Calculations and Significant Figures

The discussion in this section applies to real numbers, not integers. The accuracy of a real number depends on the number of significant figures describing the number. Usually, but not always, three or four significant figures are necessary for engineering accuracy. Unless otherwise stated, no less than three significant figures should be used in your calculations.

The number of significant figures is usually inferred by the number of figures given

(except for leading zeros). For example, 706, 3.14, and 0.002 19 are assumed to be numbers with three significant figures. For trailing zeros, a little more clarification is necessary.

To display 706 to four significant figures insert a trailing zero and display either 706.0, 7.060 × 102, or 0.7060 × 103. Also, consider a number such as 91 600. Scientific notation should be used to clarify the accuracy. For three significant figures express the number as 91.6 × 103. For four significant figures express it as 91.60 × 103.

Computers and calculators display calculations to many significant figures. However, you should never report a number of significant figures of a calculation any greater than the smallest number of significant figures of the numbers used for the calculation. Of course, you should use the greatest accuracy possible when performing a calculation. For example, determine the circumference of a solid shaft with a diameter of d = 0.40 in. The

circumference is given by C = πd. Since d is given with two significant figures, C should be reported with only two significant figures. Now if we used only two significant figures for π our calculator would give C = 3.1 (0.40) = 1.24 in. This rounds off to two significant figures as C = 1.2 in. However, using π = 3.141 592 654 as programmed in the calculator, C = 3.141 592 654 (0.40) = 1.256 637 061 in. This rounds off to C = 1.3

in, which is 8.3 percent higher than the first calculation. Note, however, since d is given with two significant figures, it is implied that the range of d is 0.40 ± 0.005. This means that the calculation of C is only accurate to within ±0.005/0.40 = ±0.0125 = ±1.25%.

The calculation could also be one in a series of calculations, and rounding each calculation separately may lead to an accumulation of greater inaccuracy. Thus, it is considered good engineering practice to make all calculations to the greatest accuracy possible and report the results within the accuracy of the given input.

READ MORE.......

Mechanical Engineering

McGraw−Hill Primis

ISBN: 0−390−76487−6

Text:

Shigley’s Mechanical Engineering Design,

Eighth Edition

Budynas−Nisbett

for STEP BY STEP GUIDE autocad simple tutorial please visit.........

www.autocadsimpletutorial.blogspot.com

---or---

www.autocad-simple-tutorial.com

Dimensions and Tolerances

The following terms are used generally in dimensioning:

• Nominal size.

The size we use in speaking of an element. For example, we may specify a 1 1/2 -in pipe or a 1/2 -in bolt. Either the theoretical size or the actual measured size may be quite different. The theoretical size of a 1 1/2 -in pipe is 1.900 in for the outside diameter. And the diameter of the 1/2 -in bolt, say, may actually measure 0.492 in.

• Limits.

The stated maximum and minimum dimensions.

• Tolerance.

The difference between the two limits.

• Bilateral tolerance.

The variation in both directions from the basic dimension. That

is, the basic size is between the two limits, for example, 1.005 ± 0.002 in. The two

parts of the tolerance need not be equal.

• Unilateral tolerance.

The basic dimension is taken as one of the limits, and variation is permitted in only one direction, for example,

• Clearance.

A general term that refers to the mating of cylindrical parts such as a bolt and a hole. The word clearance is used only when the internal member is smaller than the external member. The diametral clearance is the measured difference in the two diameters. The radial clearance is the difference in the two radii.

• Interference.

The opposite of clearance, for mating cylindrical parts in which the internal member is larger than the external member.

• Allowance.

The minimum stated clearance or the maximum stated interference for mating parts.

When several parts are assembled, the gap (or interference) depends on the dimensions and tolerances of the individual parts.

The previous example represented an absolute tolerance system. Statistically, gap

dimensions near the gap limits are rare events. Using a statistical tolerance system, the probability that the gap falls within a given limit is determined.10 This probability deals with the statistical distributions of the individual dimensions. For example, if the distributions of the dimensions in the previous example were normal and the tolerances, t, were given in terms of standard deviations of the dimension distribution, the standard deviation of the gap w¯ would be

However, this assumes a normal distribution for the individual dimensions, a rare occurrence. To find the distribution of w and/or the probability of observing values of w within certain limits requires a computer simulation in most cases. Monte Carlo computer simulations are used to determine the distribution of w by the following approach:

1 Generate an instance for each dimension in the problem by selecting the value of

each dimension based on its probability distribution.

2 Calculate w using the values of the dimensions obtained in step 1.

3 Repeat steps 1 and 2 N times to generate the distribution of w. As the number of

trials increases, the reliability of the distribution increases.

READ MORE.......

Mechanical Engineering

McGraw−Hill Primis

ISBN: 0−390−76487−6

Text:

Shigley’s Mechanical Engineering Design,

Eighth Edition

Budynas−Nisbett

for STEP BY STEP GUIDE solidwork simple tutorial please visit.........

www.solidworksimpletutorial.blogspot.com

---or---

www.solidwork-simple-tutorial.com

Design Factor and Factor of Safety

A general approach to the allowable load versus loss-of-function load problem is the deterministic design factor method, and sometimes called the classical method of

design. The fundamental equation is Eq. (1–1) where nd is called the design factor. All loss-of-function modes must be analyzed, and the mode leading to the smallest design factor governs. After the design is completed, the actual design factor may change as a result of changes such as rounding up to a standard size for a cross section or using off-the-shelf components with higher ratings instead of employing what is calculated by using the design factor. The factor is then referred to as the factor of safety, n. The factor of safety has the same definition as the design factor, but it generally differs numerically.

Since stress may not vary linearly with load (see Sec. 3–19), using load as the loss-of-function parameter may not be acceptable. It is more common then to express the design factor in terms of a stress and a relevant strength. Thus Eq. (1–1) can be rewritten as

The stress and strength terms in Eq. (1–3) must be of the same type and units. Also, the stress and strength must apply to the same critical location in the part.

READ MORE.......

Mechanical Engineering

McGraw−Hill Primis

ISBN: 0−390−76487−6

Text:

Shigley’s Mechanical Engineering Design,

Eighth Edition

Budynas−Nisbett

for STEP BY STEP GUIDE autocad simple tutorial please visit.........

www.autocadsimpletutorial.blogspot.com

---or---

www.autocad-simple-tutorial.com

Stress and Strength

The survival of many products depends on how the designer adjusts the maximum

stresses in a component to be less than the component’s strength at specific locations of interest. The designer must allow the maximum stress to be less than the strength by a sufficient margin so that despite the uncertainties, failure is rare.

In focusing on the stress-strength comparison at a critical (controlling) location, we often look for “strength in the geometry and condition of use.” Strengths are the

magnitudes of stresses at which something of interest occurs, such as the proportional

limit, 0.2 percent-offset yielding, or fracture. In many cases, such events represent the stress level at which loss of function occurs.

Strength is a property of a material or of a mechanical element. The strength of an element depends on the choice, the treatment, and the processing of the material.

Consider, for example, a shipment of springs. We can associate a strength with a specific spring. When this spring is incorporated into a machine, external forces are applied that result in load-induced stresses in the spring, the magnitudes of which depend on its geometry and are independent of the material and its  processing. If the spring is removed from the machine unharmed, the stress due to the external forces will return to zero. But the strength remains as one of the properties of the spring. Remember, then, that strength is an inherent property of a part, a property built into the part because of the use of a particular material and process.

Various metalworking and heat-treating processes, such as forging, rolling, and cold forming, cause variations in the strength from point to point throughout a part. The spring cited above is quite likely to have a strength on the outside of the coils different from its strength on the inside because the spring has been formed by a cold winding process, and the two sides may not have been deformed by the same amount.

Remember, too, therefore, that a strength value given for a part may apply to only a particular point or set of points on the part. In this book we shall use the capital letter S to denote strength, with appropriate subscripts to denote the type of strength. Thus, Ss is a shear strength, Sy a yield strength, and Su an ultimate strength.

In accordance with accepted engineering practice, we shall employ the Greek letters ó (sigma) and ô (tau) to designate normal and shear stresses, respectively. Again, various subscripts will indicate some special  characteristic. For example, ó1 is a principal stress, óy a stress component in the y direction, and ór a stress component in the radial direction.

Stress is a state property at a specific point within a body, which is a function of load, geometry, temperature, and manufacturing processing. In an elementary course in mechanics of materials, stress related to load and geometry is emphasized with some discussion of thermal stresses. However, stresses due to heat treatments, molding, assembly, etc. are also important and are sometimes neglected.

READ MORE.......

Mechanical Engineering

McGraw−Hill Primis

ISBN: 0−390−76487−6

Text:

Shigley’s Mechanical Engineering Design,

Eighth Edition

Budynas−Nisbett

for STEP BY STEP GUIDE autocad simple tutorial please visit.........

www.autocadsimpletutorial.blogspot.com

---or---

www.autocad-simple-tutorial.com