Mechanical Design

The mechanical design of an EOR HRSG is not much different from that of a horizontal tube superheater or economizer used in a power boiler, HRSG, or waste heat boiler.

From: Heat Recovery Steam Generator Technology , 2017

Applications—Solid Mechanics Problems

Zhuming Bi , in Finite Element Analysis Applications, 2018

8.1 Introduction

Mechanical design is to design parts, components, products, or systems of mechanical nature. For example, designs of various machine elements such as shafts, bearings, clutches, gears, and fasteners fall into the scope of mechanical design. Numerous criteria have been proposed in mechanical design processes, some primary design criteria include functions, safety, reliability, manufacturability, weight, size, wear, maintenance, and liability. In general, a mechanical design problem should be formulated with clear and complete statements of functions, specifications, and evaluation criteria (Mott, 2014):

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Functions are specified for what a product can fulfill. Functions are usually described by nonquantitative statements. Exemplifying product functions are to charge power on electronics (charger), clean floors (vacuum), transport objects (mobile platform), or support loads (structure).

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Specifications are detailed requirements described by quantitative statements. For example, product specifications can be defined in terms of size, weight, precision, working volume, speed, or load capacity. Specifications turn into design constraints in problem-solving processes.

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Evaluation criteria are the statements of desirable qualitative characteristics. Evaluation criteria are treated as design objectives to optimize the solutions in problem-solving processes. Evaluation criteria are set to maximize benefits and minimize disadvantages of mechanical designs.

Although the numbers and priorities of specifications and criteria vary from one product design to another; some common design considerations are applicable to any mechanical systems. These considerations include loading capability, deformation, stability, and durability. The dependence of these evaluation criteria on design variables must be modeled and analyzed to optimize products. In this chapter, the governing mathematical models of common engineering design problems are discussed, and finite element analysis (FEA) is used to analyze mechanical systems from the perspective of loading capability, stability, and fatigue life.

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Understanding and modelling fuel behaviour under irradiation

G. Rossiter , in Nuclear Fuel Cycle Science and Engineering, 2012

Mechanical design codes

Mechanical design codes are used to evaluate the mechanical behaviour of the fuel assembly. They often also have the functionality to be used for thermal analyses if the temperatures of the assembly structural components (i.e. everything excluding the fuel pins) are of interest. The standard technique employed is the finite element method, where each component of the assembly is discretised into a number of volume elements. Given suitable material properties, and any (potentially time dependent) external loading, the (potentially time dependent) stresses, strains and displacements applicable to each finite element are then determined by solution of the underlying matrix equations for the force balances, stress-strain relations and strain-displacement relations.

The primary application of mechanical design codes is to calculate the stresses imposed by the loads applied to the various assembly components (during normal operation, anticipated operational occurrences and accidents). Other uses include vibrational mode and harmonic response analysis, and buckling assessments.

Mechanical design calculations have historically been performed using in-house codes; more recently, 'off the shelf' commercial finite element software packages have tended to be employed.

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Robotic and Image-Guided Knee Arthroscopy

Liao Wu , ... Ross Crawford , in Handbook of Robotic and Image-Guided Surgery, 2020

29.2.2 Mechanical design

Mechanical design is the most important part to endow arthroscopic tools with steerability. There are generally three challenges faced in the mechanical design of steerable arthroscopic tools:

1.

Size. Restricted by the keyhole surgery setting and the confined space inside the knee joint, the size of the arthroscopic tools has to be made very small. This is especially challenging when the tools are to be added with multiple DoFs to increase dexterity. Current arthroscopic tools used in surgeries usually have a shaft with a diameter of less than 5   mm. The steerable arthroscopic tools should be designed with comparable sizes.

2.

Dexterity. In order to make the tools steerable and dexterous, complex structures and mechanisms need to be integrated with the mechanical design. There is generally a tradeoff between the dexterity of the tool and the compactness the tool can be made with.

3.

Force transmission ability. In some operations in arthroscopy, the tools are used to exert some forces on the hard/soft tissues inside the knee. In these applications, a good force transmission ability is necessary. This is, however, very challenging when the tools are made steerable. A compromising design should be taken for these situations where both the force transmission ability and the steerability are desired.

To address these challenges, researchers have proposed different mechanisms. Some of the mechanisms that have been proposed for the steerable robotic arthroscopic tools are depicted in Fig. 29.3.

Figure 29.3. Mechanisms for constructing steerable arthroscopic tools: (A) SMA-based [10]; (B) hinged-joint-based [11]; (C) lobed-feature-based [12]; (D) notched-tube-based [13]; (E) tube-and-slider-based [14]; and (F) spine-and-hinge-based [15] SMA, Shape-memory alloy.

Traditional serial-link robots have three structural components: links, joints, and motors. Links are the main body of a robot and are connected by joints; joints are actuatable mechanisms that can move the links they connect; motors are actuators that drive the motion of the joints. Steerable tools for arthroscopy usually do not have the same structures, but we can map their components to those of the serial-link robots by mimicking their functions. In this way, the mechanical structures of the prototypes shown in Fig. 29.3 can be summarized in Table 29.1.

Table 29.1. Mechanical structure and characterization of the prototypes shown in Fig. 29.3.

Fig. 29.3 Mechanical structure Characterization
Links Joints Motors Diameter (mm) DoF Applied force
A [10] Plastic disks Special arrangement of SMA wires SMA wires 8 1 1   N
B [11] Disks and spines Hinges Cables 4.2 1 At least 1   N
C [12] Disks and spines Lobed features Tendons 4.2 1 At least 3   N
D [13] Two nested tubes Asymmetric notches on the tubes Cables 5.99 1 At least 1   N
E [14] A distal link and two proximal tubes A hinge composed by two sliders Rotation of outer tube 5 1 Axial 100   N
Lateral 20   N
F [15] Disks and a central spine Space between disks and deformation of spine Cables 3.6 3 Unknown

DoF, Degree of freedom; SMA, Shape-memory alloy.

The characterization of the prototypes shown in Fig. 29.3 is also summarized in Table 29.1. As discussed previously, size, dexterity, and force transmission ability are three important factors for steerable arthroscopic tools. In Table 29.1, these factors are embodied by the diameter of the tool, the number of DoFs, and the applied force, respectively.

Generally, most of the prototypes can be made as small as less than 6   mm in diameter. The prototype in Ref. [10] was slightly larger than the others, with a diameter of 8   mm. However, according to the authors, it could be reduced to 4   mm, which is rational considering the simplicity of the design.

In terms of the dexterity, most designs chose to endow the device with only one bending DoF. Since the device is handheld, it naturally has four additional DoFs (three rotations and one translation) empowered by the motion of the hand. In consideration of this, one DoF at the tip is sufficient for some operations inside the joint. The prototype in Ref. [15] added another bending DoF to the proximal part of the tool and a pivoting DoF to the distal tip. The additional DoFs enable the device to be bent in a different plane without changing the orientation of the handle, and adjust the approaching direction of the tip without affecting the bending segment.

Most cable-driven mechanisms, which form the main part of the steerable mechanisms, have applied force less than 3   N. This is sufficient for examination purposes when the mechanism is used to deliver a camera to the surgical site for inspection. However, for some other operations such as cutting, greater force transmission capability is desired. Distinguished from the other prototypes, the design in Ref. [14] does not rely on the pulling of cables but rather uses a special mechanism to transform the rotation of a tube to the translation of a hinge at the tip. As a consequence, the force it can exert increases significantly. The tradeoff, however, is the lack of flexibility that may cause damage to the cartilage.

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Heat transfer theory

Maurice Stewart , in Surface Production Operations, 2021

9.1.2 Industry codes, standards, and recommended practices

The mechanical design of exchangers is based on the standards developed by the

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American Society of Mechanical Engineers (ASME)

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Tubular Exchanger Manufacturers Association (TEMA)

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American Petroleum Institute (API)

ASME covers

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Mechanical design of pressure vessel components

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Shells

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Channels

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Tube-sheets

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Headers and some aspects of large flanges

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Imposes restrictions on piping, instrumentation and inspection of steam boilers.

TEMA covers

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Shell-and-tube heat exchangers only

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Nomenclature

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Fabrication tolerances

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Standard tube sizes

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Standard clearances

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Minimum plate thickness

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Minimum tie rod requirements

TEMA also provides tube-sheet design rules that differ from ASME tube-sheet design rules. TEMA tube-sheet rules are used for most shell-and-tube exchangers. ASME tube-sheet rules are used for stayed fixed tube-sheet steam boilers, and air cooler headers.

API standards address mechanical design criteria of shell-and-tube exchangers, and air-cooled exchangers. There are no industry standards for double-pipe, multi-tube hairpin or plate-and-frame heat exchangers. Fouling, corrosion, tube vibration, leak tightness of flanges, and safe design for tube rupture pressure transients are also discussed in this volume because the subjects are beyond the scope of industry Codes and Standards.

Heat Transfer Research Inc. (HTRI) conducts large scale tests of heat exchangers and incorporates the results into reports and computer programs. HTRI has tested extensively shell-and-tube exchangers in boiling, condensing and single-phase service, and completed limited testing of air-coolers and plate-and-frame exchangers. HTRI technology is used by companies, under contract, for design and evaluation of thermal and hydraulic performance of heat exchangers. HTRI information is confidential and not available to the general public.

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Additively manufactured metals for medical applications

Kim Vanmeensel , ... Jan Van Humbeeck , in Additive Manufacturing, 2018

4.3 Stainless steels

Mechanical design and materials design are two straightforward approaches for the future development of AM steels for medical applications. One example of the mechanical design can be light-weight lattice cellular structures. Controlling the cell design of steel lattice structures allows to attain a broad range of mechanical characteristics. Utilization of the lattice structures in implant applications can prevent stress shielding problems and improve osseointegration between bone and implant. In vitro cytocompatibility tests with human osteosarcoma cells on stainless steel lattice structures have shown that the SLM process does not affect the cytocompatibility of 316L stainless steel after incubation.

Future materials design will focus on stainless steel alloy modification, specifically aiming at biomedical applications. The wide variety of different powder mixtures that can be applied during SLM drives the development of new biomaterials with unique combinations of properties, aiming at a combination of enhanced bioactivity and osseointegration as well as enhanced antibacterial and antimicrobial characteristics. The recent trend to use Ni-free stainless steel grades for medical applications is pushed by the fact that Ni can be toxic when released to the human body. New nickel-free nitrogen-alloyed steels are suggested as a promising replacement of conventional stainless steels in biomedical applications. Besides their high strength, better corrosion and wear resistance as well as superior biocompatibility, they are also cost effective. This combination of properties makes nitrogen-alloyed steels attractive candidate materials for laser manufacturing. However, due to the potential nitrogen evaporation, their AM manufacturability needs to be investigated.

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Slender snake-like endoscopic robots in surgery

Shumei Yu , ... Hongliang Ren , in Flexible Robotics in Medicine, 2020

1.4.2 Motion planning

Mechanical designs of snake-like surgical robots determine their workspace, which is one of the criteria of the robot's dexterity. A snake surgical robot's workspace can be derived from a forward kinematical model or backward kinematical model. Once the workspace of a snake surgical robot is known, it is essential to plan the motion of the robot to reach the operational area and manipulate the target. The anatomy of the operating environment is hard to model, which brings complexity to the robot's motion planning. Even if the organs and tissues can be reconstructed in advance, motion planning of the robot should be careful by considering tissue deformation and collision avoidance. For a snake robot with 20 linked sections for the exploration of osteolytic lesions, without modeling of the lesion's cavity, Liu et al. [69] proposed the motion planning, including collision detection based on sensor and sampling. Omisore et al. [24] proposed an inverse kinematics (IK) method for the planning of the path, with collision detection and avoidance at the assistance of virtual points. Chen et al. [70] considered less sweep area and target reachability as the motion planning criteria and proposed safety-enhanced planning based on a dynamic neural model.

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Spacecraft Structures

Tetsuo Yasaka , Junjiro Onoda , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

II.C Launch Vehicle Compatibility

Mechanical design requirements are defined for each launch vehicle so that the spacecraft can be made compatible with the launch vehicle. The mass limit is the most important consideration for the overall spacecraft system. The mass capability of a vehicle is strongly dependent on the orbit to be achieved. The available mass to the low earth orbit is the largest of all. This mass is decreased to 40% when the target orbit is a sun-synchronous or geostationary transfer orbit. The mass in the geostationary orbit is further decreased by about 50% from that in GTO.

A launch vehicle provides several choices for payload shrouds and payload adapters. The payload adapter is used to mechanically connect the spacecraft to the vehicle. A clamp band or separation nuts attach the spacecraft on to a machine shaped ring on the interface plane of the adapter (Fig. 10). The space shuttle provides multiple attach fittings as shown in Fig. 11. The spacecraft maximum dimensions should be within the specified zone of the payload shroud, leaving the clearance zone free. The clearance zone is specified so that the satellite does not touch the shroud inner surface or acoustic blankets for the maximum deflection under static and dynamic ascent loads. The position of the spacecraft mass center should be within a specified area with respect to the launch vehicle attach fitting(s). This is to avoid control problems of the launch vehicle and excessive bending loads on the satellite/launch vehicle structures.

FIGURE 10. Payload adapters: examples from H-IIA. Spacecraft attached by clump band (above) or by separation nuts (below).

FIGURE 11. Space shuttle attachment points.

Launch vehicles require spacecraft to possess rigidities above certain values, specified by the lowest vibration frequencies in the lateral and longitudinal directions under the rigidly fastened conditions at the mounting points. This requirement is imposed to limit the maximum deflection during the ascent and to avoid dynamic couplings with the launch vehicle. In many cases, the rigidity requirement poses the largest structural design constraint among the requirements specified by the launch vehicle. Typically, fundamental frequencies of 10   Hz in the lateral direction and 30   Hz in the longitudinal direction are required.

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Adult and Fetal Stem Cells

Elizabeth F. Irwin , ... Kevin E. Healy , in Handbook of Stem Cells (Second Edition), 2013

Creating Matrices with Tunable Moduli

Mechanical design parameters for artificial matrices include elasticity, compressibility, viscoelastic behavior, and tensile strength. Controlling the mechanical properties of a material at the cellular level can help elicit a desired cell response, and, in addition, the bulk mechanical properties of the matrix must be controlled such that the matrix is able to withstand loads that may be involved in downstream applications.

The mechanical properties of hydrogels can be varied and controlled via chemical synthesis and processing. Hydrogels are composed of long, hydrophilic polymer chains either physically entangled or chemically crosslinked to form a network, and their mechanical properties can be chemically altered by controlling crosslinking density (entanglements or chemical crosslinks). For the AAm gels described on page 933, the input crosslinker (bisacrylamide) concentration of the AAm gels was varied, and a linear relationship between input crosslinker density and gel modulus was found. Based on prior work (see p. 933) (Engler et al., 2006; Saha et al., 2007, 2008; Boonen et al., 2009), tuning the crosslink density of hydrogels may aid in designing systems to support stem cell self-renewal or differentiation, depending on the desired application. An increasing number of studies have illustrated a role of stiffness in regulating stem cell function in two dimensions, and initial evidence to date indicates that the mechanical properties of a material are likely to also influence stem cell behavior in 3D (Banerjee et al., 2009).

Although a direct correlation with matrix stiffness and behavior of hES or iPS cells has yet to be demonstrated, Li et al. (2006) proposed that the soft mechanical properties of their hydrogels improve the self-renewal of hES cells on their defined, synthetic hydrogels. In this work, pNIPAAm hydrogels functionalized with bsp-RGD15 and with a complex shear modulus of ~50–100 Pa (depending on the frequency of the measurement) and were able to maintain pluripotency in the short-term. Future studies are very likely to focus on analyzing the effects of stiffness and other mechanical properties on the self-renewal, lineage commitment, and differentiation of numerous cell types, providing additional key design parameters to control cell function for downstream applications.

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Development of last-stage long blades for steam turbines

T. Tanuma , in Advances in Steam Turbines for Modern Power Plants, 2017

13.4.3.1 Blade height, hub diameter, and dovetail type selection

The mechanical design starting point is the last-stage blade annulus area that was selected considering the design point and a dominant application domain in the last-stage design space. Such an annulus area can be achieved through different hub/tip radius ratios. From the mechanical design viewpoint, smaller radius ratio (larger blades on smaller hub diameter) is rather preferable for rotor design, while larger radius ratio (shorter blades on larger hub diameter) may reduce the blade stress.

Before assessment of stress levels of blades and rotors with various radius ratios, a dovetail type should be selected. A dovetail is a structure for blade–rotor connection. Standard types of dovetail for long blades are curved or straight axial entry (fir tree) type and fork (nest-finger) type. Though fork type last-stage blade design has been employed in many steam turbines for power plants, recently developed last-stage blades employed axial entry (fir tree) type (see Tables 13.1 and 13.2). This result shows the advantage of the axial-entry-type dovetail for recent very-long-blade designs. It is said that compactness of the curved axial entry dovetail results in the reduction of rotor stress.

The mechanical design provides an allowable boundary of radius ratio and the optimum radius ratio is selected considering aerodynamic and aeromechanical requirements.

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Mechanical Design

Hugh Jack , in Engineering Design, Planning, and Management, 2013

12.12 Material selection and part design

Mechanical design often centers on a number of material groups depending on the application objectives and constraints. A few of the major categories are outlined in the following list, with some of the reasons they are used. The topics behind the analysis and design of parts are covered in mechanics of materials textbooks such as Mott (2008) and Shigley and Uicker (1995).

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Steel: High strength, low cost

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Aluminum: Low mass and medium cost, decent corrosion resistance

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Thermo plastics: Low cost, complex geometries

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Thermoset plastics: Low-cost dies, complex geometries, heat resistance

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Rubbers/elastomers: Low cost, complex geometries, vibration isolation, electrical insulator

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Light metals: Complex geometries, complex properties

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Glass: Corrosion resistant, high stiffness, good insulator

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Stainless steel: Corrosion resistant

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Composites: Lightweight with high design control of strength properties

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Ceramics: Very hard, high temperatures

12.12.1 Metals

At a microscopic scale, forces in materials are described using stress and strain. The stress is a pressure calculated using the applied force and the geometry of the part. The stresses in any part are rarely uniform between any two points. Stress produces strain, a compression, or elongation, or it shears the material. The relationship between stress and strain is analogous to a spring coefficient; it is called the Elastic (or Young's) modulus (see Table 12.1 for typical values). At lower strains materials deform and bounce back elastically. At some point a material is deformed enough that it will not return to its original shape. If the strain continues to grow it eventually reaches a point where the material starts to fail and parts begin to break. The stress-strain curves in Figure 12.58 show three different types of materials (1) brittle, (2) ductile, and (3) tough. For each of the three, a "Y" is used to mark the yield point, the maximum before the material is permanently stretched. The maximum stress-strain limit for each material is marked with an "M." The point where the material breaks is marked with an "F." Material tables typically provide the ultimate tensile strength (UTS) and yield strength, as illustrated for the ductile material curve.

Table 12.1. Common Material Properties

Material Type Density (mg/m3) E (GPa) G (GPa) Poisson Tension Yield Stress (MPa) Tension Ultimate Stress (MPa) Tension Yield Strain (%) Yield Stress Compression (MPa) Ultimate Stress Compression (MPa) Thermal Expansion 106 °C−1
Aluminum Typical 2.6–2.8 70–79 26–30 0.33 35–500 100–550 1–45 23
Brass Typical 8.4–8.6 96–110 36–41 0.34 70–550 200–620 4–60 19.1–21.2
Bronze Typical 8.2–8.8 96–120 36–44 0.34 82–690 200–830 5–60 18–21
Cast Iron Typical 7.0–7.4 83–170 32–69 0.2–0.3 120 69–480 0-1 340–1400 9.9–12
Concrete Typical 2.3 17–31 N/A 0.1–0.2 10–70 7–14
Reinforced 2.4 17–31 N/A 0.1–0.2
Lightweight 1.1–1.8 17–31 N/A 0.1–0.2
Copper Typical 8.9 110–120 40–47 0.33–0.36 55–760 230–830 4–50 16.6–17.6
Glass Typical 2.4–2.8 48–83 19–35 0.17–0.27 30–1000 0 5–11
Magnesium Typical alloys 1.76–1.83 41–45 15–17 0.35 80–280 140–340 2–20 26.1–28.8
Monel 67% Ni, 30% Cu 8.8 170 66 0.32 170–1100 450–1200 2–50 14
Nickel 8.8 210 80 0.31 100–620 310–760 2–50 13
Plastic Nylon 0.88–1.1 2.1–3.4 N/A 0.4 40–80 20–100 70–140
Plastic Polyethylene 0.96–1.4 0.7–1.4 N/A 0.4 7–28 15–300 140–290
Rubber Typical 0.96–1.3 0.0007–0.004 0.0002–0.001 0.45–0.50 1–7 7–20 100–800 130–200
Steel 7.85 190–210 75–80 0.27–0.30 10–18
High strength 340–1000 550–1200 5–25 14
Machine 340–700 550–860 5–25
Spring 400–1600 700–1900 3–15
Stainless 280–700 400–1000 5–40
Tool 520 900 8
Titanium Typical alloys 4.5 100–120 39–44 0.33 760–1000 900–1200 10 8.1–11
Tungsten 1.9 340–380 140–160 0.2 1400–4000 0–4 4.3
Wood Douglas fir 0.48–0.56 11–13 30–50 50–80 30–50 40–70
Wood Oak 0.64–0.72 11–12 40–60 50–100 30–40 30–50

Figure 12.58. Typical stress-strain curves for normal tensile stress.
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Ductile and tough materials tend to deform visibly before failure. Brittle materials will often fail with no warning.

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Materials such as plastics and rubbers will "creep" when exposed to a load over a long period of time. For example, a plastic cable holding a 100   kg load will become longer over a period of time.

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When using dissimilar metals, corrosion problems occur because of the different potential voltages of metals. For example, mixing steel and aluminum parts is not a good idea.

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Temperature dependence: As temperature varies, so do physical properties of materials. This makes many devices sensitive to changing temperatures.

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Ultraviolet: Light with a frequency above the visible spectrum.

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Density: A mass per unit volume.

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Crude cost ranges:

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Basic steels: >$1/kg

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Special alloys of steel and aluminum: >$4/kg

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Basic aluminum: >$2/kg

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Plastics: <$1 to $2/kg

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Titanium, magnesium, copper: >$8/kg

Under compression, a square element of material will become shorter but also wider. The ratio is not normally 1:1, the actual ratio is called Poisson's ratio (Figure 12.59). In simple terms, under compression the volume or area of material decreases slightly. In tension an element will become longer and narrower.

Figure 12.59. Poisson's ratio for deformation.

Under static loads, material properties are generally stable. When used in dynamic loading cases, a material will weaken and may eventually fail. The graph in Figure 12.60 shows the general trend of fatigue-induced weakening. Graphs such as these are available for most common materials. The most common issues are rotating shafts. Consider a shaft rotating at 1800 RPM: it will turn 96,000 times per hour, and in a day it will have rotated over a million times. Some materials such as steel might drop from an initial strength of 500   MPa to constant 50   MPa after a few million cycles; this asymptote is called the endurance limit. Other metals such as aluminum do not have an endurance limit and will weaken until the strength goes to 0   Pa. In general, fatigue is undesirable, but techniques such as couplers can reduce bending stresses in shafts and avoid fatigue loading. Otherwise the design strength must consider the weakening by fatigue.

Figure 12.60. Fatigue loading and strength reduction.

Problems

12.65

What is the difference between UTS and yield?

12.66

Can a brittle material be tough?

12.67

An aluminum rod is carrying a mass of 1000   kg. What is the stress if the diameter is 10   mm? What is the stress if the diameter is 5   mm? What is the minimum diameter before plastic deformation?

12.68

A rubber block is 20   mm by 20   mm by 20   mm. A pressure of 100   kPa is applied to the top face. What are the new dimensions for the block?

12.69

Use the Internet to locate a price for a 5   m long I-beam with a mass of approximately 300   kg.

12.70

Why should we avoid cyclic loading in designs?

12.12.2 Stress and strain

A rigid body will naturally rotate around the center of mass. When the rigid body is forced to rotate around another point, that is not the center of mass, the increase in the moment of inertia can be predicted with the parallel axis theorem. The more mass, and the farther it is from the center of rotation, the higher the mass moment of inertia. More inertia means more torque is required to accelerate and decelerate. The center of area, or centroid, will be the same as the center of mass if the material is homogenous (all the same). A part that is twisted or bent moves around the centroid. The deformation induced by the load is a function of the area moment of inertia. Some of these mass properties are shown in Figure 12.61. For symmetrical objects, the centroid is at the geometrical center. The beam is shown with the three principal deformations. Bending caused by some combination of x or y forces will cause the tip of the beam to move sideways. Torsion at the end of the beam will result in twisting that will give the beam a helical shape. For bending, the area moment of inertia I is used. The polar area moment of inertia J is used for twisting. The mass moment of inertia and area moment of inertia are related by mass. The larger the values, the harder it is to twist. Moving material twice as far from the centroid makes it four times harder to twist.

Figure 12.61. Basic mass properties.

The mass property calculations for simple shapes are readily available in tables. 4 Masses that are a combination of simple shapes are found with simple calculations and the parallel axis theorem. More complex masses can be found using integration or with calculation tools in CAD software.

In Figure 12.62 a cantilevered beam is anchored at one side and hangs out unsupported. This configuration generates the greatest deflection at the tip and greatest stresses at the anchor. To calculate the stresses and deformations we need the material stiffness, E, area moment of inertia, I, applied load, M or P, and the length, L. Tip deflection is calculated using Equation 12.13. Mechanical engineering reference books have closed-form equations for a number of beam loading cases.

Figure 12.62. Deflection of a cantilevered beam.

When the loading cases involve multiple loads and moments, the beam bending curve will not be simple, as illustrated in Figure 12.63. The effect of the three loads are shown in the shear force, V(x), and bending moment, M(x), on the diagram. A nonlinear integral can be used to find the deflected shape. In this case the force is approximated for the bottom face. It is reasonable to use the closed-form solutions for different loading cases to estimate the maximum deflections. These results are often verified with finite element analysis (FEA).

Figure 12.63. Shear force and bending moment diagrams.

Buckling occurs when objects are in compression and bend until failure (Figure 12.64). This is normally an issue for members that are very long and/or thin. Equation 12.15 is a popular approximation of the buckling load. As long as the load is below the limit P, the beam should not buckle. Given the approximate nature of the equation it should be used with a larger factor of safety.

Figure 12.64. Euler's buckling equation.

If normal stress acts on a rectangular material element it will make the rectangle longer while compression will make it shorter, as illustrated in Figure 12.65. Shear stress pulls rectangular elements into a rhombus shape. The angled shape indicates there is tension in one direction and compression in the other. In both cases Poisson's ratio indicates that the geometries grow or shrink. For normal stresses Young's modulus, E, can be used to find the deformation. For shear, the shear modulus, G, is used. G and E are proportional and related by Poisson's ratio. When a material is in shear, the analysis needs to consider the ultimate tensile strength and the ultimate compression strength. The shear relationship is important for members in torsion (Figure 12.66). Applied torsion creates a shear force and the resulting deformation is twisting. When torsional members fail it is at a 45° angle where the tension is the greatest.

Figure 12.65. Shear stress and strain.

Figure 12.66. The stresses and deformation of members in torsion.

Problems

12.71

A larger area moment of inertia makes a beam harder to bend. How can rectangular geometry be changed to decrease the bending in one direction, keeping the cross sectional area the same?

12.72

What is the polar moment used for bending or twisting?

12.73

What is a cantilevered beam? What is tip deflection?

12.74

A rod will buckle when too much load is applied. Will the buckling load be increased by (a) decreasing the length, (b) decreasing the diameter, or (c) using stiffer material? Show the equations.

12.75

Draw shear force and bending moments for a cantilevered beam with a downward force halfway out.

12.76

Does the bending moment cause bending in the beam?

12.77

Does a beam in torsion experience stress or strain?

12.78

A 4   m long, 20   mm diameter solid steel axle has a torque of 1   kNm applied. How much does the rod twist?

12.12.3 Analysis of stresses in parts

Poisson's ratio tells us that a part under stress changes shape. If the geometry is simple, so is the part deformation. However if the geometry is complex the stresses will not be uniform. Consider what happens when pieces of fabric, paper, or plastic are cut into odd shapes. They will tend to rip at the thinnest places. When pulled they will warp or bulge in some places. Solids behave in similar ways. Consider the solids in Figure 12.67. There will be a very high stress where the force is applied, if it is at a very small point. An approximate line can be drawn to show the path for the maximum stresses. The stress around the center hole will cause deformation that will probably lead to tearing at the side of the hole. The deformation of the part will make this happen at a load below the simple normal stress. A correction factor is available. In the bottom part the forces are applied to the inside of holes. The force lines must travel through a very small area at the ends, and this will probably be the site of failure.

Figure 12.67. Stress, strain, and deformation in solid parts.

There are a large number of stress concentration factors based on geometrical features. The part shown in Figure 12.68 will fail at the top or bottom side of the hole. The graph shows that the applied force will be up to 3.0 times greater than the normal stress in the material. The total width of the part is r + d, but the part will break where the effective width is d. The stress is multiplied by the correction factor for the geometry. In this case multiplying the stress by 3 will be the safest estimate. Figure 12.69 shows a stress concentration factor for a normal stress load on a member with a stepped neck. The stress concentration can be reduced with a larger radius or smaller relative step. For torsion on a shaft, the correction graph in Figure 12.70 can be used. This would be of use when dealing with power transmission shafts that have a stepped cross section.

Figure 12.68. Approximate stress concentration correction factors for a hole in a plate in tension.

Figure 12.69. Approximate stress concentration correction factors for a step in a beam or shaft diameter in tension.

Figure 12.70. Approximate stress concentration correction factors for a step in a shaft in torsion.

The worst case for part stress and cracking is a sharp wedge shape (e.g., a knife blade one atom thick). In practical terms the stress concentration factor could be in the range of tens or hundreds. As the angle/diameter of the crack or feature increases, the stress concentration drops. The other extreme is a perfectly flat surface.

The procedure for analyzing the strength of parts is outlined in the following list. Experienced designers will be able to estimate part strength, and deflection, within a reasonable range.

(1)

If it is not obvious, identify the part geometry, and applied forces and torques.

(2)

Look for areas of the parts where there are high forces in thinner sectioning. Do a simple stress analysis for these locations.

(3)

Examine the places with changes in area or sharp internal features. If these are close to the high stress areas, calculate the stresses and use correction factors to increase the value.

(4)

Compare all of the stresses to the maximums for the material.

(5)

Consider the worst-case loads for the part and the factor of safety. Don't forget to use a larger factor (maybe 5 to 10) for dynamic loads and impacts.

(6)

Redesign the part if necessary.

Hand analysis of part strengths and deflections can be done quickly and used to do systematic design. However, as the geometry of the part becomes more complex, the accuracy of the results decrease, and the complexity increases.

Problems

12.79

Cut two identical rectangular pieces of paper. Pull one of the rectangles until it rips, and estimate the load. Cut a hole in the center of the second strip of paper. Pull the paper until it rips, and estimate the maximum force. Where did the papers rip? What was the stress (force/width)? How does this compare to the theoretical value?

12.80

Why does stress distribute unevenly in parts? Where will the part fail first?

12.81

What does it mean when the stress concentration factor is 1.0?

12.82

What features create very high stress concentrations?

12.83

Consider a d = 20   mm part that narrows to d = 10   mm. What filet radius, r, is required for a stress concentration factor of 2.0?

12.84

Use the Internet to find a trusted source for a stress concentration graph for a straight cross section with a notch on one or two sides.

12.85

Estimate the factor of safety for the part shown. Assume that the part is steel and the radius at the shaft diameter steps is 1   mm.

12.12.4 Finite element analysis

When you get results from software, reports, or people, pause and make sure it is rational.

Given that most parts are designed as solid models in CAD systems, it is possible to do a relatively fast analysis of part stresses using finite element analysis (FEA). For simpler parts these provide an estimate of stresses and deflections of a part. Advanced analyses can include three-dimensional parts, adaptive mesh refinement, and plastic/nonlinear deformation. A wise designer will recognize the limitations of the method. Accuracy is decreased as the elements become larger and have more extreme angles. It is wise to do both FEA and hand calculations to provide greater confidence in the final results. As with hand calculations, the results are not perfect. For an FEA analysis that is well done, an error of 5 to 10% might be reasonable.

Typically FEA only provides an analysis of the suitability of the part. Design often involves multiple iterations of the part geometry or materials. It is much more productive to design parts with hand calculations first and then use FEA for refinement and verification. For example, the part analyzed in Figure 12.71 could be equated to two beams with a point load. A factor of 2 could be added to compensate for the change in cross section.

Figure 12.71. Finite element analysis.

Problems

12.86

What are the advantages and disadvantages of automatic FEA mesh generation?

12.87

Why should you do hand calculations if you are doing an FEA analysis?

12.88

Use BOE calculations to estimate the strength of the part shown. Use FEA software to verify the analysis. Discuss the differences.

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https://www.sciencedirect.com/science/article/pii/B9780123971586000127