Bridging the Gap between Mechanics of Materials

Lectures and Homework with MDSolids

by

Timothy A. Philpot

tim.philpot@murraystate.edu

Department of Industrial and Engineering Technology

Murray State University

Abstract

Current educational software for the mechanics of materials course is typically presented as either tutorials, worksheets, or basic analysis packages. A new software package, called MDSolids, presents an alternative to these types of products. MDSolids was conceived as a tool to help students solve and understand homework problems typically used in the mechanics of materials course. The software is versatile, graphic, informative, and very easy-to-use. MDSolids is being used at a number of schools around the world, and feedback from users has been uniformly positive and enthusiastic.

Introduction

For many years, computers and particularly personal computers have offered the promise of a revolution in the way that traditional engineering topics are taught. In some regards, this revolution has occurred. Computer-aided drafting and design (CADD) and sophisticated analysis packages have changed the engineering curriculum, making it possible for students to analyze and design at a level of precision impossible to accomplish with hand-calculations alone. However, much of this improvement occurs at the upper-end of the engineering curriculum. At the introductory level, in courses such as mechanics of materials, the impact of computer software on the teaching of fundamental concepts has been less successful. Although educational software has been packaged with textbooks for a number of years, book company editors know that, in general, book adoption decisions are not strongly influenced by the accompanying software. Therefore, if computers and computer software hold such promise as educational tools, why isn't educational software more effective at teaching engineering fundamentals?

How Do Students Learn Mechanics of Materials

In the field of education, Benjamin S. Bloom proposed a developmental sequence for learning, commonly called the Bloom Taxonomy (1). This taxonomy is comprised of six levels, starting with the least level of sophistication. Typical examples pertaining to the mechanics of materials course are given for each level.

**Level 1 - KNOWLEDGE.** The student is able to remember either by recognition or recall information, terminology, phenomena, etc. *Example: Define the term proportional limit.*

**Level 2 - COMPREHENSION.** The student is able to know an abstraction well enough so that he or she is can correctly demonstrate its use when specifically asked to do so. *Example: Compute the normal stress in a rod given the load and cross-sectional area.*

**Level 3 - APPLICATION.** The student is able to apply the appropriate abstraction without having to be prompted as to which abstraction is correct or to be shown how to use it in that situation. *Example: Determine the elastic modulus given load-deflection data.*

**Level 4 - ANALYSIS.** The student is able to break down the problem into its constituent parts and to detect relationships among the parts and the way they are organized. *Example: Determine the maximum load that a structure can support given limits on both stress and deformation.*

**Level 5 - SYNTHESIS. **The student is able to put together elements and parts to form a complete solution. Relates concepts and processes. Able to adapt knowledge from various sources to solve problems. Creative expression with ideas being learned and with ideas already known. *Example: Design a beam, incorporating statics, shear/moment/deflections diagrams, normal and shear stresses, and combined stress analysis to determine principal stresses.*

**Level 6 - EVALUATION.** The student is able to apply standards and determine levels of quality. *Example: Design concrete beams to best satisfy several considerations.*

As professors, we seek to guide students from Level 1 up to Level 5 in the introductory mechanics of materials class. While more fundamental levels of learning (knowledge, comprehension, and application) may be addressed in lectures, time constraints dictate that in-class examples and problems focus on developing analysis and synthesis skills. Each student learns at his or her own rate, and unfortunately, the pace of lecture topics is sometimes faster than the student finds comfortable. Concepts and problem solving skills that should be firmly in place before proceeding to analysis topics are sometimes absent or underdeveloped.

Homework assignments are the primary device used to develop the student's understanding of the mechanics of materials topics. The typical assignment can be somewhat lengthy; therefore, only selected problems can be assigned. Professors may expect (or hope) that their students will work enough extra problems so that the fundamentals are firmly grasped, but students sometimes struggle just to keep up with the homework and exam schedule. To supplement the student's educational development, the self-study potential offered by software would seem to be the ideal means of filling the gap between the material presented in lectures and the understanding and skills expected in homework and exams.

Educational Benefits Unique to Software

Software can help students study mechanics of materials and develop the necessary problem solving skills in several ways that are not inherent in lectures or customary homework assignments.

**Correct Solution and Intermediate Results:**When learning a new concept, it's very helpful to use the correct solution as a benchmark. Knowing that the problem has been solved correctly gives the student confidence in their problem solving skills and thereby provides a foundation for more challenging problems. Every textbook provides answers to selected problems for this reason. Software can provide the student with the correct solution for a particular problem, but in addition to the final answer, software can provide intermediate solutions that can be used to confirm the calculations along the way. These intermediate results can be helpful in tracking down faults in the problem solving approach.**What-If Analyses:**Observing a cause-and-effect relationship can be quite helpful to students. For example, a single concentrated load placed on a simply supported beam produces a shear and moment diagram. The student can add a second concentrated load and observe the changes in these diagrams. As another example, the student can readily change the end support conditions or add intermediate supports to a column and then observe the effects on the buckled shape. Without calculating a single number, the student can learn something about the nature of structures by simply observing these changes. This can help students to develop engineering intuition that will help them know what the correct solution should be before they calculate a single number.**Availability:**In the evening hours, during weekends, or when working at home (which may be distant from the classroom), students don't have access to professors, graduate assistants, or others who can help them understand the course material. Having a versatile software tool at hand to supplement the textbook and lecture notes can be a big asset.**Repetition:**Some people must see or perform more repetitions involving a concept before they begin to fully understand it. Time limits the number of examples that can be presented in lectures, and textbooks can only present a few examples. With software, students can drill themselves, trying various numeric combinations for a particular problem type until they feel confident in their understanding of the concepts.**Visualization:**Software can depict deformations or show stress distributions produced in the problem being considered. Visualization of the material's behavior in response to the loads acting on it can help the student to understand the relevant theory and to develop engineering intuition.

Current Mechanics of Materials Educational Software

Most of the current educational software developed for the mechanics of materials course can be grouped into three categories: tutorials, worksheets, and basic analysis packages.

**Tutorials** direct the student through a series of prepared screens, each focused on a specific concept or skill. In this manner, tutorials are like lectures delivered in a different format. Recent tutorials such as the Multimedia Engineering series (2, 3) feature an impressive presentation, complete with animation, video clips, and audio files. Despite excellent presentation, however, tutorial products are limited in applicability. The student must follow the sequence of the tutorial presentation in the same way that they would follow along in a lecture. The student must master the concept presented by the tutorial and then apply that concept to the particular problems that they are asked to solve in their homework assignments.

**Worksheets** for equation-solving software such as Mathcad, MatLab, and TKSolver have also been developed to supplement the mechanics of materials course (4, 5). One drawback of worksheets is that the student must be somewhat familiar with the host software package in order to use the worksheet. In a sense, this disadvantage can also be viewed as an advantage since worksheets encourage the student to develop some command of the equation-solving software, and familiarity with the equation-solving software is a skill that is useful in later engineering courses. However, to the student whose immediate goal is learning the mechanics of materials concepts, the added burden of gaining proficiency with the equation-solving software can be daunting.

**Basic analysis packages** have been included in several widely available mechanics of materials textbooks such as Lardner/Archer (6) and Craig (7). These programs are useful as tools for assisting students in fundamental skills such as plotting shear and bending moment diagrams or performing Mohr's circle calculations. Basic analysis programs may require students to define nodes and elements and to assign section properties and material constants to the elements. While this is the way that the calculations must be organized for the computer, this approach is not user-friendly for the novice engineer. Furthermore, basic analysis programs have typically lacked a well-developed graphical user interface. Input for these programs has typically been very text-based, often requiring a user's manual to ensure that the proper data and the proper sign conventions are used and to help in interpreting the program output.

In all three categories, the software is generally developed from the professor's point of view, emphasizing lecture topics or permitting the student to perform more advanced calculations. To be successful, educational software should be developed from the student's point of view. Rather than forcing the student to solve a problem posed by the software, the software should solve the problem of interest to the student. To do this, educational software must be:

- versatile in the types of problems that can be solved,
- strongly visual to illustrate the behavior of materials,
- informative in explaining how and why the calculations are performed, and
- intuitive and easy-to-use so that the student is presented with just the right amount of information and analytical power.

The MDSolids Concept

MDSolids is an educational software package devoted to the introductory mechanics of materials course. The hypothesis of the MDSolids concept is that students are most interested in understanding the specific homework problems assigned by their professors, and that students will use educational software that helps them with their immediate course concerns. In the process, the software can help to develop problem solving skills by giving students an intuitive interface that guides them to the important factors affecting various problem types, helps them visualize the nature of internal stresses and deformations, and provides an easy-to-use means of investigating a greater number of problems and variations. Based on this premise, MDSolids was developed with several objectives in mind:

**Versatility:**MDSolids has routines pertaining to all of the topics taught in a typical mechanics of materials course. These routines are grouped in modules, similar to typical textbook chapters, and the modules can be individually accessed in any sequence. Eleven modules are presently available to handle a wide range of common textbook problems: basic stress and strain, beam-and-strut axial problems, trusses, statically indeterminate axial structures, torsion, determinate beams, section properties, general analysis (of axial, torsion, and beam members), column buckling, and Mohr's circle transformations. Within the modules, each routine solves types of problems typically found in all mechanics of materials textbooks. Some routines are fairly general (e.g., Mohr's circle analysis) while some routines are specific (e.g., plotting a stress-strain curve). The scope of MDSolids offers routines to help students at all levels of understanding, from the most fundamental knowledge-, comprehension-, and application-type problems to more complex problems requiring analysis and synthesis.

**Ease-of-Input:**Ease-of-input is an essential aspect in the MDSolids concept. Solving the mechanics of materials problems is confusing enough for students. To be effective, educational software must not add to the confusion. Ideally, the student should be able to define a problem intuitively and directly from a textbook without the need for a user's manual. Throughout MDSolids, graphic cues are provided to guide users in entering data. The illustrations can be easily adjusted so that the MDSolids input screen looks very similar to the textbook illustration. Various units (e.g., stress units, length units) are available and internal conversion factors are present to ensure dimensional consistency.**Visual Communication:**Each MDSolids routine features a picture, sketch, or plot that graphically depicts important aspects of the problem. Sketches are used to show the direction of internal stresses, applied loads, and reaction forces. Plots are given for a number of topics including critical buckling stress, beam deflections, and shaft shearing stress. As the cliché goes, "one picture is worth a thousand words."**Correct Solution and Intermediate Results:**MDSolids is an "electronic solutions manual," giving not only the correct solution for a particular problem but also providing intermediate solutions that can be used to confirm the problem solving approach step-by-step.**Text-based Explanations:**Many of the MDSolids modules provide extra explanations to describe in words how the calculation is performed. These explanations can help students develop the thought processes used in solving mechanics of materials problems. The text explanations are dynamic and context-sensitive. The message is tailored specifically to the particular problem being considered, in terms of the values and units entered for the problem. Common mistakes in equilibrium equations, unit inconsistencies, and equation manipulations become obvious when a student compares his or her hand calculations with the MDSolids explanations.**Ease-of-Modeling:**MDSolids takes advantage of mouse input to facilitate the creation of models. For example, defining a truss with 13 members and various loads can be accomplished graphically with a mouse in about 30 seconds. Various cross-sectional shapes can be defined just as rapidly. This simplicity encourages students to test out their problem solving skills on alternative configurations.**Help Files:**The MDSolids help files contain instructions for using the software, but more importantly, the help files contain theoretical background and practical suggestions for solving various types of problems. The help files also contain a number of worked example problems. These example problems describe how to solve the solid mechanics problem by hand, not through the use of MDSolids. Therefore, MDSolids users can take advantage of the software to solve a problem as well as getting a detailed step-by-step description of the solution process.

MDSolids has been used by students at Murray State University for three semesters. The software was made available free-of-charge to the engineering educational community in January, 1998 at the MDSolids website http://msumusik.mursuky.edu/mdsolids . In the first two months of its availability, over 1500 professors and students from around the world downloaded the software. The response of MDSolids users has been uniformly positive and enthusiastic.

MDSolids Exhibits

**Text-based Explanations:** MDSolids includes a wide range of routines pertaining to problems typically used in teaching the mechanics of materials course. The Stress-Strain module focuses on introductory problems used to develop an understanding of basic concepts and problem solving skills. All problems in this module have a consistent style, as illustrated by the bolted connections routine.

The user is presented with questions typically asked for this type of problem. Depending on the choice of question, the user is directed to supply the necessary input data. Upon clicking the **Compute** button, the numeric results are displayed, a simple free body diagram is shown, and a text description of the process used to solve the problem is printed.

One of the most important skills developed in the mechanics of materials course is creation of shear force and bending moment diagrams. In the Determinate Beam module, the student can quickly create a shear force and bending moment diagram, such as shown below.

While the diagrams are useful in themselves, MDSolids also provides tips on constructing the shear force and bending moment diagrams. For example, the student can put the mouse cursor on the beam supports in the load diagram to see the equilibrium equations applicable to the beam:

The student can position the mouse cursor over a region of the shear diagram to get tips on constructing the moment diagram. Clicking the mouse on this region produces further explanation on how to find the area under the shear diagram and how this area dictates the change in the moment diagram.

Calculation of cross-sectional properties is essential in many typical mechanics of materials problems. MDSolids provides a number of typical shapes that the student can select. After defining the appropriate dimensions,

After clicking the **Compute** button, the student is given a report of the section properties, but additionally, the student can see details of the calculation procedure. These details, shown below for the centroid and moment of inertia calculation, are presented in tabular format referring to the numbered shapes shown in the figure above.

**Visual Communication:** MDSolids relies on graphical depictions to help students develop an understanding of the behavior of materials in response to applied loads. For example, bending stresses in a beam (either normal stress or shear stresses) are depicted for a specified beam cross-sectional shape and at any specified point along the beam:

For torsion, the deformations occurring in a circular shaft in response to an applied torque and the shear stresses acting at a typical point are illustrated in addition to the numeric solution:

In Mohr's circle transformations, the Mohr's circle is constructed from the specified normal and shear stresses:

Elements showing the magnitude and orientation of the principal stresses and of the maximum shear stresses are also shown:

MDSolids is also useful in helping students visualize the behavior of structures. For example, the Euler buckling shape of a column braced at midheight is shown below. While the column buckling theory used to prepare this illustration is beyond the scope of the introductory mechanics of materials course, this type of active sketch can help students to develop intuition about column behavior.

**Help Files:** MDSolids also contains a number of help files. While these help files contain instructions for using the software, they also include general discussions describing skills needed to solve mechanics of materials problems. There are also a number of worked example that explain in detail the calculation procedure needed for a hand solution of typical problems.

Conclusions

MDSolids has proven to be a valuable addition to the mechanics of materials courses at Murray State University, and it is becoming known and being used by professors and students around the world. The software is conceived as a tool to help students bridge the gap between topics presented in lectures and the application of that theory in solving problems commonly used in mechanics of materials homework assignments. Using MDSolids, students get numerical, visual, and textual results and details pertinent to a wide range of problems. Since MDSolids is so easy-to-use and because it provides ample feedback, students are encouraged to attempt more mechanics problems and to explore what-if variations. Through this extra repetition, students develop engineering intuition and greater confidence in their problem-solving skills. MDSolids has been successful in helping students attain mastery of the knowledge, comprehension, application, analysis, and synthesis levels of the learning process.

References

- Bloom, B.S., ed. (1956).
*Taxonomy of Educational Objectives, Handbook I: The Cognitive Domain*, David McKay, New York, N.Y. - Gramoll, K., Abbanat, R., and Slater, K. (1996).
*Multimedia Engineering Statics*. Addison Wesley Interactive, Reading, Mass. - Gramoll, K., Abbanat, R., and Slater, K. (1996).
*Multimedia Engineering Dynamics*. Addison Wesley Interactive, Reading, Mass. - Evensen, T.C. (1997).
*Mathcad Supplement*in Gere, J.M. and Timoshenko, S.P. (1997).*Mechanics of Materials, 4*., PWS Publishing Co., Boston, Mass.^{th}ed - Turcotte, L.H. and Wilson, H.B. (1998).
*Computer Applications in Mechanics of Materials using MATLAB.*Prentice Hall, Upper Saddle River, N.J. - Lardner, T.J. and Archer, R.R. (1994).
*MECHMAT*in*Mechanics of Solids: An Introduction,*McGraw Hill, New York, N.Y. - Craig, R.R. (1996).
*MechSolid*in*Mechanics of Materials*, John Wiley & Sons, New York, N.Y.