An important property or quality of the equilibrium of a structure is the stability of the equilibrium; that is, its sensitivity to small disturbances. If, after the small disturbance has ended, the structure returns to its original position, then the equilibrium state is said to be stable; on the other hand, if the small disturbance causes an excessive response, then the equilibrium state is unstable. An important consideration is to where the unstable structure goes – this is called the postbuckling behavior. The postbuckling behavior is typically highly nonlinear, undergoing large displacements and sometimes incurring plasticity effects. Figure 6.1(a) shows an example of a collapsed frame.

In all stability analyses, there is an important parameter associated with the unfolding of the instability. For example, the axial compressive load in the buckling of a column or the velocity in an aeroelastic flutter problem. Imperfections of load or geometry also play a significant role in unfolding the instability. Identifying this parameter and observing its effect is one of the keys to understanding the stability of a system.

The explorations in this chapter consider the stability of both the static and dynamic equilibrium. The first exploration uses imperfections (of loading and geometry) to illustrate the concept of sensitivity to the unfolding parameter. The second exploration introduces eigenanalysis as a tool to determine the buckling loads and mode shapes of a perfect structure; Figure 6.1(b) shows the first three buckled mode shapes of a ring with uniform pressure around the circumference.

Wave propagation is the transport of energy in space and time. That is, the essence of wave propagation is the space–time localization of energy that moves with definite speed and amplitude characteristics. This contrasts with vibrations that set each point in the structure in motion simultaneously. Figure 4.1 illustrates some characteristic wave behaviors. It shows the velocity response of a semi-infinite two-material rod free at one end and impacted at the junction. The pulse in the lower semi-infinite part travels at a constant speed, conducting energy away from the joint. Observe how the pulse is initially trapped in the upper material (resulting in multiple reflections) but eventually leaks away after the multiple reflections.

The general wave in a structure is dispersive; that is, it changes its shape as it propagates, and so identifying the appropriate propagating entities is quite difficult. For example, Figure 4.2 shows an example of the deflected shapes of a plate transversely impacted; observe that, although “something” is propagating out from the impacted region, it is not obvious how to characterize it.

The collection of explorations in this chapter considers waves in extended media as well as in particular types of waveguides with an emphasis on understanding dispersive behavior. The first exploration uses a pretensioned cable to introduce the fundamental ideas in wave propagation; namely, the speed with which entities propagate in space and time and their amplitude variation.

There are two important concepts in the design of structures to withstand loads. One is stiffness, which relates to the ability of a structure to maintain its shape under load; the other is stress, which relates to the fact that all structural materials can withstand only a certain level of stress without failing. Stiffness is a global structural concept, whereas stress is a local concept.

The stiffness properties of structural members are greatly affected by their cross-sectional properties; this is especially true of thin-walled members. For example, Figure 2.1(a) shows a C-channel fixed at one end with an upward load applied at the other end along the vertical wall. What is interesting is that this load causes a counterclockwise rotation as shown and not a clockwise rotation as might be expected. The reason is because the shear center (the center of twist) is to the left of the wall.

Figure 2.1(b) shows an example of stress distribution in a bar with a hole. Changes in local geometry can cause significant changes in stress, giving rise to what are called stress concentrations. These are clearly visible around the edge of the hole.

The explorations in this chapter consider the stiffness properties of various structures and the stress distributions in some common components. The first and second explorations establish the stiffness properties of basic structural components and some thin-walled 3D structures.

When trying to understand a complex system, it is quite useful to have available some simple models – not as solutions per se but as organizational principles for seeing through the voluminous numbers produced by the FE codes. This chapter is concerned with the construction of simple analytical models; it gathers together many of the simple models used throughout the previous chapters and tries to illustrate the approach to constructing these. Although the models are approximate, by basing them on sound mechanics principles, they are more likely to capture the essential features of a problem and thus have a wider range of application. The models discussed are shown in Figure 7.1.

The term “model” is widely used in many different contexts, but here we mean a representation of a physical system that may be used to predict the behavior of the system in some desired respect. The actual physical system for which the predictions are to be made is called the prototype.

There are two broad classes of models: physical models and mathematical models. The physical model resembles the prototype in appearance but is usually of a different size, may involve different materials, and frequently operates under loads, temperatures, and so on, that differ from those of the prototype. The mathematical model consists of one or more equations (and, more likely nowadays, a numerical FE model) that describe the behavior of the system of interest.

Structures are to be found in various shapes and sizes with various purposes and uses. These range from the human-made structures of bridges carrying traffic, buildings housing offices, airplanes carrying passengers, all the way down to the biologically made structures of cells and proteins carrying genetic information. Figure 1.1 shows some examples of human-made structures. Structural mechanics is concerned with the behavior of structures under the action of applied loads – their deformations and internal loads.

The primary function of any structure is to support and transfer externally applied loads. It is the task of structural analysis to determine two main quantities arising as the structure performs its role: internal loads (called stresses) and changes of shape (called deformations). It is necessary to determine the first in order to know whether the structure is capable of withstanding the applied loads because all materials can withstand only a finite level of stress. The second must be determined to ensure that excessive displacements do not occur – a building, blowing in the wind like a tree, would be very uncomfortable indeed even if it supported the loads and did not collapse.

Modern structural analysis is highly computer oriented. This book takes advantage of that to present QED, which is a learning environment that is simple to use but rich in depth. The QED program is a visual simulation tool for analysis.

Suppose the existence of a very powerful computer, so powerful that it can execute any given command such as build a cantilever beam, excite the beam with this force history, record the velocity histories, and the like. Suppose, further, that it cannot answer questions such as what is elasticity? why is resonance relevant to vibrations? how are vibration and stiffness related? Here then is the interesting question: With the aid of this powerful computer, how long would it take a novice to discover the law governing the vibration of structures? The answer, it would seem, is never, unless the novice is a modern Galileo.

Now suppose we add a feature to the computer; namely, access to a vast bibliographic database (in the spirit of Google or Wikipedia) that can respond to such library search commands as find every reference to vibrations, sort the find according to the type of structure, report only those citations that combine experiment with analysis, and so on.

There are two main sources of nonlinearities in the mechanics of solids and structures. The first is geometric in nature and arises from large deflections, large rotations, and large strains. The second arises from the material behavior and is typified by elastic-plastic and rubberlike materials. Contact problems are also nonlinear even when the contacting bodies themselves remain linear elastic. Figure 5.1 is an example of a long, thin cantilevered plate undergoing large deflections and rotations because of an end moment.

The explorations in this chapter look at aspects of each of these nonlinear behaviors. The first exploration considers the concept of stress and strain under large deformation conditions because they need to be refined relative to the ideas explored in Chapter 2. The second and third explorations consider nonlinear material (constitutive) behavior in the form of elastic-plastic and rubber elasticity, respectively. The presence of a nonlinearity can affect the fundamental behavior of phenomena; the fourth exploration shows the rich, complex behaviors of nonlinear vibrating systems. Within a nonlinear context, applied loads affect the stresses, which in turn affect the stiffness and thus the deformations; this in turn affects the stresses, leading to a complicated connection between the applied loads and the responses. The fifth exploration uses vibrations under gravity to illustrate this point. The final exploration considers the contact that is due to impact of various shaped bodies and the resulting contact force histories.

When a structure is disturbed from its static equilibrium position, a motion ensues. When the motion involves a cyclic exchange of kinetic and strain energy, the motion is called a vibration. When this occurs under zero loads, it is called a free or natural vibration; whereas if the only loads are dissipative, then it is called a damped vibration. An example of a damped vibration is shown plotted in Figure 3.1; observe that the amplitudes eventually decreases to zero, but oscillates as it does so.

The explorations in this chapter consider the linear vibrations of structures. The first exploration uses a simple pretensioned cable to introduce the two basic concepts in vibration analyses; namely, that of natural frequency and mode shape. The second exploration looks at the meaning of mode shape in complex thin-walled structures. The stiffness properties are affected not only by the elastic material properties but also by the level of stress; the third exploration looks at the effect of prestress on the vibration characteristics. A significant insight into linear dynamics can be gained by analyzing it in the frequency domain. The fourth exploration introduces DiSPtool as the tool to switch between the time and frequency domains. Generally, increasing the mass of a structure decreases the vibration frequencies; however, in the presence of gravity, the mass can increase or decrease the stiffness and thereby affect the vibrations differently.

The beam theory of Chapter 4 and the corresponding finite element implementation was formulated in a fixed global frame of reference using the total displacements and rotations. In many cases it may be advantageous to consider the beam element with reference to a local, element-based, coordinate system. Motion of the beam then implies motion of the local frame of reference as well as deformation of the beam element within this frame. The separation of the motion of the element into two parts – a rigid body motion associated with the element-based frame of reference and a deformation of the element within this frame of reference – is called a co-rotating formulation. The co-rotating formulation has a number of advantages, provided it can be demonstrated that the tangent stiffness can be decomposed into the sum of a part associated with the rotation of the element-based frame and a part associated solely with the deformation of the element within this frame of reference. The first advantage is that displacements and rotations within the local frame of reference are small or at most moderate. Therefore, the deformation of the beam can be modeled by approximate beam theory. Secondly, the co-rotating formulation is closely associated with the idea of ‘natural modes’, advocated by Argyris et al. (1979a,b). The idea of the ‘natural modes’ is to consider any increment of the motion of an element as made up of a set of rigid body modes – typically translation and rotation – and a set of deformation modes – representing extension, bending and torsion of the beam element.