Nonlinear Dynamics: An Intuitive Digital Representation of Structure

Abstract

The article discusses the pedagogic advantages of structural analysis software based on nonlinear dynamic structural theory, in contrast to software most commonly used in architectural education which is based in linear static theory. Traditionally, structural education in both architecture and engineering has considered nonlinear dynamic theory to be an advanced-level tool, reserved for graduate level engineering study. The article illustrates that with an appropriate interactive interface, nonlinear dynamics can support on-screen structural models that are far more physically intuitive. The article includes examples of student work from a first course in structural design.

Keywords:

• structural design education
• non-linear analysis
• dynamic analysis
• interactive teaching tools

Introduction

Software for structural analysis has been used in architectural education for more than 20 years. The value of such software in teaching structural behaviour was emphatically argued in Black & Duff’s paper A Model for Teaching Structures: Finite Element Analysis in Architectural Education (1994). In the years since, structural analysis software has been widely adopted, and is now bundled with mainstream texts in architectural structures; e.g. the 6^th edition of Structures by Schodek (2007) is bundled with the MultiFrame software package (Formation Design Systems 2011).

Despite advances in graphics and functionality, the software most widely used in architectural education continues to be based on the same theoretical representation as software from 50 years ago: linear static theory, where linear means the response of the structure is always proportional to the load, and static means conditions do not change with time. This paper describes a new kind of structural analysis software that expands the limits of linear analysis because it is based on nonlinear dynamic theory. The software, called Arcade and developed by the author, enables modes of learning that are more physically intuitive.

Arcade is not unique in its ability to model nonlinear dynamic phenomenon. There are dozens of programs that can model such phenomena: e.g. commercial programs such as ANSYS Inc. (2011) and SAP2000 (Computers and Structures Inc. 2011) and academic programs such as OpenSEES (2011) from the University of California at Berkeley. These programs, however, are designed for advanced engineers and are difficult to learn and use. Arcade, in contrast, is intended to make the advanced-level theory of nonlinear dynamics accessible to undergraduates in a first structures course. The computation method and interface of Arcade are both derived from methods widely used in computer games, enabling interaction which is far more visual and direct than conventional engineering software (Martini 2005).

Basing computation on a different underlying theory opens new possibilities for using the computer to explain and explore structural behaviour. Applying nonlinear dynamic theory to teaching introductory structures breaks strongly with the traditions of structural education, both in engineering and in architecture. The following discussion examines those traditions, and the benefits of breaking them.

Measures of Complexity in Structural Representation

The following discussion identifies measures of complexity in structural representation, and uses those measures to characterise nonlinear dynamic analysis with respect to theoretical representations traditionally used in architecture education. There are two primary measures of complexity: linear vs. nonlinear; and static vs. dynamic.

Linear vs. Nonlinear Behaviour

The first measure of complexity in structural representation concerns the assumed relationship between the forces applied to a structure and the deformation of the structure in response. Figure 1 illustrates three types of relationship between force (P) and deformation (Δ). The first is a rigid representation, which assumes that the deformation Δ is always zero. Next is a linear representation, which assumes that Δ is directly proportional to P. Finally, a nonlinear representation assumes some more complex relationship. Rigid theory is never true, but can be useful in many cases. Linear theory applies to most conventional building structures under normal conditions. Nonlinear theory is the most realistic and most complex. Nonlinear behaviour arises from two major sources: material nonlinearity, and geometric nonlinearity.

Figure 1 Rigid, linear, and nonlinear models of force-deformation relationships.

Concerning material nonlinearity, most building materials exhibit linear behaviour at low stress levels, but eventually reach a threshold where small increases in stress produce large increases in deformation, called yielding. Furthermore, all materials eventually reach a point of breaking, called rupture or fracture. To consider the behaviour of a structure after yield or fracture, it is necessary to use nonlinear analysis.

Geometric nonlinearity concerns the following paradox: in order to determine the effects of load, it is necessary to know a structure’s geometry, but loading changes that geometry to an unknown state. Linear analysis resolves this situation by assuming that the changes in geometry are negligible. While this assumption is reasonable for a wide range of practical structures, there are many situations where it does not apply, such as flexible cable structures and frames approaching collapse.

In summary, linear analysis can be appropriate when a structure has no yielding, and when deformations are negligibly small: the expected state of a conventional structure under everyday loading. Of course, much of structural design concerns understanding structural response under extreme conditions where material yields and deformations are large. Modelling such behaviour is beyond the limits of linear analysis.

Static vs. Dynamic Behaviour

Structural response over time introduces another measure of complexity. Figure 2 shows a cantilever beam with two versions of its response to a load P. In the graph, the horizontal axis is time, and the vertical axis is the reaction force R. The curve labelled Dynamic shows the response when the load P is applied quickly; the curve labelled Static shows the response when the load is ramped up slowly. In both cases, the reaction R settles to the same value: equal and opposite the applied load P. The difference is in the early stages where the dynamic case oscillates. During oscillation, R is sometimes more, and sometimes less than P. Oscillations occur because the quickly applied load sets the mass of the structure in motion, so the reaction force must resist both the applied load and the changing momentum of the moving mass. When the load is applied slowly, the momentum of the mass is negligibly small. The effect of the moving mass is typically called an inertia effect, and is the key distinguishing characteristic between static and dynamic behaviour. Inertia effects are paramount in structural response to earthquakes and blast, and can be important in the response to wind and human activity such as dancing or walking.

Figure 2 Static vs. dynamic structural behaviour.

Representational Territories in Structural Education

Linear vs. nonlinear and static vs. dynamic define two measures of complexity in representation. Figure 3 shows these measures as perpendicular axes defining areas of theoretical study. Traditional engineering education begins with rigid statics, working to linear-static, and possibly linear-dynamic theory in undergraduate curricula, with nonlinear-static and possibly nonlinear-dynamic theory addressed in graduate curricula. Traditionally, architecture curricula in structures have used the same starting point, but have not gone beyond rigid-static and linear-static representations. The application of software in architectural curricula has been similarly limited to linear static analysis. Although most of the software packages commonly used in architectural education (e.g. RISA and Multiframe) support dynamic analysis to some degree, there is little evidence to suggest that structural dynamics is being widely taught in architectural structures. Textbooks in architectural structures address the topic only in the most conceptual terms, and recent conferences addressing structures in architecture (e.g. the Building Technology Educators Conference in 2009), have included little discussion of structural dynamics.

Figure 3 Representational territories in structural education.

The traditional approach is logical from the perspective of teaching students to do calculations, since it begins with theories that are easiest to calculate. This approach, however, is less logical from the perspective of teaching students to grasp physical phenomena, since the simple theories are the least life-like. The stark abstractions of linear static analysis make its models unintuitive. In contrast, the ability of nonlinear dynamic theory to model motion and material damage makes its models more intuitive. Even though nonlinear dynamic analysis is theoretically more advanced, it is more appropriate for approaching elementary topics.

Taking full advantage of nonlinear dynamic theory requires software with an intuitive graphic interface. The Arcade program includes interactive tools which enable interaction with a model in real time, the way that a game responds to input from a controller. These tools include the hook tool, which allows a user to click a node and then push or pull it with the mouse (Figure 4), and the bomb tool, which allows a user to remove an element by clicking it (Figure 5). These figures and others in this article approximate the animated motion using stop-motion rendering, but don’t do justice to the realistic motion of the models. Interested readers are encouraged to download the program with example files (Martini 2011).

Figure 4 The hook tool pulls a model of a cable structure by clicking a node and dragging the mouse.

Figure 5 The bomb tool removes a member from a truss, causing it to collapse.

Expanding the Boundaries of Learning

Models based on a nonlinear dynamic representation are intuitively more accessible and mathematically more sophisticated than those based on linear statics. This combination makes it possible to expand the boundaries of software application at both introductory and advanced levels. At the introductory level, the intuitive accessibility makes it possible to introduce software from the first topics in statics. At the advanced level, nonlinear dynamic analysis can model buckling, yielding, large deformations in cables structures, and other phenomena that are beyond the scope of linear static analysis.

Lowering Thresholds

At the introductory level, one obstacle in teaching with linear static software is that students must master several key concepts before they can use the software effectively, i.e. there are ‘thresholds of learning’ to pass before using the software. Considering the curriculum Black & Duff (1994) proposed, it did not introduce analysis software until the second course, following a first course in statics that did not use computers. Improvements in graphic interfaces have made analysis software more accessible since then, but the limitations of linear static theory impose thresholds that interface design cannot overcome. The only way to lower those thresholds is to change the underlying theoretical representation.

Structural supports are an example of such a threshold. A linear static model requires a stable configuration of supports, e.g. a configuration that prevents the structure from sliding away as a rigid body, or moving as a mechanism. If the configuration of supports in a model is not stable, linear-static software produces an error message; since that theory cannot calculate such a model (numerically, linear analysis forces the computer to divide by zero when attempting to analyse an unstable structure). In contrast, a nonlinear dynamic representation can analyse such models without difficulty. Figure 6 illustrates an Arcade demonstration with five models, where three are stable, and two are unstable. In an interactive lecture, students are asked to identify which are stable. Their answers are tested by tugging with the hook tool; stable configurations stay in place, unstable configurations move. In addition to the utility of teaching fundamental structural concepts in a more visual manner, the nonlinear dynamic representation also instils a different perspective on structural behaviour, a perspective that emphasises interpreting structures in terms of movement, rather than solely in terms of force balance.

Figure 6 The hook tool demonstrates stable and unstable configurations of supports and hinge releases. The models on the upper and lower left are unstable.

Raising Limits

Nonlinear dynamic software can also raise the limits of learning by modelling aspects of structural behaviour that static linear analysis cannot, this is particularly important when considering questions related to collapse, such as buckling and yielding. Linear static analysis can address neither post-yield behaviour, nor post-buckling behaviour. Most instructors use physical props, diagrams, and photographs to illustrate the general nature of these phenomena. All these tools are valuable, but nonlinear dynamic analysis can extend and amplify their lessons.

Figure 7 shows an example of frame behaviour that linear analysis cannot explain. The figure shows three versions of a structural frame where one or more columns have been removed. In part a), the frame continues to carry load after the column is removed because the beams and columns above work to span across the gap (this load-carrying mechanism is called Vierendeel action, because it is the same principle as a Vierendeel truss: a truss that works without diagonals because its connections are moment-resisting). For part b), where two columns have been removed, the frame collapses. The deformation of the structure in this case shows that the collapse is not only downward, but also inward. The unsupported beams span like cables (called catenary action), exerting large horizontal forces on the outer parts of the frame, pulling them inward. For part c), the frame includes diagonal braces in the outer bays. Adding these braces gives the structure the ability to resist the large horizontal forces that the catenary action produces. Linear analysis is unable to explain this behaviour because it can model neither the yielding of the beams, nor the large deformations of catenary action. Interactive nonlinear dynamic analysis makes it possible to explain the behaviour of the structure in terms accessible to non-engineers.

Figure 7 Nonlinear dynamic analysis illustrates collapse modes in frames.

Applications in Teaching Introductory Structural Design

As discussed above, interactive nonlinear dynamic analysis is useful in helping students gain insight into many aspects of behaviour; see Martini (2005) for examples. In the context of a design curriculum, it is important that students use that insight in generating original solutions to open-ended design problems. The following discussion describes two such exercises in the context of a first course in structural design. The two exercises share the following characteristics:

• Work in teams: students work in teams of two, which requires them to argue and reason with one another about ways to develop their design.

• Use easily evaluated performance criteria: the performance criteria are simplified so they can be evaluated quickly. This allows for rapid design cycles.

• Emphasise structural form over member-sizing: available member sizes are limited so the primary design decisions involve structural form rather than member sizing.

• Make students construct their own knowledge: the exercises give minimal advice concerning solution approaches. The exercises force students to explore the problem and construct their own knowledge about the relationships between structural form and behaviour.

• Follow-up discussion: each exercise has a follow-up ‘debriefing’ session which examines selected student-produced models. This discussion identifies characteristic patterns of errors and success that emerge from the body of student work.

In addition to building students’ skills in analysis and structural behaviour, these exercises seek to demonstrate that structural design is an iterative exploratory process rather than a linear step-by-step sequence. This lesson is often obscured in structural education.

Exercise 1: Cliff House

Overview

Figure 8 shows an abridged version of the problem statement. The exercise is undertaken as a 75 minute in-class exercise, with students working in pairs at computers. It is possible to do this exercise effectively with conventional linear elastic software; the primary benefit that nonlinear dynamic representation brings is the ability to remove members and instantly see the effect.

Figure 8 Abridged statement for the Cliff House project.

One common pattern is that students begin with structures that have far more members than necessary. As the instructor and teaching assistants talk with student teams, they can ask whether any members can be eliminated, and students can use the bomb tool to remove the members and instantly check the response. Figure 9 shows an example. The left part of the figure shows a student model in progress. The vertical members are ineffective because they have a support at each end, the horizontal members appear ineffective because there is no horizontal reaction at the middle support. Four clicks of the bomb tool quickly reveal that the members are in fact unnecessary. Removing the members makes clear the primary behaviour of the structure: two independent chevron-shaped structures, one for each load.

Figure 9 Using the bomb tool to identify ineffective members in a framework.

Follow-up Session

An essential part of the exercise is a follow-up session carried out during a lecture period. This session emphasises the significant limitations of the exercise, and examines typical design approaches. Figure 10 shows a few such examples, organised in a progression from worst to best. Part a) shows a common approach in design practice: take a conventional structural configuration, and then modify it so that it meets the criteria, resulting in an inefficient design. Part b) shows a better structure, while part c) shows a good solution that boils the solution down to an essential structural form for the situation. The lecture demonstrates that the form of the efficient solution in Figure 10.c can be derived from the moment diagram for the span and load condition. The structure’s key dimensions can be estimated using a ‘back of the envelope’ calculation where the profile of the structure is assumed to be proportional to the moment diagram, as is the structure in 10.c.

Figure 10 Examples of Cliff House projects.

Exercise 2: Digital Bridge

Overview

One of the key lessons of the Cliff House project is the pedagogic value of the question: What happens to this structure if you remove that member? This question forces thinking both locally and globally in terms of the individual member and its role in the larger system. The next project, called the Digital Bridge, brings this question to the forefront. The project includes a requirement for single member redundancy, meaning that the structure must be able to continue to carry load when any member is removed. Students work in teams of two over a one week period. Figure 11 shows an abridged problem statement.

Figure 11 Abridged statement for the Digital Bridge project.

The requirement for single member redundancy dominates the structural concerns. If a proposed design does not satisfy the strength or stiffness criteria, it is usually clear what to change, typically increasing the structural depth, in addition to increasing the size of some members. When a proposed design fails to meet the redundancy criterion, it is usually much less clear what needs to be changed. The redundancy requirement also means that conventional truss configurations typically do not work, since most will collapse immediately when a member is removed.

Follow-up Session

As with the Cliff House project, the Digital Bridge project includes a reflective follow-up session; the following discussion examines two example models from such a session. Figure 12 shows a cable-stay type structure, which has a clear strategy for providing redundancy. Under normal conditions, shown in part a), the action is like a cable-stay bridge, but if a tower is removed, as shown in part b), the structure acts with a combination of an arch across the top and an asymmetric suspension cable across the bottom. The area where redundancy is questionable is in the deck, where there are four points supported by only a single diagonal stay, raising the question of how the deck will carry load if one of the stays is removed. Part c) of the figure answers the question: the deck goes into tension resulting in fracture and failure. The tension in the deck is the result of catenary action observed earlier in the course in the frame from Figure 8.

Figure 12 A cable-stay design that fails to meet the redundancy criterion.

Figure 13 shows a trussed-arch design that appears to have the same vulnerability as the preceding design, since the points on the deck are supported by only one member. Part b) of the figure shows that the deck members are strong enough to work in catenary action when a vertical support is removed. Part c) shows that the strong deck also serves to support the structure when a member is removed from the primary arch. The design in this example has a clear strategy to address redundancy: the primary structure is the arch, using a strong deck in catenary action as a backup. Note that this particular redundancy strategy was never discussed in class. The designers of this project constructed their knowledge through their own exploration, and the follow-up session shared that knowledge with the rest of the class.

Figure 13 A trussed-arch that uses a strong deck for redundancy.

Measuring Effect

The question arises of how to measure the effect of this teaching tool. Answering definitively would require rigorous control-group studies where the teaching tool is the only variable, which is impractical. For the Cliff House project, the effect has been measured as part of a student survey distributed during the second year of Arcade’s adoption. For the Digital Bridge project, the effect has been measured by evaluating student performance on the project.

In a survey, students were asked about various aspects of Arcade in teaching, including the Cliff House exercise. The survey included Likert scale questions on each topic, with 5 corresponding to strongly agree, and 1 corresponding to strongly disagree, as well as open ended questions seeking comments and suggestions for improvement. From the 86 students enrolled in the course, 81 responded, a 94% response rate.

Concerning the Cliff House exercise, the Likert-scale question was the following:

The lab exercise using Arcade enhanced my understanding of structural behaviour and the design process.

The average score was 4.22 with 85% scoring 4 or 5. Student comments reflect broad enthusiasm for the open-ended nature of the problem:

“It demonstrated the cyclical nature of the design process - making hypotheses and checking the consequences against constraints, then making new adjustments and decisions.”

“The lab was extremely effective in understanding the conceptual behaviour of the structure and proceeding with the design process. Trial and error tests of designs worked very well.”

“The lab was helpful in being able to check your hypotheses and seeing exactly how it affected the structure. Working in partners was also essential to help work through the problem.”

A smaller number of students found the open-ended approach to be negative:

“The lab was somewhat helpful but it was a bit guess and check and stumble across an answer. In lecture going over the lab I learned more than during the lab.”

“More of the learning about a system in this came from the debriefing in lecture which followed than actually using the program: nothing about the Arcade program itself led me to the answer. However, what it did extremely effectively was to visualise what was wrong with any given solution.”

“I felt I had too much freedom to screw it up.”

These responses reflect differences in learning style. While some students welcomed the freedom of an open-ended problem, others were disoriented. The follow-up session is intended to provide an explicit conceptual framework for students who need it.

The assessment of the digital bridge project is based on a review of the performance of student designed models. Considering two recent years, the models can be organised in three categories: those that passed the performance criteria, those that did not pass the performance criteria, and those that were invalid because they failed to follow instructions. Table 1 shows a breakdown of these categories for two different years.

Table 1 Performance of student-designed Digital Bridge models.

Year	Passed	Not Passed	Not Valid
1	23 models, 59%	14 models, 36%	2 models, 5%
2	24 models, 69%	8 models, 23%	3 models, 9%

The rates of passing are less than ideal, but the scores attest to the difficulty of the project: the criteria are unconventional, and there is no step-by-step solution procedure. Although difficult, the problem was within reach for most students. One of the objectives of the project is to make students understand that structural design is an iterative, nonlinear process. Students can learn that lesson even when their models do not pass.

Limitations

While the nonlinear dynamic representation underlying the Arcade software enables useful new teaching avenues, there are clear limitations. Compared with conventional linear static software, Arcade does not include a library of standard steel sections, and it does not include functions for checking building code criteria. More importantly, the Arcade software is limited to two-dimensional (2D) representations of structure. While 2D representations are useful and appropriate for many topics in introductory structures, it is essential to get students thinking in three dimensions. Linear-based software remains the best choice for examining three dimensional phenomena.

Beyond the limitations of any particular software package are the limitations of software in general. Working with software (as well as manual calculations) instils an understanding of mathematical models, but that understanding is useless without a grasp of the imperfect mapping between a model and a real structure. Work with software and other mathematical models must be complemented with devices such as case studies and physical models in a way that emphasises the nature and limitations of that mapping.

Closure

Employing nonlinear structural analysis at the introductory stages of learning strongly breaks traditions of structural education, but brings clear benefits in providing computer-based models that are simultaneously more rigorous and more intuitive. Nonlinear dynamic theory lowers thresholds of learning by making it possible to model simple situations such as the motion of a single particle, which conventional linear static software cannot model. Nonlinear dynamic theory also raises the limits of software application, by modelling phenomena such as structural yielding and collapse, which linear static theory cannot model either.

Although software can engage architecture students in structural design, it is important to remember that an architecture student armed with structural analysis software is not an engineer. The objective is to develop students who can collaborate with structural engineering consultants in all phases of design. Most real technical problems cannot be solved by a linear step-by-step procedure, yet that is how technical problem solving is typically presented in architectural education. Solving real technical problems involves multiple cycles of developing and refining proposals. The physically intuitive models enabled by nonlinear dynamic representation of structure make it possible to engage students in such a process in their first course in structures, connecting the technical rigor of structural engineering with the intuitive creativity of architecture.

Acknowledgements

This material is based upon work supported by the National Science Foundation under Grant No. 0230573. That support is gratefully acknowledged. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.