From Doctor Lom

Strength of Materials. Basics

in general, basics of strength of materials (mechanics of materials) are even simpler than the multiplication table. The multiplication table is large, it must be blankly memorized as "Our Father". But the basics of the strength of materials can be reduced to a few basic postulates, which are quite easy to demonstrate so they are easy to remember.

However, this is only my opinion. Many people think that the strength of materials - it is very difficult thing. For coming doctors and history teachers it is simple to memorize ten heavy volumes before examinations, but for people with analytical mind it is easier to remember a few basic provisions of this or that technical discipline, and it is not necessary to remember all the formulas. Most formulas can be determine by using mathematical tools and relying on basic provisions, anyway I often did so during exams. But let's not get distracted, and get back to basics.

If very briefly, the basics of the strength of materials set out in this article are as follows:

  • Any physical body can be a building structure.
  • Any construction have supports.
  • The supports provide the static equilibrium of the structure.
  • Forces act on any structure.
  • These forces can be external and internal.
  • External forces are acting outside the structure, such forces are called loads. Internal forces are acting within the structure, such forces are called stresses.
  • But physics makes no difference between external and internal forces. We can always replace the external forces to internal and internal forces to external, if the static equilibrium of the construction isn't broken.
  • Method to determine the stresses in the various points of the structure, is called the method of cross-sections.
  • Stresses in any point of the structure must be less than the resistance of the material of construction.

That's all basics of strength of materials. Formulas and calculations are everything else, but we will try to add more clarity to these postulates by the example of the beam.

It's enough to have two school wooden rulers, few books and a table to visualize the basics of strength of materials.

1. Types of supports.

Now we put a ruler on the two books:

 

ruler on the books is as a model of a beam on supports

Figure 1.

 

1.1. Simple supports

If we look at this situation from the point of view of theoretical mechanics (we have not yet reached up to strength of materials), we can see the model of a beam on two simple (or roller) supports:

a beam on simple supports

Figure 2.1

1.1. The rings on the ends of the beam are hinges around which the beam is free to rotate. If we remove the book on the right side, the left end of the ruler will remain on the left book (left support), but the right free end of the ruler will fall on the table after rotating around left support.

1.2. Horizontal lines with oblique hatching represent some stable base. In this case it is the table.

1.3. Some distance between the base and the supports of the beam is a certain similarity of the air bag. This distance means that supports can slide along the base without friction.

Simple supports do not allow to move the ends of the beam only vertically, so simple supports create only one support connection - the vertical.

1.2. Pinned supports

In fact, our ruler has no hinges, linking it with the supports. And we can draw our ruler like this:

model of the beam on the pinned supports

Figure 2.2

In the technical literature, such image of supports (without hinges) is also common, and means that the supports do not prevent rotation but prevent the movement vertically and horizontally. Such supports are called pinned. So pinned supports create two support connections - the vertical and horizontal.

And indeed, if we try to move horizontally the ruler to the right or to the left, then we need to apply a force greater than the force of friction between the ruler and books.

1.4. On figure 2.1 hinges are at the ends of the beam, on figure 2.2 pinned supports have got a distance from the ends of the beam. Which option is the right one?

The gauge length (span) of the beam

This means that the actual length of the beam and the gauge length of the beam - these are not the same things. The actual length of the beam is always more than the gauge length.

The gauge length of the beam, also called span of beam, is the distance between upper points of supports. In this case this is the distance between the books. The length of the beam is measured on the x-axis. Gauge length - span of beam - is denoted by letter l.

The ends of the ruler, based on the book can be seen as support sectors lsup of the beam. Strength calculation of support sectors of the beam is a separate big topic. In this case we only meet the basics of strength of materials and at this stage it is enough to know that length of the beam is:

ln = l + 2lsup (1.1)

1.3. Hinged support connections of the beam

Any physical body, in this case the ruler, has three degrees of freedom in the plane of the xy: 1) the body can move along the x-axis, 2) the body can move along the y-axis, 3) the body can rotate about some point, even if the freedom of movement is limited.

So any stable and statically determinated structure must have at least three simple supports. The beam on figure 2.1 is a statically determinated, but is not stable, because it has only 2 simple supports or 2 vertical connections. The beam on figure 2.2, is stable but statically indeterminate because it has two vertical and two horizontal support connections or 4 simple supports. Meantime static equilibrium equations can determine only three unknown quantities.

Therefore, the following scheme of supports can be used for the stable and statically determinated beam:

beam scheme with a simple and a pinned support 

Figure 2.3

The physical meaning of such scheme is following:

1. A single span statically determinated beam (at once I should say that our ruler is a single-span statically determinated beam) has two vertical support connections (in figure 2.3 vertical support rods are shown in purple color) and one horizontal connection (in figure 2.3 horizontal support rod is shown in blue color). The directions of that three simple supports don't intersect in the same point, so the beam and the whole system of 4 rods is stable.

2. These support rods are hingely connected not only with the beam but with a some rigid base. This means that not only the beam can freely rotate relatively to the support rods, but support rods can freely rotate relatively to the rigid base.

3. Horizontal support rod is necessary for stable system (it seems to be unnecessary for calculations on the vertical load, but that's another story).

4. The rods, which indicate the vertical and horizontal connections - it's a big conventionality. Further we will replace them on the relevant support reactions. Conventionally, these rods are considered as infinitely rigid, ie, deformation of the rods at any load is equal to zero.

5. Thus, the vertical rod at the point B (figure 2.3) corresponds to a simple support, shown in figure 2.1. The vertical and horizontal rods at the point A correspond to a pinned support, shown in figure 2.2.

1.4. Fixed supports

If we add books:

a ruler is jammed with books

Figure 3.

then such ruler can be roughly considered as a beam on fixed supports, and then the beam scheme will look like this:

 

a beam with fixed supports

Figure 3.1.

The physical meaning of this scheme is following: fixed support prevents not only the vertical and horizontal movement of the beam, but also rotation, ie. it limits all 3 degrees of freedom of the physical body.

We can easily see this, if we remove one pile of books. Ruler, jammed in another pile, will remain in previous position.

In this case is not quite right to consider the ruler as a beam with fixed supports, if support sectors fairly short. But if we'll glue the books and ruler, and support sectors of the ruler are relatively long, after drying of the glue we can already consider the ruler as a beam with fixed supports.

1.5. Journal bearing supports (sliding supports)

Some supports prevent rotation, but don't prevent movement along the x- or y-axis. Such supports are shown on fugure 3.2:

types of sliding supports

Figure 3.2. a) the sealing is sliding relative to the x-axis; b) the sealing is sliding relative to the y-axis.

If we push a glued pile of books along the x-axis, we receive the version a) on figure 3.2. If both piles are glued and we lift one pile, we receive the version b) on figure 3.2.

Other options of supports are not considered in theoretical mechanics.

2. Loads (external forces).

If we look at figure 1 more attentively, we can see that the ruler in the middle have a little deflection. If we take a longer ruler and base it on the books, the deflection in the middle of the ruler will be more noticeable, but still not very clear.

Why did this happen?

A load - the own weight of the ruler - acts on our beam and under the influence of this load the beam sags. Moreover, the greater the distance is between supports - l, the less load-bearing capacity  of the beam is with the same cross-section and it turns out that the estimated length of the beam is a very important factor in the calculations. Since the ruler has a simple shape, and aproximately constant density, then this is considered as a uniformly distributed load and scheme for this load can be depicted as follows:

 

scheme of uniformly distributed load on the beam

Figure 5.

2.1. The load may be uniform distributed, as shown in Figure 5, and non-uniform distributed, for example triangle distributed.

2.2. If we press by a finger on the ruler in the middle, the ruler will sag more noticeably. In this case, concentrated load (a force at a point) acts on the ruler (at figure 6 distributed load from own weight is not shown):

concentrated load on beam

Figure 6.

If we press by a finger on the ruler in one of supports, the ruler will not sag.

Why is this happening?

It turns out the load creates a bending moment. The value of bending moment not only depends on the load, but on the lever arm of the force. It is not difficult to guess that the maximum bending moment occurs when the concentrated load is acting on the beam in the middle.

Sometimes a moment (as couple of forces) can act on the beam:

moment is acting on the beam

Figure 7.

This is all, there are no more options for loads which act on the beam. However, load classification is a separate big topic.

And now directly the strength of materials, because until we described the terms and concepts of theoretical mechanics.

3. Stresses (internal forces).

If we continue to press on the ruler with more force, the ruler will sag more and more, until crash (of course, instead of brute force, you can use the power of his intellect, I will not mind)

Why is this happening?

Everything has a limit, and in this case, the limit of the resistance of ruler's material (wood) has been overcome.

If we take the steel strip with the same cross-section parameters and the same length as that of a wooden ruler, put it on the book and apply to it the same load in the middle, we'll not crash steel strip by the finger, at least because of steel resistance is ten times more than wood resistance. But let us return to the consideration of a wooden ruler.

When we push by finger on the ruler, the ruler is deformed, the upper part of the ruler is compressed and thus compressive normal stresses arise in this area. The lower part of the ruler is stretched and thus tensile normal stresses arise in this area. These stresses are the reaction on material on the load.

Normal stresses are acting along the normal (perpendicular) to the considered cross-section of the beam.

Shear stresses may occur in addition to the normal stresses in these sections, so stress-strained state may be not only linear, but also flat or three-dimensional (but about this not now).

A distribution of normal stresses in various cross-sections of the beam and in various points of these cross-sections will look like this:

stresses in cross-sections of the beam

Figure 8.

So the highest possible normal stresses σ in any point must be less than the resistance of the material R:

σ < R (1.2)

But it is not all. I hope you don't have broken wooden ruler, because now the time to stand the ruler on books:

 

a ruler is standing on books

Figure 9.

In this position, it is difficult to crash the ruler not only by a finger but also by a foot. Cross-sectional area of ​​the beam is not changed, the load is not changed, the bending moment is not changed, we have all the same wooden ruler.

Why is this happening?

The strength of materials explains this miracle so that when we turned the ruler, the width b and height h of the beam were changed, and thus the moment of inertia, moment of resistance and, therefore, the bearing capacity of the beam were changed.

In general, this is the sence of the strength of materials: the correct definition of the loads and the selection of the optimum cross-section design.

That's all. As you can see, the basics of strength of materials are really easier than the multiplication table. The basic formulas are following further.



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