Masters Theses

Date of Award

12-1991

Degree Type

Thesis

Degree Name

Master of Science

Major

Mechanical Engineering

Major Professor

Remi C. Engels

Committee Members

Louis Deken, Roger Crawford

Abstract

An unrestrained strut with geometric discontinuities is subjected to a nonuniform thermal loading and is analyzed for stress. The geometric discontinuities are three holes drilled through the strut parallel to its axial direction. Two separate thermal loadings are applied to the strut. The first thermal loading is a uniform temperature distribution due to the continuous application of cooled air along the outside surfaces of the strut. The second thermal loading adds an instantaneous passing of heated air through the three holes in the strut. The superposition of these thermal loadings produces complicated stress patterns in the strut which may be the cause of structural failure. The objectives of this thesis are: (1) to compute the axial stresses in the strut as functions of position and time; (2) to study the effects of these stresses on the structural integrity of the strut. The analysis of the strut consists of two parts: a heat transfer analysis and a stress analysis. The commercially available ANSYS finite element program is used to first determine the nonuniform temperature distribution in the strut due to the thermal loading. The temperature of the entire strut is initially at -42°F due to the cooled air. In addition, the temperature of the heated air in the holes is held at a constant 350°F. This heat transfer analysis is a transient analysis and the changing nonuniform temperature distribution is tracked in time until the final steady-state is reached. The temperature results at each time step and at each node location are saved for future use in the stress analysis. Because the thermal loads are applied consistently in the axial direction such that the thermal loads are the same on each cross section normal to the axial direction, it is only necessary to perform a two-dimensional analysis on a typical cross section. The results of the heat transfer analysis are. then used to determine the corresponding stress distribution in the strut. A linear elastic stress problem is solved at each of the time steps previously established in the thermal analysis. The stress problem is three-dimensional. The strut is unrestrained and can grow in any direction. This permits stresses and strains to occur both in and out of the plane of the axial cross section. Thus, the problem is neither a plane stress nor a plane strain problem. Because the stress patterns are highly irregular, a very large number of elements are required to accurately model the stress problem. Consequently, a straightforward three-dimensional analysis of the strut would require a very large finite element model with several thousand degrees of freedom. This would require a large mainframe computer and be very expensive due to many hours of computer processing time. A simpler and more cost effective approach must be used. It is important to note that all cross sections of the strut normal to the axial direction are deformed and stressed in the same manner. This is because the thermal loading and resulting thermal profiles are the same for each cross section normal to the axial direction of the strut. The problem is thereby essentially reduced to a two-dimensional one. The axial stresses normal to the cross section are calculated from the two-dimensional thermal and geometric profiles also normal to the cross section. The procedure to determine the axial stresses is based on the superposition of many stress fields, each of which produces compatible strain fields in the strut. Three different types of stress fields are used: the first type together with the applied thermal loading produces zero strain; the second type is a uniform axial stress distribution resulting in a uniform strain field; and the third type is a linear stress distribution corresponding to pure bending. All three distributions are known to produce compatible strain fields. In general, the procedure superimposes four such stress distributions per finite element: one of the first type, one of the second type and two of the third type. The solution for the axial stresses is approximate and is subject to the usual assumptions and limitations adopted in the elementary linear theory of elasticity. In addition, the solution also depends on Saint Venant's principle, and is therefore only valid for cross sections relatively far away from the ends of the strut. The proposed technique uses the same finite element grid as the heat transfer analysis. A computer program was developed to perform the stress analysis following the discretized grid procedure. Although the procedure is applied to a strut, it is also useful for the analysis of other structural components such as composite laminates, marine ship hulls, airplane wings, and nuclear reactor components. This procedure assumes that the cross sections are the same geometrically and thermally at any location along and normal to the axial direction of the strut. The procedure cannot be used to calculate the in-plane stresses because the cross sections and thermal profiles are not uniform for the other directions. No closed form solutions are known to calculate the in-plane stresses. It is assumed that the magnitudes of the in-plane stresses be equal to or less than the out-of-plane stresses since the temperature differential in the non-axial cross sections is equal to or less than the temperature differential in the axial cross section. The calculation of the in-plane stresses is beyond the scope of this thesis. These analyses show that the stress and temperature profiles are almost perfectly in phase. For a particular time, the locations of extreme stress are identical to those of extreme temperature and occur almost at the same time. However, these locations do change with time. The overall extreme stresses occur when the temperature gradient in the strut is the largest, and not when the temperature of the material is at its peak. The minimum overall stress is associated with the trailing edge and the maximum overall stress with the bottom of the upper hole. The strut analyzed in the present study appears to be overstressed but is expected to relieve itself thereby avoiding permanent damage. The relief is due to local plastic deformation and allows the stress to become redistributed in the strut in a more uniform profile. The magnitudes of the maximum and minimum stress are expected to decrease toward zero while the lower stressed regions are expected to increase. There is no known way to calculate the change in stress magnitudes due to thermal stress relief. The stresses reported in this thesis are based on no change in magnitude due to thermal stress relief.

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