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Composite Structures 32 (1995) 659-666 0 1995 Elsevier Science Limited

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Logarithmic stress singularities at clamped- free corners of a cantilever orthotropic beam

under flexure

Nerio Ibllini & Marco Savoia University of Bologna - Istituto di Tecnica delle Costruzioni, Viale Risorgimento 2, 40136, Bologna, Italy

The rectangular orthotropic beam under flexure is studied by decomposing the problem into an interior problem and a boundary problem. The interior problem is required to satisfy field equations and boundary conditions at the lateral faces, whereas the boundary conditions at the beam ends are imposed in the average sense. The boundary problem, which reestablishes the pointwise boundary conditions at the beam ends, is solved via eigenfunction expansion under the assumption of transverse inextensibility. It is shown that logarithmic stress singularities are present at the comers of the clamped end section and the corresponding stress-intensity factor is computed.

1 INTRODUCTION

Singular solutions typically arise in mixed boundary value problems at the transition points between displacement and stress bound- ary conditions. For instance, the elastic wedge with clamped-free edges has been investigated closely. Williams* used the Airy stress function and eigenfunction expansion in terms of powers of the radial coordinate 1. Kuo & Bogy employed complex function representation and generalised Mellin transform to analyse an anisotropic wedge, so obtaining also a stress sin- gularity of the form lnr as r+O. Dempsey & Sinclair3 extended Williamss approach to derive conditions for the existence of logarith- mic stress singularities.

Singular solutions at the corners of clamped semi-infinite beams have been studied widely, e.g. Ref. 4, mainly with reference to extension and bending problems; in only a few cases the stress intensity factor has been evaluated. With reference to the cantilever isotropic beam under flexure, Gregory & Gladwell used a projection method to add the singular solutions to the Papkovich eigenfunction expansion and evalu- ate the stress intensity factor. For the orthotropic case, Lin & Wan6 reduced the flex-

ure problem to the numerical solution of a singular integral equation.

In this paper, the 1-D theory recently pro- posed by the authors7 is used to investigate the stress field at the right-angled clamped-free corners of a cantilever orthotropic beam under flexure. The solution procedure is based on the decomposition of the problem into an interior problem, defined in terms of the loading condi- tion with average boundary conditions at the beam ends, and a boundary problem which re- establishes the pointwise boundary conditions, i.e., null displacement at the clamped end. The boundary solution is obtained in terms of eigen- function expansion and is based on the only kinematic assumption of transverse inextensibil- ity of the beam. It has been shown in Ref. 7 that this assumption is particularly indicated for the analysis of strongly orthotropic materials. In fact, in this case the 2-D lower-order wide boundary-layer obtained in Ref. 8 and the asymptotic estimate of the characteristic decay length of end effects given in Ref. 9 can be reobtained exactly.

In Sections 3 and 4, particular attention is devoted to the analysis of the stress field near the corner points of the clamped end section of orthotropic beams under end shear force and

659

660 N Tullini, M. Savoia

uniformly distributed load. First of all, the results are compared with analytical and numer- ical results reported in the literature, showing excellent agreement, in terms of normal and shear stress distributions, in the neighbourhood of stress singularities. Moreover, it is shown that in both cases the axial normal stress presents a logarithmic singularity at the clamped-free cor- ners. The stress-intensity factor is computed analytically and turns out to be of the order (E,/ Gv)~~ for strongly orthotropic materials. In passing, the existence of the logarithmic singu- larity implies that the eigenfunction expansion in terms of powers like ra, as typically done in the literature, may not be complete.

2 THE INTERIOR PROBLEM

Let a rectangular beam be referred to a Carte- sian reference frame Oxlxz, where x1, x2 axes are chosen in the axial and transverse direction, respectively (Fig. 1). The beam length and total height are denoted by 1 and H = 2h. The beam is made of homogeneous, orthotropic, linearly elastic material, with orthotropy axes coinciding with the reference axes. A uniformly distributed transverse load q2/2 is applied at the top and bottom faces; moreover, the two end sections are, respectively, clamped (at x1 = 0) and loaded by a bending moment &f and a transverse shear- ing force E (at x1 = Z), see Fig. 1. Consequently, the following boundary conditions must be sat- isfied:

c&l, +h) = +qJ2, &x1, &h) = 0 (1)

&(o,x2)=&(O,-$)=O (2)

M(I) =A?, Q(Z) =P (3)

where ati and u,(a = 1,2) are the stress and displacement fields and M(xr), Q(xl) are (for a unit thickness) and bending and inner shear resultants, respectively.

Introducing the dimensionless variables x = xi/l, y = xdh, the beam domain reduces to [0, l] x [ - 1, 11. The exact stress field satisfying the stress boundary conditions (l), (3) is the following: l*l

0% - hY

-r M(x)-&+2&J 1 2 3-5y2 3. 1 (4) a;* =

3Q (4 7 (1 -y>, CT dy = 0 (8)

in order tildefine the rigid body terms. Hence, the displacement field takes the following form:

- q*/*

I- e 4 x2 t Fig. 1. Cantilever orthotropic beam.

3-sy* 2 3. +5mh I)

Logarithmic stress singulanties of a cantilever orthotropic beam under fkxure 661

l2 u

662 N. Tullini, M. Savoia

where fi is the traction prescribed at the end section x = 1 (odd function with respect to y) and superscripts p and r refer to the principal and residual part of the solution, respectively. Horgan & Simmonds showed that eqn (8) is a necessary condition for the decay of the solu- tion of the residual problem (12-15) when displacements are prescribed at the beam ends.

The boundary problem (12-15) is solved by the separation of variables method. Substituting the displacement field:

Ul =h@(x)U(y), u2=44 (16)

in eqns (12, 13a), the following eigenvalue prob- lem is obtained:

d2U, -++;u, = 0, @:,-y;(& = 0 dY2

(17)

where An is the eigenvalue and yn = A,JG is the dimensionless decay rate. Equation (17a) yields two sets of orthogonal eigenfunctions, odd and even functions of y. For flexure problems, only the odd eigenfunctions are required, given by:

Un(y) = sin Any/sin ;1, (18)

Substituting the corresponding residual shear stress (12b) in the null shear resultant condition (13b), the lateral deflection of the boundary solution is obtained:

q = - f Q&C) (19) n=l

Making use of eqns (18, 19), the boundary con- ditions (14) yields the characteristic equation:

tan ;1, = ;2, (20)

which gives the eigenvalues A, corresponding to the odd eigenfunctions (18). Finally, the resid- ual problem is solved by integrating the differential equation (17b), so obtaining:

0, (x) = A, cash y,x +B, sinh y,x (21)

The integration constants A,, B, can be deter- mined making use of Fouriers method by expanding boundary conditions (15) in terms of eigenfunctions, so obtaining:

(I&(O) = u: and @L(l) = C$ (22) where:

+_A_ l s 2 &6Q(O) h _ uf(O,y)Udy=- 1 A, A

Rlll 1 gp = - n

h s Lfi(Y)-~~l(l~Y)l un dY _l RII~ 1 =-

h s 2 R66qd flU,dy-t~ A (23) -1 II are the Fouriers coefficients of the residual part of the right hand side of eqns (15). It is worth noting that this solution procedure can be used since eigenfunctions (18) constitute a com- plete set of functions. In fact, eqn (17a) represents a standard Sturm-Liouville eigenva- lue problem, see Refs 7, 8 for details.

4 CANTILEVER ORTHOTROPIC BEAM SUBJECTED TO AN END SHEAR FORCE

(A) Comparison with 2-D elasticity solutions

An orthotropic cantilever beam is considered here, subject to a shear force p and a bending moment ti =FI acting at the free end (x = l), such that the applied bending moment vanishes at the clamped section (x = 0). The bending moment ti is applied through a linear variation 1 of the tractionf,(y), so that no boundary effects arise from the end at x = 1. The procedure described in the preceding section yields the following axial normal and shear stresses at the clamped section:

Q(O,Y) = - 7

%1(&Y) =;. (24)

In order to compare the solution given by the proposed 1-D theory with 2-D elasticity solu- tion, consider a semi-infinite beam; in this case, eqn (24a) reduces to:

as llh-tm. (25) It is worth noting that the term tanh ?/n

appearing in eqn (24a) attains quickly the value of unity also for beams of usual slenderness.

Logatithmic stress singulatities of a cantilever orthotropic beam under flexwe 663

For instance, an orthotropic beam with E,/ G12 = 8.94 has y1 = 3.02 l/H, and for l/H = 2 gives tanh y, = 0.999988. Figure 2 shows that the normal stress at the clamped section given by the proposed 1-D theory is in excellent agree- ment with the elasticity results obtained by Gregory & Gladwell and Lin & Wan6 for iso- tropic and orthotropic beams. In particular, the different singular stress fields near the corners of the cross-section when different orthotropic materials are considered (see Table 1) are very well described. It is worth noting that due to the loading condition adopted, the bending moment vanishes at the clamped end, so that Fig. 2 shows the only contribution to the normal stress distribution due to the restrained warping.

(B) Estimate of stress singularity near the corner of the clamped end section

It is easy to verify from eqn (25) that a stress singularity for the normal stress cl1 occurs at the corners of the clamped section, and the order of the singularity can be computed analy- tically.

First of all, for y + + 1 the eigenfunction (18)

Fig. 2. Cantilever beam in plane stress under flexure at infinity. The normal stress distribution at the clamped section (x = 0) given by proposed theory (- ) is com- pared with the 2-D solution obtained in Ref. 5 for the isotropic case (ooo) and in Ref. 6 for the orthotropic

cases (000).

takes the form U,(y) = y + 0 (( 1 - 1 y 1 )2), and an investigation of eqn (20) reveals that &~,

664 iV Tullini, M. Savoia

where:

R &f 66. -J Rll In the neighbourhood of

can be rewritten as follows:

(31)

the corner, eqn (30)

oli(O,y)g +&[2-In 2-crc+ln(l- ]y ])I

(32)

and & becomes the logarithmic stress-intensity factor. This estimate seems to be new to the literature. Equation (31) shows that the stress- intensity factor KI is of the order (E1/G12)12 for strongly orthotropic materials.

Figure 3 shows the axial stress distribution at the clamped section given by the proposed 1-D theory (ooo), see eqn (25), and compared with the estimate reported in eqn (30) (---). It is worth noting that the value of & (that is the slope of the diagram in the semi-log scale of Fig. 3) predicted by eqn (31) agrees excellently with the analytical solution of eqn (25). Never- theless, the figure shows that the non singular term (2-crc)y in eqn (30) is not very accurate, since a constant shift of the diagram occurs. In order to improve this term, a different proce- dure is developed in the following.

First of all, it can be verified that the eigen- values 1, given by eqn (20) are very close to the

l-x#z

10 -8 -6 -4 -2 0

Fig. 3. Axial normal stress near the corners of the clamped section of a semi-infinite cantilevered beam given by the proposed theory (ooo), see eqn (25), is compared with the estimates of (eqn (3;) (---) and eqn (34)

-.

values a, of eqn (26), so that the series appear- ing in eqn (25) can be approximated as

c +y)E 1 2(-1) sin a,y. (33) n=l n n=l 42

Since the series at the right-hand side of eqn (33) can be computed analytically, eqn (25) may be rewritten in the following form:

I 1 -sinty \

1 olr(O,y)zKI 2 sinSy+Tln

\ 1 +sinqy

as llh+co (34)

Finally, Taylors expansion of eqn (34) near the corners of the clamped cross-section yields:

all(O,y) = f& 2+ln$+ln(l- ]y]) i

llh+co as

y+fl (35)

where KI still represents the logarithmic stress- intensity factor, which coincides with that obtained in the previous expansion, see eqn

(31). The solid line in Fig. 6 represents the axial

normal stress given by eqn (34), which turns out to be very close to the analytical solution.

It is worth remembering that, for the iso- tropic case, Gregory & Gladwell showed by means of a 2-D analysis that no singularity of the type ra can occur for null value of the Pois- sons ratio; the 1-D analysis performed here shows that a stress singularity of logarithmic form is present, with KI = 0.90 F/A. This result suggests that additional investigations in the 2-D elasticity field are necessary.

5 CANTILEVER ORTHOTROPIC BEAM SUBJECTED TO UNIFORM LOAD

In this section an orthotropic beam subject to a uniformly distributed transverse load split in two equal parts (g2/2) acting at the top and bottom faces is considered. The procedure

Logatithmic stress singulatities of a cantilever orthotropic beam under jkxure 665

described in Sections 2 and 3 yields the follow- ing expression for the axial normal and shear stresses:

4J2 3 g11 = - Ah

[ j- (1 -+Y+; (3Y -5Y3)

Q=+ $1~X)(1-y2) I

+c * 32n,, cos Any - cos An n=l n sin 1, 1

where:

fin(X) = sinh y,x - tanh yn cash y,x

cash y,x -

in cash in

f2n Cd = cash y,x - tanh yn sinh 7/nx

sinh 7/,x -

in cash in

(36)

(37)

In order to assess the accuracy of the pro- posed solution, a thick orthotropic beam with length-to-height ratio l/H = 4 is analysed. The elastic properties of the beam, typical of a fibre-

reinforced composite material, are El = 175 GPa, El/E2 = 25, E11G12 = 50, v12 = O-3. Since no exact solution to this problem can be found in the literature, the accuracy of the proposed model is assessed through comparison with results obtained via FEM. A 12 x 48 mesh of S-node square isoparametric elements in plane stress is adopted (CPS8, Abaqus 4/6), and stres- ses are computed at Gauss points of finite elements. Figure 4a,b shows that both normal and shear stress distributions given by the pro- posed 1-D model are in very close agreement with FEM results even in the neighbourhood of the clamped cross-section (x,/H=O*O417). The dashed line in Fig. 4a refers to Euler-Bernoulli solution, based on the assumption of cross-sec- tions remaining plane after deformation. As is to be expected, Euler-Bernoulli model is not able to capture the actual stress distribution near the clamped end, which is strictly related to boundary effect.

In order to investigate the stress field due to the warping restrained at the clamped end, an additional moment A? = q2Z2/2 is applied at the free end (X = l), such that the bending moment vanishes at x = 0. In this case, the axial normal stress of eqn (36a) at x = 0 reduces to:

q2l R66

%1(&Y) = -7 R 6 \i[

3y-5y3

11 1o

(4 @I

Fig. 4. Thick cantilever orthotropic beam (l/H = 4) under uniformly distributed transverse load. Axial normal stress (a) and shear stress (b) given by the present analysis ( -) are compared with FEM 2-D results (000) and Euler-Bernoulli

solution (- - -).

666 N. Tullini, M. Savoia

For a semi-infinite beam, yn -+ co and p+O, and eqn (38) is rewritten in the form:

Oll(O,Y) =

as llh+co (39)

The series appearing in eqn (39) is the same as that of eqn (25), and the procedure of the pre- vious section can be used to study the stress singularity. In this case, the axial normal stress gll presents a singularity of logarithmic form at the corner and

(40)

is the corresponding logarithmic stress-intensity factor. It is worth observing that, as in the case of the cantilever under end shear force, eqn (40) holds also for beam of usual slenderness. In fact, with reference to eqn (38), the cubic polynomial and the term x,2/y,& = 2 fi Z, l/J: in bracket give rise to non singular terms, and tanh yn attains the value of unity even for very thick orthotropic beams.

ACKNOWLEDGEMENT

The financial support of the (Italian) National Council of Research (CNR contract No. 94.00034.CTO7) and of the Human Capital Pro- gramme (contract No. CHRX-CT93-0383-DG 12 COMA) is gratefully acknowledged.

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