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Forces  Primer

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                                       What is a force?

                                       Combinations of forces

                                       Resolving forces into components

                                       Forces in equilibrium

 

What is a force?

Like many other words, "force" has a specialist meaning in engineering and science, different from the meanings it has in everyday usage.

It may be easier to discuss what forces do than what they are.  The diagrams below show simple examples, using the convention red for compression and blue for tension.  Compression shortens a strut, while tension lengthens a tie.  Note that these diagrams are not quite right.   If we connect a wide strut or tie to a narrow one, the forces will spread out rather gradually.  The fourth diagram shows the sort of thing that happens.  Many effects in nature are subject to continuity; that is, the quantity being studied is found to vary smoothly rather than suddenly.  But what sort of smoothness occurs?  Many variations are possible.  The one that is found is the one that minimises the strain energy in the object, strain being a measure of the deflection which will be explained in another page.

This diagram is a little more realistic, though the intensity of the blue should decrease as the force is spread out.

So here is an even more realistic diagram, but even here, the variation in stress has been greatly scaled down to keep the colour visible.  Although these diagrams are idealised, they do illustrate a very important aspect of design - the need to avoid stress concentrations when joining parts together.  A means of spreading the forces across a large area is very desirable. 

Tension stretches an object: compression shortens it.  In both cases the forces are present throughout the object.  Sometimes it is convenient to represent force by an arrow which shows the direction and magnitude of the force.  The direction of the arrow has meaning only at a boundary: within a region, the force is not pointing one way or the other.  Note that two forces or more are needed to obtain a change in dimension, in a static situation.

The dimensional changes are greatly exaggerated in the diagram above; with commonly used structural materials we can barely detect any changes as loads are applied and removed.  But they do occur.  If removal of the forces brings the material back to its original condition, the material is said to be elastic.  Otherwise it is plastic.  For some structural materials, the deflections are proportional to the forces.

The one force that acts on all objects on earth is gravity, which pulls each object with a force that is precisely proportional to its mass.  It is this force that creates many of the problems involved in structural engineering on earth.  Compare the weight of a bridge such as the Firth of Forth railway bridge with the weight of the traffic it can carry.  The Forth bridge includes about 55000 tonnes of steel.  What do you think would be the weight of a typical railway train on the bridge?  The problem is that the bridge has to hold the weight of the train (and of course parts of itself) at distances of up to 850 feet/259 m from the nearest support.  It's a reasonable supposition that the further the load is from the support, the greater the forces you will need.

Sometimes you see diagrams that seem to show forces "flowing" through a structure.  It is true that lines can be drawn that do show the directions of tension or compression at any point within an object, but we must not draw an arrow.  There is no flow, one way or another.  The places where we can draw arrows is at the edge of an object, or at the edge of a part of an object.  In the diagrams already shown, each object is subject to a pair of forces which are in equilibrium.

If we want to investigate the forces between objects, or between parts of objects, that are in equilibrium, we can pretend that the objects are separate, and draw the forces between them, as in the example below, where we have split the objects into two halves, to look at the forces between the halves.

 

Sometimes you see diagrams in which arrows are drawn as if forces flow into, through, and out of, objects.  Those diagrams are only a visual guide: arrows must not be drawn within an object.  We can subdivide an object in imagination right down to atomic level, and we will always find these pairs of equal and opposing forces between the parts.

Combinations of forces.

A quick look at a truss bridge tells you that structures often include places where several forces are acting.  So we need to look at the way forces combine.  The following diagrams show how two equal forces (blue) combine to form a resultant force (black).  Can you see the general method for working this out?

           

You can see that the bigger the angle between your two initial forces and the one you are trying to create, the bigger these two forces must be.  So parts of a structure may be subjected to forces that are far greater than the actual weights involved, if those forces are at large angles from the vertical.  You can see this by running this program.

A similar construction works for unequal forces.

The rule for the resultant is that the three arrows must form two sides of a parallelogram and its included diagonal.  Another way is to think of the arrows as the sides of a triangle.

Most structures are three dimensional.  The method used here will still work, but it has to work when the forces are projected on to nay plane.  More generally, just as we have seen here how to add two forces, we can also create two forces from one, by an inverse process.  If we define three perpendicular axes, we can resolve each force in a system into three components, one on each axis.  We can then for each axis add all the components.  We can then recombine the three results into the final resultant force.  It sounds complicated, but it is simpler than trying to do complicated three dimensional maths with the original forces.

Resolving Forces into Components

In a structure with forces pointing in many directions, it is often more convenient to work with three directions at right angles, one of which is usually vertical.  The other two would usually be along the axis of the bridge and at right angles.  In two dimensions we have something like this -

Using this construction we can work out almost anything by measuring the lines.  But a drawing is no good to a computer: we need to be able to calculate.  The next diagram shows how to do it.

Cosine and sine are functions that are used in trigonometry, which means measuring triangles, though in fact these functions are used in vast number of applications that have little or nothing to do with triangles.  As far as we are concerned, they are just a shorthand way of writing more complicated things.  If we now go back to the original diagram - 

we can write -

Horizontal Component = HC = F cos(A)

Vertical Component = VC = F sin(A)

Mathematical tables, technical calculators, and many computer programming languages, all include these functions to allow for simple calculations.

One of the benefits of components at right angles is that they are completely independent, and you can go right through a calculation, dealing with V and H components separately, combining the results only at the end.  Quantities that are independent in this way are called orthogonal.  As mathematicians so often do, they have extended this concept to cover all sorts of other things that are not forces, and not even things that point in certain directions.

Let's try some examples of components.  The next picture shows a cable hanging from two supports.  If you don't like mathematics you can click here to skip this section.

If the weight of the cable is W, then the vertical component V at each support must be 0.5W in order to hold the cable.  Now we need to calculate the horizontal component H.  Let us call the tension in the cable at the support T.

We know the following - 

V = 0.5W

V = Tsin(A)

H = Tcos(A).

So 0.5W = Tsin(A) and therefore T = 0.5W / sin(A)

Since H = Tcos(A) we see that H = 0.5Wcos(A) / sin(A)

and H / W = 0.5cos(A) / sin(A)

The next diagram is a graph of H versus A, for a fixed length of cable, so that the horizontal span varies with the angle.

When A is near 90 degrees, the cable is hanging almost vertically, and there is only a small horizontal pull, but when A tends to zero, the horizontal pull tends to infinity.  You cannot pull a cable into a horizontal position.  In the same way, a flatter arch generates more thrust than a higher one.  Instead of a very flat arch you might as well make a beam, which removes the thrust from the abutments and contains it within itself, by generating tension as well as compression.

The next graph shows the tension in the cable, rather than just the horizontal component.  When the angle is 90 degrees, that is, the cable is hanging vertically, the tension is one half of the weight.

On the other hand, if the horizontal span of the cable is fixed, the length of the cable (and therefore its weight) varies with the angle, and the graph is slightly different.  Whenever we state that a variable Y varies in a certain way with a variable X, we must specify the conditions exactly, in particular we must say what other variables remain the same or are allowed to vary.  The previous curve is shown in grey for comparison.  For the case where the angle is 81 degrees, the horizontal pull is more than twice what it was for the case of a constant length of cable.

We see that in a very flat arch, or in a very flat suspension bridge, such as the Millennium bridge in London, the forces must be very great.  For a sufficiently flat suspension bridge, the cables would be so thick that they would be stiff enough to act as beams.

Finally, we look at the tension in the cable versus angle, for a fixed span.

There is an angle, for a given span, for which the tension is a minimum.  The angle is about 57 degrees, or about one radian.

UWEProp9DecS.jpg (109775 bytes)Here is a practical example of forces and resultants.  The footbridge is supported on two inclined struts.  The vectors show that the force in the strut is rather bigger than the vertical force from the bridge.  Because of the slope, the bridge is in tension across its width at this point, to resist the outward push from the strut.

UWEFB.jpg (120913 bytes)UWEFBTriangS.jpg (93018 bytes)An advantage of this arrangement is that only one foundation is needed.  The disadvantage is that the bridge receives no support against torsion at this point, and it has to be stiff enough to carry torsion back to the abutments, in the case where numerous people all walk along the same side.  The second picture shows that some ties have been added, giving a measure of triangulation, and therefore stiffness.

You will have realised that unless the ties are pre-stressed, one or the other will not be in tension, depending on where the people are.

Having looked at combinations of two forces, and the the way that a single force can be considered as a combination of two components, we turn now to combinations of three forces in equilibrium.

Forces in Equilibrium

In any static structure, such as a bridge, there will be forces at any point, again in three dimensions, that have to be in equilibrium.  Here are some examples of three forces in equilibrium, in a plane.  Remember that blue means tension and red means compression.

        

Most of this looks much like what you would expect.  As before, the rules use the parallelogram and triangle.  It is all very academic, so here are some photographic examples.  As usual in this website, red will represent compression, blue will represent tension, and green will represent forces due to gravity, that is, weights.

Clifton7XA.jpg (82621 bytes) M42TiedVA.jpg (87235 bytes) ForthRailA.jpg (23536 bytes) ForthRailB.jpg (23611 bytes) OrangUtanSmallA.jpg (58555 bytes) OrangUtanSmallB.jpg (58509 bytes) OrangUtanSmallC.jpg (58711 bytes)

What  About  Rotation?

So far, all the forces have been acting at a point, but you don't find many points in real life.  Here's a small block of metal with two forces pulling or pushing on it.

    

In the first picture, although the two forces are equal, they are not aligned, and the block will begin to rotate.  The two forces form a torque.  In the second picture they are not even equal, and the block will begin to translate (move along) as well as rotate.  The product of the force and the perpendicular distance between it and the axis of potential rotation is called a moment.  Two forces as shown above are called a couple.

A cantilever is an object for which the self-weight and the load are not above the support, and a moment must be supplied to counteract the moment of the weights.  A beam is supported at each end.  The weight and load produce moments such as to produce rotation about the supports.  The necessary counteracting moments must be provided within the beam.  This need to produce internal moments is one reason why beams can never be as long as arches and suspension bridges.  It's almost as if there were an internal conflict, some of the strength being used up in creating opposing forces.

How do the structural members produce internal forces?  They do this by deflecting until the forces thus generated exactly balance the applied ones.  If a member reaches its limit of strength before the necessary force is attained, in will bend, snap, twist, or break.

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What about the drawings below?  The structures depicted cannot possibly obey the rules given above, because the members at the tops of the piers are collinear.  What is wrong?  One fallacy is that we must not assume that for each straight member we have one force.  All these beams will be subject to internal stresses which are not even parallel to their axes.  Another is that the rules given earlier apply to the forces acting on an object or a part of an object.  It is not obvious that this is the case here.

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Here is an example of forces in equilibrium.  The next picture shows a series of arches, apart from the lowest one, which is a beam with sloping supports.

Working up from the bottom, the spans become more and more like continuous curved arches.  In the next picture we see the weight of one block or voussoir and the two forces which balance it.

The horizontal thrust in any arch is the same throughout: only the vertical forces change.  If we imagine going to a very large number of small voussoirs we see that there are forces throughout every part of the spans.  That is true of the examples shown; there are forces throughout each block.

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The next picture shows a series of chains made of rigid links.

The horizontal thrust in any freely hanging chain or cables is the same throughout: only the vertical forces change.  If we imagine going to a very large number of small links we see that there are forces throughout every part of the spans.  That is true of the examples shown; there are forces throughout each link.

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Here are two rockets, one far out in space, one on a launch stand.  There are similarities between the two cases, especially if the thrust of the rocket is equal to the weight of the rocket.

Let's look first at the moving rocket.  If it is far enough from any star or planet, we can ignore all forces on it except the thrust of the engine.  What is a force?  Perhaps it is easier to consider what a force does.  In this case it will change the velocity of the rocket, which is called acceleration.

A single force on an object causes acceleration.

It seems reasonable that accelerating a massive object needs more force than accelerating a light object, and that is what happens.  The diagram of the rocket has been divided into slices.  We now have to think about the fact that each slice needs much less force than the whole rocket.  Look at the small piece at the front of the rocket.   It needs a very small force to accelerate it.  But the only force we can find in the diagram is the big one from the engine.  If we think of the top half of the rocket, it needs about half the total force.

So how does the force on the parts reduce as we go up through the rocket?  The same question can be asked about a railway train.  The coupling between locomotive and first wagon has to pull the whole train, while the last coupling has to pull only the last wagon.  This leads us to another important property of forces. 

Forces can change the shapes of objects.

If you pull the ends of an object, it gets slightly longer: if you push each end, it shrinks.  These deflections are not visible in most structural materials, but they can be measured.  The larger the forces, the bigger the deflections.  So in the rocket we looked at, all the parts shrink slightly until every part experiences just the right force to accelerate it at the same rate as the rest of the other parts.  How long does this adjustment take?  The effects travel at the speed of sound in the material, which is very fast.  Note that these deflections require at least two forces.  The speed of setting up of the deflections and the forces is so fast that it isn't usually noticed.  But when oscillations occur, this speed plays an important role.

Some materials deflect very little, while others deflect at lot, that is, materials differ in rigidity.  If an object returns to its original shape and size after the forces are removed, it is called elastic.  There are several types of responses to forces.  These are -

Compressing and stretching

Bending

Torsion or twisting

And any combination of these.

Any elastic object which reaches equilibrium ends up in a state in which all the forces it exerts on the outside world are exactly opposite to those which the outside world is imposing on it.

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Here is the accelerated rocket again, with the colours reversed.  If we imagine that we could look inside the metal and measure the forces within, we would see something like the effects shown in the next diagram.

The forces vary along the rocket because as we go from engine to nose, the mass of rocket being accelerated gets less.  

By drawing all the slices of the rocket separated, we can draw the forces with which they act upon each other.

In every case the arrows match in strength, because as Newton's third law says - "Action and reaction are equal and opposite."  Note that the law is true for both accelerated systems and non-accelerated systems.  However finely we subdivide an object, we will always find equal and opposite forces.

Here is the rocket with some sections separated to show the forces increasing down the rocket.  The rocket should have been coloured red as it is entirely in compression, to be consistent with the rest of this website.

If there is a satellite in the nose-cone, it will experience no internal stresses when it is floating in orbit around the earth, but it will have to be strong enough to withstand the acceleration during the launch.

One of the highest accelerations in nature is that of a jumping flea, which may attain 200 g.  No muscle can achieve this: the trick is to store energy in elastic material, and then release a trigger.  Click beetles and springtails leap using similar mechanisms.

We have not yet discussed the rocket that is still sitting on the launch pad.  The weird thing is that the distribution of forces is exactly the same as in the accelerated rocket that is in empty space.

Einstein was so taken by the idea that the mass used in calculating weight has exactly the same value as that used in calculating acceleration that he wondered whether weight and acceleration could be equivalent.  After a mighty struggle with the mathematics, with which he was helped by Michael Besso, he produced the general theory of relativity, which does not mention "force of gravity", but refers instead to curvature of space-time.  Einstein's theory gives virtually the same results as Newton's theory, except for some small discrepancies, all of which were verified later.  Nobody has yet found any observations that are not in accord with Einstein's general relativity.  The equivalence principle means that if you sit in a spacecraft on the launch pad, and if you sit in one that is being accelerated at 1 g in empty space, you will feel exactly the same in both cases.

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A very common structure is the masonry arch, so let's use that as our final example.

Because the bridge is symmetrical, we will look at one half only.  Green represents gravity (weight) and red represents compressive forces.

The horizontal component of all the forces is the same throughout the arch, but the vertical components increase towards the abutments, because each voussoir has to carry the weight of all those above it.  This of course is the reason why the arch is curved.

There is a lot about forces in this website.  But how do we measure them, especially inside an object?  There is no way of measuring the forces inside objects: the best we can do is to measure the strain at the surface.  From the value of the elastic modulus, we can calculate a stress from the measured strain, but it will only be an average.  We can never measure the stress at points inside.

Measuring forces

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