ABSTRACT
This paper presents an effective way to determine the individual stresses in arbitrarily-loaded multiply-perforated finite plates whose various size cutouts are randomly distributed. Thermoelastic stress analysis (TSA) is used such that the recorded temperature data are processed utilizing an Airy’s stress function. Advantages of TSA include full-field, non-contacting, nondestructive, no surface preparation other than a paint is needed, unnecessary to differentiate the measured information and has a resolution comparable to that of commercial foil strain gages. The present analysis was inspired by previous studies involving the TSA stress analysis of an incline-loaded clamped plate containing a single cutout [1] and a tensile plate containing multiple holes which were located collinearly to the external load [2]. The individual stresses in vertically- and incline-loaded aluminum plates containing two side-by-side circular holes are determined here. The holes are sufficiently close together that their stress fields interact with each other. The absence of any universal loading fixturing causes the incline-loaded plate to be subjected to some unknown in-plane bending as well as the tension. This unknown loading of the inclined plate hampers stress analyzing the plate numerically, and theoretical solutions to finite geometries are extremely difficult. Notwithstanding the aforementioned statement, TSA results agree well with those from commercial strain gages and ANSYS. Load equilibrium is also satisfied.
Introduction
Many engineering structures involve multiple perforated finite plates and it is imperative to be able to evaluate the stresses reliably for cases where theoretical or numerical approaches are not available. Although illustrated here for the particular situation of vertically- and incline-loaded plates containing neighboring holes, the general approach described is applicable for stress analyzing arbitrarily-loaded, multiply-perforated or -notched finite plates whose various-size cutouts are randomly distributed. For example, locating auxiliary holes about an original hole can influence the stresses at the latter compared with those in the absence of the auxiliary holes. Unlike theoretical or FEM analyses, the beauty of this technique includes it requires neither accurately knowing the boundary/loading conditions, figure 1(b), nor the constitutive properties, nor differentiating the measured data. In addition to those from strain gages, the present TSA results are compared with those from an approximate FEM prediction. Recognizing that the thickness of the plate exceeds the diameter of the small hole, a 3-D FEA validates the plane-stress assumptions. The two cases of figures 1 actually involve the same plate, the plate being vertically loaded in figure 1(a) but inclined in figure 1(b).
Figure 1: Schematic of ( a) symmetrically-loaded (b) unsymmetrically-loaded (~15o w.r.t. loading) plates
The circular holes in the plate of figure 1(a) are horizontal with respect to the vertical direction of loading whereas in figure 1(b) they are inclined at -15o to the horizontal axis. In each case a two dimensional thermoelastic stress analysis was conducted to determine stress distribution around the neighboring holes, the latter being sufficiently close together that their respective stress fields interact.
Relevant Stress Functions
Equations (1) and (2) are two specific Airy stress functions for the symmetrically-and unsymmetrically-loaded plate of figures 1. Derivatives of the stress function provide the individual components of stress. These are omitted here due to lack of space. Having derived the individual stress components, the traction-free conditions on the boundary of the hole were imposed analytically. This reduced the number of independent Airy coefficients and consequently simplified the stress functions from the original more complicated form. Reference [1] provides the detail derivation and expressions for individual components of stresses and isopachic stress.
where r is the radius measured from the center of a hole, angle d is measured counter-clockwise from the horizontalx-axis (figures 1) and N is the terminating value of the above series (N can be any positive integer greater than one). An individual coordinate system is employed for analyzing each hole of each plate of figures 1. TSA-wise, the doubly perforated plates of figures 1 are stress analyzed by employing a separate stress function associated with individual coordinate systems having their origins at the center of the respective holes.
Experimental Details, Analyses and Results
The aluminum (E = 68.95 GPa and v = 0.33) plate of figures 1 was sprayed with Krylon Ultra-Flat black paint, loaded in a MTS testing machine using hydraulic grips and the data recorded using a DeltaTherm DT1410 camera as shown in figures 2. The symmetrical plate of figure 1(a) was clamped vertically making the plate symmetrical about x-axis, figure 2(a). The unsymmetrical plate of figure 1(b) was inclined at an angle of 15o with respect to the vertical hydraulic loading grips, figure 2(b). These plates were sinusoidally loaded between 3560 N (800lb) ± 2224 N (500lb) at a frequency of 20 Hz. Figures 3 show an actual TSA image (256×256 = 65,536 data values) for each of the symmetrical and unsymmetrical loadings. Since the plate of figure 1(a) is symmetrical about the line through the holes (x-axis), the recorded temperature data were averaged about that axis. A separate uniaxial tension calibration specimen was used to determine the thermomechanical coefficient, K = 406 U/MPa (2.8 U/psi). Some vertical strain gages were bonded along lines CD and C’D’ of figures 1.
The large and the small holes of the symmetrically-loaded plate of figure 1(a), were analyzed individually i.e., when analyzing the large hole, the temperature data near that large hole were used along with the corresponding stress function, equation 1, and similarly for the small hole. A total of m1 = 4,378 (small hole) and m2 = 2,031 (large hole) input values were used to evaluate the corresponding unknown Airy coefficients (equation 1) for the large and the small hole, respectively. Their respective source locations are shown in the figures 4. For the symmetrically-loaded plate, k1 = 9 was found to be an appropriate number of Airy coefficients for each of the small and large holes.
Figure 2: Specimen loading and TSA recording: (a) symmetrically-loaded (b) unsymmetrically-loaded plate.
Figure 3: Actual recorded TSA images, S*: (a) symmetrically-loaded (b) unsymmetrically-loaded plate.
The large and the small holes of the unsymmetrically-loaded plate of figure 1(b) were similarly analyzed individually. In this case a total of m3 = 9,372 (small hole) and m4 = 4,094 (large hole) input values were used to evaluate thermoelastically the corresponding unknown Airy coefficients for the large and the small hole, respectively. Their source locations are illustrated in the figures 5. For unsymmetrically-loaded plate, k2 = 17 was found to be an appropriate number of Airy coefficients for each of the small and large holes.
Using the TSA input information, S*, all the unknown Airy coefficients of equations 1 and 2 were evaluated for each hole of the symmetrically- and unsymmetrically-loaded plate and thereby provided the individual components of stress i.e., the expressions for individual components of stress involve the now-known Airy coefficients.
TSA results are compared with those from finite element analysis (ANSYS). For the symmetrically-loaded plate of figure 1(a), the symmetrical boundary condition along the horizontal line of symmetry and a uniform end tensile stress of 9.19 MPa (1333.33 psi) were applied. The end loading conditions of the unsymmetrically-loaded plate are not well known, which challenges the modeling of finite element analysis. Nevertheless an approximate finite element model was made for the unsymmetrically-loaded plate in which, in addition to applying a far-field stress of 9.19 MPa (1333.33 psi) at the ends of the plate, the inside ends of the clamped plate were restrained horizontally, compatible with the physical situation that the ends of the plate do not move in the x-direction (horizontally) when loaded.
Figure 4: TSA source locations for symmetrically-loaded plate.
Figure 5: TSA source locations for unsymmetrically-loaded plate.
Tangential stresses, a00, normalized with respect to the far field stress, a0 (= 9.2 MPa (1333.33 psi)), are plotted in figures 6 around the boundary of each of the holes for the symmetrically-loaded plate and similarly for unsymmetrically-loaded plate in figures 7. Figures 8 compare the strains along the line CD (symmetrical-loading) and C’D’ (unsymmetrical-loading) extending from the large hole figures 1) from TSA, strain gages and finite element analysis (ANSYS).
Figure 6:
on the edge of the holes of symmetrically-loaded plate from TSA (k1 = 9 coefficients) and ANSYS.
Figure 7:
on the edge of the holes for unsymmetrically-loaded plate from TSA (k2 = 9 coefficients) and ANSYS
Figure 8: Strain eyy along CD of figure 1(a) (left) and e00 along C’D’ of figure 1(b) (right) obtained from TSA (k1 = 9 and k2 = 17 coefficients; m1 = 4,378 andm4 = 9,372TSA values), strain gages and ANSYS
Summary and Conclusions
A major objective of the present research was to evaluate the individual stress components associated with two neighboring holes which are either perpendicular or inclined to the direction of the loading and whose respective stresses interact and so as to influence the stress concentration factor at the boundary of the original larger hole alone, figures 1. The TSA technique utilized is not confined/restricted to specific geometry/loading conditions, location/shape of holes, specific arrangement of multiple holes, hole spacing, finite/infinite geometries or number of holes. Here, two different cases were conducted: one in which a finite plate is symmetrical about horizontal x-axis, and the other does not enjoy such symmetry.The stress functions are based on the geometry and traction-free conditions, and are irrespective of far-field loading conditions.
TSA results agree with those from strain gages and as predicted by an approximate FEA. The uncertainty of the ANSYS model (results) for the unsymmetrically-loaded plate of figure 1(b) raises the question of the validity or usefulness of comparing TSA and ANSYS results for that case. However, the excellent agreement between TSA and measured strains for the unsymmetrical situation substantiates the reliability of TSA for stress analyzing practical engineering members. Load equilibrium from TSA-based stresses agrees with the applied load (= 4715 N (1000 lbs)) within 6%, further supporting the reliability of the TSA results. Although the thickness of the plate exceeds the diameter of the small hole, a separate 3-D FEA justifies the present plane stress assumptions. Other results illustrate that whether or not compatibility of the respective individual stress components is enforced between the holes has little consequence. While demonstrated here for neighboring circular holes, the general approach is applicable to non-circular holes or notches, or combinations thereof.
