Upper and lower bounds for the shear stress in the Saint-Venant theory of flexure

Journal of Elasticity, Vol. 6, No. 4, October 1976 Noordhoff International Publishing Leyden Printed in The Netherlands

Upper and lower bounds for the shear stress in the Saint-Venant theory of flexure

LEWIS W H E E L E R and C O R N E L I U S O. H O R G A N Department of Mechanical Engineering, University of Houston, Houston, Texas 77004, USA

(Received June 1975)

ABSTRACT

Upper and lower bounds are derived for the shear stress as it is determined by Saint-Venant's theory of flexure, and used to establish the asymptotic character of the classical Strength of Materials formula in the limit of vanishing thickness.

RI~,SUMI~

On d6rive des limites sup6rieures et inf6rieures des contraintes tangentielles suivant la th6orie de la flexion de Saint-Venant, que l 'on utilise aux fins d'6tablir le caract6re asymptotique de la formule de la R6sistance des Mat6riaux darts le cas limite d 'une 6paisseur extr~mement petite.

1. Introduction

The purpose of this investigation is to arrive at upper and lower bounds on the maximum shear stress in the Saint-Venant theory of flexure. Attention is directed exclusively at the case when the load is parallel to a principal axis and passes through the center of flexure, so that the stresses receive no torsional contribution. Particular emphasis is placed upon cross sections of uniform thickness having symmetry about an axis perpendicular to the load.

Our pr imary interest is in supplying an asymptotic characterization for the classic Strength of Materials result for shear stresses in slender beams. We establish that the value t o given by Jouravsky 's formula 1

PQ to = I h ' (1.1)

in which P denotes the load, (~ the area moment about the axis of symmetry of the part of the cross section lying to one side of this axis, I has its customary meaning, and h stands for the thickness, furnishes a lower bound on the max imum shear stress and

1 This formula is found in most elementary Strength of Materials books. See, for instance [1, p. 230].

Journal of Elasticity 6 (1976) 383403

384 L. Wheeler and C. O. Horgan

at the same time an asymptotic approximat ion appropriate to the limit h ~ 0. The fact that t o itself is a lower bound on the maximum shear stress is shown very easily in Section 4. The major part of the paper is devoted tt, obtaining an appropriate upper bound. Although the results are somewhat complicated, the analysis used to establish the asymptotic character of (1.1) may be used to get a bound on the error. This procedure is illustrated by considering as an example the case of a rectangular cross section.

The work reported here has many points in common with the work on St.-Venant torsion given in [-2, 3], where the asymptoti~ character of well-known approximations was established with the aid of upper and lower bounds, and explicit error bounds were derived. It is clear that the results of the present paper can be combined with those on torsion to get upper bounds for the case where torsion is present, provided the center of flexure can be determined or suitably estimated. In the latter connection, the results of Payne [4] appear to be promising. Although it is certain that an upper bound can be found by synthesizing the findings of [2], [4] and the present paper, it remains to be seen whether they lead to interesting approximations.

2. Formulation and mathematical preliminaries

Consider a beam in the form of a right cylinder or prism of length L, and let (x, y, z) stand for rectangular cartesian coordinates in a frame situated such that one end of the beam lies in the plane z = 0 and the other in the plane z = L. We assume that the x and y axis are principal for the section z = 0, and since we are concerned exclusively with bending in the absence of twisting, we take the origin to lie at the center of flexure for this section. In order to speak of the cross-section without ambiguity, we will let this term apply to the domain D occupied by the interior of the projection of the beam onto the plane z = 0. We assume D to be simply connected.

For the case where the load, P, acts parallel to the x-axis, Saint-Venant's theory z gives for the normal stress components,

P txx = t,r =- O, tzz = - ~ ( L - z ) x , (2.1)

where

I = fDX2dA.

As for the components of shear stress, we have

tx, =- O, t= --- 7 ~b , - , tz' = - 7 4~, x, (2.2) 3

where ~b = 4~(x, y), the stress function, is determined to within an additive constant by

2 See, for instance, Chapter 11 of the book [5] by Timoshenko and Goodier. 3 The subscripts preceded by a comma refer to partial differentiation.

Upper and lower bounds for the shear stress in the Saint-Venant theory o f f lexure 385

V A~ = y on D,

l + v

81~ ) X 2 dy

8s 2 ds on 8D,

(2.3)

in which A denotes the Laplace operator in the variables (x, y), v stands for the Poisson's ratio, and s designates arc length measured along the boundary, 8D, of the cross section D. We assume from here on that

e ~ 3 ( D ) 63 ~1( /3) . (2.4)

In other words, • is taken to be three times continuously differentiable on the (open) domain D and once so on its closure /3. The reasons for assuming this degree of smoothness will become apparent shortly.

The relations (2.2) and (2.3) suggest that our goal of finding bounds for t~ and tzy might be reached by getting bounds on the derivatives of solutions to a suitably general class of boundary-value problems. We turn now to a result intended to serve this purpose.

THEOREM 1. Let F c c~3(D) ~ q~l(/3), let Z ~ ~3(/3), and assume

A F = f on D, F = Z on 8D.

Suppose co E 'N2(D) c~ c~1(/3), and

Aco= 1 on D, co = 0 on 8D.

Then, on D,

IF, ~l < Iz, ~1 + N sup IVo I + Nxlcol, 0D

If,,] _-< IZ y l + g s u p IVcol+N,,Icol, 0D

where

N = s u p I f - A z l , N x = sup [ f , , x - A Z , ~ l , D D

(2.5)

(2.6)

(2.7)

N, = sup If , , -AZ, ,I . (2.8) D

PROOF. We present here only the proof of the first of (2.7), since the second has a completely analogous proof. Let

0 = F - z + N c o on /3.

In view of (2.5), (2.6) and (2.8)

AO = f - A z + N >= 0 on D

0 = 0 on 8D.

Accordingly, the maximum principle implies for the normal derivative of 0 in the outward direction,

386

S0 - - > 0 on 0D. 0 n -

Therefore,

~n ( F - x ) => -U--on

L. Wheeler and C. O. Horgan

on ~D. (2.9)

The same reasoning applied to the function - F + )~ + N o gives

0 &0 ~ ( F - Z) < N ~nn on 0D.

Since F - X and o~ are constant on 0D, we infer from (2.9) and (2.10),

IV(F-z)I < N s u p ]Vo)] on 0D. 0 D

Consider now the function

go = F , x - Z , x + N x C O on /).

By (2.5), (2.6) a n d (2.8),

Ago = f ~ - A z ~ + N ~ > 0 on D.

Therefore, the maximum principle and the vanishing of ~o on 0D furnish

F, x - Z, x + NxC° < sup (F, x - Z, x) on D. 0D

Similarly, the choice

go = -F,x+)~,x+NxCO

leads to the inequality

F x - )~ x - N x c° > inf(F x-)~,x). ' ' ~ D '

Combine (2.12) and (2.13) to get

]F,x-Z,x] <= sup f x-)~,:, +Nxlc°[ on D. OD

Therefore, and since

sup~F,x-Z,x] < sup [V(F-)0[, OD OD

(2.11) implies

[F,x-Z,x[ < Ns~ p ]Vco[+NxlCO[, O D

(2.10)

(2.11)

(2.12)

(2.13)

from which the desired conclusion follows at once. The foregoing theorem sets up a scheme for getting bounds on the derivatives of the

Upper and lower bounds for the shear stress in the Saint- Venant theory of flexure 387

solution to a Dirichlet problem for the inhomogeneous equation AF = f. The domain D need not be simply connected for the theorem to apply. Of the two auxiliary functions co and X that enter, the first plays the role of a "shape function", i.e., it is determined by the domain D. Its properties (2.6) are closely related to those possessed by barrier functions. The usefulness of such functions in getting gradient bounds appears to be well known (see e.g. [6], p. 343). We will turn shortly to some results on co and its gradient for strip domains. The function X is a ~g3 continuation of the boundary-values of F, and will be dealt with in the next section.

In view of (2.2), (2.3), the application of Theorem 1 to t=y is straightforward. On the other hand, the term px2/(21) in the expression for t=~ may lead to an undesirably crude bound if Theorem 1 is applied in the obvious way. Thus, in connection with tz~, rather than attempt to utilize a bound on [q~,y[, we prefer to introduce the function T defined by

xZy T = ~ - - on /). (2.14)

2

From (2.2) there follows

P t== = 7 T , (2.15)

and the first of (2.3) implies

1 A T - y on D. (2.16)

l + v

Turning now to the results on co, we assume that D is a strip domain as indicated in Figure 1, and introduce a system of curvilinear coordinates (a, 2) consisting of arc length

¢=1

c\

I y

X = - h / 2

h i

x

Figure 1.

388 L. Whee le r and C. O. Horgan

along the midcurve C and the distance of a point from its project ion onto C. We assume that C is a simple arc of class ¢g4, so that

C = {(x ,y ) lp(a) = x e x + y e y , 0 <- a < I}, (2.17)

where ex, ey denote unit base vectors, and the posi t ion vector p is in ¢g4 [0, lJ. This degree of differentiability ensures that the curvature k(o-) of C is in class oK2[0, l]. Addi- tionally, we conclude that (o-, 2) represent coordinates in an orthogonal curvilinear system which is free of singular points if

tch/2 < 1, where tc = max ]kl (2.18) [0, t]

and h stands for the thickness of D. The posi t ion vector x of points in D is related to points (o-, 2) in the rectangle

R = (0, l) x ( - h / 2 , h/2) (2.19)

through

x = p(o-) + 2v(o-), (2.20)

where v denotes the unit normal vector to C,

dp v = (e x x er) x v, v = - - . (2.21)

do-

The Frenet formulas for plane curves (see [7]),

• ' = kr, v' = - k~, (2.22)

which are valid in the present context, are useful in establishing the or thogonal i ty of the curvilinear system associated with the coordinates (a, 2).

In terms of the coordinates (o-, 2), the equat ion

A c o = 1 on D

takes the form

Y[oo] = p ( o - , 2 ) ~ + q(a, ~)~2-J = 1-2k(o-) for (a, 2 ) e R , (2.23)

where 1

p(a, 2) - q(o-, 2) = 1 -2k(a ) . (2.24) 1 - , ~ k ( o - ) '

The coefficient functions p and q are in class c~2(/~) since k ~ ~2[0, 1], and satisfy

1 P > =- Po, q >~ 1 - h~c/2 = qo on /?. (2.25)

1 + h~:/2

We assume that (2.18) hold, which guarantees that the opera tor 5o is uni formly el l ipt ic

on the rectangle /~. A s t ra ightforward applicat ion of the max imum principle gives

0 < - co < - (1 + h~c/2)go, (2.26)

Upper and lower bounds for the shear stress in the Saint- Venant theory of jlexure 389

where c5 is the solution of the problem

•[(5] = 1 on R, c5 = 0 on OR. (2.27)

We may apply the results of a recent paper [8] to get a suitable bound on (5, and as a consequence of (2.26), for co. Let R be part i t ioned into the three domains -@i:

~1 = {(a, 2)10 < o" < h/2, 121 < hi2-a},

~3 = {(~r, 2)ll-h/2 < a < l, [2[ < ~r-l+h/2}, (2.28)

~2 = R - ( ~ 1 w ~ 2 ) "

The domains ~1 and ~3 are bounded by isosceles triangles 4 having in the case of ~1 the end o- = 0 as a side and in the case of ~3, a = l as a side.

Let a and b be positive constants such that

(qo; } a < rain {Po, qo}, b > max sup IP,~J, sup Iq.)j , (2.29) R R

where

p = sup p, g /= sup q, (2.30) R R

and consider the function w defined o n / ~ by /a a + g [e <b/")(h/27 ~ - ehb/2a], (a, 2) ~ ~1

bw(o-,2)= - [ 2 [ + h / 2 + (e(b/~)lXt--ehb/2~), (a, 2) S N a (2.31)

a _ ehb/2a]~ l-- o" + ~ [e (b/~)(h/2 + ~- 0 (a,)~) e ~3"

This function vanishes on 0R and satisfies

£~°[w] > 1 (2.32)

on each of the subdomains Nz, but fails to be differentiable on the parts of 0 ~ 2 that lie within R. If it were not for this lack of differentiabitity, one could conclude at once from the minimum principle that

w < (5 on R. (2.33)

In Section 3 of [8], we demonstra ted with the aid of Li t tman's results [9] on weakly 2 ' - subharmonic functions that (2.33) is true. Combining (2.26), (2.31), and (2.33), we arrive at

1 ( ,¢h)(aem,/2~_a ~) lcol ~ ~ 1 + 2 ) \b b -- f2. (2.34)

We assume I > 2h.


Another term on the right-hand side of (2.7) for which a bound is needed is the supremum over 0D of [Vc0[. A suitable bound, which is easily deduced from the results of [2], is

h teh 2 sup IVcol < + = F. (2.35) oD = ~ 4(2 - tch)

From [8], Section 3, we get an alternative bound

sup IV(o[ =< (i + h~c/2)(e hb/2a- 1)/b. (2.36) OD

Both of these bounds have the feature that they imply

h sup IV~ol < + o(h) as h --+ 0, (2.37)

which is important to our present objective, but (2.35) is somewhat simpler. It is similarly of importance that the bound ~2 on leo[ given by (2.34) conforms to

(2 = o(h) as h ~ 0. (2.38)

3. U p p e r bounds for Itzxl and Itzyl

In attempting to construct suitable auxiliary functions )~ for use in connection with Theorem 1, we have found it expedient to assume that the domain D is symmetric about the y-axis. We are thus interested for the remainder of this investigation with domains like those of Figure 2.

_--y

. . . . . . . ~=0 X

F i g u r e 2.

Our immediate task will be to express the boundary-values of the stress function ~b in terms of the variables a and 2. F rom (2.3), there follows s for 0 _< a _< I

5 W e t a k e ~ (0 , - h/2) = 0, t hus f ix ing the a d d i t i v e c o n s t a n t .

Upper and lower bounds for the shear stress in the Saint-Venant theory o f f lexure 391

q~(a, - h/2) = ~ [x(~, - h/2)] 2y, ~(~, _ h/2)d~.

Therefore, and since D is symmetr ic about the y-axis,

q~(l, - h/2) = O.

Consequently,

= Ix(l, t/)]2y,,(/, ~l)d~l ( - h / 2 < 2 < h/2).

Similarly,

~b(a, h/2) = ¢(l, h/2)+ ~ [x(~, h/2)]2y, ¢(~, h/2)d~ (0 < a < l),

1 fl)2Ex(O ' 4)(0,)~) = ¢(1, h/2)+ ~ t/)]2y ,(0,~?)d~/ ( - h / 2 < ). < h/2).

In view of (2.20), (2.21) and (2.22),

x, ~ = ( 1 - ; t k ) ~ , x , = v .

Therefore, and by (2.20), we may write (3.1) as 1; q~(a, - hi2) = ~ [px(#.) - hv~(~)/212[1 + hk(~)/2]'cy(~)d~

whereas (3.2), (3.3), and (3.4) give, respectively

1 f_h/2[Px(1)+

q~(a, hi2) = eP(1, h/2) + ~ [px(~) + hv~(~)/2l 2 [1 - hk(~)/2]zy(~)d~,

1i/ [p~(0) + t/v (0)] 2vy(0)&/. eP(O, 2) = eP(l, h/2) + ~ /2

Let

(3.1)

(3.2)

(3.3)

(3.4)

(3.5)

(3.6)

(3.7)

(3.8)

(3.9)

i f ~ = [px(/) + t/vx(/)] ;Vy(1)dtl for - h/2 < 2 <_ h/2. (3.10)

Since vx(O) = -v~(l), vy(O) = vy(l) and px(O) = p~(l), a change of variables leads to

lfh~ = [px(0) + ~v~(0)]2v,(0)dq + 9(h/2). (3.11)

Accordingly, and in view of (3.7), (3.9), (3.10),

• (l, ,~) = ~b(0, 2) -- g(2). (3.12)


Define

X(O-, 2 )= ~(~ -2)~b ( 0 3 - ~ ) + ~(~ + 2 ) [ ~ (a, ~ ) -g (h /2 ) ] +g(2) (3.13)

for every (~r, 2)e/~ = [0, I] x [-h/2, h/2]. It follows from (3.6), (3.8) and (3.12) that

(3.14) Z = ~ on 0D.

With a view toward casting X in a more convenient form, we define

Q(a) = [" xdA (0 <-or<- 1),

where ~ is the subdomain

{ ( ~ = xlx = p(~)+2v(~),(~,2)e(~r,/)x - ~, .

Since the centroid lies on the y-axis, we have

Q(0) = O.

Further, it is clear that Q has the properties

Q < 0 o n [0,1], Q$ on I0 ,~] , QT on I ~ , l ] ,

max IQI = Q(I~ =-Q_., [0,1/21 \ z /

and because of the divergence theorem, may be written in the form

,£ Q(a) = ~ Xanx ds,

where n~ denotes the x-component of the outward unit normal. It follows from the symmetry of D about the y-axis that

_ h 2 f i [ x ( { , ~)] Y.e.({, -h2)d{ : 0 .

(3.15)

(3.16)

(3.17)

(3.18)

(3.19)

(3.20)

(3.21)

Therefore, and since n x = dy/ds on ~ , (3.20), (3.1), (3.3) and a change of variables yield

Q(a)= ~b (a, h 2 ) - ~ (o',-h2)+½q(a), (3.22)

where

~ -h/2 q(~) = [X(~, ~)]2y ,(G, ~)&.

,dh/2

From (2.20), there follows

~ -- h /2

q(o-) = [px(cr) + t/Vx(O-)] 2vy(o-)d~. (3.23) ,Jh/2

Upper and lower bounds for the shear stress in the Saint- Venant theory o f f lexure 393

As one may readily verify with the aid of (2.21),

~ = Vy and "cy = - v ~ on [0,1], (3.24)

and as a result, (3.23) furnishes

{ h2 } q(a) = - h'cx(a ) [px(a)] 2 + ~-~ I-v~(~r)] z (3.25)

upon carrying out the integration. Combining (3.22) and (3.13), we get the desired expression for Z,

1 Z ( ~ , 2 ) = ~ [ ~ ( a , ~ ) _ q ) ( ~ , _ ~ ) l + ~ [ Q ( ~ ) _ ½ q ( ~ ) ] + g ( ~ . ) - 1

In view of (2.2), (2.3), (2.34), and (2.35), Theorem 1 yields

(3.26)

I ! I%1 = [ - ~ l ~ Ix x + MrF + MrY2, (3.27)

where

My sup IAz,~I, (3.28) D

sup v M r = A Z - 1-~vv y ,

D

and ;g is given by (3.26). As we shall see in the sequel, the term I){ J is dominant in the behavior of t~y for small values of h. We turn now to the problem of expressing this term in explicit form. By the chain rule,

Z,:, = Z ,a , x+ Z,z)~,,,.

From (2.20), (2.21), and (2.22) there follow

"/7 x 27y

a'x = 1 - 2 k ' '~'x = vx' a'r - 1 - 2 k ' "~,r = vy,

so (3.29) and (3.24) furnish

Z,~ - l_2/~Z,~+vx)~ z,

which reduces the problem to one of calculating )~, and Z, z-

By (3.13), (3.6) and (3.8),

h z

Z,~ = 2 1 h \ 2

h 2

which, after a brief computation leads to

(3.29)

(3.30)

(3.31)

(332)


PZ~Zr +½zy{2[2Pxv _k(2p~+h2_~) j h2v~ (3.33)

F rom (3.26) there follows

and (3.10) yields

i f ' = l [ px ( l ) -~- ,~l~x(1)]2ljy(1)

= h h z

Therefore, and by (3.25),

Q z~f2 h 2 = -}- - - V 2 -~- Z;., ~ -~-kPx+ 12 ~)v~(1)v '( l )[a2px(l)+(22-h~--~)v~(1)] .

Combining (3.31), (3.33) and (3.35), we get

Q v~ + y(o-, ;.),

where

(3.34)

(3.35)

(3.36)

J(o, 2) -- 2kp~ z~v~, k 2 2(l_2k) Z~zy+ 2(l_2k) {2[2p~vx- ~(2P.~+ h~'2--)]

h2v ~ - 1 % + - - ~ ,~

+ -2-- 24

2vxv ~(t)vr(l ) h 2 + 2 [2p~(/) + 2vx(/)] - ~ [v~(1)]2v,(1)v~. (3.37)

Turning now to It=l, we see from (2.14), (2.15), (2.16), (2.35), (2.36) and Theorem 1 that

I r ~t t~ t = f~rl --< ~P , I + M , +M~O, (3.38)

where

M x = sup IA(p- ff, M• = sup IAcpy- f , I , (3.39) D D

1 x2y f = 1 + v y' q~ = Z-- ~ - , (3.40)

and Z is given by (3.13). As in the case of It~v], the leading term on the right in (3.38) is dominant. F rom (3.40), (3.24), (3.30),

27y X 2

q0y = l_21¢Z,~+%Z,~ 2 (3.41)


Therefore, and by (3.33), (3.35), (3.24), (2.20),

Q ~o,y = ~ vy + K(a, ,~),

where

(3.42)

2k K(o, 2) - 2G (2 G + 2G) + -2"c2 2 2(1 - 2k) p~ y

2 k 2 + z ~ { 2 [ 2 p , ~ V x - ~ ( 2 p , ~ + h g ~ ) J + h 2 v x ( vx 2 ( 1 - 2 k ) 2 , 2 - k p x / ,

h2% 2z ~G(l)vy(1) + ~ - {%v~ - [G(/)] 2vy(l)} + 2 [2G(/) + 2vx(/)]" (3.43)

Combine (3.27), (3.38), (3.36) and (3.42) to get

-~~< ~ + ~ + M x r + M ' x e + h x = -- +J+MyF+My ) ~ ,

where

Y - sup I J] and / ( = sup ]K]. (3.44)

Since Iv] = 1, this gives

I ~ < ~ + [g + Y+ (M~ + M,)F + (M" + M;)~]

, 2} ~- + (K + MxF + M2Q) 2 q- (Jq- MrF + MyO) (3.45)

Accordingly, and in view of (3.19), there follows

[Y+ + (M x + M,)r + (M'~ + M'y)(2] 2h

t/t o <__ 1 + -~ K

h2 }~ t 2 + ~ [(/( + M S + M'xf2) 2 + ( J + MyF + Mr2) ] , (3.46)

~g

where

P~ t = sup ~ , t o - (3.47)

D Ih

The behavior of(3.46) as h ~ 0 is an impor tan t part of our objective, and in particular, we would like to have

lira ( t ~ < 1 (3.48) h~O \to~

as a consequence of (3.46). This feature is readily confirmed with the aid of the order condit ions


h/Q = O(1), (3.49)

Y = o(1), R = 0(1), (3.50)

M x = O(1), M, = 0(1), M'x = O(1), M, O(1), (3.51)

F = o(1), f2 = o(h) (3.52)

as h --+ 0. The assertion (3.49) is easily demonstrated on the basis of(3.19), (3.15), whereas (3.50) follow at once from (3.37), (3.43). As for (3.52), these may be established by means of (2.34) and (2.35) without much difficulty. On the other hand, (3.51) are much more involved, and as a result, the calculations have been relegated to the Appendix. It is worth mentioning that all of the foregoing order conditions - and hence the right-hand side of (3:46) - can be cast into fully explicit form by means of straightforward calculations, but the resulting estimates are too unwieldy to warrant their inclusion here.

4. Asymptotic characterization of the Jouravsky approximation

From (2.2), (2.3), there follows

Px t . . . . q-tzy, y - I on D (4.1)

which is of course, just one of the equations of equilibrium upon which the Saint-Venant formulation is based. Integrating both sides over @,, and applying the divergence theorem, we get

fo (tz~nx + tzrnr)ds - (4.2) PQ(a)

~ I '

having used the definition of Q(a) in (3.15). Since

tzxn x + tzyny = 0 on ~D,

(4.2) yields

f_h/2 [tzx(a, 2)Zx(a)+ t~,(a, 2)zy(a)]d2 - (4.3) PQ(a)

hi2 I

for 0 < a < I. Accordingly, and in view of (3.47), we have

PQ(a) - - - _ < t for 0 < a < l . (4.4)

Ih -

Therefore, and since Q < 0 on (0, 1), (3.19) and (3.47) furnish

t > t o. (4.5)

This, together with (3.46), implies

{ , 2h [ j + R + (M~ + M,)F + (M'. + M,)f2] l <=t/to<= 1+~


h2 + Qy [(K + M~F+ M'Y2) 2 + (J+ M f + M',O) 2] ,

and as a result of (3.47) and (3.49)-(3.52), it is seen that

P0 t = ~ - + O ( 1 ) as h ~ 0 .

(4.6)

In the present generality, an explicit form of (4.6) would be extremely complicated and not particularly revealing. An example where a manageable form can be arrived at is furnished by the case where D is the rectangle

D = (x, 2 < x < 2 , 2 < y < .

In this case we have

~ t/2 hl 2 Q = h xdx - (4.8)

,J0 8

Furthermore, k - 0, and as a consequence, (3.44), (3.37) and (3.43) yield

J = / ~ = 0. (4.9)

With the aid of (A.1), (A.3), and (A.17), which appear in the Appendix, one finds

Mx, My < l+h+ 2(1 +v~" (4.10) hv

In view of (A.2), (A.4), (A.5), (A.26) and (A.31), there follows

v - - M; < 3. (4.11) M' x < 3 + l + v

The factor F appearing in (4.6) is simply

h F = 2 ,

as one can see from (2.35). As for ~2, (2.34), (2.29), (2.30), (2.25) and (2.24) yield

l[aehb/a" a h21 (2(a, b) = ~ ~ b

for every a e (0, 1] and every b > 0. Choose a = 1, and pass to the limit as b ~ 0 to get

(2 = h2/8. (4.13)

Substitution from (4.8)-(4.13) into (4.6) yields

1 < __< 1+84 2(1+5)+ +252 6 + to

I 3v5 ] 2 [ 2v5 3512} ~, (4.14) +52 4(1+5)+ l + v +35 +51 4(1+5)+ ~ +

(4.12)

(4.7)

398 L. Wheeler and C. O: Horgan

where

= h/l.

Tak ing v to lie in the range

- l < v < ½ ,

we see tha t

(4.15)

V

l + v < ½ •

Therefore, and because of (4.14), we get

t 1 = - - < { 1 + 8e[2(1 + ~) + ½~] + 2~2(6 + ½)

to

+ e214(1 + e) + e + 3el 2 + ~214(1 + e) + 2e + 3el 2}~,

which m a y be wri t ten as

t 1 < - - < f (e) = {1 + 8 e ( 2 + ~ e ) + ~ e 2 + 16e2(1 +2e)z+ez(4+~3e)2}~.

to

A brief table of values for f(e) appears below.

f(e)

10 4 1.00079 10 -3 1.00799

5 × 10 -3 1.03999 10 -2 1.08002

5 × 10 2 1.40526 10 -1 1.83601

Appendix

Appendix. Verification and explicit expressions for (3.51).

F r o m (3.39) and (3.40), we have

IAzI + v Mx < sup - - s u p [y[, o l + v D

Y M'x < sup IAz, r ]+

D l + v

(A.1)

(A.2)


whereas (3.28) imply

My _< suplAz] + - - s u p [Yl, (A.3) D l + v D

t My =< sup IAg, x[. (A.4) D

Since

AZ, x = (Az), ~a, x + (Az), ~,~, x

and

AZ,, = (Az),~a ,+(AZ) ,~2 , , ,

we see from (3.30) and (2.18) that

1 IAz, xi - < 1 - hK/2 i(Az), ~l + [(Az), zl,

1 Az ~ < I(dz),~i+l(Az) xl.

= 1 - h~c/2

(A.5)

t Thus, our objective of finding suitable bounds for M x, My, M'~, arid My is reduced to

(A.6)

finding bounds on IAzi, i(Az),~[, and I(Az),zl. An upper bound for lAx1. We have

1 l { ~ ( - 1 Z 2 k ~ ) + ~ [ ( l - 2 k ) ~ l }, AZ - 1 - 2 k

which we may write as

1 ] - 1 (~2/~ (~2 z A Z - l ~ 2 k L~I --2k 8a 2 +(1 -2k) ~-~a +

By (3.15), (3.19), (2.20)

2k' 8Z k ~ . (A.7) ( 1 - ~k) 2 6~O "

(A.8) hl h l ( h2) Q = 1Q(1/2)1 < ~ - s u p Ixl _-< ~ x + , D

where

/?x = sup ]Px]. [0, z]

Consequently, (3.35) furnishes

l ( h 2 ) h_ h 2 = +~px+ ] p x + ~ . I)~z < i Px+ 1-2

Next, we see from (3.32), (2.18), and (A.9) that

IZ, o[ < ~ /~+ 1+ .

(A.9)

(A.IO)

(A.11)


Turning now to Z ... . we see that (3.32), (2,21) and (2.22) furnish

+ ( ~ + 2 ) { ( P , ~ + 9 ) 2 [ - ~ - + ( 1 - h ~ - ) k v y ] + 2 ( P x + 9 ) ( 1-hk'~2 ]

(A.12)

Therefore, and because of (2.18) and (A.9),

1 h + 2 (A.13)

where

t¢' = sup [k'[. (A.14) [o, t] As for i~,z;,, we get from (3.35)

7, ~ = vx(1)vy(l)[px(1) + 2Vx(1)], (a.15)

SO h

[Z,~[ ~/3=+ ~. (A.16)

Combining (A.7), (A.10), (A.11), (A.13), and (A.16~, we get the desired bound for [Azr

(Pxq-½h) (Xq-7_) hff' 1 (1-+-~)2} IAzI < 2(1 _½Kh)2 {(/3= + ~)[K + + 2 + @=+ ~)

hKt(I_}_½hK) (, :)2 K, I ( ~ ) h9~41 + 4(1-½htc) 3 \ x+ + 2(1-½tch) l i0=+ +i02+/~xh+ • (A.17)

It is now clear from (A.1) and (A.3) that

M = = O(1) and M y = O(1) as h--*0,

i.e., the first two assertions in (3.51) are confirmed. Bounds for I(Az),~I and I(Az), 41. By (A.7)

1 32k' (A)0,~ - (1 -2k) e Z . . . . +7~,a~+ (1 -2k) 3 Z,~

k 1 k' - l _ 2 k Z , Z~+ (l_2k)4[(1-2k)2k"+3(2k')2]Z,o ( l_2k)2Z, z. (A.18)

With the exception ofz . . . . . Z ~a~ and X, z~, we can utilize previous calculations for bounds on the derivatives appearing in (A.18). F rom (A.15) there follows

)~,~ - 0, (A.19)

Upper and lower bounds for the shear stress in the Saint-Venant theory of flexure 401

whereas (3.13), (3.1) and (3.3) furnish

, + g x lY .. . . I}. (A.20) LZ . . . . l < sup{[(x,~)2 +lx[Ix,~l]ly,~l+2x x ~ y ~1 ~ 2

By (2.21), (2.22)

X

X,a

X,~G

Y,Ga

= p~ + 2vx, y = py + 2v~,

= r~(1-2k), y,~ = ~y(1-2k)

= kv(1 - 2k)- 2k'zx

= kvy(1 - 2k)- 2k'zy

y . . . . = k'vy(1 - 32k) - kZz~(1 - 2k) - 2k"~y.

Let

K" = sup Ik"[

and use (A.21), (A.20), (A.9), (A.14) to get

(A.21)

(A.22)

....

+1 (A.23)

Next, we get from (3.33),

k 2 7~ ~ = l~ , [2p~vx- ~(2px + h 2 ~ ) l , (A.24)

SO

]Z, ;.~] < / 3 + ~ p~ + . (A.25)

Accordingly, (A.18), (A.23), (A.19), (A.13), (A.11) and (A.10) furnish

< (l+½~ch) ~ch'~ 2 ~ ) + ~ } ,(Az),~I = (l_½tch)Z {(1 + ~- ) +3 (iOn+ ~)[~c(1 +

3(/3:, + ½h)ht¢' + 4(1_½~ch)3 {@~+ ~ ) [ ~ ( 1 + ~ ) + ~ ' 1 + 2 ( 1 + h~) 2}

+ 2 ( ~ 4 1 + - ~ + ~ J ~ P x +


+ ~Px + ~ Px + • (A.26) + (1 --½Kh) 2 p~+ 2-2

Turning to [(Az). ~[, we see from (A.7) that

1 k 2k (Az),~ = Z,~z+ (l_2k) 2 ;g,o~- l_2kZ,~+ (l_2k) 3 Z,o~

2k' k 2 k' ( 32k "~ + ~ Z . ~ ( 1 ( 1 - ~ ;g'z+ (1 _2k) ~ \1+ ~ ) Z ~ . (A.27)

By (A.15), Z,~zz = [vx(1)]2v,(1) ,

SO

I)~, ~x?~[ < 1. (A.28) A brief computation based on (A.24) yields

Z, , ,~a=-3pxk 'cX'cx-(k2"Cx+k"c,) (~+~) +zSrv~(1- k4~h2) • (A.29)

Consequently,

', (2/32 ~ ) ( 4h~. (A.30, Iz~l _-< 3KPx+~t¢2+ x x+ + 1- ~c

Therefore, and in view of (A.27), (A.28), (A.16), (A.13), (A.25), (A.10), and (A.I1),

1 [12t@~ +(1¢2 + to')( -22p~ + h2) + 4 - tc2h2] I(Az),~I < 1+ 4(1 ~ 2 -~:h)

(1._-~Kh, (/3x + ~ ) + (1 _½~ch,3 {(/3 + ~)ft¢ (1 + +2 +

[ ( ~ ) 1 to2 [ £ ( : ) h821 hx' tc p2_~_ _}_ j0x_~_ 1-2 h + 2(1-½xh) 3 /Sx+ 2 (1 ~ 2 -g~ch)

+ 2(1_½~ch) 3 2_~)~p~+ ~)2(1+ . (A.31)

The assertions t M'~= 0(1) and My O(1) as h ~ 0

follow from (A.2), (A.4), (A.5), (A.26), and (A.31).

Acknowledgment

We gratefully acknowledge the National Science Foundation, Washington, D. C., for the support provided by Grants ENG-74-01152 and ENG 75-13643.

Upper and lower bounds f o r the shear stress in the Saint- Venant theory o f f l e x u r e 403

REFERENCES

1-1] Popov, E. P., Introduction to Mechanics of Solids. Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1968 1-2] Wheeler, L. and Fu, S.-L., Stress bounds for twisted bars of strip cross section, Int. J. Solids Structures,

10 (1974) 461468 [3] Wheeler, L., Stress bounds for the torsion of tubes of uniform wall thickness, J. of Elasticity, 4 (1974)

281-292 [4] Payne, L. E., Upper and lower bounds for the center of flexure, J. of Research of the National Bureau of

Standards, (1960) Vol. 64B, No. 2, pp. 105-111 1-5] Timoshenko, S. P~ and Goodier, J. N., Theorie of Elasticity, 3rd edn. McGraw-Hill Book Co., New York,

1970 1-6] Courant, R. and Hilbert, D , Methods of Mathematical Physics, Vol. 2. Interscience Publishers, New

York, 1962 1-7] Goetz, A.0 Introduction to Differential Geometry. Addison Wesley Publishing Company, Inc, Reading

Massachusetts, 1970 1-8] Wheeler, L. and Horgan, C. O., A two-dimensional Saint-Venant principle for second-order linear

elliptic equations, Quart. Appl. Math. (in press) [9] Littman, W., A strong maximum principle for weakly L-subharmonic functions, J. Math. and Mech.,

8 (1959) 761~70

Upper and lower bounds for the shear stress in the Saint-Venant theory of flexure

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