Materials and cross-sectional shapes for bending stiffness

Materials Science and Engineering, A 163 (1993) 51-59

51

Materials and cross-sectional shapes for bending stiffness J. S. Huang Department of Civil Engineering, National Cheng Kung University, Tainan 70101 (Taiwan)

L. J. Gibson Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139 (USA) (Received October 8, 1992; in revised form November 9, 1992)

Abstract A strategy has been developed for selecting the materials and cross-sectional shapes of elastic beams for bending stiffness. A dimensionless shape factor is used to measure the efficiency of a cross-sectional shape; the maximum shape factor is limited either by local buckling or by yielding. The maximum shape factors for tubes, I sections and sandwich sections have been examined and compared with one another. The optimum cross-sectional shape, which gives the highest shape factor, has been identified for various materials: metals and alloys, glasses and ceramics, and rigid polymers. For a minimum weight beam, the choice of material and cross-sectional shape depends on the solid material properties and the design parameters, including the required bending stiffness, loading configuration and minimum load capacity.

1. Introduction The stiffness of an elastic beam, its resistance to deformation under an applied load, depends on the Young's modulus of the solid from which the beam is made and on the shape of its cross-section. Materials which minimize the weight of the beam for a given bending stiffness are those with a high value of E 1/2/p, where E is the Young's modulus and p is the density [1, 2]. Cross-sectional shapes which minimize weight are those with a high value of the shape factor (1)B e = 4JrI/A 2, where I is the moment of inertia and A is the area of the cross-section [3]. The factor is defined such that a solid circular section has a shape factor of one (Fig. l(a)); sections with higher shape factors are lighter for the same bending stiffness. The shape factor of a thin-walled tube, for example, is the ratio of its radius to its wall thickness r/t; if bending stiffness were the only consideration, the shape factor would be maximized (and the weight minimized) by selecting the tube of the largest radius and thinnest wall thickness. In practice, such a tube would buckle or yield locally at very small loads; in general, there is also some minimum load capacity that the member must resist without failing in some way. As a result, the maximum shape factor of the tube depends on the material properties of the solid from which it is made. In this paper, we have examined the selection of the cross-sectional shape for a flexural member which I)921-5093/93/$6.00

minimizes weight for a given bending stiffness, load capacity and span; tubes, I beams and sandwich beams have been compared. The results suggest that these shapes can give high shape factors; the maximum shape factor for each beam depends on the mechanical properties of the solid material and on the design parameters. For beams made from metals and alloys, glasses and ceramics, or rigid polymers, a maximum possible shape factor for a given bending stiffness can be identified.

2. Maximum shape factor The bending stiffness of an elastic beam, SB, is the ratio of the applied force, P, to the central deflection of the beam, 6:

P SB-6-

B1EI

BIE~BeA 2

~3 -

4jrg3

(1)

where g is the span of the beam and B l is a constant which depends on the loading configuration (Table 1). In eqn. (1), the area of the beam can be expressed in terms of the design parameters and the shape factor:

A=

4 7tSBg -~ 1/2 BIE~B ~

(2)

Using eqn. (2) to represent the area of the beam gives © 1993 - Elsevier Sequoia. All rights reserved

52

J. S. Huang, L. J. Gibson

/

Materials and cross-sectional shapes for bending stiffness

TABLE 1. Constants for bending of beams

factor is:

Mode of loading

B1

B2

B3

B4

Cantilever, end load Cantilever, uniform load" Simply-supported, central load Simply-supported, uniform loada Fixed ends, central load Fixed ends, uniform loada

3 8 48 76.8 192 384

1 2 4 8 4 8

1 2 4 8 8 12

1 1 2 2 2 2

~Be

r

A2

t

(4)

Since A = 2 n r t = 2Zq~Bet2, the thickness and the radius of the tube can be expressed in terms of the design parameters and the shape factor of the tube (from eqns. (2) and (4)):

"P= qg; here q is a uniform distributed load per unit length of

beam.

4~I

t

=[sJ311" LnB,E(~ e)3

(5)

( SB~ 3~)Bel 1/4 r= - ~1E

O

(a)

]

(6)

The maximum shape factor of the tube for a given material is limited either by buckling or by yielding. The required moment for buckling failure is [4]:

(b)

Mbuckling __ Pg B3 _ ( 1 -E v z) rt2

(7)

b

b

hi L

(c) Fig. 1. Cross-sectional shapes of beams.

Here B 3 is a constant which depends on the loading configuration (Table 1), and v is the Poisson's ratio of the solid material. Substituting eqns. (5) and (6) into (7) gives the shape factor (assuming v = 0.3): *B3=0.543EI/5 / ~

yield is:

Myielding -- /19e -- 7~tr2ay B3

g=

(8)

Similarly, the required moment to cause the tube to

(d)

the mass of the beam:

[ 4~SBg3 ] 1/2

-]

p2 ]lj2 J L -?J (3)

The lightest beam is that with the highest value of E ~ a e / p 2 for a given set of design parameters (SB, g and B 1) [3]. The maximum shape factor of a beam, however, is limited by the onset of local buckling or yielding. Different cross-sections might have different failure mechanisms, and consequently different maximum shape factors. We begin by studying the shape factors for tubes, I sections and sandwich sections, for a required bending stiffness. The optimum shape for a beam of given stiffness is then identified to minimize its mass.

(9)

Here Oy is the yield strength of the solid material; failure occurs when the extreme fiber stress of the tube reaches the yield strength. Substituting eqns. (5) and (6) into (9) gives the shape factor for yielding failure:

a 4 (SB3B34~5

(I)Be = Yg~''~ / ~

(10)

The maximum shape factor of the tube depends on the design parameters (SB, P, ?, B 1, B3) and the material properties (E, ay) (eqns. (8) and (10)). The functions a and f of the design parameters and the material properties are defined as:

4 O'y a - - E3

(11)

and 2.1. M a x i m u m shape factor of a tube

For a tubular beam (Fig. l(b)), of cross-sectional area A = 2 n r t and moment of inertia I = Jrr3t, the shape

3

4 5

SB B3 l

(12)

J. S. Huang, L. J. Gibson / Materialsand cross-sectionalshapesfor bendingstiffness The smaller value between eqns. (8) and (10), for a given set of ct and f, is the maximum shape factor of the tube. A design map for the maximum shape factor of tubes is shown in Fig. 2. The maximum shape factor for candidate materials for a tube can be found from Fig. 2. For example, an aluminum tube ( E = 6 9 GPa, a = 7 . 8 x 10 - 3 N m -2) has OB~=318 for buckling failure and q~B~=25 for yielding failure for f = 103 m 2 N-~; the maximum shape factor of the aluminum tube is 25. Commercially available metal tubes have shape factors in the range of 20-70.

2.2. Maximum shape factor of an I beam For an I beam (see Fig. l(c)), with A =2btf+ htw and

53

metrical factors of Q and h/tw. Flange yielding, flange buckling, web yielding and web buckling are the possible failure modes which should be considered.

2.2.1. Flange yielding Since the cross-sectional area of the 1 beam is A =2btf+ htw =(2f2 + 1)(h/tw)tw 2, the thickness of the web can be expressed as (from eqns. (2) and ( 13)):

-647r3Sj3 ( ~ + ~ 1 1 2 ( 2 f 2 + 1 ) 6 ] 1/4 tw= BIE(O,e) 3 ~2 12]

(14)

The required moment to cause the I beam to yield can be calculated from the elementary strength of materials:

I = btfh2/2 + twh3/12, the shape factor is: OB~

/t 2 - 4 ~

M =--B3= Oy b6 h

6+

(o 1)

tf+l tw ] 2 ] [ h tw

=4 r 2+12 (2n+l

(13)

where if2 = btf/h&; b and t~ are the flange width and thickness; h and t,~ are the web height and thickness. The maximum shape factor is related to the geo-

6

= °t Q + -6)ltw)-- tw

(15)

Substituting eqns. (13) and (14) into (15) gives the shape factor:

Oy (SB-'g"B3

o , e = 16~r Q +

(2Q+ 1)-2 ~

(16)

2.2.2. Flange buckling The critical flange buckling stress is [4]:

10 4

°cr=KE/I(1-v2)(b/2]]2,

tf ] J

E-10 3 GPa

(17)

E=I0 2 GPa E=101

GPa

gzl0 0 GPa

I03

E = 1 0 -1 G P a E~I0

2 GPa

10 2

where K is a buckling coefficient depending on the boundary conditions. A conservative buckling coefficient, corresponding to one free and three simplysupported edges (K=0.35), is assumed. Flange buckling occurs when the flange compressive stress exceeds the critical buckling stress; the required applied moment is (assuming v = 0.3):

M=B~=Oc, b t f h + ~ - } : l . 5 4

Q + 6 a2E

t~3

m

(18) 10 1

Substituting eqn. (14) into (18), gives OB~=3.05~

1000-5

10-2 10-1 10 0 DESIGN

10 I

102

103

PARAMETER,

104 f

105

106

107

108

[( :)° Q+

f ~ ( 2 Q + l ) -la

Ihl. . ×[-|\o,

E

/ B~p~ ]

(19)

(m2/N)

Fig. 2. Maximumshape factor of tubes. The lines of higher slope correspond to failure by plastic yielding while those of lower slope correspond to failure by elastic buckling.

2.2.3. Web yielding in shear The required applied shear force to cause the web to

54

J. S. Huang, L. J. Gibson / Materialsand cross-sectionalshapesfor bendingstiffness

yield is: =0

B4

(20)

2

~tw]

where B 4 is a constant which depends on the loading configuration (see Table 1). The shape factor is obtained by substituting eqns. (13) and (14)into (20): 1.34~ Oy2 (SB~p3B42/ OBe--(2ff2+ 1)2 E / ~ }

(21)

yielding and flange buckling. The shape factor increases with the geometrical factor ff~. In practice, 1/3 < Q < 1; the maximum shape factor for flange yielding is obtained by assuming if2 = 1 in eqn. ( 16):

/

e;--7.6p/

8 e4 /

Similarly, the maximum shape factor for flange buckling is obtained from eqn. (19)(if2 = 1 and h/b = 3):

[S 3gSB 4~1/9 2.2.4. Web shear buckling The web shear buckling strength for the boundary condition of four simply-supported edges which gives a conservative buckling coefficient, is [4]:

OBe=N'9EI/9 / ~113~3 )

4.4E rc~-(1_ v2)(h/tw)2

oBe=0.47 Oy2

(22)

Web shear buckling occurs when the web shear stress reaches the critical buckling stress, giving:

dpBe=4.N23r(Q+l)4/5(2Q+l)-2E1/5(~)

~/5 (23)

2.2.5. Design maps of maximum shape factor Figure 3 shows the maximum shape factor for the cases of Q = 0.1, f~ = 1 and f~ = 10 for both flange

<241

(25)

The maximum shape factor for web yielding is

(Q=I):

[SBe3B421

--El Bl P2 }

(26)

The maximum shape factor for web buckling is (~ = 1 ):

( SB•3 B4211/5

OBe=I'7E'/5 k B ~

]

(27)

Since the maximum shape factor of I beams for web yielding and buckling has a different dependence on design parameters and material properties, two new functions of the design parameters and material properties are defined: 2

18 4

fl=ay E

// a - l O -I

N/m s

/

.~

E-100 GPa

/

SBB42g 3

t st

g ,,,'" **w,ww

Buckling ~=i0 ............yielding ~

~

Buckling n=l Yielding

~0 2

m

~

Buckling

Q-O.1 .....

J t

I

Yielding

~=0.I

// /J ,0~0:s.;.;:~;-;:i.;0/!'7; i7; <;; ~T;~~; ~;; ~~; ~;o8 DESIGN PARAMETER,

f

(29)

w , s , "w

a , ' ' ' I ' ~ ~

101

B1p2

e

t 105

(28)

(m2/N)

Fig. 3. Shape factor of I beams with Q = 0.1, 1 and 10 for flange yielding and buckling.

Figure 4 shows the maximum shape factor for the cases of ff~= 0.1, 1 and 10 for web buckling and yielding; the shape factor increases with decreasing f~. A design map of the maximum shape factor of I beams for flange yielding and buckling (eqns. (24) and (25)) in terms of the design parameters, f, and material properties, a and E, is shown in Fig. 5. A design map of the maximum shape factor of I beams for web yielding and buckling (eqns. (26) and (27)) is shown in Fig. 6. For a given material, the shape factor for each failure mode can be obtained from Figs. 5 and 6; the smaller value is the maximum shape factor of the I beam.

2.3. Maximum shape factor of a sandwich beam A sandwich beam is composed of a lightweight foam core and two stiff faces (see Fig. l(d)). Gibson [5] proposed an optimum design for sandwich beams

J. S. Huang, L. J. Gibson

/

Materials and cross-sectional shapes for bending stiffness

55

10 4

/ (/

~=10 5 N/lll 2 E=I00

/

GPa

/

• ;=103 GPa

/

]=i0 ~ GPa =i01 GPa /

i~

~F"

:/.-/

l,i,,,

=i0 e GPa

Buckling

o.o

-10 : GPa =I0 2 GPa

10 2

~f I

s-/I

" I I I I,"

/

/// ,,

..: .:/"

/

/.,' t

°°"

........

..........

o.o

,~,, """ :~

.....

Yieldtng O=0. 1

.:: .:" / .'

10 7 I0 6 10 5 1 0 - 4 1 0 - 3 1 0 - 2 10-1 10 0 DESIGN

--

/ ...'"

I0 I

t °" I °-° °~&° ,o" I °° t

...-'-

/..

I

Buckling

PARAMETER,

g

I0 1 10 2

10 3

10 4

(m2/N)

Fig. 4. Shape factor of I beams with if2=0.1, 1 and 10 for web yielding and buckling.

,v

,~

,~

,v

DESIGN 10 4

,v

,~

pARAMETER,

,v

,u

g

lu

,v I

(m2/N)

Fig. 6. Maximum shape factor of 1 beams for web yielding and buckling.

,i03 G P a ~i02 G P a =i01 GPa

10 3

=10 ° GPa =10 -I GPa =i0 2 G P a

t=O'32g

BIO.42B22 ~tor/

(31) (32)

10 2

where B 2 is again a constant which depends on the loading geometry (see Table 1 ); E,., pf, Es and ps are the Young's modulus and density of the face and the solid core materials respectively. The shape factor of the sandwich beam, ¢~B~=Ztc2/2bt, can be rewritten as (from eqns. (30) and(31)):

m

10 I

*~=90.8

[ B' 3 \PJ TEf iSBb4) ]

(33)

100 Ic DESIGN

PARAMETER,

f

(m2/N)

Fig. 5. Maximum shape factor of I beams for flange yielding and buckling.

giving the core thickness, c, the face thickness, t, and the core relative density, Pc /Ps, which minimize the weight for a given stiffness: c = 4"3g [ B ~ - 2-

El2

(30)

Note that there is no consideration of failure mode in the optimum design and the beam width is regarded as a given design parameter. From eqn. (33), it is seen that the shape factor increases with decreasing beam width. The maximum shape factor, however, is limited either by face yielding, by face wrinkling or by core shear yielding. In this study, b is taken as a variable to obtain a maximum shape factor at which possible failure modes occur for the smallest loading capacity. It is found that the minimum beam width is obtained when face yielding occurs; detailed results are given in

J. S. Huang,L. J. Gibson / Materialsand cross-sectionalshapesfor bendingstiffness

56

Appendix A. The minimum beam width is:

bmin=O.O8P'B,3B2EsEf3 (pfy B35 Oyf5~5 \Ps]

1

10 4

(34)

SB 4

where ayf is the yield strength of the face material. Substituting eqn. (34) into (33), gives the maximum shape factor of the sandwich beam:

10 3

.Be= 329 °y~ {SB3B34g5I

Ef3 / B3p4 ]

(35)

In deriving the maximum shape factor of sandwich beams (eqn. (35)), only the mass of faces and the bending deflection are considered. If the mass of the core and the core shear deformation have been taken into account, an equivalent shape factor should be used in order to make a comparison between the sandwich beam and other beams. It is found that the equivalent shape factor is 1/75 of the shape factor in eqn. (35); a detailed calculation is shown in Appendix B. The equivalent maximum shape factor of the sandwich beam is:

(dPBe)eq=4.4~

{SB3B34~pS'

I B13P4

®m

10 2

,e, d

o

10 1

10~0-3

10-2

10-~

10 0

10 1

10 2

10 3

10 4

10 5

10 5

10 7

10 8

(36) DESIGN PAP./kNETERf

A design map of the equivalent maximum shape factor of sandwich beams in terms of the functions of the design parameters, f, and material properties, a, is shown in Fig. 7.

3. Selection of materials and cross-sectional shapes

From the above studies, it is found that the maximum shape factor of an elastic beam for bending stiffness depends on design parameters (SB, P, g, B1, B2, B3, B4) and material properties (E and Oy); the maximum shape factor for tubes, I beams and sandwich beams are summarized in Table 2. The maximum shape factors of tubes, I beams and sandwich beams can be compared for particular classes of materials. The results are shown in Figs. 8-10. For metals and alloys, the Young's modulus is between 70 GPa and 350 GPa; E 1/9 is between 16 and 19 (N m-a) 1/9. In constructing Fig. 8 we take a value of 17.5 (N m-e) U9. The best shapes for metals and alloys are 1 beams and sandwich beams. For ceramics and glasses, we use the fracture strength instead of the yield strength in maximizing the shape factor as they fracture before yielding. A similar map is constructed and shown in Fig. 9 (assuming E 1/9= 17.5 (N m-2)1/9); again I beams and sandwich beams are the best shapes.

f

(rrt2/N)

Fig. 7. Equivalent maximum shape factor of sandwich beams.

The maximum shape factor for rigid polymer tubes and I beams is shown in Fig. 10; sandwich beams with polymer faces are uncommon and are not considered here. For rigid polymers, 1 GPa< E < 5 GPa; as a result, 10 (N m-2)U9
4. Discussion

For beams in which the maximum shape factor is limited by local buckling, the shape factor might be increased by suppressing local buckling. A tube, for

J. S. Huang, L. J. Gibson

/

57

Materials and cross-sectional shapes for bending stiffness

TABLE 2. Maximum shape factor for tubes, 1 beams and sandwich beams Shape

Maximum shape factor

Tubes

yielding

qb,~'=0-543E"' k

Bl3p4

=0'543E'::f':5

]

buckling

ov4 [S 3/5B 4\ I beams

flange yielding flange buckling

qbB~=8"9E'/'~ k Bi3p4 ] ¢~.~ = 0.47 £

web yielding

(SB ~P3B42/ B , P " /=0.47fig

E k

web buckling

~ B, p2 J

• ~'=l.7E':

4

(qb~)~q - 4.4 ~

Sandwich beams

10 4

,

=l.7El'~g I':

4 S

3

[5. B3 g l . . . .

10 4

I / I

/

i /

I

/ t ,

t

/

,

i

/ l

/

10 3

t

t

,' i

/

o: ,v:

: •

~u

~: ~:

~ :

:

:

"

•

~-

7: u:

~: o:

: .

~ : :

: : :

:

:

:

:

:

• "

:

• -

%: \:

• :

]0 1 1

:

~ :

.:

:

:

: "

: :

:.: ~I ,~"

®m o

: :

10 2

:

: :

: ~:

do

:

:

: : :

: .

........

:

........

:

: :

.. ........

: -

.......

.

~:

-

~-

:

7: U:

......

: .

i f: :: /

I0 1

:

:

,.

Sandwich Beam-Fracture

~:

:

........................

...... ~ B...~.~d~g

f

(m2/N)

Fig. 8. Maximum shape factor for metals and alloys.

I Beam-Bucklinq

i

: ~:

:

:

="

:

:

......

="

I Beam-Fracture

:

...... : ......• .......: ........ • .........................................................

-

PARAMETER,

:

:

..............

'°°,o-. ,o-. ,o-, ,oO ,o-, ,o-. 7o~ 7o~ ?o~ 7o6 7o~ ;o. DESIGN

.:

•

7,:

:

:

:

# • U : o : ~ ,

: :

: :

!!

: :

~. : ~: ~ : 7" ~ " o.." ~.."

i

i

:

: .-"

: :

o. 7,:

:

i

i i

.~

i

i i

:

~m

,e,

i i

i

;

:

i I

t

~'

,

~'

102

11 I

/

'

,

'

#

i

,

,,

i

I

10 3

/

,

/ ,

/ t

iI

iI

10

0- 3

i'OZ_~" 10-_1

i0- 0

DESIGN

i0- T

10 2

PARAMETER,

10 3

f

10 4

10 5

10 6

(m2/N)

Fig. 9. Maximum shape factor for glasses and ceramics.

10 7

TO 8

J.S. Huang, L. J. Gibson

58

/

Materials and cross-sectional shapes for bending stiffness T A B L E 3. Maximum shape factor for beams subject to a dimensional constraint Shape

Maximum shape factor

Tubes

qbae --

I Belm'Bac~llng

I~am-Yieldlng

• mm,,,

Tuba- Buc~lln~

xBiE(rmax) 4 Sag 3

10 3

I beams

Sandwich beams

BiE(hmax) 4 qbBe

8.42SBt3

qbBe = 75(qbae)eq = 0.266

Bi Ef( cmax)4 SBL 3

10 2

Note: rmax is the maximum radius of a tube; hmax is the maximum height of an I beam; and Cmax is the maximum core thickness of a sandwich beam.

c .,./ 10 1

around 1. The charts indicate that the shape of the cell walls could be dramatically improved (and the efficiency of the cellular material increased) if the shape of the cell walls could be controlled. I section cell walls may be difficult to manufacture. Tubular and sandwich cell walls can be made, however.

2: :

~:

: 7:

10010-3

i0--2

I0-I

I0 0

DESIGN

10 1

10 2

PARAMETER,

10 3

f

10 4

10 5

10 6

10 7

10 8

5. Conclusions

(m2/N)

Fig. 10. Maximum shape factor for rigid polymers.

example, may be filled with a foam of high enough density to suppress the buckling failure; the maximum shape factor is then only limited by yielding (eqn. (10)). However, it can be shown that sandwich beams are more efficient than foam-filled tubes in a minimum weight design for bending stiffness (from eqns. (10) and (36)). The maximum shape factor of a beam could be limited by manufacturing and dimensional constraints in addition to the failure modes of local buckling and yielding. Steel beams with shape factors of 30 are common. But such high shape factors cannot be achieved in wood: in practice, manufacturing techniques can set an upper limit to the shape factor. There may also be constraints to the maximum size of the beam; dimensional constraints should be taken into account in maximizing the shape factor. Table 3 lists the maximum shape factor for beams subject to a dimensional constraint. The design charts (Figs. 6, 8-10) are useful in the selection of cross-sectional shapes for a given set of design requirements. They can also guide the microstructural design of cellular materials, which deform primarily by bending of the cell walls [6]. Currently available foams have cell walls which are triangular or rectangular in cross-section; both have a shape factor

The selection of a material and cross-sectional shape for the minimum weight design of a beam of a required bending stiffness has been considered. A dimensionless shape factor is used to measure the efficiency of a cross-sectional shape. The maximum shape factors for tubes, I beams and sandwich beams, limited by the onset of local buckling and yielding, have been examined. The analysis suggests that the maximum shape factor of each beam depends on the design parameters and the mechanical properties of the solid material from which the beam is made. A series of design charts of maximum shape factors for various materials, including metals and alloys, glasses and ceramics, and rigid polymers, has been developed. The optimum shape for a given material can be easily identified from the design charts. Acknowledgment Financial support of the Army Research Office Program in Advanced Construction Technology (Grant Number D A A L 03-87-K-0005) is gratefully acknowledged. References 1 J. A. Charles and F. A. A. Crane, Selection and Use of Engineering Materials, Butterworth, London, 2nd edn., 1989.

J. S. Huang, L. J. Gibson / Materialsand cross-sectionalshapesfor bendingstiffness 2 M. F. Ashby, On the engineering properties of materials, Acta Metall.,37(1989) 1273-1293. 3 M. E Ashby, Materials and shape, Acta Metall. Mater., 39 (1991) 1025-1039. 4 R. N. White, P. Gergely and R. G. Sexsmith, Structural Engineering,Vol. 3, Wiley, New York, 1974. 5 L.J. Gibson, Optimization of stiffness in sandwich beams with rigid foam cores, Mater.Sci. Eng., 67 (1984) 125-135. 6 L. J. Gibson and M. F. Ashby, CellularSolids: Structureand Properties,Pergamon, Oxford, 1988.

Substituting eqns. (30), (31) and (32)into (A2), gives

Pf,,-

b,13 x ~

0.4B,3B:

\,0d

bfw=

\Pd ~ }

(A8)

0.914g 3B33ps2SB4 x 3 2

(A9)

P B~B[ pf E.

If bfy/bfw < 1, the minimum beam width at which face yielding occurs will give the maximum shape factor of the sandwich beam. From eqns. (A7) and (A9), the ratio bfy/bfwis obtained:

.{BIB20,414(p__f]4(Ks]2 (gf)5

bfy bf~-3"4°k

{ G"-v'3/t~3°clTjtzf .

.

.

.

.

.

.

tzs k Ps ]

[,, ,\3/~ P~=O.31B4bcaysl~,, )

(A2)

(A3)

1 l,

~]

The maximum beam width at which face wrinkling occurs is obtained from eqn. (A8):

Appendix A Three different failure modes are considered: face yielding, face wrinkling and core shear yielding. For each possible failure mode, the maximum applied load is related to the geometrical parameters and material properties:

59

~

) \Pd ~E,} \ayZ!

(~_)8

(A10)

(0.4BIBz/B32) < 8.5, pf ~--p~, Es/E f ~ 0.02, Ef/Oyf~lO00 and 6/g =0.01; the ratio bfy/bfw is less

Here, 1.2 < than one.

Appendix B where Oyf and O'ys a r e the yield strength of the face and the solid core respectively. In order to determine the dominant failure mode which gives the minimum beam width, comparisons between these three possible failure modes are made. The ratio of the loading capacity for face yielding to that for core shear yielding is:

The deflection owing to shearing deformation is twice that owing to bending deformation for the opti-' mum design [5]; the stiffness of the sandwich beam can be expressed in terms of the applied force and the deflection owing to bending deformation:

Ply --

SB .

B3( t / ~ )O'yf

(A4)

P,~ 0.31B4Oy~(p~*/p~)3/2 Substituting eqns. (31 ) and (32) into (A4), gives

P~,

~

\Or's] \tof/\Ef]J

Here, 0.33 < (B3-Bz/B 42 Bl) < 1.6, Oyf ~'~ Oys, Ps "~'Pf and Es/Ef= 1/50; the ratio Pfy/Pcs is less than one. Therefore, face yielding occurs before core shear yielding. Substituting eqns. (30) and (31 ) into (A1), gives:

P'v= l'38gB3°yt b~/5

SBa

(p~]2 1 ]~/s 0.4B 3B2 \Pfl ~ }

(A6)

The minimum beam width at which face yielding occurs is obtained from eqn. (A6): bf,.•

O.08PSB,3B2E~Ef3 (pf]2 1 ~

ILJ'~ Oyf ~"

"

\Ps]

SB 4

(A7)

P

Bl EfI

. . . 3(~bending 3~,3

Bj Ef

~

12~,3 ~B AI=

(B1)

where Af is the cross-sectional area of the faces. It ts also found that the mass of the core is four times that of the face in the optimum design of sandwich beam [5]. Hence, the mass of the sandwich beam is m = 5pfAfg. The mass can be rewritten in terms of the design parameters and material properties from eqn. (B 1 ):

[4;rSBgS]~/2175pf2 ]l/2 m = L ~

j

LE,¢, el

(B2)

Note that there is a difference between eqns. (3) and (B2). In order to make a comparison between sandwich beams and other beams (tubes and I beams), an equivalent shape factor of sandwich beams is introduced and defined as: (IDBe

(~J)~q - 75

(B3)