Perforated Shell Structure Analysis
6.1 Previous Results
Chapter 6
reactor technology [60]. Without exact theoretical solutions for the displacements and stresses occuring within a tubesheet under some specified load, Gardner [61] introduced the concept of an equivalent solid plate with elastic properties such that, under the same load, this solid plate would undergo the same displacements as the tubesheet. Stress concentration factors then assist in predicting the maximum stress that occurs at the hole edges [60]. Research since then has focussed on measurements and theories that predict these equivalent elastic properties. A thorough review of the historical development of tubesheet theory and experimentation can be found in [62].
One of the seminal works in tubesheet theory was the thesis by Meijers [63]. In this work, Meijers primarily dealt with determining analytical solutions for the equivalent elastic properties of doubly- periodic perforated plates. The plane stress condition is typically only encountered when there are in-plane loads acting on a thin plate. A plate is considered to be thin whent/L1, wheretis the thickness of the plate and L is any characteristic dimension of the plate. Plane strain conditions, on the other hand, are typically encountered when dealing with thick plates under the influence of in-plane loading. Conversely, a thick plate is one for which t/L 1, and traditionally has been defined to be any plate for whicht/L≥2 [64].
In the past, researchers [65] have asserted, and Duncan [66] demonstrated, that the equivalent elastic properties are very similar for thick perforated plates in tension or under bending loads.
Meijers argued that, while this assertion may be supported by experimental data, the reasoning is not soundly based in theory. He then proceeded to derive approximate relations between the effective elastic constants applicable to thick plates under plane stress and bending. Using these relations, effective elastic properties for perforated plates in the bending state could be found in the two limits:
thin and thick. Figure 6.1 shows the effective properties, under bending, found from Meijers’ theory for the triangular hole pattern (see Figure 3.2) in thin and thick plates. Data obtained by Duncan [66] for thick perforated plates (t/R= 2.29) is also shown. The results are plotted as a function of the total open-area fraction, Θ. For a triangular hole pattern,
Θ =
√3π 6
R p/2
2
, (6.1)
-0.5 0
0 0 0 0.2
0.3 0.3
0.4
0.5
0.6 0.6
0.6 0.8
0.9 0.9
1
1 1.5
EffectiveYoung’sModulus(E∗ /E) EffectivePoisson’sRatio(ν∗/ν)
Open Area Fraction Θ Open Area Fraction Θ
Duncan [66] t/R= 2.29 Thin Theory - Meijers [63]
Thick Theory - Meijers [63]
Figure 6.1: Effective moduli as a function of open-area fraction. Effective moduli predicted from theory and measured from experiments for varying-open area fractions of plates in the thin (t/R1) and thick (t/R1) limits. Poisson’s ratio used to obtain the data from theory isν = 0.3. Poisson’s ratio for the material used in the physical experiments isν = 0.27. The open-area fractions of the NSTAR screen and accel grids, Θs= 0.67 and Θa= 0.24, are indicated.
where R is the hole radius and p is the pitch. The open-area fraction has a maximum value of Θmax = 0.907 for this hole pattern. With Ra = 0.57 mm, Rs = 0.955 mm, and p = 2.22 mm, the open-area fractions of the NSTAR accel and screen grids are Θa= 0.24 and Θs= 0.67, respectively.
The NSTAR open-area fractions are indicated in the figure.
It can be seen that theory predicts a large difference in the effective properties depending on whether the plate is considered to be thin or thick. Note that for thick plates under bending, the effective Poisson’s ratio, ν∗, is expected to be larger than the undrilled bulk value, ν, for all open- area fractions. Duncan’s results seem to verify this. On the other hand, the effective Poisson’s ratio for thin plates under bending is expected to always be less than the undrilled bulk value, and may even be negative for sufficiently large open-area fractions.
While Meijers’ initial work is useful in the situation of dealing with perforated plates that are within either of the two limits, it provides no information in cases where the thickness is of the same order as the other dimensions in the plate, such as t/R ∼1. Later work of his [67] extended the
00 00
0.2 0.2
0.2
0.4 0.4
0.4 0.5
0.6 0.6
0.6
0.8 0.8
0.8
1
1 1
EffectiveModulusE∗ E 1+ν 1+ν∗ EffectiveModulusE∗ E 1−ν 1−ν∗
Thickness-Pitch Ratio p+tt Thickness-Pitch Ratio p+tt
Θ = 0.735 O’Donnell [64]
Θ = 0.580 O’Donnell [64]
Θ = 0.227 O’Donnell [64]
Θ = 0.510 Meijers [67]
Θ = 0.403 Meijers [67]
Figure 6.2: Effective moduli as a function of plate thickness. Effective moduli predicted from theory and measured from experiments for various open-area fractions. Poisson’s ratio used to obtain the data from theory is ν = 0.3. Poisson’s ratio for the material used in the physical experiments is ν = 0.33. The data points fort/(p+t) = 0 and t/(p+t) = 1 in the measured data sets were not obtained by measurement but from theory. The thickness-pitch ratios of the NSTAR screen and accel grids,t/(p+t) = 0.15 andt/(p+t) = 0.19, respectively, are indicated. Dotted lines through the measured data sets are simply to guide the eye.
previous to include terms of O(t/R), which, in addition to yielding the limit values fort/R= 0 and t/R→ ∞, allows for computation of the effective properties ast/R→0. If the effective properties are viewed as a function oft/R, the additional analysis yields the slope of this function att/R= 0.
Using finite element analysis, Meijers was able to fill in the range between the two limits for all possible values oft/R. Physical experiments were carried out by O’Donnell [64] on perforated metal plates of varying thickness under bending. The effective properties found by theory, finite element analysis, and experiment are shown in Figure 6.2. The first quantity shown is the first Lam´e constant, normalized by the undrilled bulk value, or shear modulus. It has been argued [64, 67] that, under equi-biaxial bending, the quantity (1−ν)/E should be relatively constant for all plate thicknesses.
It is this quantity, again normalized by the bulk value, shown in the second figure. The limit values for the experimental data are taken from [64], which originally were obtained from Miejers’ theory [63]. The NSTAR accel and screen grids have thicknesses oft= 0.51 mm andt= 0.38 mm, yielding
thickness-pitch ratios,t/(p+t), of 0.19 and 0.15, respectively. The NSTAR thickness-pitch ratios are indicated in the figure. Figure 6.2 enables us to interpolate the effective elastic constants,E∗ andν∗, for a plate of any thickness, and with an open-area fraction 0.227≤Θ≤0.735.