Dr. Beam

BEAMS 1: What if a support does not match its idealized conditions? Physical beam supports do not match perfectly the idealized conditions we use in our models. In this worksheet, some of the ways this influences the accuracy of analyses based on such idealized models are investigated.

BEAMS 2: How does load distribution affect a beam's response? It is important to understand how different load types influence a beam's behavior. It is also important to realize that in real applications it is very difficult to predict what the actual physical loads will be on a given structure. This worksheet will help you think about how different load distributions influence a beam's response.

BEAMS 3: Is there a general way to interpret the moment sign convention? The goal here is to broaden your understanding of the deformation-based moment sign convention shown above so that you can use your physical intuition to interpret moment diagrams and their relation to beam displacements.

BEAMS 4: What general process is used to solve problems with analysis tools like Dr. Beam? Dr. Beam is designed to solve a special class of structural mechanics problems, but the basic steps and ingredients associated with setting up and solving a problem are similar to what one would need to solve any problem in structural mechanics. In this exercise the basic steps are illustrated in the context of a simple example.

BEAMS 5: How do shears and moments plot in the case of distributed loading? We have seen in beam analysis that the nature of the functions describing loads, moments, and shears have relatively complex continuity properties. Kinks and jumps are common, and this requires special attention from a mathematical point of view. In this and the following two worksheets, we will take a closer look at these issues.

BEAMS 6: How do the shear-load and moment-shear relations plot in the case of concentrated loading? In this worksheet, the effect of point loads on shear and moment diagrams is explored.

BEAMS 7: How do the shear-load and moment-shear relations plot in the case of concentrated moments? This worksheet, similar to the previous two, examines the relationship between concentrated moments and moment-shear plots.

BEAMS 8: How can I determine maximum moments in the case of moving loads? In many applications applied loads are not fixed&emdash;i.e., the locations and/or magnitudes of the loads change. One of the simplest examples is a bridge, in which the loads due to cars and trucks move across the span. Determining maxima in these cases can be trickier than for cases of single, fixed load cases.

BEAMS 9: How special a case is pure bending? Pure bending is often used as a starting point for deriving beam bending equations. To get a feel for how special this case is, various configurations in this exercise have been identified as giving rise to pure bending over some or all of each beam's span. Using Dr. Beam files, the fragility of these pure moment configurations is revealed.

BEAMS 10: Can a beam deflect without curvature? Curvature is a key geometric concept in understanding beam behavior. Dr. Beam provides a helpful environment for picturing curvature in various situations, and in particular can help us answer the title question.

BEAMS 11: Can reduced loads on a beam increase stresses? One might guess that increasing the load on a structure increases the stresses, while decreasing the load reduces the stresses. Broadly speaking this is true, but in general one must be careful, as shown in this worksheet.

BEAMS 12: How do different support conditions effect an actual beam design? The design of a beam depends not only on the loading conditions but how it is supported as well. The relationship between support conditions and design is explored in this worksheet.

BEAMS 13: How do stiffness efficiency considerations enter a design? 0A design might be safe but is it practical. In this worksheet, the role of beam stiffness is examined.

BEAMS 14: How do I design a beam if the loads can move? In most real structures, loads will move within or on structure over time. This exercise deals with how to design a beam that must support several loading conditions.

BEAMS 15: Do shear stresses ever govern a design? Often normal stresses arising from bending and axial loading dictate a beam's design. This worksheet demonstrates how shear can also be the limiting design factor.

BEAMS 16: How do I determine displacements for beams with composite cross-sections? Designing a beam of a homogenous material is hard enough. This worksheet investigates how beams constructed from multiple materials can be analysed and designed.

BEAMS 17: How does stiffness depend on materials choice in a composite cross-section? In this worksheet, a doublysymmetric cross-section is presented and used to illustrate the influence of different material combinations on bending stiffness.

BEAMS 18: What happens when a beam's maximum moment reaches the yielding point? The behavior of a beam that reaches its yielding point due to bending depends on the beams material. This exercise explores the effects fof materials with different stress-strain curves on beam behavior at yielding

BEAMS 19: What can an analytical solution tell me that Dr. Beam doesn't? Dr. Beam provides fast, accurate results for beam analysis. This worksheet illustrates that analytical solutions are still useful under certain circumstances.

BEAMS 20: What does Dr. Beam tell me that an analytical solution doesn't? Analytical solutions are concise but sometimes esoteric. Visual and numerical presentations of results such as those generated by Dr. Beam is this worksheet provide very useful design information at a glance.

BEAMS 21: What can Dr. Beam tell me that an analytical solution doesn't II? This worksheet illustrates how Dr. Beam's flexibility has further advantages over analytical solutions.

BEAMS 22: How could I verify superposition experimentally? Superposition ultimately relies on the linearity of the governing equations for a beam, but it is useful to consider what an experimental verification of superposition would look like. The beam in this exercise is a good candidate just such an investigation.

BEAMS 23: Are there any other interesting features of multiple loads beyond superposition? Superposition is a useful principle for generating hand solutions to problems. In this worksheet another interesting property of beams and loads will be revealed.

BEAMS 24: Is it possible to have zero moment at a point but no inflection? A good question! This worksheet takes a look at if and when this could happen.

BEAMS 25: How do statically indeterminate beams relate to statically determinate beams? Because statically indeterminate systems tend to require more involved analysis, they are normally categorized distinctly from determinate systems on that basis alone. Programs like Dr. Beam remove the work-load difference between determinate and indeterminate problems, and so one can focus on the more important practical distinctions of how indeterminate beams behave.

BEAMS 26: What happens if fixed supports are not truly fixed? In practice it is difficult to obtain truly fixed supports, so it useful to consider what happens if the idealized conditions are not realized.

BEAMS 27: Are there any special design concerns for statically indeterminate beams? In previous worksheets, we have seen that there are design advantages associated with indeterminate beams. In this exercise one of the problematic design issues associated with indeterminate beams is illustrated.

BEAMS 28: How does a beam suspended by a cable behave? This worksheet examines one way of modeling a cable support and how loads are carried in general in a structure.

BEAMS 29: How do neighboring spans influence one another in a continuous beam? Multispan continuous beams are common in practice and also provide a good starting point for understanding the behavior of more complex frame structures. In this worksheet one aspect of the interaction between neighboring spans is presented.

Dr. Stress

STRESS 1: What happens on inclined planes in axial loading? Even simple loads can create non-intuitive stress patterns in materials. This worksheet takes a look at how stresses in inclined planes behave during axial loading.

STRESS 2: What happens on inclined planes in torsional loading? We have seen in earlier exercises that an apparently simple uniaxial state of stress actually induces varying amounts of normal and shear stress in a material, depending on the orientation of the material block one considers. In this worksheet what happens in the case of pure shear, such as arises in torsional loading, is investigated.

STRESS 3: What happens on inclined planes with combined torsional and axial loading? Having seen what happens in the cases of pure shear due to torsion and pure tension due to uniaxial loading, you might be curious about what happens if both loadings are present simultaneously. The combined case is considered in this worksheet.

STRESS 4: What is a tensor? Stress states are more complex than vector quantities like velocities, and are represented by things called tensors. Tensor mathematics itself is relatively abstract, but fortunately there are many physical interpretations associated with stress which help ease the task of figuring it all out.

STRESS 5: What do traction vectors look like? The standard transformation equations for stress states are expressed in terms of the normal and shear scalar components acting on each face of an infinitesimally small block of material. These stress states are described with traction vectors, the focus of this worksheet.

STRESS 6: What happens if I rotate a plane stress state out of its plane? Having gained some familiarity with plane stress analysis, you might wonder if that's the end of the story. In particular, having seen that even uniaxial stress states exhibit relatively complex behavior from a 3-D point of view, it is worth considering what happens with a plane stress state in 3D.

STRESS 7: How does Mohr's circle relate to stress block depictions? Both Mohr's circles and stress block figures are common ways to represent stress states. In this exercise a quick visual look at how these two depictions are related is presented.

STRESS 8: Are there limits to plane stress analysis? It is straightforward to determine principal stresses and directions for plane stress states using simple equations or Mohr's circle. It is important to realize, however, that these in-plane results tell only a part of the story. In this worksheet some important aspects of out-of-plane behavior are examined.

STRESS 9: Are there any handy ways of thinking about triaxial stress states? The three Mohr's circles that correspond to stress states in 3D provide a useful summary of the main quantities of interest. In this worksheet additional ways to interpret this summary information are considered.

STRESS 10: What kinds of combined loadings cause triaxial stress states? Triaxial stress states are important in many applications, and they arise naturally in regions localized around geometric, material, and loading irregularities. The types of analysis necessary to calculate such 3-D stress states, however, typically require more advanced analysis techniques than what you are likely to have encountered to date. In this worksheet a relatively simple case that requires triaxial consideration is considered.