2.4 STRESS ANALYSIS

Stringent reliability and weight requirements call for rigorous and complete stress analyses during the mechanical design of engine components. Stress analyses predict the manner in which a mechanical part is likely to fail under anticipated working conditions. They also generate means to prevent failure. The goal is to design a part with sufficient, but not excessive, strength in every detail.

In conjunction with the engine design layouts, the approximate shape of the parts will be established, based on functional requirements and on similar satisfactory designs of the past. First, a preliminary design sketch is made. Subsequently, the determination of probable working loads, environmental effects, deflections, stresses, and determination of the final dimensions is made step by step, together with the selection of materials. The following steps are typical for stress analyses: (1) Analyze and determine the loads and environmental effects to be expected during the useful life of the part. (2) Evaluate the various possible modes of part failure from stress and strain induced within the part by the working loads and from other effects. (3) Select the materials and establish their mechanical properties under anticipated working conditions. Applying a likely theory of failure, arrive at the final dimensions of the part. (4) Apply experimental stress analyses to refine the above procedure. Redesign the part if necessary, depending on the results.

In certain cases, such as with highly stressed lightweight members, further stress analysis refinement must be obtained. The greater the refinement desired, the more nearly the methods of stress analysis must indicate the true strength of the member. This requires consideration of complex states of stress, residual stresses, stress concentrations, dynamic effects, inelastic effects, and other influences which determine the true stresses within the member. The following is a discussion of the four steps of stress analysis enumerated above.

Analysis of Working Loads and Environmental Effects

In stress analyses for mechanical design, working loads and environmental effects should be considered jointly. Stresses and strains induced in mechanical parts by external forces, which we will call loads, are affected significantly by environmental effects such as temperature, chemical reactions, corrosion, etc. Furthermore, the mechanical properties of most materials are affected by temperature. Sometimes, thermal stresses are induced as a result of temperature gradients within the part. Chemical reactions or corrosion can change the mechanical properties of the material, as well as the size and shape of the part.

For the analysis of working loads and environmental effects, determination of the following is essential: (1) The type of load: constant, impact, or repeated (2) The maximum value and duration of a constant load; the maximum and the minimum value for repeated or varying loads (3) The nature of load application: concentrated or uniform: rate per unit of time and, for repeated loads, total number of working cycles (4) Vibration load effects (5) Load effects with respect to the nature of material: ductile or brittle (6) Load effects with respect to the shape of a part: effect of geometry on stress concentration (7) Temperature effects: thermal stresses, high-temperature creep and reduction of strength, low-temperature embrittlement (8) Chemical reaction or corrosion effects: embrittlement, stress concentration A part will have a proper margin of safety if it is designed with a design limit load larger than the maximum expected working load. The design limit load in turn should be smaller than the calculated damaging loads because of the uncertainty and inaccuracy involved in stress analyses. Damaging loads include: endurance limit load, yield load, and ultimate load, which are defined below. The more accurate the analysis, the smaller the allowable margin between the design limit loads and damaging loads.

Below are given typical recommended criteria for the working loads, the design limit loads, and the damaging loads (yield and ultimated loads). The proof-testing loads applied to component design are also defined. (1) Design limit load: Select the largest of the following:

$$\begin{align*} & 1.2 \times \text { load (A) } \\ & 1.2 \times \text { load (B) } \\ & 1.1 \times \text { load (C) } \tag{2-8}\\ & 1.0 \times \text { load (D) } \end{align*}$$

where

Load (A) = Working load under normal steady operating conditions Load (B) = Working load under normal transient operation conditions, such as during normal engine start and stop Load $(\mathrm{C})=$ Working load under occasional transient operation condition, such as load during irregular starts Load (D) = Mandatory malfunction load which must be taken into account. For example, in a clustered engine configuration, certain mount members may carry the greatest load when one engine ceases to fire while the others are still operating (engine-out capability). In certain instances it is mandatory that an individual rocket engine continue to operate when a given component fails. If this causes significant structural loads, these are considered mandatory malfunction loads. (2) Yield load $=1.1 \times$ design limit load

Yield load is the load which will induce a stress equal to the yield strength of the material used under rated ambient conditions. (3) Ultimate load $=1.5 \times$ design limit load (2-10) Ultimate load is the load which will induce a stress equal to the ultimate strength of the material used under rated ambient conditions. (4) Proof test load $= 1.0 \times$ design limit load (2-11) Proof test load is the load which is applied to test the part during the acceptance inspection. Its value can be adjusted for material properties if the rated ambient conditions cannot be duplicated for the test.

When a part is subjected to an indefinite number of cycles during service life, such as in rotating machinery, the endurance limit of a material should be applied instead of the ultimate strength. The endurance limit is the stress which can be repeated an infinite number of times without causing failure of the material from progressive fracture or fatigue. The endurance limit of metals, depending largely on range of stress variation, is as low as between 20 to 60 percent of their ultimate strength in tension. An additional design margin of safety should also be allowed for dynamic impact loads. When the shape of a part changes abruptly, as with a groove, a notch, a hole, or where a small section joins a large one, the value of unit stress at points close to the abrupt change or discontinuity increases steeply. The amount of stress increase generally ranges from 100 to 300 percent of the mean stress in the section.

Sample Calculation 2-2

The hydraulic accumulator of a large liquidpropellant rocket engine has the following design parameters: (a) Required volume (fluid capacity), $7238 \mathrm{cu} \mathrm{in.;} \mathrm{(b)} \mathrm{working} \mathrm{pressure} \mathrm{(load)} \mathrm{under}$ normal steady and transient operating conditions, 2000 psia ; (c) occasional surge pressure, 2200 psia; (d) mandatory malfunction pressure, 2450 psia; (e) maximum ambient temperature, $300^{\circ} \mathrm{F}$; (f) material selected, AISI 4340 H.T.- 180 . (Strength at room temperature: Ultimate, 185000 psi ; yield, 170000 psi . Strength at $300^{\circ} \mathrm{F}$ : Ultimate, 178000 psi ; yield, 150000 psia .)

Determine the following: (a) Lightest possible configuration and resulting dimensions; (b) required proof test pressure at room temperature.

Solution

(a) Since a sphere is the lightest pressure vessel for a given volume and pressure, we will use this configuration. For a $7238-\mathrm{cu}-\mathrm{in}$. volume

Required inside diameter of the sphere

$$\begin{aligned} & =\sqrt[3]{\frac{6}{\pi} \text { volume }}=\sqrt[3]{\frac{6}{\pi} \times 7238} \\ & =24 \text { inch } \end{aligned}$$

From equation (2-8), design limit pressure $=$ largest of the following:

$$\begin{aligned} & 1.2 \times 2000=2400 \mathrm{psia} ; \\ & 1.1 \times 2200=2420 \mathrm{psia} ; \\ & 1.0 \times 2450=2450 \mathrm{psia} \\ & \text { Selected: } 2450 \mathrm{psia} \end{aligned}$$

From equation (2-9), yield pressure $=1.1 \times 2450=2695 \mathrm{psia}$

Thickness of sphere wall

$$\begin{aligned} & =\frac{\text { Yield pressure } \times \text { diameter of sphere }}{4 \times \text { yield strength at } 300^{\circ} \mathrm{F}} \\ & =\frac{2695 \times 24}{4 \times 150000}=0.108 \text { inch } \end{aligned}$$

or from equation (2-10): Ultimate pressure $=1.5 \times 2450=3675 \mathrm{psia}$ Thickness of sphere wall

$$\begin{aligned} & =\frac{\text { Ultimate pressure } \times \text { diameter of sphere }}{4 \times \text { ultimate strength at } 300^{\circ} \mathrm{F}} \\ & =\frac{3675 \times 24}{4 \times 178000}=0.124 \text { inch } \end{aligned}$$

We will use the higher value 0.124 inch. Therefore, the sphere dimensions $=24-\mathrm{inch}$ inside diameter $\times 0.124-$ inch wall thickness. (b) From equation (2-11), nominal proof test pressure at $300^{\circ} \mathrm{F}$

$$\begin{aligned} & =\text { Design limit pressure } \\ & =2450 \mathrm{psia} \end{aligned}$$

Proof test pressure corrected for room temperature conditions:

$$\begin{aligned} & =2450 \times \frac{\text { Yield strength at room temperature }}{\text { Yield strength at } 300^{\circ} \mathrm{F}} \\ & =2450 \times \frac{170000}{150000}=2780 \mathrm{psig} \end{aligned}$$

Evaluation of Failure Modes

There are three basic types of failure modes: elastic deflection, permanent plastic deformation, and fracture. Although the first type is not a material failure, it may cause a part to perform improperly with resulting malfunction of a component or system. The other two-plastic deformation and fracture-are material failures influenced by material properties, load and environmental conditions, and by the shape of the part.

Each of the three failure modes is characterized by certain criteria. For elastic deflection, strain is the criterion. For plastic deformation and fracture, the criterion is stress. In the process of stress analyses, following load determination, the possible modes of failure of the part can be established in relation to the criteria induced by the loads. Failure cause can thus be determined, and the failure prevented through design changes. Some of the possible combinations of failure modes and criteria are listed in table 2-3 together with suggested design remedies.

Selection of Materials and Dimensions

For the process of finalizing the dimensions of a part to endure all working loads and environmental conditions without failure, the strength or ability of the selected material to withstand these loads must be known. Material properties are determined through materials tests conducted with specimens. In these tests, all conceivable loads such as tension, compression, torsion, and shear are applied, often with simultaneous application of temperature, vibration, or chemical environment. The results are compiled in graphs and tables. From these tables, materials with properties most suitable for a particular application can be selected.

Experimental Stress Analyses

A rocket engine part may be of such shape, or may be loaded in such a way, that design based on theoretical analysis alone is difficult and unreliable. In such cases experimental stress analyses can supplement the theoretical methods. Many recent advances in stress analysis can be attributed to the development of effective experimental methods.

Applying loads simulating as closely as possible those expected to occur in actual use, measurements of strains and stresses are made. These loads can be applied to full-size prototype

Table 2-3.-Failure Modes and Their Criteria

Failure mode	Conditions	Criteria	Design remedies
1. Elastic deflection:
a. Stable equilibrium...	Loads within elastic limits	Strain; linear or angular displacement (stretch or bending)	Change of shape or dimensions (stiffening): material selection
b. Unstable equilibrium.	Loads within elastic limits	Buckling: ratio of applied vs. critical load	Change of shape or dimensions
c. Vibration.	Within elastic limits: abrupt changes of loads; repeated application of load at or near natural frequency	Amplitude, frequency transmissibility, resonance	Stiffening, change of natural frequency: damping
2. Plastic deformation:
a. Yield..............	Loads exceed elastic limits	Stress: permanent set	Change of dimensions and or material
3. Fracture:	Loads may or may not exceed elastic limits: elevated temperatures	Stress; slow permanent set	Change of dimensions and or material
a. Overload	Load increase beyond yield point to ultimate strength	Stress: elongation; area reduction; rupture	Change of dimensions and or material
b. Brittleness	Load above ultimate strength	Stress; rupture with little or no yield	Change of dimensions and or material change of heat treatment; change of contour
c. Impact or shock.....	Abrupt load application to ductile materials	Stress; behavior like brittle materials	Selection of most ducrile material: increased margin of safety
d. Fatigue	Many repeated load applications within elastic limits	Stress: number of load applications	Change of shape and dimensions; change of material; increase of endurance limits

parts, to scale models made from the real material or from special plastic material, or to portions of full-scale parts. Not infrequently, applied loads are intentionally increased beyond rated levels, until failure of the part occurs. These "tests to failure" will establish the actual margin of safety achieved in the design.

The tools used in experimental stress analyses include electrical, mechanical, and optical strain gages; photoelastic plastic models, lacquers, and paints.