Advances in the Bonded Composite Repair o f Metallic Aircraft Structure phần 7 pdf

Chapter 1 I. Thermal stress analysis 323

derive an expression for E, in terms of these quantities. From equilibrium

considerations we have:

hence:

where:

(11.22)

(1 1.23)

The expression for the stress state in the plate just outside the patch is given by

Eq. (11.16) for r = RI:

c3 1 c2 o=El --

[(I -v) (1 +v)R; (1 1.24)

We will now derive the expressions for the stress state in the plate beneath the patch

and in the patch. From Eq. (1 1.15) with r 5 RI, under a uniform temperature, the

displacement is given by:

Since the displacement is the same in both the patch and plate we have:

(1 + V)QTI + - U

+c2=-

2 I-RI ’

RI ’

( 1 + v)a2 TI U

where the displacement u corresponds to the location r = RI.

The radial stresses for the plate and patch are given by:

(11.25)

(11.26)

(11.27)

(1 1.28)

(11.29)

Using Eqs. (1 1.26-1 1.29) we have the expressions for the radial stresses in the plate

324 Advances in the bonded composite repair of metallic aircraft structure

beneath the patch and in the patch:

(1 1.30)

(11.31)

To obtain the residual stress in the plate beneath the patch it is necessary to sum

Eqs. (11.13) and (11.30), but with TI = -TI in Eq. (11.30). Hence the final

expression for the residual stress beneath the patch is:

( 1 1.32)

Since the initial stress in the patch is zero, then the residual stress in the patch is

given by Eq. (11.31), but again with TI = -TI hence:

(1 1.33)

and the final expression for the residual stress just outside the patch is given by the

summation of Eqs. (1 1.13) and (1 1.24), hence:

( 1 1.34)

These equations now give the residual stress in terms of the cure temperature T.

The displacement u at r = RI for Eqs. (1 1.17, 11.18) and integration constants C2,

C3 are given in the appendix.

These equations now give the residual stress in terms of the cure temperature T.

For operating temperatures different from room temperature, Eqs. (1 1.24), (1 1.30)

and (1 1.31) can be used to calculate the stresses. In this case TI= TO = uniform

temperature change from room to operating temperature. The final stresses are

obtained by superimposing these stresses on the residual stresses.

11.2.1. Comparison of F.E. and analytic results

The solution of these equations has been carried out for AT = lOO"C,and the

following quantities have been evaluated for the comparison with F.E. results, the

mesh is shown in Figure 11.4:

1. residual stress just outside the patch at r = RI

Chapter 11. Thermal stress analysis 325

Fig. 11.4. Finite element mesh of circular patch on circular plate. Here RI = 162 mm and

Ro = 500 mm.

2. residual stress in the skin beneath the patch (01)

3. residual stress in the patch (02)

As an example a circular patch and plate are considered whose mechanical

properties are shown in Table 1 1.1. These properties are representative of a quasi￾isotropic boron patch reinforcement of an aluminium plate (although the value of a

used here for boron corresponds to uni-directional boron and should have been

Table 11.1

Material properties used for study of circular repairs on circular plates, AT = 100°C.

Young's Coefficient of

Thickness modulus Poisson's thermalexpansion Conductivity

Component (mm) (MPa) ratio (1°C) (J/ms"C)

Plate(A1uminium) 1 .O 71016 0.3 23 x IOs6 13.2

Patch(boron) 0.5 156000 0.3 **4.1 x 10-6 0.294

*Adhesive (FM300) 0.254 3460 0.35

* Only used for a 3D run.

In this instance the value for the laminate is taken to be equal to the unidirectional value. **

326 Advances in the bonded composite repair of metallic aircraft structure

3.76 x 10@ for the laminate). While this is not a perfect representation of an actual

repair it is acceptable for estimating residual stresses. Also this assumes that

bending is restrained, e.g. by stiffeners or thick plates, and also the edge restraint

exists when the repair is bounded by structural elements such as spars and ribs.

Consider the case in which the plate edge is restrained in the radial direction. Firstly

consider the heating up process to the cure temperature, given by Eq. (11.13) and

shown in Figure 11.5. The comparison between theory and F.E. results is in

agreement to four significant figures. In the case of no edge restraint the initial

stresses would be zero.

The second stage of the process involves cooling down from the cooling

temperature alone. The analytical and F.E. results are shown in Figure 1 1.6 where

the curves are from Eqs. (11.24), (11.30) and (11.31), and the points on the curves

are F.E. results. In all cases very good agreement between analytical and F.E.

results are obtained, (to four significant figures). The final solution for the residual

stresses is given by Eqs. (1 1.32-1 1.34) in Figure 11.7 with the corresponding F.E.

results. Again very good agreement between analytical and F.E. results is obtained.

Note that the residual stresses in the plate shown in Figure 11.7 are significantly

lower than those during the cooling process, Figure 1 1.6. This is simply due to the

lack of initial stresses which arise as a result of the restrained edges of the plate

when heated up to the cure temperature. The assumption of edge restraints is

important. Typically a repair to an aircraft wing plate can be considered as fully

restrained in the radial direction if the repair is bounded by significant structural

elements such as spars and ribs.

Returning to the residual stresses shown in Figure 11.7. It is evident that for large

values of Ro/R, asymtotic values occur for all stress components and may be

considered as limiting values.

-100,

-105-

-110-

2 -115- A

5

cn -120- cn

a, 3 -125- F.E.

-145(. , . I I I., . , . , , I

0 5 10 15 20 25 30 35

RdRi

Fig. 11.5. Initial stresses in plate due to heating to cure temperature.

Chapter 1 1. Thermal stress analysis

0-

-20-

-40-

-60-

327

rr n 0

0

Stress in patch, Theory/F.E.

Stress in plate outside patch,Theory/F.E.

1

Stress in patch,Theory/F.E.

0

Stress in plate beneath patch,Theory/F.E.

0 501 Stress in plate just outside patch, Theory/F.E.

-504

0 5 10 15 20 25 30 35

R,IRi

Fig. 1 1.6. Comparison of theory and F.E. results for cooling down process only.

So far, the adhesive has not been considered in the analysis. However F.E. results

have been obtained in which the patch and plate have been coupled using 3D

adhesive elements. To make a useful comparison with the previous analytical work

the bending of the plate has had to be restrained. The introduction of the adhesive has

328 Advances in the bonded composite repair of metallic aircraft structure

Table 11.2

2D isotropic circular plate, patch and 3D adhesive elements. Residual displacements and

stresses, al = 23 x 10-6/"C, a2 = 4.1 x 10-6/"C (analytic values in parenthesis), [5].

Residual stress Residual stress in

Displacement at just outside patch skin beneath patch Residual stress

Ro (mm) edge of patch (mm) (MPa) (MW in patch (MPa)

500 - 0.1 1551 128.31 161.91 - 65.532

(-0.1 1499) (127.92) (1 6 1.33) (-66.813)

resulted in an error of only 2% in direct stresses, shown in Table 11.2, and indicates

that the use of a closed form solution is sufficiently accurate for patch design.

The main concern in this chapter is the equations for direct stresses in the repair.

It is important to know the direct stress in the plate beneath the patch in order to

predict crack growth rate or simply the residual strength of the repaired structure.

The adhesive itself has no effect on the maximum value of the direct stresses.

Some F.E. programs have the capability in which the material properties can be

temperature/time dependent. In this case a simulation of the bonding process can

be carried out. The adhesive properties change during the curing process. At the

end of the curing process the adhesive has developed a shear stiffness and as the

repair is cooled to room temperature residual stresses develop. If the simulation

capability is available, then residual stresses are directly obtained from the analysis.

If this capability is not available then a superposition procedure can be used. The

analysis is carried out in two steps. The first analysis is equivalent to heating up of

the plate to the curing temperature (without the patch, since the adhesive has no

stiffness at this stage). Secondly, another analysis is carried out with the patch

included, subject to a cooling temperature equal to the cure temperature.

( TI = -TI). In the work presented here this two stage procedure has been shown

to be very accurate. The superposition of these two analyses gives the residual

stresses in the repair. Since the adhesive shear modulus is temperature dependent,

an arithmetic average value of the shear modulus should be used during the cooling

process.

I I .2.2. Orthotropic solution

Recent work by [7,8] has extended the analysis of residual stresses to circular

orthotropic patches on isotropic plates. The solution of this problem is based on an

inclusion analogy, which refers to the inner region of the repair where r 5 RI in

which equivalent properties of the inclusion can be made without altering the stress

or displacement state. Exact solutions are presented for both residual stresses and

thermal coefficients of expansion.

Results of this work are shown in Figures 11.8 and 11.9 for stresses due to

cooling only. For plate stresses beneath the patch both oXx and oYY are predicted, as

shown in Figure 11.8 and correspond to clamped edge conditions. As a

comparison, the isotropic solution is included and gives close results when