Applied Engineering Analysis - slides for class teaching * Author: Tai-Ran Hsu, Ph.D .

1. Applied Engineering Analysis - slides for class teaching* Author: Tai-Ran Hsu, Ph.D. Chapter 1 Overview of Engineering Analysis * Based on the book of “Applied Engineering Analysis” John Wiley & Sons, 2018 (2018 version) Chapter 1 Overview

2. Chapter Learning Objectives ● To learn the concept and principles of engineering analysis, and the vital roles that engineering analysis plays in professional engineering practices ● To learn the need for the application of engineering analysis in three principal functions of professional engineering practices in creation, problem solving and decision making. ● To illustrate the critical needs for engineers solving problems that relate to protection of properties and public safety, and also the needs for engineers making decisions in real-time situations that often involve grave consequences. ● To learn the roles that mathematics play in engineering analysis, and the ability to use mathematical modeling in problem solving and decision making in dealing with real physical situations.

3. 1.1 Introduction: What is Engineering Analysis? Engineering analysis involves the application of scientific principles and approach that often use mathematical modeling as a tool to reveal the physical state of an engineering system, a machine or device or structure under study. Application of Engineering analysis also include electric circuit design, derivation of algorithms for computer programming, etc. It is an integral part of the professional practice of ALL engineering disciplines. Engineering analysis is a vital TOOL for practicing engineering professionals in performing their principal duties in: Creations Decision making Problem solving

4. 1.2 Engineering analysis and Engineering Practices 1.2.1 ENGINEERS CREATE: “Scientists DISCOVER what it was, Engineers CREATE what it is not” (a quote of a great scientist, Albert Einstein) Engineers create “what it is not” in DESIGN to satisfy human needs and sustain high living standard for all human kind: Greatest Engineering Achievements of the 20th Century asselected by the US Academy of Engineering 1. Electrification* 11. Highways 2. Automobile* 12. Spacecraft* 3. Airplane* 13. Internet 4. Water supply and distribution 14. Imaging 5. Electronics 15. Household appliances* 6. Radio and television 16. Health technologies 7. Agriculture mechanization* 17. Petroleum and petrochemical technologies 8. Computers 18. Laser and fiber optics 9. Telephone 19. Nuclear technology* 10. Air conditioning and refrigeration* 20. High performance materials * With significant mechanical engineering involvements

5. 1.2.2 Engineers solve Problems – often in ways like fire-fighting: Examples relating to: Design fault and ambiguity Manufacturing disorder Malfunction of equipment Inferior quality control in production Run-away cost control Resolving customer complaints and grievances Public grievances and mistrust

6. 1.2.3 Engineers make DECISIONS – at all times, and often crucial ones: Decisions are required in: ● Design – Configurations of engineering systems Selection of design methodology, materials and fabrication methods Assembly, packaging and shipping ● Manufacturing – Tools and machine tools Fabrication processes Quality control and assurance ● Maintenance – Routine inspections and Procedures ● Unexpected cases with potential grave (catastrophic) consequences – Change of customer requirements Malfunctioning of machines and equipment Defections in products Examples on Critical Decisions by Engineers on what to do if flaws or cracks appearon the surfaces of: ●Pressurized pipelines, e.g. the Alaska Pipelines and San Bruno Gas Pipeline burst or in ●A jumbo jet airplane Knowing fully well that crack of small size on the surface of loaded structure can “grow” to a critical size and become uncontrollable in its growth!!

7. 1.3 Toolbox for Engineering Analysis Toolboxes for tradesmen: Toolbox for Engineering Analysis: A. Traditional toolbox: Implementers: Math modeling: B. Modern toolbox: Computers: New Engineers’ toolbox: Programs: Algorithms: Math modeling:

8. We thus realize from the last slide that the “tool” engineers may use in modern-day engineering analysis is Mathematic Modeling involving the “TRNSLATION OF PHYSICS To MATH, and VICE VERSA.” Because All TASKS relating to: Creation Decision making Problems solving in engineering practices are of PHYSICALnature; The required ACTIONS to these tasks are of PHYSICAL naturetoo

9. Engineering Analysis by Mathematical Modeling Engineering Problems (Physical) Mathematical Modeling Mathematical Formulation Engineering Analysis Mathematical Analysis Translate engineering problems into math formby: 1) Idealizing physical situations. 2) Identifying idealized physical situation with available math representations 3) Formulate math models, e.g., expres- sions, equations. Desirable direct approach Not Possible! Mathematical Solutions Unavoidable Approach Solution to Engineering Problems (Physical) Translation Math to Physical Situation Conclusion:Math plays a principal role as a servant to Engineering (the Master) in engineering practices

10. 1.4 The Four Stages in Engineering Analysis In general, engineering analysis involves the following four (4) stages: Stage 1:Identification of the physical problem: Engineer must first have a clear understanding of the problem for the analysis, whether it is a design problem or an existing problem that requires a solution or decision. For many design problems, this stage of work relates to the understanding on the specification of the product or engineering system. Stage 2:Idealization of actual physical situations for mathematical analysis: This often is a very critical stage of the analysis; In most cases of the analysis, engineers will find that many of the stipulated requirements and constraints in the case that they are dealing with cannot be met with the available analytical ‘tools’ that he or she is aware of or has the necessary access available to him or her. He or she would thus be compelled to make necessary idealization on many of the physical situations of the problem through a number of assumptions and hypotheses on the required conditions that are stipulated in Stage 1 of the analysis so that he or she can handle the analyses by using the available analytical tools. Idealization may be required in the following specific areas: (1) On the geometry, (2) On the loading conditions, (3) On the boundary or support conditions, and (4) On other required conditions and constraints.

11. The Four Stages in Engineering Analysis-Cont’d Stage 3:Mathematical modeling and analysis: At this stage of analysis, the engineer is ready to apply whatever available analytical tools he or she has acquired either from what he or she previously learned from schools or from available engineering handbooks such as in reference [Avallion et.al. 2006] to solve the problem. Major tasks involved in the stage of analysis may involve the following actions: Develop a suitable mathematical model based on the idealized physical situations of the problem Derive applicable mathematical expressions, either in simple algebraic, or using existing or self-derived differential equations, or using empirical formulas developed by his or her own, or empirical equations developed by his or her employer’s handling the similar problems in the past. Establish mathematical representation of loading and boundary conditions. Solve the equations for numerical solutions with idealized loading and boundary conditions. State 4:Interpretation of results: This is another critical stage of any engineering analysis. Because whatever the analytical method the engineer used in the analysis, the results are most likely in the forms of raw numbers or in graphs or charts. Major effort in this stage of the analysis is to translate these forms of solution into physical sense required for the solution to the problem. We will observe the above four stages of analysis applied in a case of general “engineering design” Analysis.

12. 1.5GENERAL ENGINEERING DESIGN PROCEDURE with 4 stages of engineering analysis Stage 1: Understand the physical problem Stage 2:Idealization for math modeling Stage 3: Math modeling & analysis Stage 4: Interpretation of results: CHECK IF ALL REQUIRED CRITERIA* ARE MET * May involve non-technical requirements, e.g.,space and financial (costs) consraits, or If the product is a structure, or compo- nents requiring sufficient strength, then the concept of safety factors will be Included in the criteria, as will be demonstrated in the Example.

13. Example 1.2 on Application of Engineering Analysis: Design of a Bridge across a Narrow Creek: Stage 1: Description of the Physical Conditions: The required short bridge is specifically designed to handle limited local traffic over a narrow creek. The span of the bridge is 20 feet, and the maximum load for the bridge is 10 tons (or 20000 pounds). The simplified geometry and dimensions of the bridge are illustrated in: Engineers decide to use two (2) I-beams made of steel to support of the bridge structures for the following two reasons: (1) the bridge is built across a narrow creek and it is mainly subject to bending loads. I-beams are best suited for carrying this type of loads, and (2) the primary concern for this structure is “public safety.” Steel is common structure material with reliable strength, and I-beams are widely available with moderate cost. However, design engineers need to find the adequate size of these beams from suppliers with STANDARD dimensions. Their first trial is to use the I-beams with the following dimensions set by the supplier (available from manufacturer’s catalog) to be: H1=12”, H2=10.92”, b1=5” and b2=0.35”. Also: Max. tensile strength= 75,000 psi and max. shear strength=25,000 psi for steel (available from materials handbooks)

14. Stage 2:Idealizations of physical situation in order to use the formula available in “simple beam theory” for the subsequent mathematical analysis: Because the primary design consideration of the structure is its capability of carrying bending loads from the intended traffic. Engineers realize that “simple beam theory” that they learned From their “strength of materials” class could be used to calculate the maximum bending stresses induced by the expected loading in these I-beams. But the formula that they had learned from simple beam theory only deal with the following simple cases: P w lb/length Simply-supported: ● ● ● ● w lb/length Rigidly held support: None of the above available conditions resembles (and is therefore applicable) to the real situation that he (she) needs to deal with in the real situations Therefore there are needs for him (her) to idealize the real situations in present case, so that he (she) can use the available formula in textbooks or reference books derived for simple-beam bending as show above.

15. Stage 2:Idealizations of physical situation in order to use the formulations available in “Simple Beam theory” for mathematical analysis – cont’d: Following are some idealizations that engineers need to make in order to use the “simple-beam theory in his (her) design analysis: • The maximum load applied to the bridge is equally carried by two identical I-beams • The weights of the I-beams and other materials are neglected (This is obviously a serious violation of reality!).We make this idealization nonetheless in order to simplify the subsequent mathematical manipulations and calculation). • Each I-beam carries half of the total load, i.e. 20,000 lbf. • Assume that each of the 4 wheels carries equal amount of load (another questionable assumption!). • By judgment, the worst condition to be considered is when the vehicle is at the mid- span position . (appears to be the only reasonable assumption!) • One end of the beam is “hinged” to a fixed anchor and the other rests on a roller. The I-beams can thus be considered to be simply supported (as engineering mechanics textbooks often indicate).

16. Stage 3: Mathematical analysisUse available formula from “Simple Beam theory” for the present engineering analysis: One-side front wheel load: One-side rear wheel load: Idealized to simple beam in bending Simulated simple beam bending Real problem The maximum bending stress in one I-beam is calculated using the available formula: Maximum normal stress along the longitudinal direction of the I-beam at the center of the I--beam cross section, Where M = maximum bending moment c = half the depth of beam cross section I = area moment of inertia The maximum shearing stress is: Numerical results: σn,max= 13,370 psi, and σs,max = 1,372 psi The maximum deflection at the center span is: 0.42 in. – not a critical factor in this design Where V = maximum vertical shearing force b2 = width of the rib of an I-beam Q (0) = statical moment at the center of beam cross section

17. Stage 4: Interpretation of Results: The max. induced tensile stress in the I-beam to be: σn,max= 13,370 psi, and The max. induced shearing stress in the I-beam to be: σs,max = 1,372 psi We obtained the numerical results from Stage 3 using simple beam theory: What shall we do with these numbers for our design problem??? Because this analysis deals with a structure and public safety is a major concern, we need to translate these numbers obtained from our engineering analysis to satisfy the criterion of “Safety to the public.” The use of “safety factor” would be relevant to the present case. Recall: We were given in Stage 1 of this analysis that the maximum tensile strength of the steel I-beams to be: 75,000 psi, which is much higher than the numerical results we computed. So, the selected 12” I-beams should be safe enough for the intended loading because both the computed σn,max and σs,max are LESS than the maximum strength of the material, or is it? Question: What would you do if you were the engineer working on this design project??

18. It would have been all right with the above assessment. However, because of the IDEALIZATIONS that we made in Stage 2 of the analysis, and some of these idealizations Or assumptions that we made are not REALISTIC, the maximum induced stresses in the beam computed on these idealized condition may NOT represent the REAL case. So, the induced stresses as we calculated above may not be real!!! How would we interpret the results of our math analysis??? In order to COMPENSATE our analytical results obtained from the “convenience” in our analysis in Stage 3 analysis based on some “UNREALISTIC” idealizations, we may include – the Safety Factor (SF) concept in our interpretation of the fresults for the “SAFETY” of the structures that we are dealing with. 1.6 The Safety Factor in Engineering Analysis of Structures: Safety Factor (SF) is often used to “compensate” the many assumptions that engineers are compelled to make in their analysis. It is defined as: SF = Ultimate tensile strength of the material (uts)/computed maximum induced stress in the structures (σmax) where the uts of the material is obtained from material strength testing (see the diagram next slide), or is available from materials handbook for common engineering materials.

19. Recollect what you learned about Typical Material Strength Testing: The tensile testing machine: Engineering stress, σ = F/A Engineering strain, ε = ∆L/L where F = Applied tensile force A = Original cross-sectional area of the specimen ∆L = Elongation of the gage length due to F L = Original gage length of the specimen ● UTS σ So, UTS is the maximum strength of the material under tension Physically, it means the structure will fail if any part of the structure has σmax ≥ UTS!! Elastic deformation of material ε

20. On Safety Factors (SF) in Engineering Analysis Dealing with Structures Safety Factors (SF) is defined as: SF = Ultimate tensile strength (of the material), UTS/maximum (tensile) stress σmax induced in the structure There is NO set rules to determine SF in engineering analysis of structures. In general, the more sophisticated analyses (with less idealizations in Stage 2), lower SF would be assigned and thus more RELIABLE analytical results. Public safety is also a factor in determining the SF in the analysis. It explains why lower SF is used in aircraft design because of more through design analyses are conducted with more material properties tested, in sharp contrast in handling pressure vessel design analysiswith higher SF. Sophisticated design analyses are costly, but for aircraft industries, lower SF means less materials are used, thereby allows more revenue in higher payloads. using A Guideline for setting Safety Factors for Engineering Analyses with Structures

21. Let us get back to the query on whether: The max. induced tensile stress in the I-beam to be: σn,max = 13,370 psi, and the max. induced shearing stress in the I-beam to be: σs,max = 1,372 psi Would be safe for the intended traffic loads: We realize the fact that distribution of the normal bending stress σn across the depth (the height) of the beam structure of the I-beam follows a linear variation from the Maximum tensile value of as:σn,max = +13,370 psi at the bottom to a maximum value of compressive stress σn,max = -13,370 psi at the top of the beam. We thus conclude that The I-beams that support the bridge has a maximum tensile stress of 13,370 psi at its bottom Surface. We may use this maximum tensile stress in the I-beams to compute the SF for this analysis to be: SF = UTS of the beam material, 75,000/ σmax = +13,370 psi = 5.61 which is well in the range of SF = the range of 5.0-7.0 as shown in the above Table. We may thus conclude that the adopted I-beams with the sizes ( H1=12”, H2=10.92”, b1=5” and b2=0.35”) described in the analysis will be adequate for the expected traffic loads. 0.35” 12” 10.92” 5”

22. What More Can be Done to Make the Design Analysis of This Bridge More Realisticaly? Making the Stage 2 of the analysis more realistic (and thus less idealistic) by doing the following: • Include the weight of the steel beams in the analysis. • Include the weight of the concrete pavement road surface, say: 20 feet long x 12 feet wide and 6 inches thick. (3) With realistic load distributions on front and rear wheels You may still use other formulas derived from “simple beam theory” but your analytical solution will be much more realistic. What differences in the numerical solutions will you get with the inclusion of conditions in (1), (2), and (3) in your analysis?

23. Suggested Engineering Case Studies Relating to Public Safety for Engineering Analyses “Engineering is a profession, not just an occupation.” Engineers are responsible for public safety and social welfare and justice too. Negligence in their duties may result in devasta- tion to the public, as will demonstrated by the following selected case studies: Only sound engineering analysis can mitigate such tragic happenings. Case 1:Research and analysis on this engineering case study by response to the following particular queries on the devastating “gas pipeline burst in San Bruno, California on September 9, 2010 : • What really happened in this accident? • What are the consequences of this engineering disaster? • What was the cause of the accident relating to engineering negligence? • What were the loss of properties and human lives by this accident? • What the owner of the pipeline – PG&E has been doing in compensations to the victims and remedial actions taken to avoid future similar occurrence? • If you were a senior engineering manager at PG&E, what would you do to prevent the recurrence of the tragedy? • How can engineering analysis help in preventing future happening? Case 2: Crash of a DC 10 jet passenger plane over the Chicago O’Hare airport on May 25th, 1979 as shown in the picture: What would you, as an engineer or engineering manger do to avoid recurrence of such tragic happenings in the future?

24. Self-Studies on Chapter-End Assignment • Read the Example on Application of Engineering Analysis on a bridge on P. 9. • Conduct an engineering analysis on the above example but include the weights of the steel structure and the required concrete road surface for the bridge. Remind you that you do not always have the information and conditions given in your design analyses. You, as an engineer, needs to make reasonable and logical assumptions on these missing information based on available reference “tools” available to you. 3. Be prepared to answer the question on the significance of “Safety Factor” used in a design analysis of a structure or machine component. What are the fundamental principles for determining the numerical value of this factor? Explain why a SF = 4 is used in pressure vessel design by ASME design code, yet SF = 1.2 is used in aircraft structure design. • Be prepared to offer example of engineers making decisions and solve problems based on your personal experience.