Properties Needed for the Design of Static Structures Mahmoud M. Farag, The American University in Cairo Introduction ENGINEERING DESIGN can be defined as the creation of a product that satisfies a certain need. A good design should result in a product that performs its function efficiently and economically within the prevailing legal, social, safety, and reliability requirements. In order to satisfy such requirements, the design engineer has to take into consideration a large number of diverse factors: • Function and consumer requirements, such as capacity, size, weight, safety, design codes, expected service life, reliability, maintenance, ease of operation, ease of repair, frequency of failure, initial cost, operating cost, styling, human factors, noise l evel, pollution, intended service environment, and possibility of use after retirement • Material-related factors, such as strength, ductility, toughness, stiffness, density, corrosion resistance, wear resistance, friction coefficient, melting point, therma l and electrical conductivity, processibility, possibility of recycling, cost, available stock size, and delivery time • Manufacturing-related factors, such as available fabrication processes, accuracy, surface finish, shape, size, required quantity, delivery time, cost, and required quality Figure 1 illustrates the relationship among the above three groups. The figure also shows that there are other secondary relationships between material properties and manufacturing processes, between function and manufacturing processes, and between function and material properties. Fig. 1 Factors that should be considered in component design. Source: Ref 1 The relationship between design and material properties is complex because the behavior of the material in the finished product can be quite different from that of the stock material used in making it. This point is illustrated in Fig. 2, which shows that in addition to stock material properties, production method and component geometry have direct influence on the behavior of materials in the finished component. The figure also shows that secondary relationships exist between geometry and production method, between stock material and production method, and stock material and component geometry. The effect of component geometry on the behavior of materials is discussed in the following section. Fig. 2 Factors that should be considered in anticipating the behavior of material in the component. Source: Ref 1 Reference 1. M.M. Farag, Selection of Materials and Manufacturing Processes for Engineering Design, Prentice Hall, London, 1989 Properties Needed for the Design of Static Structures Mahmoud M. Farag, The American University in Cairo Effect of Component Geometry In almost all cases, engineering components and machine elements have to incorporate design features that introduce changes in their cross section. For example, shafts must have shoulders to take thrust loads at the bearings and must have keyways or splines to transmit torques to or from pulleys and gears mounted on them. Under load, such changes cause localized stresses that are higher than those based on the nominal cross section of the part. The severity of the stress concentration depends on the geometry of the discontinuity and the nature of the material. A geometric, or theoretical, stress concentration factor, K t , is usually used to relate the maximum stress, S max , at the discontinuity to the nominal stress, S av , according to the relationship: K t = S max /S av (Eq 1) The value of K t depends on the geometry of the part and can be determined from stress concentration charts, such as those given in Ref 2 and 3. Other methods of estimating K t for a certain geometry include photoelasticity, brittle coatings, and finite element techniques. Table 1 gives some typical values of K t . Table 1 Values of the stress concentration factor K t Component shape Value of critical parameter, K t Round shaft with transverse hole d/D = 0.025 2.65 = 0.05 2.50 = 0.10 2.25 Bending = 0.20 2.00 d/D = 0.025 3.7 = 0.05 3.6 = 0.10 3.3 Torsion = 0.20 3.0 Round shaft with shoulder d/D = 1.5, r/d = 0.05 2.4 r/d = 0.10 1.9 r/d = 0.20 1.55 d/D = 1.1, r/d = 0.05 1.9 = 0.10 1.6 Tension = 0.20 1.35 d/D = 1.5, r/d = 0.05 2.05 Bending r/d = 0.10 1.7 r/d = 0.20 1.4 d/D = 1.1, r/d = 0.05 1.9 r/d = 0.10 1.6 r/d = 0.20 1.35 d/D = 1.5, r/d = 0.05 1.7 r/d = 0.10 1.45 r/d = 0.20 1.25 d/D = 1.1, r/d = 0.05 1.25 r/d = 0.10 1.15 Torsion r/d = 0.20 1.1 Grooved round bar d/D = 1.1, r/d = 0.05 2.35 r/d = 0.10 2.0 Tension r/d = 0.20 1.6 d/D = 1.1, r/d = 0.05 2.35 r/d = 0.10 1.9 Bending r/d = 0.20 1.5 d/D = 1.1, r/d = 0.05 1.65 Torsion r/d = 0.10 1.4 r/d = 0.20 1.25 Source: Ref 1 Experience shows that, under static loading, K t gives an upper limit to the stress concentration value and applies it to high-strength low-ductility materials. With more ductile materials, local yielding in the very small area of maximum stress causes some relief in the stress concentration. Generally, the following design guidelines should be observed if the deleterious effects of stress concentration are to be kept to a minimum: • Abrupt changes in cross section should be avoided. If they are necessary, generous fillet radii or stress- relieving grooves should be provided (Fig. 3a). • Slots and grooves should be provided with generous run-out radii and with fillet radii in all corners ( Fig. 3b). • Stress-relieving grooves or undercuts should be provided at the end of threads and splines (Fig. 3c). • Sharp internal corners and external edges should be avoided. • Oil holes and similar features should be chamfered and the bore should be smooth. • Weakening features like bolt and oil holes, identific ation marks, and part numbers should not be located in highly stressed areas. • Weakening features should be staggered to avoid the addition of their stress concentration effects ( Fig. 3d). Fig. 3 Design guidelines for reducing the deleterious effects of stress concentration. See text for discussion. Source: Ref 1 References cited in this section 1. M.M. Farag, Selection of Materials and Manufacturing Processes for Engineering Design, Prentice Hall, London, 1989 2. R.E. Peterson, Stress-Concentration Design Factors, John Wiley and Sons, 1974 3. J.E. Shigley and L.D. Mitchell, Mechanical Engineering Design, 4th ed., McGraw-Hill, 1983 Properties Needed for the Design of Static Structures Mahmoud M. Farag, The American University in Cairo Factor of Safety The term factor of safety is applied to the factor used in designing a component to ensure that it will satisfactorily perform its intended function. The main parameters that affect the value of the factor of safety, which is always greater than unity, can be grouped into: • Uncertainties associated with material properties due to variations in composition, heat treatment, and processing conditions as well as environmental variables such as temperature, time, humidity, and ambient chemicals. Manufacturing processes also contribute to these uncertainties as a result of variations in surface roughness, internal stresses, sharp corners, and other stress raisers. • Uncertainties in loading and service conditions Generally, ductile materials that are produced in large quantities show fewer property variations than less ductile and advanced materials that are produced by small batch processes. Parts manufactured by casting, forging, and cold forming are known to have variations in properties from point to point. To account for uncertainties in material properties, the factor of safety is used to divide into the nominal strength (S) of the material to obtain the allowable stress (S a ) as follows: S a = S/n s (Eq 2) where n s is the material factor of safety. In simple components, S a in the above equation can be viewed as the minimum allowable strength of the material. However there is some danger involved in this use, especially in the cases where the load-carrying capacity of a component is not directly related to the strength of the material used in making it. Examples include long compression members, which could fail as a result of buckling, and components of complex shapes, which could fail as a result of stress concentration. Under such conditions it is better to consider S a as the load-carrying capacity that is a function of both material properties and geometry of the component. In assessing the uncertainties in loading, two types of service conditions have to be considered: • Normal working conditions, which the component has to endure during its intended service life • Limited working conditions, such as overload ing, which the component is only intended to endure on exceptional occasions, and which if repeated frequently could cause premature failure of the component In a mechanically loaded component, the stress levels corresponding to both normal and limited working conditions can be determined from a duty cycle. The normal duty cycle for an airframe, for example, includes towing and ground handling, engine run, take-off, climb, normal gust loadings at different altitudes, kinetic and solar heating, descent, and normal landing. Limited conditions can be encountered in abnormally high gust loadings or emergency landings. Analyses of the different loading conditions in the duty cycle lead to determination of the maximum load that will act on the component. This maximum load can be used to determine the maximum stress, or damaging stress, which if exceeded would render the component unfit for service before the end of its normal expected life. The load factor of safety (n l ) in this case can be taken as: n l = P/P a (Eq 3) where P is the maximum load and P a is normal load. The total or overall factor of safety (n) that combines the uncertainties in material properties and external loading conditions can be calculated as: n = n s · n l (Eq 4) Factors of safety ranging from 1.1 to 20 are known, but common values range from 1.5 to 10. In some applications a designer is required to follow established codes when designing certain components, for example, pressure vessels, piping systems, and so forth. Under these conditions, the factors of safety used by the writers of the codes may not be specifically stated, but an allowable working stress is given instead. Properties Needed for the Design of Static Structures Mahmoud M. Farag, The American University in Cairo Probability of Failure As discussed earlier, the actual strength of the material in a component could vary from one point to another and from one component to another. In addition, it is usually difficult to precisely predict the external loads acting on the component under actual service conditions. To account for these variations and uncertainties, both the load-carrying capacity S and the externally applied load P can be expressed in statistical terms. As both S and P depend on many independent factors, it would be reasonable to assume that they can be described by normal distribution curves. Consider that the load-carrying capacity of the population of components has an average of and a standard deviation S while the externally applied load has an average of and a standard deviation P . The relationship between the two distribution curves is important in determining the factor of safety and reliability of a given design. Figure 4 shows that failure takes place in all the components that fall in the area of overlap of the two curves, that is, when the load-carrying capacity is less than the external load. This is described by the negative part of the (S - P) curve of Fig. 4. Transforming the distribution ( - ) to the standard normal deviate z, the following equation is obtained: z = [(S - P) - ( - )]/[( S ) 2 + ( P ) 2 ] 1/2 (Eq 5) From Fig. 4, the value of z at which failure occurs is: z = - (S - P)/[( S ) 2 + ( P ) 2 ] 1/2 (Eq 6) Fig. 4 Effect of variations in load and strength on the failure of components. Source: Ref 1 For a given reliability, or allowable probability of failure, the value of z can be determined from cumulative distribution function for the standard normal distribution. Table 2 gives some selected values of z that will result in different values of probabilities of failure. Table 2 Values of z and corresponding levels of reliability and probability of failure z Reliability Probability of failure -1.00 0.8413 0.1587 -1.28 0.9000 0.1000 -2.33 0.9900 0.0100 -3.09 0.9990 0.0010 -3.72 0.9999 0.0001 -4.26 0.99999 0.00001 -4.75 0.999999 0.000001 Knowing S , P , and the expected , the value of can be determined for a given reliability level. As defined earlier, the factor of safety in the present case is simply S/P. The following example illustrates the use of the above concepts in design; additional discussion of statistical methods is provided in the article "Statistical Aspects of Design" in this Volume. Example 1: Estimating Probability of Failure. A structural element is made of a material with an average tensile strength of 2100 MPa. The element is subjected to a static tensile stress of an average value of 1600 MPa. If the variations in material quality and load cause the strength and stress to vary according to normal distributions with standard deviations of S = 400 and P = 300, respectively, what is the probability of failure of the structural element? The solution can be derived as follows: From Fig. 4, ( - ) = 2100 - 1600 = 500 MPa, standard deviation of the curve ( - ) = [(400) 2 + (300) 2 ] 1/2 = 500; from Eq 6, z = -500/500 = -1. Thus, from Table 2, the probability of failure of the structural element is 0.1587 (15.87%), which is too high for many practical applications. One solution to reduce the probability of failure is to impose better quality measures on the production of the material and thus reduce the standard deviation of the strength. Another solution is to increase the cross-sectional area of the element in order to reduce the stress. For example, if the standard deviation of the strength is reduced to S = 200, the standard deviation of the curve ( - ) will be [(200) 2 + (300) 2 ] 1/2 = 360, z = -500/360 = -1.4, which, according to Table 2, gives a more acceptable probability of failure value of 0.08 (8%). Alternatively, if the average stress is reduced to 1400 MPa, ( - ) = 700 MPa, z = -700/500 = -1.4, with a similar probability of failure as the first solution. Experimental Methods. As the above discussion shows, statistical analysis allows the generation of data on the probability of failure and reliability, which is not possible when a deterministic safety factor is used. One of the difficulties with this statistical approach, however, is that material properties are not usually available as statistical quantities. In such cases, the following approximate method can be used. In the case where the experimental data are obtained from a reasonably large number of samples, more than 100, it is possible to estimate statistical data from nonstatistical sources that only give ranges or tolerance limits. In this case, the standard deviation S is approximately given by: S = (maximum value of property - minimum value)/6 (Eq 7) This procedure is based on the assumption that the given limits are bounded between plus and minus three standard deviations. If the results are obtained from a sample of about 25 tests, it may be better to divide by 4 in Eq 7 instead of 6. With a sample of about 5, it is better to divide by 2. In the cases where only the average value of strength is given, the following values of coefficient of variation, which is defined as ' = S /S can be taken as typical for metallic materials: ' = 0.05 for ultimate tensile strength, and ' = 0.07 for yield strength. Reference cited in this section 1. M.M. Farag, Selection of Materials and Manufacturing Processes for Engineering Design, Prentice Hall, London, 1989 Properties Needed for the Design of Static Structures Mahmoud M. Farag, The American University in Cairo Designing for Static Strength The design and materials selection of a component or structure under static loading can be based on static strength and/or stiffness depending on the service conditions and the intended function. [...]... Heiser, Analysis of Metallurgical Failures, John Wiley and Sons, 1987 N.H Cook, Mechanics and Materials for Design, McGraw-Hill, 1985 F.A.A Crane and J.A Charles, Selection and Use of Engineering Materials, Butterworths, 1984 M.M Farag, Materials Selection for Engineering Design, Prentice Hall, London, 1997 Design for Fatigue Resistance Erhard Krempl, Rensselaer Polytechnic Institute Introduction FATIGUE... load and/ or overestimation of the material property and service conditions could lead to failure in service Properties Needed for the Design of Static Structures Mahmoud M Farag, The American University in Cairo References 1 M.M Farag, Selection of Materials and Manufacturing Processes for Engineering Design, Prentice Hall, London, 1989 2 R.E Peterson, Stress-Concentration Design Factors, John Wiley and. .. particular, such joints tend to negate alloy and composition effects More detailed information on fatigue of welds and mechanical joints is contained in Ref 1, 3, and 4 References 1 Fatigue and Fracture, Vol 19, ASM Handbook, ASM International, 1996 2 M.R Mitchell and R.W Landgraf, Ed., Advances in Fatigue Lifetime Predictive Techniques, STP 112 2, ASTM, 1992 3 H.O Fuchs and R.I Stephens, Metal Fatigue in Engineering,... on weight Reference cited in this section 1 M.M Farag, Selection of Materials and Manufacturing Processes for Engineering Design, Prentice Hall, London, 1989 Properties Needed for the Design of Static Structures Mahmoud M Farag, The American University in Cairo Selection of Materials for Stiffness Deflection under Load As discussed in the section "Design of Beams" in this article, the stiffness of a... decreases Source: Ref 11 It is possible to treat fatigue as a probabilistic process and to use methods of probability theory for design However, a probabilistic design of components is prevalent if a large number of components are involved and is then part of a reliability analysis In the majority of cases, a probabilistic fatigue design is not performed because of the expense and time involved in getting... behavior, and a mechanical hysteresis loop develops between stress and strain Low-cycle fatigue testing is performed at low frequencies, usually below 1 Hz The demands of the application and the development of clip-on axial and diametral extensometers and of servocontrolled testing machines have established strain control as the standard practice in low-cycle fatigue Low-cycle fatigue investigation was in part. .. and L.D Mitchell, Mechanical Engineering Design, 4th ed., McGraw-Hill, 1983 4 W.C Young, Roark's Formulas for Stress and Strain, 6th ed., McGraw-Hill, 1989 Properties Needed for the Design of Static Structures Mahmoud M Farag, The American University in Cairo Selected References • • • • V.J Colangelo and F.A Heiser, Analysis of Metallurgical Failures, John Wiley and Sons, 1987 N.H Cook, Mechanics and. .. ductile wrought metallic materials, the tensile and compressive strengths are very close, and in most cases only the tensile strength is given However, brittle materials like gray cast iron and ceramics are generally stronger in compression than in tension In such cases, both properties are usually given For polymeric materials, which usually do not have a linear stress-strain curve, and whose static properties... burrs, identification marks, and deep scratches due to mishandling could lead to • • failure under fatigue loading Exceeding design limits and overloading If the load, temperature, speed, and so forth, are increased beyond the limits allowed by the factor of safety in design, the component is likely to fail Subjecting the equipment to environmental conditions for which it was not designed also falls under... triangular, or other) is imposed, and test results are reported in terms of stress In reference to Fig 1, the maximum and minimum stress is designated by max and min, respectively The mean stress are given by (a tensile stress is introduced as a positive quantity and a compressive stress is mean and the stress range defined as a negative quantity): mean =( max + min)/2 and =( max - min) (Eq 1) respectively . Cook, Mechanics and Materials for Design, McGraw-Hill, 1985 • F.A.A. Crane and J.A. Charles, Selection and Use of Engineering Materials, Butterworths, 1984 • M.M. Farag, Materials Selection for. M.M. Farag, Selection of Materials and Manufacturing Processes for Engineering Design, Prentice Hall, London, 1989 2. R.E. Peterson, Stress-Concentration Design Factors, John Wiley and Sons,. function and manufacturing processes, and between function and material properties. Fig. 1 Factors that should be considered in component design. Source: Ref 1 The relationship between design and