Emirates Journal for Engineering Research, 13 (2), 73-78 (2008)

(Regular Paper)

ANALOGY BASED LEARNING (ABL) FOR MECHANICS OF MATERIALS SUBJECT (MMS)

M. Isreb School of Applied Sciences and Engineering, Monash University Gippsland Campus Northways Road, Churchill, Victoria 3842, Australia Email: [email protected]

(Received May 2008 and accepted August 2008)

ال يوجد دليل في المراجع واألبحاث المنشورة على استخدام التعليم بواسطة النمذجة للعنصر الرياضي في مادة الھندسة الميكانيكية للمواد المتشكلة (والمعروفة أيضاً في أوساط معملي الھندسة باسم ميكانيكا المواد (MMS)). ويھدف تطبيق التعليم بواسطة النمذجة المقدم في ھذه الورقة إلى محاولة تسھيل استيعاب الطالب لتلك المادة الصعبة. وقد تم استخدام الطريقة القياسية لنمذجة العزوم باسلوب مبتكر. وقد اختبر الباحث ھذه الطريقة لمدة عشرين عاماً في جامعة موناش باستراليا. وأثبتت الطريقة نجاحھا بغض النظر عن طريقة الطالب وقدرتھم على التحصيل طبقاً المكانياتھم العقلية. وتميزت الطريقة بقابلية التطوير والمرونة. There is no evidence in the literature of using Analogy based-learning (ABL) education for the mathematics components of Engineering Mechanics of Deformable Bodies subject (also known widely amongst engineering educators by the name Mechanics of Materials subject, abbreviated as MMS). The motive of using the ABL for MMS in the present paper is to attempt to make, such already difficult subject, easy one to get across to students. The method used in the paper is based on standard ABL methodology of coupling, in an innovative way, each ABL source with an ABL target of the MMS. The author has tested the new approach for two decades at Monash University, Australia. The approach has proved to bring the students potentials to its fullest regardless of their style of learning, within the eight brain sectors. The approach has proved both feasible and extensible.

Keywords: Analogy-Based Leaning (ABL), Mechanics of Materials Subject (MMS), Innovative Learning.

1. INTRODUCTION learning (PjBL) in MMS is another extension of PBL. It is the MMS type of project work which can mimic Mechanics of Materials subject (MMS), also some professional reality, taking from one class session to educators call it Engineering Mechanics of several weeks to complete, Perrenet et al.[3]. Inquiry- Deformable Bodies subject is concerned with the based learning (IBL) in MMS is a flexible holistic mathematical formulations of stresses and adaptation of PBL, which can be seen as any deformations for engineering deformable bodies. pedagogy that poses a problem or investigation first, MMS is the core engineering subject for almost all and subsumes other instruction under the pursuit of the engineering disciplines. Mathematically speaking, quest, Edelson et al.[4]. MMS is one of the most difficult core engineering Learning by analogy in MMS is the central theme subjects. It contains heavy dose of mathematical of the paper keeping in mind that ABL could be used derivations of stresses and deformations leading to its with PBL, PjBL, and IBL as shown in Figure 1. constitutive laws and their related boundary conditions Learning by analogy typically involves finding a set of and other related problems. systematic correspondences (a mapping) between a Numerous attempts have been detected in literature better-known source analogy and a more novel target. to make such difficult subject easy to get across to In the present paper, the word analogy in Analogy- students using well known learning approaches, but based learning (ABL) of MMS refers to the relation none of such approaches utilises learning by analogy. between the source and the target themselves, e.g. the Problem-based learning (PBL) in MMS is one of such definition of Wikipedia Encyclopedia[5]. It should be approaches already used in which the MMS’ problems noted that it is the first time in literature an ABL is drive the learning. That is, before students learn some specifically applied it to the mathematics of MMS. knowledge in MMS they are given a problem. The Furthermore, the paper introduces an innovative problem is posed so that the students discover that educational tool to facilitate learning of MMS by they need to learn some new knowledge before they [1] [2] incorporating major mechanics constitutive laws into can solve the problem. Goulet and Margetson specially developed analogical relations between the show some of the challenges in PBL. Project-based sources and the targets of MMS mathematics.

73 M. Isreb

Table 1. Sample topics of a typical MMS syllabus No. Topic Description PBL PjBL (Problem-based (Problem-based 1 TENSION, COMPRESSION, AND SHEAR learning) learning) Introduction to MMS / Normal Stress and Strain / Mechanical ABL in MMS Properties of Materials / Elasticity, Plasticity, and Creep / Linear Elasticity, Hooke's Laws, and Poisson's Ratio / Shear Stress and (Analogy based Strain / Allowable Stresses and Allowable Loads. Changes in learning in Mechanics Lengths of Axially Loaded Members (Axial Deformation) / Changes of Materials Subject) in Lengths of Non-uniform Bars 2 TORSION Introduction / Torsional Deformations of a Circular Bar / Circular Bars of Linearly Elastic Materials / Non-uniform Torsion / Stresses and Strains in Pure Shear IBL 3 STRESSES IN BEAMS (Inquiry-based Introduction / Pure Bending and Non-uniform Bending / Curvature of learning) a Beam / Longitudinal Strains in Beams/ Normal Stresses in Beams (Linearly Elastic Materials) / Shear Stresses in Beams 4 DEFLECTIONS OF BEAMS Introduction / Differential Equations of the Deflection Curve / Figure 1. Relationships between ABL, PBL, PjBL and IBL. Deflections by Integration of the Bending-Moment Equation / Deflections by Integration of the Shear-Force and Load Equations / The ABL of MMS is introduced in the present Method of Superposition papers to engineering and sciences students by means of handy tables especially tailored for flexible ABL Table 2. ABL source-target notations for Hooke’s laws within topic 1 delivery by science and engineering faculties. The of table 1 rational behind such development is the following: ABL SOURCE ABL TARGET The MMS as a subject has always been perceived by Axial Loading Shear Loading students to be a very difficult (i.e. mathematically) Symbol Meaning of Rhyming-analogy Meaning of Rhyming- Symbols Symbol analogy Symbol oriented subject to learn. Consequently, one can σ Normal stress τ Shear stress clearly see the need for an appropriate methodology in E Modulus of G Modulus of elasticity in relation to developing an innovative and educationally elasticity shear simplified ABL mathematical approach for MMS. The ε Normal strain γ Shear strain methodology of the paper is discussed next. Table 3. ABL source-target for Hooke’s laws within topic 1 of table 1 2. METHODOLOGY as per the analogical notations of table 2 ABL SOURCE ABL TARGET The paper’s methodology of ABL for MMS is Axial Loading Shear Loading summarised in the following three parts: Part (a) σ = Eε Hook’s Law τ = Gγ Hook’s Law in shear introduces ABL for MMS framework: i.e. formulates, implements and identifies the differences between ABL and the classical/conventional teaching of MMS; Part (b) of step 1, of the paper’s methodology, is Part (b) presents examples of ABL for MMS in support best explained through an example. The example is of (as well as to illustrate) the methodology used; and shown in Table 1. The Table shows sample topics of a Part (c) shows the reasoning of the ABL methodology typical MMS syllabus. This is a conventional /classical approach. MMS teaching which involves the teaching of one Part (a) of the paper’s methodology consists of the topic after another (sequentially) with no regard to their following three steps: Step 1, as a standalone step, analogical mathematical connections at all (i.e. no represents nothing but the classical (conventional) regards to analogical relations between the sources and approach to MMS teaching; while steps 2 and 3 are the the targets of MMS mathematics). actual analogical steps within the three-step procedure Step 2 of part (a) of the paper’s methodology calls on of part (a). the lecturer to identify and teach the analogy based Step 1, of part (a) of the methodology, calls for the mathematical elements within the components of the lecturer to teach the MMS selected topics as in a topic at hand while the topic is being taught (e.g. classical way, i.e. as described in the syllabus of the analogy within topic i, analogy within topic j, analogy subject (e.g. teach topic i, then teach topic j, then topic within topic k etc). Part (b) of step 2, of the paper’s k etc.). A typical MMS syllabus of the subject can be methodology, is best explained through the following found in many textbooks, e.g. Gere[6]. The author of three example: the present paper had the privilege of reviewing Gere[6] - Example One: Table 2 is the first example of by invitation, as acknowledged on page Xiii. It should step 2 of ABL framework. Specifically it is be noted that the author of the present paper restricted about the ABL source-target notations for his revision of Gere[6] to step 1of part (a) of the paper’s Hooke’s Laws within topic 1 of Table 1. ABL methodology. source is the axial type of loading and ABL target is the shear type of loading.

74 Emirates Journal for Engineering Research, Vol. 13, No.2, 2008 Analogy Based Learning (ABI) for Mechanics of Materials Subject (MMS)

Table 4. ABL source-target notations for boundary conditions’ mathematical equivalency within topic 4 of table 1

ABL SOURCE ABL TARGET Symbol Meaning Equivalence Equation’s Order (of differentiation) Symbol Meaning Equivalence Equation v Deflection v = v Deflection

( dv ) dv v' Slope of the deflected curve θ v'≡ θ = arctan dx dx d 2v d 2v v'' Proportion to Bending Moment M ( v''≡ ) M = EI = − EI v'' dx2 dx2 3 3 d v d v = v''' Proportion to Shear Force V ( v'''≡ ) V = EI − EI v''' dx3 dx3 4 4 d v d v = EI v'''' Proportion to Distributed Load q ( v''''≡ ) q = −EI − v'''' dx4 dx4

Table 5. ABL source-target for loadings-deformations mathematical notations between topic 1 and topic 2 of table 1 ABL SOURCE ABL TARGET Deformations Due to Torsion Loading Deformations Due to Axial Loading Rhyming-analogy Symbol Meaning of Symbols Meaning of Rhyming-analogy Symbol Symbol

φ Angle of Twist of a bar in torsion δ Elongation of a bar or spring L Length of bar L Length of a bar or spring Polar Moment of Inertia A Cross Sectional Area I p G Modulus of Elasticity in Shear Ε Modulus of Elasticity Τ Torque (Twisting Couple) N Axial Force

Table 6. ABL source-target for loadings-deformations laws between topic 1 and topic 2 of table 1 as per the analogical notations of table 5 ABL SOURCE ABL TARGET Deformations Due to Torsion Loading Deformations Due to Axial Loading Equation Meaning of Equation Rhyming-analogy Equation Meaning of Rhyming-analogy Equation TL NL φ = Angle of Twist of a bar in torsion δ = Elongation of a bar or spring GI p EA n n N L n n T L i i i i δ = ∑δi = ∑ φ = φi = Bar Consisting of Prismatic Segments Bar Consisting of Prismatic Segments ∑ ∑ i=1 i=1 Ei Ai i=1 i=1 Gi (I p )i

L L Tdx L L Ndx φ = dφ = Bar with Continuously Varying Dimension δ = dδ = Bar with Continuously Varying Dimension ∫0 ∫0 ∫0 ∫o GI p (x) EA(x) L L T(x)dx L L N(x)dx φ = dφ = Bar with Continuously Varying Torque and δ = dδ = Bar with Continuously Varying Axial Load and ∫0 ∫0 ∫0 ∫o GI p (x) Dimension EA(x) Dimension

Table 7. ABL source-target for loadings-stresses mathematical notations between topic 1 and topic 3 of table 1 ABL SOURCE ABL TARGET Stresses Due to Torsion Loading Stresses Due to Bending Loading Rhyming-analogy Symbol Meaning of Symbols Meaning of Rhyming-analogy Symbol Symbol Shear stress at the outer surface of τ ( or ) Maximum normal stresses at the top and bottom fibbers of the beam max the bar σ σ1 σ 2 T Torque M Bending moment Distance from the Neutral Axis (NA) to the extreme elements in the positive r Radius of the bar c ( c or c ) 1 2 and negative y direction Radial distance in polar ρ y Distance in rectangular coordinates coordinates 2 2 I p = ρ dA Polar moment of Inertia I = y dA Moment of inertia about Neutral Axis (NA) ∫A ∫A

Emirates Journal for Engineering Research, Vol. 13, No.2, 2008 75 M. Isreb

Table 8. ABL source-target loadings-stresses laws between topic 1 and topic 3 of table 1 as per the analogical notations of table 7 ABL SOURCE ABL TARGET Stresses Due to Torsion Loading Stresses Due to Bending Equation Meaning of Equation Rhyming-analogy Equation Meaning of Rhyming-analogy Equation Tr Mc τ = Maximum shear stress at distance r from the Maximum bending stress at distance c from the max centre of the shaft σ max = ± Ip I Neutral Axis (NA) of the beam Tρ My Bending stress at distance y from the Neutral Axis τ = Shear stress at distance ρ from the centre of σ = ± I p the shaft I (NA) of the beam πr 4 πd 4 bh3 Moment of inertia of the beam cross-section about I = = Polar moment of inertia of the shaft I = p 2 32 12 its Neutral Axis (NA)

- Example Two: Table 3 is the second example - Example Four: Table 8 is the fourth example of step 2 of ABL framework. Specifically it is of step 3 of ABL framework. Specifically it is about the ABL source-target for Hooke’s Laws about the ABL source-target loadings-stresses within topic 1 of Table 1 as per the analogical Laws between topic 1 and topic 3 of Table 1 as notations of Table 2. Again, in this example, per the analogical notations of Table 7. ABL ABL source is the axial type of loading and source is stresses due to torsion loading and ABL target is the shear type of loading. ABL target is the stresses due to bending. - Example Three: Table 4 is the third example Part (c) of the paper’s methodology for ABL for of step 2 of ABL framework. Specifically it is MMS is about reasoning behind the ABL approach. about the ABL source-target notations for Reasoning involves a set of challenging problems, boundary conditions’ mathematical equivalency namely: how to accumulate MMS syllabus topic within topic 4 of Table 1. ABL source is the problem solving experience, how to define and decide symbol meaning and ABL target is the when two syllabus topics problem solving situations equivalence equation’s order (of are similar. Veloso[7] describes the integration of differentiation). analogical and case-based reasoning into general Step 3 of part (a) of the paper’s methodology calls problem solving and planning as a method of speedup on the lecturer to identify and teach the analogy based learning. mathematical elements between topics after such topics have been taught (e.g. analogy between topic n 3. RESULTS and topic q). Part (b) of step 3, of the paper’s methodology, is best explained through the following The Author has tested the ABL for MMS project for four example: more than two decades at Monash University, - Example One: Table 5 is the first example of Australia. The performance of two groups of students step 3 of ABL framework. Specifically it is enrolled in MMS course has been repeatedly about the ABL source-target for loadings- compared by the Author. The first group received a deformations mathematical notations between conventional training (sample topics of a typical MMS topic 1 and topic 2 of Table 1. ABL source is syllabus are given in Table 1). The second group of the deformations due to torsion loading and students used the ABL in their training (Tables 1-8). ABL target is the deformations due to axial The impact of the ABL on the student learning attitude loading. showed enthusiasm and commitment. Due to the fact - Example Two: Table 6 is the second example that ABL recognised a wide range of learning of step 3 of ABL framework. Specifically it is activities, it reduced the emphasis on lecturing by an about the ABL source-target for loadings- estimated minimum of 50%. The ABL project allowed deformations laws between topic 1 and topic 2 self-paced learning enabling students to become active of Table 1 as per the analogical notations of learners. The students developed an ability to identify, Table 5. ABL source is the deformations due to synthesise and analyse problems, by observing torsion loading and ABL target is deformations analogical cause and effect relationships. The author due to axial loading. has found that ABL of MMS has the following - Example Three: Table 7 is the third example educational outcomes for the following four types of of step 3 of ABL framework. Specifically, it is learners within the eight brain sectors (left and right about the ABL source-target for loadings- brain sectors for each type of learner): - stresses mathematical notations between topic 1 Imaginative learner - and topic 3 of Table 1. ABL source is about the Analytical learner - stresses due to torsion loading and ABL target Common sense learner - is the stresses due to bending loading. Dynamic learner

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The main outcomes of are detailed next: have been published in numerous fields such as - Connect students directly to the subject, begin Chemistry, Economics, Epidemiology and Ecology, with a level that is familiar to students and build e.g. Matthias and Hanno [8] and Newby et al.[9] but on what they already know. none has been published in relation to MMS. In fact, - Explore with students the needs to develop a the analogical models of (for examples) Tables 2 and 3 broad introductory understanding of MMS. of the present paper (i.e. Hooke’s Laws) are even - Summarize and review similarities and analogical to “Box 3” of Matthias and Hanno[8] of differences in basic concepts. chemistry law of mass action model, of economics - Connect what students are learning at the Cobb-Douglas production function model, of analogical stage to what students already taught epidemiology infection rate model, and Matthias and in a traditional lecture with emphasis on Hanno[8] of ecology fraction change. engineering judgment, engineering modelling in The paper, in a nutshell, applies for the first time analogical context. the creative capabilities of analogy to MMS to come - Establish analogical road maps of solving MMS up with the following MMS teaching principle: Many problems from the initial step of establishing MMS problem solving strategy relies on analogy and the FBDs (Free Body Diagrams) in analogical involves analogical reasoning as well as analogical way (e.g. the FBDs and the solution roadmaps deduction. of problems related to Topic 1 of Table 1 is analogical to the FBDs and the solution 5. CONCLUSIONS roadmaps of problems related to Topic 2 of the same Table). The paper’s most important result is the following: - Establish analogical components of, for - The paper, in its presented methodology, example, representations of applied loadings enables students to gain crucial understanding and internal loading (e.g. axial loading versus of the analogical interactions within each topic torsion analogy). and between different topics of the MMS - Establish connection between the mechanics syllabus. concept and its relationship to the students' - The above main result sets the scene for a professional activities such as stresses and departure mechanism away from filling students strengths of MMS. with conventional teaching of MMS. ABL of - Draw students’ attention to MMS analogical MMS allows optimal thinking capacity of characteristics such as area, moment of inertia, students to be prevailed due to MMS stunning length, modulus of elasticity. analogy, within the mathematics of the MMS as - Set ABL exercises based on reinforcement and a core engineering subject. Additional by- manipulation of experimental work in relation product benefit of the new approach is to train to ABL elements (e.g. Hook’s Law versus its students to think, learn and to exercise analogical counterpart, i.e. analogical Hook’s competent engineering judgment in an Law in shear). analogical manner throughout their professional - Highlight usefulness of ABL approach through life. analogical iterations of MMS concepts: All - The paper introduces a new approach which students have reached the ability to analyse transforms MMS from being an already problems in an ABL mode much faster than in difficult subject to be an interesting one to conventional mode. students. - Transform the gained knowledge into an - The goal of the new approach is to train integrated understanding with engineering students to think, learn and to exercise quality assurance of skills and stimulate competent judgment in an innovative analogical students thinking in highlighting new way tailored specifically to MMS’ mathematics. discoveries in ABL for MMS. The approach has proved to bring the students - Allow students to identify various structural potentials to its fullest regardless of their style mechanics failures in analogical set of patterns. of learning within the eight brain sectors, e.g. Isreb et al. [10,11]. - 4. DISSCUTIONS ABL for MMS has no limitations of significant value to upset the analogy pattern within MMS According to Matthias and Hanno[8]: “If necessity is constitutive mathematical equations within the mother to invention, then analogy is father.” field of Solid Mechanics. On the contrary, ABL Generalisations of mathematical analogies in literature has the potential to be applicable to other is not new, however, applications of ABL to MMS is “Mechanics” fields such as Geo-Mechanics, new. In fact, applications of mathematical analogies Soil Mechanics, and Fluid Mechanics. Further research is needed to establish such potential.

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ACKNOLEDGEMENT 5. Wikipedia Encyclopedia, (2008). Analogy, URL:http://en.wikipedia.org/wiki/Analogy, retrieved The author wishs to thank his employer Monash March, 2008. University, Australia, for allowing him to further 6. Gere, J. M. (2001). Mechanics of Materials, Fifth SI develop the MMS for civil, mechanical, electro- edition, Nelson Thornes. mechanical, electrical, mechatronics, interdisciplinary 7. Veloso, M.M. (1994). Planning and Learning by (civil), interdisciplinary (mechanical) and civil and Analogical Reasoning, Springer. environmental engineering bachelor's degree programs 8. Matthias, R. and Hanno, B. (1999). Creating for the last twenty two years. Knowledge by Analogy, http://www.bu.edu/cees/ research/workingp/pdfs/9914.pdf, retrieved March, REFERENCES 2008. 9. Newby, T.J., Ertmer, P.A. and Stepich, D.A. (1995). 1. Goulet, R.U. and Owino, J. (2008). Experiential Instructional analogies and the learning of concepts. Problem Based Learning in the Mechanics of Educational Technology Research & Development, Materials Laboratory, http://www.ni.com/pdf/ 43, 1, 5-18. academic/us/journals/lv02_26.pdf, retrieved April 10. Isreb, M. et al, (2000). An Internet-4MAT-CAL 2008. system for adaptive finite element mesh refinement”, 2. Margetson, D. (1997). Why is Problem-based Finite Elements in Analysis and Design, The Learning a Challenge? in The Challenge of Problem- International Journal of Applied Finite Elements and based Learning, 2nd Ed, Vol. eds. D. Boud & G. Computer Aided Engineering, ELSEVIER, 34, 1-24. Feletti, Kogan-Page, London, 36-44. 11. Isreb M. and D. Nag D. (2006). An innovative 4- 3. Perrenet, J.C., Bouhuijs, P.A.J and Smits, J.G.M.M. MAT-based system for geo, solid and fluid (2000). The suitability of problem-based learning for mechanics education with geo-mechanics engineering education: theory and practice, teaching applications, World Transactions on Engineering and in Higher Education, 5, 3, 345-358. Technology Education, 5, 1, 163-166. 4. Edelson, D.C., Gordin, D.N. and Pea, R.D. (1998). Addressing the challenges of inquiry-based learning through technology and curriculum design. Journal of the Learning Sciences 8 (3&4), 391-450.

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