Advances in the Bonded Composite Repair o f Metallic Aircraft Structure phần 4 docx

144 Advances in the bonded composite repair of metallic aircrafi structure

with a step change in stiffness from ~ptp/( 1 - v’,) to Eptp/( 1 - v’,) + ERtR/( 1 - vi)

over a central potion ly( 5 B - b, as indicated in Figure 7.4(e), with b given by

(7.13)

This equivalence will be exploited in Section 7.4 to assess the redistribution of stress

due to a bonded reinforcement.

The prospective stress in the plate directly underneath the reinforcing strip,

00 = op(x = 0), can be readily determined by integrating Eq. (7.4),

1 1 b = - tanh j?B- (BB< 1) B “P ’

with S denoting the stiffness ratio given below,

(7.14)

(7.15)

which is an important non-dimensional parameter characterising a repair. As will

be shown in the following section the actual prospective stress 00 is somewhat

higher than that given by Eq. (7.14). This under-estimation is primarily due to the

ignorance of the “load attraction” effect in a 2D plate associated with reinforcing.

7.4. Symmetric repairs

We return to the solution of the problem formulated in Section 7.2, assuming

that the repaired structure is supported against out-of-plane bending or the cracked

plate is repaired with two patched bonded on the two sides. The analysis will be

divided into two stages as indicated in Section 7.2.

7.4.1. Stage I: Inclusion analogy

Consider first the re-distribution of stress in an uncracked plate due to the local

stiffening produced by the bonded reinforcement. As illustrated in Figure 7.2(a),

the reinforced region will attract more load due to the increased stiffness, leading to

a higher prospective stress than that given by Eq. (7.14). The 1D theory of bonded

joints (Section 7.3) provides an estimate of the load-transfer length j?-’ for load

transfer from the plate to the reinforcement. If that transfer length is much less

than the in-plane dimensions A, B of the reinforcement, we may view the reinforced

region as an inclusion of higher stiffness than the surrounding plate, and proceed in

the following three steps.

1. Determine the elastic constants of the equivalent inclusion in terms of those of

2. Determine the stress in the equivalent inclusion.

the plate and the reinforcing patch.

Chapter 7. Analytical methodsfor designing composite repairs I45

3. Determine how the load which is transmitted through the inclusion is shared

between the plate and the reinforcement, from which the prospective stress 00

can be calculated.

Step (2) is greatly facilitated by the known results of ellipsoidal inclusions [3]: the

stress and strain within an ellipsoidal inclusion is uniform. as indicated

schematically in Figure 7.2(a). The uniform stress state can be determined

analytically with the help of imaginary cutting, straining and welding operations.

The results are derived in [SI for the case where both the plate and the reinforcing

patch are taken to be orthotropic, with their principal axes parallel to the s - y

axes. We shall not repeat here the intermediate details of the analysis but simply

recall the results for the particular case where both the plate and the reinforcement

are isotropic and have the same Poisson's ratio, vp = VR = v. The prospective stress

in the plate along y = 0 within the reinforced region (1x1 I A) is

60 = &T= ~ (7.16)

where

(7.17) BA 4~- 4+2-+2-+S Z '[ A B

with

2 = 3( 1 + S)2 + 2( 1 + S)(B/A + A/B + vS) + 1 - v2S2 (7.18)

It is clear that the stress-reduction factor 4 depends on three non-dimensional

parameters: (i) the stiffness ratio S, (ii) the aspect ratio B/A, (iii) the applied stress

biaxiality i.. The parameters characterising the adhesive layer do not affect 00, but

we recall that the idealisation used to derive Eq. (7.17) relies on B-' < A, B, and p-'

is of course dependent on adhesive parameters.

To illustrate the important features of Eq. (7.17), we show in Figure 73a) the

variation of the stress-reduction factor 4 with aspect ratio for two loading

configurations: (i) uniaxial tension (i = 0), and (ii) equal biaxial tension

corresponding to pure shear (A = -l), setting S = 1 and v = 1/3 for both cases.

It can be seen that there is little variation for aspects ratio ranging from B/A = 0

(horizontal strip) to B/A = 1 (circular patch), so that for preliminary design

calculations, one can conveniently assume the patch to be circular, to reduce the

number of independent parameters. It is also noted from Eq. (7.17) that for v = 1/3

and a circular patch (A/B = I), the stress-reduction factor cp becomes independent

of the biaxiality ratio A. As illustrated in Figure 7.5(a) the curves for 1 = 0 and

;L = -1 cross over for B/A = 1, indicating that, for a circular patch, the transverse

stress gFX does not contribute to the prospective stress, so that this parameter can

also be ignored in preliminary design estimates. In this particular case, the stress￾reduction factor 4 depends on the stiffness ratio S only, as depicted in Figure

7.5(b), together with the first-order approximation given by Eq. (7.14). It can be

seen that the first-order solution ignoring the load attraction effect of composite

146 Advances in the bonded composite repair of metallic aircraft structure

Uniaxial tension(X= 0)

Pure shear(X=-1) 0.9

". .

0 0.2 0.4 0.6 0.8

Aspect ratio of patch B/(A+B)

(a)

1 .o

0.9

0.8

0.7

8 0.6

2 0.5

2 0.4

2

0.3

0.2

0.1

0

a

-

o Exactsolution

- += (1+0.277S-O.O712S2)/(l+S)

- - -&= 1/(1+S)

1 .o

0 0.5 1 .o 1.5 2.0

Stiffness ratio S

(b)

Fig. 7.5. Variation of reduced stress with (a) aspect ratio for an elliptical patch of serni-axes A and B

under uniaxial tension and biaxial tension equivalent to pure shear; (b) stiffness ratio S for a circular

patch.

patch overestimates the reduction in plate stress. An improved solution can be

obtained by constructing an interpolating function based on the exact solution,

, (7.19)

1 + 0.277s - 0.0712S2

l+s $=

which is shown by solid curve in Figure 7.5(b), indicating a very good fit to the

exact solution.

The inclusion analogy also gives, as a natural by-product, the stress in the plate

outside the reinforced region. The stress at the point x = 0, y = B+ is of particular

interest, because this stress represents the increased stress due to the so-called load

Chapter 7. Analytical methods for designing composite repairs 147

attraction effect; a load attraction factor QL can be defined as the ratio of the plate

stress just outside the patch to the remote applied stress

(7.20)

It is clear from Figure 7.5(a) that for the case of a balanced patch (S = 1) under

uniaxial tension, this load attraction factor ranges between 1 for patch of infinite

width to 2 for patch of zero width. For the typical case of circular patch, the load

attraction factor is approximately 1.2.

7.4.2. Stage 11: Stress intensity factor

Once the stress at the prospective crack location is known, one can proceed to the

second stage of the analysis in which the plate is cut along the line segment

(1x1 5 a, y = 0), and a pressure equal to go is applied internally to the faces of this

cut to make these faces stress-free. Provided that the load transfer to the

reinforcement during this second stage takes place in the immediate neighbourhood

of the crack, the reinforcement may be assumed to be of infinite extent. Thus the

problem at this stage is to determine the stress intensity factor K, for the

configuration shown in Figure 7.3(a).

Without the reinforcement, the stress-intensity factor would have the value KO

given by the well-known formula,

KO = ~ofi (7.21)

This provides an upper bound for K,, since the restraining action of the patch

would reduce the stress-intensity factor. However, KO increases indefinitely as the

crack length increases, whereas the crucial property of the reinforced plate of

Figure 7.3(a) is that K, does not increase beyond a limiting value, denoted by K,,

as will be confirmed later. That limiting value is the value of the stress intensity

factor for a semi-infinite crack. It can be determined by deriving first the

corresponding strain-energy release rate as follows. Before we proceed, let us first

determine the deformation of the reinforced strips shown in Figure 7.3(b). The

adhesive shear stress ZA is governed by the differential Eq. (7.9), which has the

following solution for the particular case of semi-infinite strip,

~A(Y) = ZA,rnaxe-’.’ , (7.22)

where rmax can be determined from the simple equilibrium condition,

QOtf = so“ ZA (YPY,

ZA,max = P~P~o (7.23)

Recalling Eq. (7.4), the opening displacement of the plate at y = 0 can be readily

148 Advances in the bonded composite repair of metallic aircraft structure

determined,

(7.24)

Let us denote the total opening as 6 = 224,. The above equation can be rewritten as,

with

(7.25)

(7.26)

Consider the configuration shown in Figure 7.6. If the semi-infinite crack extends by

a distance da, the stress and displacement fields are simply shifted to the right by da.

The change in the strain energy UE is that involved in converting a strip of width da

from the state shown as section AA' in Figure 7.6 to that shown in section BB', as

depicted in Figure 7.7. Consequently the change in the potential energy for a

crack advancement ha, which is defined as the difference between the strain energy

change UE (= 1/2ootp6) and the work performed by the external load W (= GotpJ),

1 n=uE-w=--rJ OtPd

The crack extension force, Le. the strain-energy release rate G, is given by

which can be re-written as, recalling Eq. (7.25),

+Z

B B4 A A

B' B'+ * A' A'

Fig 7 6 A patched crack subjected to internal pressure

(7.27)

(7.28)

(7.29)