Failure mechanism and mechanism boundaries

6.3. I-beams in torsion 7. MINIMUM MASS RELATIONSHIPS 8. SUMMARY AND CONCLUSIONS

ACKNOWLEDGEMENTS REFERENCES

61 61 62 62 62 62 63 65 66 66 70 70 71 73 73 81 87 94 94 98

102 106 106 III III I13 II4 123 123 125 127 127

1. INTRODUCTION

1.1. Material and Shape

Shaped sections carry bending, torsional and axial-compressive loads more efficiently than solid sections do. By shaped we mean that the cross section is formed to a tube, a box section, an I-section or the like. By efficient we mean that, for given loading conditions, the section uses as little material, and is therefore as light, as possible. Tubes, boxes and I-

61

62 Progress in Materials Science

sections will be referred to as simple shapes. Even greater efficiencies are possible with sand- wiches [thin load-bearing skins bonded to a foam or honeycomb interior()) and with structures (the Warren truss, for example2].

There are practical limits to the thinness of sections, and these determine, f0r.a given material, the maximum attainable efficiency. These limits may be imposed by manufacturing constraints: the difficulty or expense of making an efficient shape may simply be too great. More often they are imposed by the properties of the material itself, which determine the failure modes of the section. Here we explore these ultimate material limits on shape efficiency. This we do in two ways. The first is empirical: by examining the shapes in which real materials are available, recording their limiting efficiencies. The second is by the analysis of the mechanical stability of shaped sections.

1.2. The Perspective of This Study

Techniques for analysing and optimising load-bearing sections have emerged largely from the disciplines of mechanics and of aircraft and civil structures. The range of materials of interest here is limited: structural steel and aluminium alloys predominate. The emphasis, understandably, is on the mechanics of the problem: how to design most efficiently with a single material, and, particularly, how to predict the failure load of efficient shaped sections. The approach of this paper is materials-centred rather than mechanics-centred; it is to examine how the properties of the material itself limit the efficiency of the shapes that can be made from it. There have been many structural materials that have been introduced in the recent past (such as polymer composites, metal-matrix composites and new alloy metals) and this work is intended, at least partially, to grade the efficiency of these materials in structural applications. We draw heavily on the mechanics literature, but focus on limiting shape, and the influence of material on it, rather than limiting loud (circumstances are such that this allows a certain simplification in dealing with the otherwise difficult problem of failure-mode interaction: the limiting load is much more sensitive to this interaction than in the limiting shape). The novel feature of the present work, then, is that of a materials perspective of efficient load-bearing sections.

2. THE LIMITING EFFICIENCY OF STANDARD SECTIONS

2.1. Shape Factors

A material can be thought of as having properties but no shape; a component or a structure is a shaped material. A shape factor is a dimensionless number which characterises the efficiency of a section shape, regardless of scale, in a given mode of loading. Thus there is a shape factor, & for elastic bending of beams, and another, &, for elastic twisting of shafts (the superscript e means elastic). These are the appropriate shape factors when design is based on stiffness; when, instead, it is based on strength (that is, on the first onset of plastic yielding or on fracture) two more shape factors are needed: & and & (the superscript f meaning failure). All four shape factors are defined so that they equal to 1 for a solid bar with a circular cross-section.

2.1.1. Elastic extension

The elastic extension [Fig. I(a)] or compression of a tie or strut under a given load depends on the area A of its section, but not on its shape. No shape factor is needed.

Material Limits for Shape Efficiency 63

\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

(a) TENSION : TIE

(b) BENDlNd : BEAM MOMENT I,

(c) TWISTING : SHAFT

POLAR MOMENT J

(d) COMPRESSION : COLUMN

MOMENT I,,

Fig. I. A tie, a beam, a shaft and a column. Efficient shapes are shown for each.

2.1.2. Elastic bending and twisting

[Fig. l(b) and (c)] If, in a beam of length 1, made of a material with Youngs modulus E, shear is negligible, then the bending stiffness of the beam (a force per unit displacement) is

where CI is a constant which depends on the details of the loading. Shape enters through the second moment of area, [,,, about the axis of bending (the x axis)

Ir\- = Lection JJ* d (2)

where y is measured normal to the bending axis and dA is the differential element of area at I.


The first shape factor-that for elastic bending-is defined as the ratio of the stiffness Sa of the shaped beam to that, S;, of a solid circular section (second moment I) with the same length, 1, cross-section, A (and thus mass per unit length) and material (and thus modulus E). Using equation (1) we find

Now I for a solid circular section of area A is just

from which

Note that it is dimensionless-Z has dimensions of (length)4, and so does A2. It depends only on shape: big and small beams have the same value of C& if their section shapes are the same. Solid equiaxed sections (circles, squares, hexagons, octagons) all have values very close to l-for practical purposes they can be set equal to 1. But if the section is elongated, or hollow, or of Z-section, or corrugated, things change: a thin-walled tube or a slender Z- beam can have a value of c#& of 50 or more. These shapes are efficient in that less material (and thus less weight) is needed to achieve the same bending stiffness*: a shape factor of 50 means that the section is 50 times stiffer in bending than a solid beam of the same weight. In subsequent sections we shall use 4; as the characterising shape factor, simply calling it 4.

Shapes which resist bending well may not be so good when twisted. The stiffness of a shaft-the torque T divided by the angle of twist, 6 (Fig. lc)-is given by

ST =$ (5) where G is the shear modulus. Shape enters this time through the torsional moment of area, K. For circular sections it is identical with the polar moment of area, J

J = Ssection dA (6) where dA is the differential element of area at the radial distance r, measured from the centre of the section. For non-circular sections, K is less than J; it is definedt3) such that the angle of twist 0 is related to the torque T by

where 1 is length of the shaft and G the shear modulus of the material of which it is made. The shape factor for elastic twisting is defined, as before, by the ratio of the torsional

stiffness of the shaped section to that of a solid circular shaft of the same length, 1, cross- section A and material (and thus G)

*This shape factor is related to the radius to the radius of gyration, Rs, by $JL =4rRi/A. It is related to the shape parameter, k , of Shanley w by &, =4nk,. Finally, it is related to the aspect ratio LX and sparsity ratio I of Parkhouseti7*2 by &=ia. None of these authors extends the concept to describe torsion, or to include strength, as here, though their approaches would allow it.


The torsional constant K for a solid cylinder is

giving

f#+~. (8)

It, too, has a value very near 1 for any solid, equiaxed section; but for thin-walled shapes it, too, can be large. As before, sets of sections with the same value of c$; differ in size but not shape.

2.1.3. Failure in bending and twisting

Plasticity starts when the stress, somewhere, first reaches the yield strength, a,; fracture occurs when this stress first exceeds the fracture strength, (Trr. Either one of these constitutes failure. We use the symbol of for the failure stress, meaning the local stress which will first cause yielding or fracture. One shape factor covers both.

In bending, the stress is largest at the point y, in the surface of the beam which lies furthest from the neutral axis; it is

where M is the bending moment. Thus, in problems of failure of beams, shape enters through the section modulus, Z = Z/ym. If this stress exceeds of the beam will fail, giving the failure moment

Mf = Zaf (10)

The shape factor for failure in bending, &, is defined as the ratio of the failure moment Mf (or failure load Ff) of the shaped section to that of a solid circular section with the same length 1 and area A

&J$=$ f

The quantity Z is the section modulus of a solid circular cylinder with the same area A as the shaped section

~312 r=ar3=_

of

giving

4&Z &=F. (11)

Like the other shape factors, it is dimensionless, and therefore independent of scale and its value for a beam with a solid circular section is 1. For thin-walled sections the shape factor


can be large and indicates the amount by which the section can resist more bending moment than a circular solid beam of the same weight.

In torsion, the problem is more complicated. For circular tubes or cylinders subjected to a torque T [as in Fig. l(c)] the shear stress z is a maximum at the outer surface, at the radial distance rm from the axis of bending

Tr, z==-.

J (12)

The quantity J/r,,, in twisting has the same character as Z = Z/J,, in bending. For noncircu- lar sections with ends that are free to warp, the maximum surface stress is given instead by

z=- e (13) where Q, with units of m3, now plays the role of J/r,,, or Z. (details in Youngc3)). This allows the definition of a shape factor, & for failure in torsion, following the same pattern as before

Tf Q 2JstQ &=F=Q"= ~312 (14)

Fully plastic bending or twisting (such that the yield strength is exceeded throughout the section) involve a further pair of shape factors. But, generally speaking, shapes which resist the onset of plasticity well are resistant to full plasticity also. New shape factors for these are not, at this stage, necessary.

2.1.4. Axial loading and column buckling

A column loaded in compression [Fig. l(d)], buckles when the load exceeds the Euler load

FC = n2z2EI m,n

12 (15)

where n is a constant which depends on the end-constraints. The resistance to buckling, then, depends on the smallest second moment of area, IminT and the appropriate shape factor (&) is the same as that for elastic bending [equation (4)] I replaced by Zmin.

2.2. Practical Limits for the Eficiencies of Standard Sections

Standard sections for beams, shafts, and columns are generally prismatic; prismatic shapes are easily made by rolling, extrusion, drawing, pultrusion or sawing. Figure 2 shows the taxonomy of the kingdom of prismatic shapes. The section may be solid, closed-hollow (like a tube or box) or open-hollow (an I-, U- or L-section, for instance). Each class of shape can be made in a range of materials. Each combination of material and shape has a set of attributes: they are the parameters used in structural or mechanical design. They include the dimensions, the section properties (moments Z, K and the section moduli Z and Q) defined in the last section, the structural properties (the stiffnesses EZ and GK; the strengths Zo, and Qzy, etc.) and the general properties: mass per unit length (m/l) and cost per unit length (C/l). Each record in the taxonomy describes a combination of a material, a size and a shape.


General properties weight/length (kg/m)

Hot rolled Ic steel

/

Cold rolled Ic steel

L Extruded rjg& wall thickness (m)

\

\

aluminium

Pultruded GFRP moment of area I (m4) Ton properties

section modulus 2 (m3 ) Sawn softwood (....same. for torsion)

HOLLOW< L-section Complex

Structural properties bend stiffness EI (Nm*) bend strength a,.Z (Nm) (....same, for torsion)

Fig. 2. A taxonomy of prismatic shapes

A database of real sections* allows the limits of shape to be explored. Figure 3 show Z, K, Z and Q plotted against A, on logarithmic scales for standard steel sections. Consider the first, Fig. 3(a). It shows log(l) plotted against log(A). Taking logs of the equation for the first shape factor (& = 47rI/A*) gives, after re-arrangement

iogz = 2logA +logZ

meaning that values of & appear as a family of parallel lines, all with slope 2, on the figure. The data are bracketed by the values 4: = 1 (solid circular sections) and 4: = 65, the empirical upper limit for the shape factor characterising stiffness in bending for simple structural steel sections. An analogous construction for torsional stiffness (involving &=4nK/ A*), shown in Fig. 3(c), gives a measure of the upper limits for this shape factor. Note that the open sections group into a different band characterised by a lower 4% because they have poor torsional stiffness.

The shape factors for strength are explored in a similar way. Taking logs of that for failure in bending (using & = 4fiZ/A3*) gives

3 &? logZ=ZlogA+log4Jj;.

Values of #f, appear as lines of slope 312 on Fig. 3(b), which shows that, for steel, real sections have shape factors with an upper limit of about 13. The analogous construction for torsion (using &=2J;;Q/A3*), shown in Fig. 3(d), suggest an upper limit for this shape

*The selection of standard structural sections using section and structural properties based on this taxonomy is discussed elsewhere4.

JPMS 41+--C


Shape factor cp: =%

x stedbox-beana

0 st6acolvms

If steei&cukxtubecSS484B(1991)

l coklrolbdl-BeamsBs2994

.&livemi~

r l.oEtoz l.tso3 l.CE+04 l.OE+05 Section Area, A (l@ m2)

(b)

Shape factor 41: = 1

l.oE+o3 1 .cEto4

Section Area, A (la6 m2)

Fig. 3(a,b)

ec l.a3c9 E

51 b = 'DE+08

p3 l.OEto7

$

2

d 'DE+06

ii

ClD '.OE+O5 3

2

g '.OE+O4

b.3

Material Limits for Shape Efficiency

Shape factor $5 = 13

- C

0 l.OE+07 .a

s ._ 2 lOE+O6 8

0 - l.OE+O5

ii '3 x lOEt04

E

5 '.OE+O3 .I

'.OE+O2

(d)

Section Area, A (le m*)

Shape factor I$I\ = 7

Shape factor 4: = 1

Data: as for I-A plot for Steel

1 .M+o2 l.oE+O3 l.OE+M 'S&O5

Fig. 3. (a) log(l) plotted against log(A) for standard steel section. (b) log(K) plotted against log(A) for standard steel sections. (c) log(z) plotted against log(A) for standard steel sections.

(d) log(Q) plotted against log(A) for standard steel sections.

69

70

Material

Progress in Materials Science

Table 1. Limiting shape factors for a range of materials

(GJnl,, (&),.x (&I),*, (&xix

Structural steels 65 25 13 7 Aluminium alloys 44 31 10 8 GFRP and CFRP 39 26 9 7 Polymers (e.g. nylon) 12 8 6 5 Woods (solid sections) 8 I 4 I Elastomers ~6 3 -

factor of about 7. Here, again, the open sections cluster in a lower band than the closed ones because they are poor in torsion.

Similar plots for extruded aluminium, pultruded GFRP, wood, nylon and rubber give the results listed in Table 1. It is clear that the practical upper-limiting shape factor for simple shapes depends on material. The obvious questions are: what sets the upper limits seen in Table 1; and why do they depend on the material? They are answered in the next section.

3. MATERIAL LIMITS FOR SHAPE FACTORS

3.1. The Approach

The range of shape factor for a given material is limited either by manufacturing constraints, by a dimensional constraint (e.g. maximum height) or by a balance between com- peting failure modes. Steel, for example, can be drawn to thin-walled tubes or formed (by rolling, folding or welding) into other efficient I-sections; shape factors as high as 50 are common. Wood cannot so easily be shaped; ply-wood technology could, in principle, be used to make thin tubes or I-sections, but in practice, shapes with values of 4 greater than 5 are uncommon. That is a manufacturing constraint. Composites, too, can be limited by the difficulty in making them into thin-walled shapes, though techniques for doing this now exists.

When efficient shapes can be fabricated, the ultimate efficiency is set by a balance between failure modes: commonly, global buckling, local buckling and plastic collapse or (in brittle materials) fracture. Studies of this topic date from the 1940s when improvements in structural efficiency were crucial to the success of high-performance aircraft. Much of the work was undertaken by the US National Advisory Committee for Aeronautics (NACA), with significant contributions in Germany() and the UK@. The studies, which focussed on wings and fuselages, are summarised by Gerard, (7) who compares the relative merits of simple shapes, sandwich panels, stringer panel-rib and multi-cell construction. From this work emerged a formulation for minimum weight structures, expressed in terms of three groups of variables. One, which we call the material index, contains the material properties. The second, known as the structural loading coeflcient, (8) has the dimensions of stress and combines the given loading and specified geometrical details of the problem into an index which has applicability to a wide range of loads and given dimensions. The last contains nondimensional groups of geometric parameters (which are a function of the shape factor). The expressions listed for the mass of structures that appear in later Sections of this paper all have this form. This result is extended in the current work. The existence of an optimal shape allows us to substitute for the last group (geometric parameters) in terms of the material index and the structural loading coeficient so that the expressions for minimum weight are written in terms of material index and structural loading coefficient only.


The aim of the present work is that of developing criteria for optimising the coupled selection of material and section-shape. To do so, we explore the ways in which material properties influence shape efficiency, and derive general expressions for the efficiency of sections which meet specified constraints on stiffness and strength. This requires us to general- ise the results obtained by aircraft engineers to a broader range of conventional structures and modes of loading: circular hollow tubes, hollow square tubes and I-sections subjected to axial compression, torsion and bending. The results for a particular structure and loading type (e.g. circular hollow tube carrying axial compression) are presented as efficiency charts which show the range of parameters over which a particular failure mechanism is dominant. The charts plot a non-dimensional form of structural loading coefficient (the Ioad factor, fl on the abscissa against a non-dimensional measure of structural efficiency (the shape factor, $I), as the ordinate. Mass contours are superimposed on the chart, identifying the value for the shape factor that minimises mass for a given value of the load factor.

Solid sections, when loaded to failure, generally fail by yield or fracture, or-if axially compressed-by general (Euler) buckling. As sections are made more efficient (measured, say, by increasing 4) new failure mode appear, almost all associated with one or another mode of local buckling. It is a characteristic of efficient sections that the loads required for two or more failure modes are almost equal. When two failure modes occur at nearly the same load, they interact, reducing the actual failure load to a value below that for either one of them acting alone. This interaction, crucially important in the design of safe, light, structures, has been the focus of intense study in between 1940 and 1970, leading to pre- scriptions, some empirical, some based on analysis, for dealing with it. A brief summary of this important topic is given in the next section, and further reference is made to it in subsequent sections. The brevity is justified by our perspective (Section 1.2): it is that of explor- ing the ways which material properties influence the efficiency of shape, measured by the maximum useful value of 4; it is not that of assessing the safety of structures. This maximum useful 4 is insensitive to details of failure mode interaction, as shown below.

3.2. Failure Modes and their Interaction

A prismatic bar (a tube, box or I-section), loaded in compression, bending or torsion, can fail by global buckling (buckling with a wavelength related to the length of the bar), by local buckling (buckling with a wavelength related to the section width or thickness), or by plastic collapse (general yield). It is well known, (9*10*) that, in minimum weight structures, failure mechanisms interact, and this has another consequence: they can be highly sensitive to defects.(2) As an example, the axial buckling loads of thin-walled cylindrical tubes are highly sensitive to imperfections in the wall thickness:0.3 an imperfection with an ampli- tude of 20% of the wall thickness can reduce the general buckling load by as much as 50%, and the sensitivity increases with r/t (or c/t* for plated structures).(24) On the other hand plate-structures such as I-sections and box-beams show little interaction between local and global buckling.(5*6) The individual plates of these structures show significant postbuckling strength, making them relatively insensitive to imperfections. In principle, unstable interaction could occur between local buckling of a plate structure and the global failure modeV6.7) because, if a plate buckles locally at a stress well below that for global buck-

*r = radius of tube, c = plate width, I = thickness


ling, the resulting loss of stiffness can diminish the flexural stiffness of the structure as a whole, triggering global buckling-an effect which can become severe in sandwich structures.() Experiments with box and I-beams, however, suggest that the loss in flexural stiffness after a flange has buckled is small because the webs provide longitudinal support. For a similar reason, coupling between global and local buckling is negligible in plate structures for which torsional buckling is not a consideration. (15) Despite this, designs which lead to simultaneous failure modes give potentially ill-conditioned structures that may exhibit unstable load-deflection response, even though response to each failure mechanism, when acting alone, is stable.

Furthermore, Thompson (lo) shows that for some structures (e.g. a sandwich without a core) with moderate imperfection the optimum shape is completely wiped out due to interaction effects between local and global buckling. For tiny imperfections, the resulting optimum shape (with much reduced gain due to interaction) shifts towards global buckling; implying that global buckling occurs at loads just below that to cause local buckling. Our design charts reflect this behaviour. Indeed, the shape factor is also found to be useful in assessing imperfection sensitivity. Other researchers (summaries in Brush and Almrotho4) and Allen and Bulson() have noted the strong dependence of imperfection size with r/t ratios for cylindrical shells and c/t for plated structures; which are simply the respective shape factor.

Design codes address the interaction of column buckling with general plasticity.(*) The approaches are semi-empirical in the sense that an imperfection parameter, t], is varied until the expression matches experimental data. A standard approach for steel columns uses the Perry-Robertson formula. (3) In the absence of local buckling the failure stress, CT, can be written as

0 1 2 112

-=- - CY * [ 1+(1+n)Z 1 [( ; l+(l+?;)z ) -2 1 (16)

where (Tn and oY are the Euler buckling stress and yield stress, respectively. The imperfection parameter, q is approximated as

which gives a 40% reduction in failure load for an = ay. Plates with supported sides withstand larger stresses because of a redistribution of axial

stress across the plate width after buckling. Near the sides the plates take more stress due to membrane effects whilst in the middle of the plate, bending effects predominate and the stress acting is less than that to cause initial buckling. () This leads to the idea of an effective-width: the actual plate is thought of as a plate of reduced width in which the membrane stress is uniform. Ultimate failure would then occur when this equivalent uniform stress reaches the yield stress of the material. Then

beay = barnax (17)

where b, b, and a,aX are actual plate width, effective plate width and postbuckling strength, respectively. The local buckling stress, a,,, can be expressed in terms of plate widthC3

a,,xb-* and a,ccb;* (18)


giving, for the post-buckling stress, dmax

flmax flCr

0

112 -=- . (19)

CY CY

The expression matches experimental data well except when r~,, and cy are close in magnitude; under these conditions imperfections cause yield and plate buckling to interact, reducing the maximum stress. Corrections for thisc2) take the form

or (20)

urnax -= 1 for 2 > 2.2

gY OY

leading to a 22% reduction in strength when ccr = cY. Interaction effects between all three failure modes (local buckling, global buckling and

plastic collapse) are treated by a modification of the Perry-Robertson formula [equation (16)] (BS449, Allen and Bulson*)

2 0 CT 1 cmax -- = - [ _+(1+q)Z 1 - ay 2

(21) 0~

R ; %+(1+9)Z ) Y

-2 1

giving a reduction in failure load of 50% when ccr = cY = au. The picture, then, is this: failure modes interact: the interaction reduces the failure load,

sometimes drastically; and there are ways of allowing for this in the design of safe structures. Our concern is different: it is that of comparing alternative combinations of material and shape. While interaction must not be forgotten, it can be treated here in an approxi- mate way because it has very little influence on this comparison.

4. SHAPE LIMITS FOR TUBES

4.1. Axially-Loaded Tubes

We start with the analysis of the efficiency of a tubular column because it best illustrates the method. The column [Fig. 4(a)] is loaded in compression. If sufficiently long and thin, it will first fail by general elastic (Euler) buckling. The buckling load is increased with no change in mass if the diameter of the tube is increased and the wall thickness correspond- ingly reduced. But there is a limit to how far the load-bearing capacity can be increased: it is determined by the onset of local buckling or by general yield. Thus there are three com- peting failure modes: general buckling, local buckling (both influenced by the modulus of the material and the section shape) and plastic collapse (influenced by the yield strength of the material and-for axial loading-dependent on the area of the cross-section but not on its shape). The most efficient shape for a given material is the one which, for a given load,


(a)

Fig. 4. (a) Tube loaded in compression. Local (chessboard) buckling of tube walls in compression (right). (b) Tube loaded in bending with chessboard buckling on compression side. (c)

Tube loaded in torsion showing local buckling mode.

uses the least material. It is derived below. In doing so, three equations are used frequently. The first is that for the cross-sectional area A of a thin-walled tube

A = 2mt (22)

Its second moment of area, I, is

I = nr3t (23)

and its shape factor, 4 [the short-hand we shall use for & of equation (4)] is

47tl r =A2=i (24)


from which

(25)

General buckling of a column of height 1, radius r, wall thickness t and cross-sectional area A, with ends which are free to rotate, occurs at the load

IGEI E=----

12 (26)

where E is the value of Youngs modulus for the material of which the column is made. Dividing equation (26) by A, substituting for Z/A2 from equation (24) and writing F/A = (T where c is the axial stress in the tube wall, we obtain an expression for the value of the stress uI at the onset of general buckling

(mechanism 1) 0, = (~.Ec#J.;)~. (27)

Local buckling, the second failure mode, occurs in a thin-walled tube when the axial stress exceeds, approximately, the value (Young, 1989, [Ref. (3), pp. 262-2631

62 = 0.6ciEi (28) Y

and is characterised by a chessboard pattern of Fig. 4(a). This expression contains an empirical knockdown factor, a, that YoungC3 takes to be equal 0.5 though at large r/t ratios this becomes unconservative because of the interaction of different buckling modes. NASA14 advocates the use of an empirical formula that recognises that the knockdown factor increases with r/t ratio, but there is apparently no universal agreement on how to deal with this. For example, ESDUC2 remain loyal to Koiters theoretical approach but still need to fit an imperfection parameter to match experimental data. Allen and Bulson recommend a modification of equation (28) with a given by

0.83 a=JmTrlt ~212

and

0.7 a=- JO.1 + r/loot

r/t > 212.

The choice of a influences the failure stress, but-as shown below-is has almost no effect on the optimal shape, our present interest. Replacing r/t by C$ [equation (2411 allows the stress at which the second failure mode occurs to be written as

(mechanism 2) a2 = 0.6a 4

(29)


The final failure mode is that of general yield. It occurs when the wall-stress exceeds the value

(mechanism 3) q = Q,, (30)

where cY is the yield strength of the material of the tube. Consider now the range of loads and shapes for which a given mechanism is dominant,

meaning that it takes place at a lower stress than any other. Mechanism 1 is dominant when the value of crl is lower than either o2 or 03, mechanism 2 when c2 is the least, and so on. The boundaries between the three fields of dominance are found by equating the equations for (rl, c2 and (r3 [equations (24) and (28)-(30)], taken in pairs, giving

l.OE-01

l.OE-02

l.OE-03

l.OE-04

l.OE-05

l.OE-08

l.OE-07

(1 - 2 boundary) F 1.44~~ E 1

- = -. - . - @ 7c 0 GY 43

(31)

(1 - 3 boundary) -$ = z. e) . j (32) Y

(a)

-.

?- TUBES-AXIAL LOADING

1 10 100 1000

Shape Factor (I Fig. 5(a)


1 .OE-01

1 .OE-OZ

1 .OE-O3

1 .OE-O4

1 .OE-06

1 .OE-07 I I,, , I I I,, I III,,,

10 100 1000

Shape Factor (I

Fig. 5(b)


(slight change in 0 J

l.OE-01

l.OE-02

ro" ol l.OE-03 7 IL

b 'i

II l.OE-04

'c3

8 A i.OE-05

l.OE-06

l.OE-07

- fTBES-AXIAL LOADING \I

j I

IL,_. 1 I I rl-I- - -

__!._!.__- I I Ii-

8 I I I I III, cm-- = t =I= c e tl

..ZZ Z,ZFFF, __---_ ___ w !A_---, L LI pn_--_+_ -+ - rn----+- + t-1 r Ll-l____d-_ -8-L !-I I I , I I I I I

1 10 100 1000

Shape Factor $

Fig. 5. (a) A plot of structure loading coefficient F/a,12 against shape factor 4 for by/ E = 3 x 1O-3 for axially loaded tubes. The field boundaries appear as full, boldlines: the shaded bands show the extent of their shift when by/E varies from 10e3 (mild steel, 1000 series aluminium) to IO- (ultra-high strength steel, GFRP). The contours show the quantity m/13p. (b) A plot of structure loading coefficient F/a,12 against shape factor 4 for aY/E3 x IO- for axially loaded tubes. Superimposed are standard aluminium tubes for slenderness ratios (I/D = 10-100). (c) A plot of structure loading coefficient F/c#J,~* against shape factor 4 for cry/E = 1 x IO-* for

axially loaded tubes, for two different interaction expressions.

2 -3 boundary) 4 =0.6ue (33)

Here we have grouped the variables into the dimensionless groups F/a,12, a,/E and C#I (dividing both sides of the equation by gY when necessary to achieve this), thereby reducing the number of independent groups to three.

This allows a simple presentation of the failure-mechanism boundaries, and the associated fields of dominance, as shown in Fig. 5(a). Here the axes are the normalised structural loading coefficient or loadfactor F/;/a l2 and the shape factor I$. The diagram is constructed for specific value of a,/E of 3 x lo-. The field boundaries appear as full, bold lines; the


shaded bands show the extent of their shift when CT,/E varies from lop3 (typical of mild steel and 1000 series aluminium alloys) to lo-* (the value for ultra-high strength steel and GFRP) .

To explore efJi&ncy we require a measure of the mass of the column required to support the load F without failing by any of the three mechanisms; we aim to find the value of $J which minimises this. The mass, m, of the column is

m = Alp (34)

where A is the area of its cross section and p is the density of the material of which it is made. Within the general-buckling regime 1, the minimum section area A which will just support F is

Al =E. 01

Inserting this into equation (34) and replacing ol by equation (27) gives for regime

*E (massinregime 1) A= lll. 6-4.

Y ( > 4 Pp

Within the local-buckling regime 2, equation (29) for (TV dominates and we find instead

(mass in regime 2)

and for the yield regime 3, using equation (30) for c3

(mass in regime 3) F

- = -!YY . c ) o+* Pp

(35)

As before, the variables have been assembled into dimensionless groups; there is one new one: the mass is described by the group (m/f3p). For a chosen value of this quantity and of a,/E, each equation becomes a relation between the load factor F/o,l*, and the shape factor, 4, allowing contours of mass to be plotted on the diagram, as shown in Fig. 5(a).

As mentioned earlier, failure modes can interact-a consideration near the failure field- boundaries. The European Convention for Constructional Steelwork(21) recommend the following expression for interaction between local buckling and yield

-=I for Q


n 1

u ave = ( 1 -l/2

c -1 i=l bi

(39

where g,,, is a smeared stress. We have used this, calculating the cross-section A (and from it the mass m) via

AL. Gave

(40)

The smearing effect is seen in Fig. 5(a) as the gradual change of slope in mass contours from one failure zone to another as they cross a field boundary. Although more difficult to justify on theoretical grounds it has found favour with some researchers (e.g. Ref. 22, p. 347) and it is conservative: there is a 30% reduction in the load factor for interaction between any two modes and a 43% reduction for all three modes occurring at the same stress. Of more interest here is that mode-interaction has very little effect in the optimum value of 4 (corresponding to the peaks in the m/13p contours), as shown in Figure 5(c). So, in seeking this, the choice of interaction expression is not important.

We can now approach the question: what is the most efficient shape, measured by 4, for the cross-section of the column? Tracking across Fig. 5(a) from left to right at a given value of the load factor, the mass at first falls and then rises again. In the lower half of the diagram the minimum mass lies at or near the l-2 boundary; higher up it lies to the left of the 2-3 boundary. If the column is designed for a specz$c value of the load factor, the optimum C$ can be read from the diagram. But if the column is intended as a general-purpose component, the load factor is not known, though all reasonable values lie within the range shown in the vertical axis of Fig. 5(a). Then the safest choice is a value of 6 a little to the left of the 2-3 boundary since this ensures that, if the column were to fail, it would fail by yield rather than the more catastrophic local buckling. This boundary lies at the position given by equation (33). Allowing a reserve factor on shape of 1.5 (that is, reducing C#J by a factor of 2/3) to the we find the optimal shape factor for the tubular column to be

which for a = 0.5 is

#I opt = - 0 ; 0plzo.2$

When the load factor is known, we can do better. The optimal shape factor is given by equation (41) for load factors in excess of

(42)

as found by substituting (41, 42) into (31). For load factors less than this the optimal shape factor follows the l-2 boundary in Fig. 5(a). Allowing for a shift to the left by a factor of


2/3, its value is found by rearranging (31) in terms of C#I to give

(43)

It increases slowly as the load factor decreases. It is instructive to examine where real sections lie on Fig. 5. Available section data can be

plotted by realising that the variables of the chart can be expressed in terms of section properties of the tube: area A, diameter D, second moment of area Z, and wall thickness t

A n=E I

(l/D)2DZ Or ay D4(l/D)4

(the last expression dependent on whether yield or buckling is the active constraint). Tubes, when used as columns, typically have slenderness ratios in the range

Using this range, and values of A, D, E, I and a, for real tubes, taken from suppliers catalo- gues, (29) the positions they occupy on the Figure can be plotted; they all lie in the shaded region of Fig. 5(b). The most efficient sections possess shape factors with values approach- ing 30 which is sufficiently far left in the chart to preclude local buckling for the entire spec- trum of load factors. The chart also shows that designs are limited by global buckling considerations in all cases except short columns.

In the sections which follow we repeat the exercise, more briefly, for two further modes of loading: bending and torsion. We then examine symmetric box-sections, and the more complex I-sections.

4.2. Tubes Loaded in Bending

Figure 4(b) shows a tube loaded in bending. Three failure-modes are possible: its stiffness may be insufficient (leading to unacceptably large elastic displacement); it may fail by local buckling; or it may fail by the onset of plasticity. As in Section 4.1, we calculate the maximum stress in the tube wall at which each mechanism cuts in, and identify the dominant mechanism as the one with the lowest such stress. The tube-wall stress, of course, is largest along the fibres which lie furthest from the neutral axis, so it is this stress which is of interest. The method follows that of Section 4.1 but is made slightly more involved by the need to relate this outer fibre stress a to the bending moment, M. This relationship is

Mr a=-

I

where, as before, I is the second moment of area of the tube wall and r is the tube radius.


The first failure mode (insufficient stiffness) is described by

where K is the maximum acceptable elastic curvature of the beam when acted on by a moment M. Writing 4 = 4nI/A2 = r/t [equation (24)J and with A = 2nrt we have

and eliminating the section area A between equations (44) and (45) gives

(46)

(47)

The second failure mode-local buckling-requires a little discussion. A long thin-walled tube tends to ovalise under flexural loading. (23) As the moment applied to the tube increases, ovalisation makes its second moment of inertia decrease until, as it approaches instability, local buckling or kinking cause collapse. (lo) The critical stress in the outer fibres of an isotropic cylinder at the point of collapse(24) is found to be close to

(mechanism 2a) o&&r = ---

By contrast, short thin-walled cylinders in which the ends are held circular, loaded in bending, fail by a short-wave instability on the compression side at approximately the same stress as a cylinder under uniaxial compression. Because of this, they are more imperfection sensitive than long tubes. The critical local buckling stress is

1 t ucr = Y31/2(l _ V2)E;

where y is an empirical knock-down factor to allow for the presence of imperfections, which can reduce the buckling stress by as much as 50% for relatively small imperfections.(2) Design codes formulated by NASA9 recommend either

or, for a,,120,/3

y = 1 - 0.731(1 -e-&d) (50)

0.83 Y=

2/l + r/lOOt r/t < 212.

The ECCS suggest instead

y = 0.7 &l + r/lOOt

r/t > 212

and the ESDU22 advocate

y = 3.76 $ 0

-0.18

(53)

(51)

(52)


(Allen and Bulsonc2) recommend further correction: that of multiplying the ECCS y by 1.3 to allow for the stress gradient across the bent cylinder.)

We shall assume that the ends of the long cylinder are free to deform such that ovalisation of the cross-section is always a consideration. It follows that the maximum stress can not exceed the Brazier buckling stress given by (48). Short cylindrical shells with larger/t ratios are more imperfection sensitive so that local buckling occurs with less ovalisation and [equation (49)] applies. For v = 0.3, [equation (48)] becomes

(mechanism 2a) a2 z 0.6~ f (54)

and its buckling pattern is shown in Fig. 4(b). The lesser of the stresses given by equations (48) and (53) is used in plotting the charts, effectively limiting the upper value of y, in (52), to 0.5. Otherwise, we use y from (50) and conservatively ignore the effects o the stress gradient through the section.

The third failure mode-yield-is described, as before, by

(mechanism 3) as = aY. (55)

Regime-boundaries are defined by equating equations (47) and (48) or 53) and (55) and taken in pairs. The results are

(1 - 2 boundary) -$= (().6yJ4n E ." Y 0 aY ~'t~~3

(1 - 3 boundary) -$ = TC~$) . f . -&

(2 - 3 boundary) I$ =0.67(z) 90.3(:).

(57)

The groupings of variables are, as before, dimensionless. The mass contours in each regime are found by calculating the section-area A which will

just support the moment M without failure, and substituting this into equation (34). Within regime 1 (the stiffness-limited regime)

Mr a=-=al.

I

Expressing I and r in terms aY of 4 and A through equations (29) and (24) gives

A = (!$3.(k!)23 (5% Substituting equation (47) for al and inserting the resulting value of A into equation (34) gives

(regime 1) ml - = 2g.p . 13P

Repeating this for the local buckling and a2 and a3 in equation (59) gives

(60)

the yield regimes by replacing a, successively by


(regime 2)

(regime 3) m3 - = (8rc) 113. M .- PP ( )

2~3 1

GYP cl/3 *

(61)

(62)

Interaction between mechanisms, smoothing the transition between regimes, was treated, as before, by using equation (39).

The results are displayed in Fig. 6(a) which shows (A4/Z30,) plotted against (&) with contours of (m/Pp). Following the line of argument used in Section 4.1, we identify the optimum value of #I with the peaks in the mass contours, multiplying it by a factor of 2/3 to give a safety margin against local buckling. For designs for which the load factor is unknown, the optimum lies at a value a little to the left of the 2-3 boundary, that is, at

(a) l.OE-03

l.OE-04

l.OE-05

l.OE

l.OE-06 I

l.OE-OB

l.OE-10

l.OE-11

Klr0.12 -

Shape Factor 4 Fig. 6(a)


l.OE-03

l.OE-04

l.OE-05

l.OE-06

l.OE-07

l.OE-08

l.OE-09

l.OE-10

l.OE-II 1 10 100

Shape Factor 4

1000

Fig. 6. (a) A plot of structure loading coefficient M/u,13 against shape factor C$ for uYyl E = 3 x 10m3 for tubes in bending. The field boundaries appear as full, bold lines; the shaded bands show the extent of their shift when cr,/E varies from 10m3 (mild steel, 1000 series aluminium) to lo-* (ultra-high strength steel, GFRP). The contours show the quantity m/lp. (b) A plot of structure loading coefficient M/uYi3 against shape factor 4 for ay/E = 3 x lo- for tubes in bending. Superimposed are standard aluminium tubes for two slenderness ratios (i/D = 10-100). (c) A plot of structure loading coefficient M/Q3 against shape factor 4 for uy/E = 3 x 10e3 for tubes in bending. The shaded bands show the extent the field boundaries shift as the stiffness constraint changes from KI = 0.036 to KI = 0.12 corresponding to maximum deflection/length ratio

of l/l00 to l/333 for a beam in three-point bending.

which for y = 0.5 is

4 [email protected] 2 0

.

CY (63)

As in Section 4.1, there is a critical value of load factor below which optimal designs follow the l-2 boundary, and greater efficiency can be achieved by using this fact. Substituting


(58) into (56) gives

M n 54 1

~YGossover = 0.3 E (~1)~ (>- (64)

as the crossover point. For values of load factor below this, the optimal shape factor is given by rearranging (56) and allowing for a shift to the left, as

&@ w f ((0.3)%(~) --$g (65) Available section data are plotted on the charts by noting that

A A

= P = (//D)2D2

(Kl)EZ l4o

Y

(the last expression dependent on whether stiffness or yield is the active constraint). As before, we assume that tubes used as beams will generally have length-to-diameter ratios in the range

This allows the regime in which real tubes lie to be plotted as a shaded band on the chart. Figure 6(d) shows the results for aluminium tubes. They all lie comfortably to the left of the 2-3 boundary.

Figure 6(c) shows the effect of changing the stiffness constraint. As expected, this changes the size of the stiffness zone and the positions of the mass contours within it, but not elsewhere. The effect on the critical value of 4 is small.

4.3. Torsion of Tubes

A thin-walled tube loaded in torsion [Fig. 4(c)] has three possible failure modes: inadequate stiffness, local buckling and general yield. The cross-sectional area of the tube, assumed thin-walled, is

A = 2xrt (66)

and its polar moment of the section, K

K = 2rrr3t (67)

The torsional shape factor & was defined in equation (8). It can be expressed in terms of 4, using equations (23) and (67)

Progress in Materials Science

that is, the shape factors for torsion and for bending, for thin-walled tubes, are equal. The first failure mode-inadequate stiffness-is approached as follows. The shear stress

in the wall of a tube carrying a torque T is

Tr (27c)li2T 7=z=m* (69)

The stiffness of the tube is measured by the angle of rotation of its loaded end, 0. This is related to T by

T=KG;+G$. (70)

Eliminating A between the last two equations gives an expression for z which contains only quantities specified by the design and the shape factor 4

(mechanism 1) ~1 = (71)

The second failure mode-local buckling-is made complicated by the fact that the wall- stress to cause it depends on the length of the tube itself (unlike bending). For practical purposes, it is given byC3):

dE ~2 = KS---

12(1/?)2

with

Assembling the last two equations with equation (70) and grouping the variables into appropriate dimensionless groups gives

(mechanism 2) 72 = yq&)8fy--& (72)

The final failure mode-yield-is simply described by

(mechanism 3) ~3 = zy (73)

where zY is the shear-yield strength. To make the diagrams for torsion directly comparable with those for other modes of loading, we replace G and ~~ by

E=;G (74)

and

CsY = 22,. (75)


The field boundaries, obtained as before, are

89

(1 - 2 boundary) $ = 347 t $$ Y 0

(1 - 3 boundary) 5 = 7.5(sJ3& (77) Y

(2 - 3 boundary) $ = 0.16(976$9. Y

(78)

The mass of the tube is (as before) given by equation (34). Inverting equation (68) gives an equation for the cross-sectional area which will support a shear stress 7 with a shape 4

1 &E-O2

l.OE-03

7 l.OE-05 I-

u ([I

3 l.OE-07

l.OE-08

1 .OE-O9

l.OE-10

TUBES-TORSION tFo.2 -OY / I-9 = 1x1o-3 - by / m 3x1 PbY

--= .lxIo_2 /

1 10 100 looo

Shape Factor 41

Fig. 7(a)


1 .OE-O2

1 .OE-O3

1 .OE-O4

1 .OE-O5

1 .OE-O5

1 .OE-O7

1 .OE-O8

1 .OE-O!J

l.OE-10

1 10 100

Shape Factor 4 Fig. 7(b)

1000


I.OE-02

I.OE-03

I.OE-04

I.OE-05

I.OE-06

I.OE-07

I.OE-08

I.OE-09

I.OE-10

field

.-. ..... _. ....

........ ....... ...... _ ......

.._ ....... .... ...... : ....... .

, . .... , ... * * ... _ .,

1 IO 100

Shape Factor 0

1000

(a) ,4 plot of structure loading coefficient T/Q3 against shape factor 9 for +/ E = 3 x lo- for tubes in torsion. The field boundaries appear as full, bold lines; the shaded bands show the extent of their shift when u,/E varies from 10e3 (mild steel, 1000 series aluminium) to 10e2 (ultra-high strength steel, GFRP). The contours show the quantity m/lp. (b) A plot of structure loading coefficient T/u+' against shape factor C$ for by/E = 3 x IO- for tubes in torsion. Superimposed are standard aluminium tubes for two slenderness ratios (l/D = 10-100). (c) A plot of structure loading coefficient T/u& against shape factor C$ for by/E = 3 x lO-3 for tubes in torsion. The shaded bands show the extent the field boundaries shift as the stiffness con-

straint changes from 0 = 0.1 to ff = 0.2.

Replacing z by zl, z2 and z3 in turn, and substituting in equation 934) and using equation (73) gives equations for the mass contours


(regime 1) $=4.05[($) .$]

m2-25 La, ( > l/12

(regime 2) pp- . oY13 E w S/l2 .e1/12

(regime 3) m3 pp = (?;)i3.($Cy.

630)

(81)

For similar reasons as those given for bending we use the interaction formula of

equation (36).

Figure 7(a-c) show the behaviour. They plot (T/130,) against (#I) with contours of (mf

Pp). Available section data are plotted on the charts by realising that

m

( >

A - pp =ji

and 4 =y

T

( >

K 3E K Pa, = D4(1/D)4 Or 80, 04(1,D)4

Using the limits on slenderness

105 ; I100

allows the area occupied by real tubes to be plotted on the charts. The optimum shape fac-

tor, when the load factor is unknown, lies at the left of the triple point, and is found by

equating any pair of equations (76~(78):

4 E 415

opt x0.34 - 0 CY

(82)

When the load factor is known, it is possible to pick an optimal shape to match the load

factor. It follows the 2-3 boundary for load factors greater than

T by '715 1

~YC*ossover =5.1LE -

( > @3/S

with the value

$0,~ = 1.2(-$79-& (&)9 Below, it follows the l-2 boundary, with the value

k :: .:

. : 6 ---r- C

Fig. 8. (a) A box-section loaded axially. Local buckling mechanism of sides walls under compression (right). (b) A box-section loaded in bending. Local buckling of compression flange (right). (c) A box-section loaded in torsion. Local shear buckling of side walls under torsion

(right).


c#+ = 1.7(-y-.& (qy . (83)

5. BOX SECTIONS

Before examining I-sections, the optimisation of which is rather more complex, it is help- ful to analyse square box sections (Fig. 8). The steps follow those for tubes.

Four equations are used repeatedly. The section-area of a square box section of height and depth c with a uniform wall thickness I, assumed small, is

A = 4ct. (84)

The second moment of area is

and the shape factor for a square thin-walled box loaded in bending is

from which

(87)

5.1. Axial Loading of Box Sections

General buckling occurs, as with tubes, at the Euler load [equation (26)]. The stress in the tube wall at the point of buckling is then [using equations (84x86)]

(mechanism 1) 61 = (t.g+@).

The stress for local buckling of the box faces [Fig. 8(a)] is

(mechanism 2) 02 = 3.6EciJ2= 3.6E(Gj2. (8%

It does not lead immediately to instability because membrane stresses appear which stabilise it, giving a post-buckling reserve. Finally, the stress for face yielding is

(mechanism 3) 03 = by. PO)

The field boundaries are found by equating these three results in pairs, giving

(1 - 2 boundary) -$ = 1.3(E) $ Y

(91)


(1 - 3 boundary) -j$ = i (2) $ Y

.

(92)

(93)

The mass contours are found by substituting 01, u2 and 03, successively, for IT in the equation

AZ 0

and entering this value of area into equation (34). The results are

(regime 1) !$= (i. (6) .@!J2 (94)

l.OE-01

i.OE-03

l.OE-04

l.OE-05

l.OE-OE

l.OE-07

(a)

10 100

Shape Factor 4

Fig. 9(a)


l.OE-02

l.OE-03

l.OE-04

l.OE-05

l.OE-07

(b)

1 10 100 1000

Shape Factor Q,

Fig. 9(b)


\

aqns21and20 -postbuckiing strength

- eqn39elastic

l.OE_@

10 100

Shape Factor $I

Fig. 9. (a) A plot of structure loading coefficient F/a,,? against shape factor I#J for uY/ E = 3 x lo- for axially loaded box sections. The field boundaries appear as full, bold lines; the shaded bands show the extent of their shift when a,,/E varies from 10V3 (mild steel, 1000 series aluminium) to 10-t (ultra-high strength steel, GFRP). The contours show the quantity m/l3 pi. (b) A plot of structure loading coefficient F/u,? against shape factor 4 for +/E = 3 x lo- for axially loaded box-sections. Superimposed are standard aluminium box-sections for two slenderness ratios (L/D = 10-100) (c) A plot of structure loading coefficient F/o,12 against shape factor

(regime 2) z= (&).(%J@

(regime 3) - = ?- m3

FP PGY

(95)

The problem of mode interaction was discussed in Section 3.2; if precise failure loads are required, the interaction equations listed there should be used. Here we focus not on failure load but on optimum shape, and for this the simple interaction equation (39) is adequate. It is this which we have used to construct the figures.

The results are shown in Fig. 9(a-b). The first point to note is the similarity between these charts and those for tubes [Fig. 5(a,b)]. Boxes with the larger values of shape factor


(up to 25), at low slenderness ratios (l/D x lo)*; extend into the local buckling regime. The optimum shape factor, 4, for an all-purpose box structure, lies slightly to the left of the 2-3 boundary at about

(97)

For designs in which the load factor is specified the optimal shape factor follows the l-2 boundary for load factors below the crossover value

F by 312

~Y%ossover =1.3 jj

( )

Above, it follows the 2-3 boundary. Allowing for a shift to tors are given by

(98)

the left the optimal shape fac-

(99)

Fig. 9(c) compares mass contours and optimum shape computed from the more accurate interaction equations (16) and (21). The peak in the mass curves, identifying the optimum &J is influenced only very slightly.

5.2. Box Sections in Bending

The three failure modes of tubes are relevant here. The stress in the face of the box-beam when it carries a bending moment A4 is given by equation (44) with r replaced by CO/~. The stiffness condition is given by equation (45). Using equations (86) and (87), and eliminating the section area A between the modified equation (44) and equation (45), gives

(mechanism 1) ~TI = (100)

The second failure mode is that of local buckling [Fig. 8(b)]. The beam face is thought of as a plate of width co, simply supported along its two long edges. Such a plate buckles when the compressive stress it carries reaches the value

(mechanism 2) t 2

02 w 3.6E c, 0 En2 E = 3.6 . # x z. The final failure mode is that of yielding, when the stress is given as before by

(mechanism 3) a3 = ay.

The field boundaries are

(1 2 boundary) A4 1E 1 1

- - a,P

= ---a~- 30, 4 WI3

(101)

(102)

*Here D is used to indicate total depth of beam.


(1-3boundury) s=i.e).j.-& Y

(2 - 3 boundary) 3 6/n

4 = +.

The mass contours are found, as before, by substituting for A in equation (34). The equations for the contours in each of the fields are as follows. The section area is found from

(4

BOXES-BENDING

i-

KI=0.12

OY / = lxlo-2

-bY / E = 3x1o-3

OY E / E = 1x1o-3

l.OE-03

l.OE-04

l.OE-05

l.OE-06

l.OE-07

l.OE-08

l.OE-09

l.OE-10

l.OE-11

JPMS 41/l-2--D

10 100

Shape Factor 4

Fig. 10(a)

l.OE-03

l.OE-04

l.OE-05

l.OE-06

l.OE-07

l.OE-08

l.OE-09

l.OE-10

l.OE-11


c Li:: ;..i :,;;_p__...Gi_.._. _> 1 i : : .I..:.iL_.~ . _.;_ __.__._.__.__. __*_T = 1 o-4 I

1 10 100

Shape Factor (I

1000

Fig. 10. (a) A plot of structure loading coefficient M/e,,l against shape factor 4 for a,/ E = 3 x lo- for box-sections in bending. The field boundaries appear as full, bold lines; the shaded bands show the extent of their shift when a,/,!? varies from lo- (mild steel, 1000 series aluminium) to lop2 (ultra-high strength steel, GFRP). The contours show the quantity m/l p. (b) A plot of structure loading coefficient M/cQ against shape factor C#I for uy/E = 3 x 1O-3 for box sections in bending. Superimposed are standard aluminium box-sections for two slenderness ratios (l/D = 10-100). (c) A plot of structure loading coefficient M/uy13 against shape factor 4 for a,/E = 3 x 10m3 for box-sections in bending. The shaded bands show the extent the field boundaries shift as the stiffness constraint changes from Kl = 0.036 to KI = 0.12 corresponding to

maximum deflection/length ratio of l/l00 to l/333 for a beam in three-point bending.

with c replaced successively by ol, ~7~ and g3. giving the following results

(regime 1) 3 = 2z12 pp ($2(~)2&& (105)


(regime 2) $4.2. (g3(!yo

(regime 3) $ = (24*)/3 . ($_y3.(_!!13.

(106)

(107)

Fig. lO(a-c) shows the results and similar comments to before apply. When the load factor in not specified, the optimum shape factor is

4 E i/2

opt z - 0 CY

W)

It is lower than that for tubes because local buckling starts at a lower fibre stress. Box-sections for specified load factors have this optimal shape factors above the triple point (the cross-over point) defined by

M i 2 70 i 13a,=j E 0 - (K1)3 .

Below the cross-over point the optimal shape factor is

(109)

allowing for a shift to the left of the l-2 boundary.

5.3. Box Sections in Torsion

The shear stress in the wall of a box section carrying a torque T is approximately

Tc T Z=s=G (110)

where K, defined in Section 2.1, is given by K = tc3. The torsional stiffness is

T=KG; (111)

and the shape factor 4; for torsional loading was defined in equation (8). For square box- sections it can be expressed in terms of r#~ [using equation (86)] as

(112)

Combining equations (1 10)(112) with (86) and (87), following the method used for tubes gives an expression for r which contains quantities specified by the design and the shape factor f$

(mechanism 1) zr = (113)

and is the same expression as that for tubes (71). The critical shear stress for local buckling


of the faces of the box, assumed simply-supported (5) is

103

(mechanism 2) ~2 = 4.84E f 2= 1.33;. 0

(114)

The final mode, yield, is simply

(mechanism 3) ~3 = zy. (115)

Making the substitutions for G and (TV from equations (74) and (75) allows the field boundaries to be expressed as

(1 - 3 boundary) -$ = 10(y)3J-- Y 3 Cpe3

l.OE-02 BOXES-TORSION fj=o.2 ffv

/ E = 1x1o-z - b* / E = 3x10

l.OE-04 a9

b ij l?

l.OE-08

u

8 l.OE-07 J

l.OE-08

l.OE-03

l.OE-IO

(117)

Shape Factor I$

Fig. 1 l(a)


1 .OE-02

1 .OE-03

1 .OE-O4

1 .OE-O5

1 .OE-O6

1 .OE-OB

l.oE-10 1 ! ! ! ! !h ,. &I I I II!, 11-r .I

1 10 100 1000

Shape Factor (I

Fig. 11(b)

l.OE-02

i.OE-03

l.OE-04

l.OE-05

l.OE-06

l.OE-07

l.OE-08

l.OE-09

l.OE-10


(c)

1 10 100 1000

Shape Factor 4

Fig. 1 I. (a) A plot of structure loading coefficient T&I3 against shape factor 4 for aY/ E = 3 x 10M3 for box-sections in torsion. The field boundaries appear as full, bold lines; the shaded bands show the extent of their shift when uy/E varies from 10V3 (mild steel, 1000 series aluminium) to lo- (ultra-high strength steel, GFRP). The contours show the quantity m/l p. (b) A plot of structure loading coefficient T/+1 against shape factor C$ for u,/E = 3 x 10-j for box- sections in torsion. Superimposed are standard aluminium box-sections for two slenderness ratios (l/D = 10-100). (c) A plot of structure loading coefficient T/ay13 against shape factor 4 for o,,/ E = 3 x 1O-3 for box-sections in torsion. The shaded bands show the extent the field boundaries

shift as the stiffness constraint changes from 0 = 0.1 to 0 = 0.2.

(2 - 3 boundary) 4 = 1.6 ; 0

112

105

(118)

The mass of the box-section is found by substituting for A in equation (34). Combining

equations (110) and (112) and inserting into (84) gives

(119)

as an expression for area. The mass contours for each dominant mechanism is simply


found by substituting for the relevant z into equation (119)

(regime 1) !$2fi[($) .-&L/3]

(regime 2) z = l.65(&)23($)23~

(regime 3) 2 = ($)3(+)23*

(121)

(122)

Fig. ll(a-c) show the results. Unlike that for tubes, the 2-3 boundary for box-sections in torsion is independent of shape factor. This arises because the wavelength of local buckling for boxes is dictated by the web depth, whereas that for circular tubes scales with tube length. The optimal shape factor for an all-purpose box (load factor not prescribed) is

(123)

allowing, as before, for a shift to the left. Boxes made for specific load factors have this optimal shape above the cross-over point

Below, the optimal shape is

(124)

having allowed for a shift to the left in shape factor in design space.

6. I-SECTIONS

I-sections are more complex than tubes or boxes because their shape now involves not one, but three dimensionless ratios: B/T, c/t and T/t, shown in Fig. 12. We will use c/t as the main shape-characterising ratio, and seek criteria which allow the other two ratios to be re-expressed in terms of c/t. This allows failure charts to be constructed using, as before, a shape factor (proportional to c/t) and the relevant structural loading coefficient F/a,12 as axes.

6.1. Axial Compression of I-Beams

When I-sections are used as columns [Fig. 12(a)] supporting a compressive axial load, it is desirable that the two principal section moments [r.X and Zyy should be equal:

JXX = IyI

where


(b) , ,v

107

(cl

X

X

i) Hexural Buckling of Web ii) Shear Buckling of Web

Fig. 12. (a) Dimensions of an l-beam. (b) I-Beam in compression. Local buckling modes of flange and web (right). (c) An I-beam loaded in bending (above). Two efficiency limiting web failure

mechanisms (below).


and

(ignoring second order terms and higher in t and T), giving the first relationship between dimensions. A second relationship is found by noting that the material in the I-section is most efficiently used if the local buckling of flanges and web occur at the same load, whenc3

(125)

with k,= 3.62 and kh = 0.46. Using these results we find that the shape factor for the I-section column can be expressed in terms of c and t only

We can now express the failure stress for each mechanism in terms of 4 as was done for tubes and boxes. The first failure mode is that of general buckling. Here the critical stress is identified by the same expression as that for tubes:

(mechanism 1) rrr = ( ) i.$.Ef#J

r/2 (127)

The second failure mode is that of local buckling, which can occur with equal likelihood in the flanges or web. The stress, F/A, at which this occurs is, using equation (125)

(mechanism 2) ~2 = 0.33 5. 4

(128)

The third failure mode, as before, is face yield, when

(mechansim 3) 03 = a,.. (129)

The boundaries of the failure mechanism fields are found, as before, by equating pairs of the mechanism equations

(1 - 2 boundary) ; =0.14 ; -j 5

Y (>O

4 C (1 - 3 boundary) k = ; -f -J

Y (,(I

(130)

(131)

(2 - 3 boundary) C#I =0.57 t 0

I/2 . (132)

The mass contours are found by substituting cl, 02 and u3 of equations (127)-( 129) succes-


sively, for c in the equation

The results are:

(regime 1) $ = (i.(&).(s) i)

(regime 2) 3 = ;; 3(&)*@~

(regime 3) $ = &

(133)

(134)

(135)

The results are shown in Fig. 13(a,b); the similarity between these charts and those of tubes and boxes [Fig. 5(a,b) and Fig. 9(a,b)] is evident. Postbuckling strength has again been neg-

(a)

-AXIAL LOADING

--l-T-P+*

r-,-l----f--r-T-l-l-C,1

---c--c-I-t--c,-4 P-T7-l-t-I-l r -r -7-I_T,-l

l==~~El~=:E3 ___Cffftlzl:z -4-c+++c,-----A

---i---1-+-C+t++,-

-_-_-I-_-i-~-

1 10 100

Shape Factor $

1000

Fig. 13(a)


l.OE-01

1 .OE-02

l.OE-03

1 .OE-04

l.OE-05

l.OE-06

l.OE-07 1 10 100 1000

Shape Factor t$

Fig. 13. (a) A plot of structure loading coefficient F/cry/ against shape factor 4 for uY/ E = 3 x 10e3 for axially loaded I-section. (b) A plot of structure loading coefficient F/CT,? against shape factor 4 for uy/E = 3 x 10m3 for axially loaded I-section. The field boundaries appear as full, bold lines; the shaded bands show the extent of their shift when +/E varies from 10m3 (mild steel, 1000 series aluminium) to lo-* (ultra-high strength steel, GFRP). The contours show the

quantity m/13 p.

lected because its inclusion results in no significant shift in the positions of the field boundaries or the optimum values of 4, The optimum shape factor for an all purpose I-column lies slightly to the left of the 2-3 boundary (to give a safety margin) at about

4 E r/2

opt zo.3 - . 0

(136) OY

For columns for specific applications the optimum shape factor is given by equation (136) for load factors above the cross-over value


Below, it follows the l-2 boundary, giving

111

6.2. I-Beams in Bending

We consider an I-section consisting of two rectangular flanges of width B supported at a spacing c by a web, as shown in Fig. 12(a). It is subjected to a bending moment, M, and a shear force, V; its length, I, is specified, and so is its stiffness, measured by the permissable deflection 6. For maximum efficiency, the I-beam is proportioned to ensure simultaneous web failure and flange yield for strength-limited design and to ensure simultaneous web failure and stiffness constraint satisfaction for stiffness-limited design. Four web failure mechanisms are considered: yield, shear buckling, flexural buckling and compression buckling, giving a total of eight potential failure combinations. Two complete analyses (out of the possible eight) are given, the first strength-limited design flange yield and web shear buckling), and, the second for stiffness-limited design (stiffness constraint satisfaction and web shear buckling).

6.2.1. Relationships between dimensions for strength-limited I-Sections

The stress in the outer fibres of a beam in bending is related to its section modulus Z and applied bending moment A4 by equation (34). The largest fibre stress the beam can withstand safely is identified with the elastic limit, oY, of the material of which it is made, giving the requirement

(137)

The section modulus can also be expressed in terms of the dimensions of the section as

z=!g+$ (138) (ignoring the bending stiffness of the flanges about their own centroids), where Ar is the total area of the two flanges and other variables are defined in Fig. 12(a). Substituting for Ar from A = Ar+ ct and rearranging for A gives

A =z+$t. c 3 (139)

In the literature on the optimisation of I-sections a value is generally assumed for the web slenderness

True optimisation requires that a is not predetermined but has a value determined by web buckling considerations-that is, the web stress must be maintained below the web-buckling stress o,,. The web slenderness, a, is written in terms of c by using the equationt3) for the shear buckling of a plate of modulus E with simply supported sides under a shear


load V (a worst case scenario):

from

cl = !?f and ~2 = va3 kE kE

(140)

where k is a constant. Often, the web is designed to resist shear loads (see, for example, BS 5950). Under these circumstances we find k = 6.89 when Poissons ratio v = 0.3 (Young3, Table 35, Case 4). Substituting for c2 and ct from equation (140) into (139) gives

A = 22 E ( ) 3/2 1 V p+g a2- (141) The section, and thus the mass, is minimised by setting dA/da = 0, giving the optimum slenderness ratio

with K=-&. W

(142)

It is now convenient to introduce a new parameter o defined as the ratio of web area to total area

which can also be written in terms of equations (139)-(141) as

Ku2

w= 22 2Ka2 * - - ~112~3/2 + 3

(144)

Rearranging to extract a gives

y217p17 6w =K3/7 with y =3- (145)

This expression is of the same form as that of equation (142). Comparing the two we find that for the I-beam of least weight

y = 914 (146)

from which

0 = g/14.

The minimum cross-sectional area Amin is found directly from inserting a back into (142) to give

(147)

The shape factor can now be calculated. Substituting expressions for A and a into the ex-


pression for shape factor

4&Z

= f/2~3/14 (148)

14sy* &(C) 0

= (3 + y)32

which for the optimum I-section reduces to

4; = 1.6(f)

again showing that shape factor is determined by web depth/web thickness ratio, c1 and web area to total area ratio. The analysis has reduced the number of variables from three (B/T, B/c and c/t) to one (c/t).

6.2.2. Relationships between dimensions for stiflness-limited I-Sections

Stiffness limited design is analysed in a similar way. Here EZ, rather than Z, is defined by the design. The area of the section becomes

A =$+;ct.

Substituting for c2 and ct from equation (6). 10) gives

(149)

Minimising this with respect to c1 gives an expression for optimal web slenderness ratio a when local buckling of the web and the stiffness constraint are just met

91 I5 a= - 0 K? with K = !- kE, (1W

For non-optimal beams we again introduce the web area as a proportion of total area o as a new variable. Then

KCC2

CJJ = 41 2Ka2 (151)

KCr3+7-

Rearranging for CI gives

(152)

with

60 -- -3-20


Once again, this expression for cz is of the same form as that given for the optimal value given in equation (150); the difference lies in the value of the numeric coefficient. At least weight

y = 912 (153)

giving

w = 9/10

and the minimum cross-sectional area is

A = Ai2j5Kf5 with A = 25( 1 + y/3) y3/ (154)

The shape factor for elastic bending [equation (4)] which we here call 4 has the value

= F = A21:,;;2,5

nyr

=2(1+/3)2

= Rr

where I 8ri

=(6

which, for the optimal I-section reduces to

f#J = 1.13;. (155)

As before, the efficiency can be written in terms of just one variable c/r. The ratio of elastic to plastic shape factor is simply found by dividing the expression in (148) by that in (155) as

(#)pg = (4(3 ; 4) 2(p2. (156)

The gain from shape in stiffness, always larger than that in strength, increases with slenderness ratio c/t.

This analysis has shown that the efficiency of I-beams is identified in terms of one variable a = c/r for a given flange/web mass ratio o. Flanges are proportioned to resist local buckling loads and torsional-lateral buckling failure. Similar results are given below for other combinations of simultaneous flange failure and web failure.

6.2.3. Failure mechanism and mechanism boundaries

In addition to the mechanisms considered already, there are four other failure mechanisms of importance: flange yield, web yield, web flexural-buckling and web-compression buckling. Of these, the first three dominate over most of the design space covered by the failure charts. Flange buckling depends on B/T, but, because I-beam efficiency depends strongly on BT but only weakly on B/T, the value of B/T can be chosen independently to


suppress flange buckling without compromising overall efficiency; we shall not considered it further.

The first failure mode is that of inadequate stiffness. The stress in the flange of the beam carrying a bending moment M is given by

MC

a=21* (157)

The stiffness condition is given by equation (45). Combining equations (143) (155) and (156) gives c in terms of A

et CJj=-

A

from which

c = 5!y! = (dy2= (u/!;)2.

The section area is eliminated by combining equations (157) (45) and (158) to give

(mechanism 1) o] = (($) $Eb(KI)I)4.

(158)

(159)

The second failure mode is that of web shear-buckling considered in Section 6.2*. Defining fi as the ratio of bending moment to shear, M/V = l//3, and substituting equations (155) and (156) into equation (140) and rearranging gives

(160)

Combining this with equation (158) eliminates the section area A. By substituting the resulting expression into (157) gives a value of flange stress at which buckling of the web occurs

(mechanism 2) (~2 = . (161)

The third failure mode is flange yield and occurs when the flange stress exceeds the yield stress

(mechanism 3) ~3 = q. (162)

Web yield occurs when the stress (TV caused by the shear force transmitted by the web exceeds the yield stress G,,

from which

*There may be some considerable postbuckling reserve in the web under shear loading as the redistribution of stresses effectively leads to a diagonal tension field @). The level of postbuckling reserve then depends on the material properties and the support conditions. For consistency between comparison of materials the postbuckling reserve of strength has been neglected in this analysis.


A&& Y

(163)

Combining this with equation (158) eliminates the section area A. Substitute the resulting

expression into (157) o give a value of flange stress at which web yield occurs,

(mechanism 4) 04 = . 064)

In addition to shear buckling of the web under shear load, V, compression buckling of the web can occur due to the component of flange stress acting downwards (curvature effects).

Ignoring shear, beams under a constant bending moment, M, bend into an arc of radius, R.

Then the stress in each flange, 0, is given by

EC

c=2R*

The component of force tending to compress the web is

Fc = aAf. 1 -sm 2R 2 ( )

or, for small angles of bending

where l/2 Af is the area of each flange. Then the compressive stress through the web is

oh.& _ 02Af oW=--Ect* t1/2

Using standard results for the buckling of plates,(3) the web will buckle when

712 t2 a,>-E -

0 12 c

from which

02Af x2 t 2 -=12Ec Ect 0

giving an expression for the stress at failure by compressive web buckling of

(mechanism 5) 05 = (C?($-)($)2(f).

(166)

(167)

For web area/total area ratios between 0.3 and 0.9 the multiplier of E/c#J in equation (165) differs little from unity. This implies that the boundary between flange yield and buckling lies at a shape factor of


which is of the order of 300 for most metals. Buckling of the web under bending loads tends to dominate at small shape factors. It

occurs if the stress at the flange/web interface exceeds the value

(mechanism 6) t=

rs6 = 21.5E ; 0

= 21.5!A=5

(169)

42 .

This is also the stress in the flange.= There is a third buckling mode in which the flange buckles into the web. It is analogous

to the case of a plate buckling into an elastic foundation. Allen gives the value of stress for this as

o7 = 0.51 E13 E;

where EC, the modulus of the foundation, is approximated by

(170)

E,=E-f- B

Substituting equation (171) into (170) gives

(mechanism I) UT = 0.57Q2/3E

(171)

(172)

as an expression for the buckling stress. Typically, depth/breadth ratios (c/B) of I-beams lie within the range 2-3 and for a shape factor of 100 equation (172) gives a buckling stress orders of magnitude greater than the yield stress. Indeed, for most metals the flange wrink- ling/yield boundary occurs at shape factor in excess of 7000. Clearly, other failure mechanisms dominate over the entire range of practical shape factors.

Of the last three buckling modes, mode 6 (web buckling under bending loads) is the only mechanism of potential importance. This allows us to write the tield boundaries in terms of

4 as

(1 - 2 boundary)

(1 - 3 boundary) (j$) = Q(&)3(:)3v

(2 - 3 boundary) ($3 = 4&4*5(~)y-3-$

(1 - 6 boundary) ( >

E Y

(173)

(174)

(175)

(176)


(3 - 6 boundary) E 10

4 = 5.1R - 0 OY (2 - 6 boundary)

w4 El = 0.006,0k3 - ?.

P 0 aY 4

(177)

(178)

(1791

The mass contours are found by substitution in equation (34) as before. The equations for the contours in each of the fields are as follows. Combining equations (158) and (157) and using the definition of shape factor gives

A = (!??&3(~~ (1811

as an expression for section area with a replaced successively by aI, 62, as, a4, as and 66, giving the following results

(regime 1) (2) = ((j$&;;)2 (182)

(regime 2) ($) =($)%&Pi

(regime 3) (z) = (&J3&3(j$-

(regime 4) &) =i(+$)

(regime 5) ($) = (48($)2Q&y4)3

(183)

(184)

(185)

(186)

(regime 6) ($) = (0.24(g) $f2)23. (187)

Fig. 14(a) shows the failure mechanism map for I-beams, with data for aluminium sections superimposed. Interactions between failure modes were based on equation (39), and the chart neglects postbuckling reserves of strength because they do not influence the optimum value of shape factor. The complexity of the chart arises because of the co-existence of two

i.OE-01


1 .OE-03

I .OE-O4

0

2 1 .OE-O5

lfyq

1 .OE-06

--t 1 .OE-O7

i.OE-08

119

Shape Factor 4,

Fig. 14(a)

sub-structures within an I-beam: the web and the flanges. As found with tubes and box-sections, standard sections lie in the stiffness and yield-limited regions, precluding local buckling. The boundaries between fields is complicated by their dependency on an additional variable, w, web-area/total area, and are plotted for two values, o = 0.9 and w = 9114, corresponding to optimal values for stiffness/web shear buckling and strength/web shear buckling.* The web failure mechanisms tend to dominate at large values of shape factor, as would be expected with high slenderness ratios, indicative of local buckling. Data for extruded aluminium I-sections is superimposed on the chart for two stenderness ratios (i/! D = 10 and l/D = 1OO)t. Figure 14(b-c) shows the shift in boundaries as uy/E varies and similar conclusions to those for tubes and box-sections apply here. Figure 14(d) shows the dependency of field boundaries on stiffness constraint, which, as expected, affects the size of the stiffness field only.

*The optimal value of o for other combinations of mechanisms differs little from these extremes. Mere D is the total depth of the beam (D = c + 223.


1 .oE+oo

1 .OE-ol

(b)

l.OE-03

l.OE-04

l.OE-05

l.OE-06

(bending) )

Shape Factor Cp

Fig. 14(b)


(4 1 .OE+OO

1 .OE-Ol

1 .OE-O2

1 .OE-03

1 .OE-O4

1 .OE-05

1 sOE-06

1 .OE-08

1 .OE-1 0

Web 1

buckling 1

iooo

Shape Factor (I Fig. 14. (a) A plot of structure loading coefficient A4/aYP against shape factor 4 for aY/ E = 3 x low3 for I-section in bending. The field boundaries appear as full, bold lines. The contours show the quantity m/Z3 p. Superimposed are standard aluminium box-sections for two slenderness ratios (I/D = l&100). (b) A plot of structure loading coefficient M/Q3 against shape factor 4 for aJE = 3 x 10F3 (mild steel, 1000 series aluminium) for I-section in bending. The field boundaries appear as full, bold lines. The contours show the quantity m/l3 p. (c) A plot of structure loading coeflkient M/a,J3 against shape factor C#I for aJE = 3 x lo- (ultra-high strength steel, GFRP) for I-section in bending. The field boundaries appear as full, bold lines. The contours show the quantity m/Z3 p. (d) A plot of structure loading coefficient M/aY13 against shape factor #J for uy/E = 3 x lo- for I-section in bending. The field boundaries appear as full, bold lines. The contours show the quantitym/Z3p. The shaded bands show the extent the field boundaries shift as the stiffness constraint changes from M = 0.036 to Kl = 0.12 corresponding

to maximum deflection/length ratio of l/l00 to I/333 for a beam in three-point bending.


Web yield/flange yield is the most stringent failure combination in terms of structural efficiency. But, as Fig. 14(a) shows, it is unusual that existing sections are used at such high levels of load factor that web yield becomes a consideration. Indeed, real section data are made to preclude web shear buckling [left of the 2-4 boundary, equation (177)], corresponding to a value of shape factor given by

cp E l/2

Opt z:2.3 G . 0 ww

This is a useful guideline for design of general purpose I-beams. For I-beams with specific applications in mind the optimal shape factor is given by

E j2 (.) M #%5Q 0, Z> Y

6.3. I-Beams in Torsion

Open sections are much less efficient in torsion than closed sections. If efficient shapes to carry torsion are sought, I-sections are not competitors. They will not be considered further.

7. MINIMUM MASS RELATIONSHIPS

For each combination of loading geometry and section, there exists a characteristic relationship between mass, structural loading coefficient and material properties. It is found by substituting the expression for optimal $I back into the expression for mass. The resulting expressions are shown in Table 2. Each is of the form


Table 2. Expressions for minimum weight in terms of structural loading coefficient and material index

Tubes l-2 Boundary 2-3 Boundary

Compres

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Material Limits for Shape Efficiency

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