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Bending Moments

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Words are often used in engineering, mathematics and science with precise meanings that do not correspond to anything in everyday life.  "Moment" is such a word.  We can speak of "momentous events", say that something is of "no moment".  Engineers speak of the moment of a force.  if we consider two forces that are equal, but working in opposite directions at the same point in an object, we could certainly say that they are of no moment in that nothing will result from them.  But if they are acting at different points, the object will tend to change its rate of rotation if no other forces are present.  The moment of force about a point is the product of the magnitude of a force, multiplied by the perpendicular distance from the point being considered.

Thus a given quantity of moment could result from a large force at a small distance or a small force at a large distance.  Spanners and wrenches are designed to achieve exactly that.

 

BendingMom267S.jpg (89038 bytes)Here is a photograph of a flexible strip being pulled by a string, and consequently being bent, and below are some diagrams of a similar arrangement.

In the left hand diagram above we see that the bending moment that is doing the bending is P x L, being the product of the force P and the perpendicular distance to the gripped end. We can think of bending moment as what is often called leverage. A moment in mathematics, engineering and physics is simply the product of some physical quantity and a distance raised to some power. The physical quantity might be an element of area, or a mass, or as here, a force. We are using the first moment of the force. For the second moment we would use the square of the distance, and for the third moment the cube. The first two moments are the most commonly needed.

To get back to the main point, for other points at distances represented by X, the bending moment is P x X, which gets smaller as X gets smaller.  That is why the rod becomes less curved towards the top.  In fact, just as the top is reached, the curvature reaches zero.  You see also a pair of forces S which squeeze the rod at the bottom.

Sadly, this diagram is wrong.  Because the forces S cancel each other, there is a net sideways force P on the rod, which should therefore accelerate.  Furthermore, the moment of P is unopposed, and the system should begin to rotate.

To understand better, the second diagram shows the jaws of the grip slightly opened.  We can now achieve two things.  We can make S2 bigger than S1, by just the amount to cancel P.  We can also make the anticlockwise moment of S1 and S2 cancel the clockwise moment of P.  That's better.  Now we understand.  We can even start imagining what happens to the size and position of S1 and S2 as we close the jaws.

Sadly, we still don't understand.  When the jaws are fully closed, where exactly do they grip the strip?  Each jaw exerts an inward force S on the rod, which of course exerts an equal outward force S on the jaw.  These forces ought to move the jaws outwards, but they don't, because the jaws are held by another part of the system, which itself is held by a bench, which rests on the ground.  And what happens tp the forces in the ground?  In fact, we cannot have a completely balanced system of forces until we have considered as many objects as we need to encompass all the forces.

The next diagram shows one end of a simple arch, with the thrust spreading out into the ground.

 

We can see that the effects of the bridge extend well into the ground, and it is the job of abutments or foundations to ensure that where the stresses reach the native rock, they are small enough for that rock to bear indefinitely.  In order that there be no net force on the whole planet because of the bridge, there must be some volumes which experience tension somewhere under the bridge, but these will be so diluted that they seldom need to be considered.  What actually happens in the design of a real structure is that the calculations either stop where the foundations reach hard, load bearing rock, or they stop at a point where soil mechanics determines that the load has been sufficiently spread to be borne safely.

The example given here of a rod bent by a force at one end is much simpler than those in most practical structures.  Distributed self-weight in two and three dimensions in complex shapes, together with live loads, and the effects of wind and other natural forces, mean that calculation is often very complicated.

Here is a simplified diagram showing compression and tension in a beam bridge.  As with the bent rod, there is no bending moment at the end.  Were the spans to be rigidly joined, the moment distribution would be quite different.  Robert Stevenson joined the spans of his Britannia bridge in such a way as to introduce some pre-stressing, to make the stresses in the bridge more uniform.

Do you wish to know how to calculate a bending moment? Here's how. Imagine a uniform beam of weight W and length L. Let's calculate the bending moment at a point P that is at a distance D from the centre. What forces do we need to consider? We need to look at all the forces to the right, or all the forces to the left of our chosen point. Let's go right. 

At the end of the beam we have a force from the support of 0.5W acting vertically upwards, at a distance D from P, and so the moment of that force about P is 0.5WD anticlockwise. The other force is the weight of the section from P to the end. This force is spread out uniformly along the beam, but for calculations like this we can imagine that it is concentrated at a point called the centre of gravity. For a uniform section, the centre of gravity is at the centre of the section, that is, 0.5D from P. The weight of the section is W x D/L, so the moment of the section about P is 0.5DWD/L = 0.5WD2/L, acting clockwise. So the net moment is M = 0.5WD - 0.5WD2/L = 0.5WD(1 - D/L). This equation defines a parabola. At the end of the beam, D = L, and the quantity in the brackets is zero, so there is no bending moment. At the centre, D = 0, and M = 0.5WD. 

The bending moment at any place has to be resisted by the forces within the beam, which when added together must make an equal and opposite moment. The beam will sag until the forces have built up sufficiently. The engineer must make the beam stiff enough to build up the forces without bending too much.

The same kind of calculation, with suitable modifications, can be used for other shapes.

Bending moment exists when a solid object is subjected to forces that make it bend. At any point where an object or structure is connected to something else only by a pin or a hinge, or by simply resting on a support, there can be no bending moment. Such places include central or terminal pin of an arch, pivot of guyed mast, end of beam, pivot of jib, derrick, etc. The same is true at the end of any object, such as a leaf, insect antenna, pylon, cantilever, and so on. Click here for a set of pictures that illustrates this.

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