Yetanotherwriter
Introduction edit
I am an Aeronautical Engineer, interested in topics like Newton's Laws of Motion, Strength of Materials, Continuum Mechanics, Aerodynamics and similar topics. I intend to use this page as a blog to share my thoughts on some of these topics. Thoughts related to these topics, and new ones are welcome, and I'd be delighted to take part in a healthy discussion to learn and understand concepts better. And, yes, constructive criticism is always welcome.
Forces Acting on Sections of a Non-Equilibrium Member edit
Anyone familiar with basics of strength of materials will be able to draw free body diagrams of sections of a structural member subjected to loads, but in equilibrium. Equilibrium, I'd say, is one of the 'taken for granted' assumptions of Structural Mechanics subjects and one hardly wonders why such an assumption is even considered. Is it not possible to think of stresses and deformation in accelerating bodies? In fact, are we not using structural mechanics concepts to design members which accelerate- an airplane, for example?
Yes, we are familiar with problems where we talk of stresses and strains in a rotating member, but we acknowledge rotation only to include a partially understood 'centrifugal force' as a force which varies along the length of the member and acts as a tensile load. But I believe that (please correct me if I am wrong) few venture to think further than that, and I feel that a rigorous discussion on the 'internal' forces acting on a (let's just take a linearly elastic) deformable member will reveal that the assumption of equilibrium simplifies several things- to our disadvantage.
Sections of a member in Equilibrium edit
For the sake of continuity, I'd like to start this discussion with a simple sectioned view of a prismatic bar subjected to axial tension. To keep things simple, let's consider simple sections whose cross sections are identical to that of the bar itself, as shown in the figure.
Initially, let's consider this bar as a single entity. We have two forces, equal and opposite in nature being applied on it, and we may boldly state that the member is in equilibrium. The energy spent in applying the forces - well, at least some of it, is spent in the deformation that the member undergoes. In spite of all this, the member, and hence, every section of the member is in equilibrium (considering the member as a system)
Now, let us look at sections of the member (standard assumptions of the material being linearly elastic, its properties being homogeneous and isotropic, and loads being applied within elastic limits are made). Section A has an external force of P units acting on it. But from the argument made above, we know that Section A is in equilibrium. This means that at least a single force (in this case, that can be the only way) is acting on it, such that the net external force and moment acting on Section A is zero. Bluntly speaking, the only section that can apply such a force is Section B. We may boldly state that Section B applies a force whose magnitude equals that of P in a direction opposite to it, along the same line as the applied force (so that the net moment is zero). It is to be noted that the force that B applies is not because of Newton's 3rd Law. It is not a Reaction (physics) to the applied force as Newton's Law defines it, but is simply an explanation of how Section A is in equilibrium.
Now that the free body diagram of Section A has been completed, we can move on to Section B. It is here that Newton's 3rd Law comes into play. Since Section B applies a force of magnitude P in the direction of FrA, according to Newton's 3rd Law, an "equal and opposite" force is applied by Section A on Section B. SO, FlB acts on Section B and has a magnitude of P. The argument continues with Section B, and we mark FrB on Section B, claiming that it is applied by Section C on Section B, and has a magnitude equal to P. According to Newton's 3rd Law again, Section B applies a force of magnitude P in the direction of FlC as marked in the figure. Now, Section C has two forces, each of magnitude P and acting in opposite directions, hence keeping it in equilibrium.
Splitting the member into several sections in this manner gives similar results, with "internal" forces acting on every section, and this gives rise to the concept of (mechanical) stress. However, it must be remembered that the generation of these "internal" forces is not a "reaction" (there's nothing wrong in calling these forces as reactions as long as one doesn't confuse it with the reaction force defined by Newton's 3rd Law) as defined by Newton's 3rd Law, but is a consequence of the interactions between different sections of a body. The propagation of these internal forces, however, may be loosely stated as a consequence of Newton's 3rd Law of Motion.
Sections of a member not in Equilibrium edit
Now let's repeat the exercise for a body with an unbalanced set of forces. For the sake of simplicity again, let's consider only forces and no moments, and all other assumptions (except equilibrium of course) which had been made for the previous discussion. Subscript text To look at both tensile and compressive effects in a member due to a unidirectional force, let us apply our force in the centre of a prismatic member as shown. Similar to the previous case, let us start drawing free body diagrams starting from the section, say, on the extreme left. We know that the entire member is accelerating at a constant value of acceleration (denoted by 'a'), whose magnitude is equal to the applied force divided by the mass of the member. Further, since the member is accelerating at this value of acceleration, each section of the member is accelerating at the same rate, namely, 'a'. To "feed" this physical concept into the mathematics, we apply the same value of acceleration. i.e, 'a', to every section of the member.
Now, for the section under consideration (let us call it mn) to accelerate at 'a', a force equal to the product of the mass of the section and its acceleration, 'a', must act upon it. The only object that can apply this force on the section is the section to the immediate left of the section considered, i.e, the section with mass mn-1. This force is marked as mna in the figure, and acts towards the right side.
Now we move on to the section with mass mn-1. A reaction force equal in magnitude to the mna on mass mn, but opposite in direction is applied by the section with mass mn on that with mass mn-1, according to Newton's 3rd Law of Motion. However, we know that this section (whose mass is mn-1) accelerates at 'a' towards the right side, as per the figure. This must mean that mn-2 exerts a force of magnitude equal to mn-1a + mna towards the right side, on mn-1.
Similar arguments result in cumulative tensile forces on sections on the left side of the member, as we proceed towards the centre of the specimen. Since forces act in opposite on both sides of each section, there is compression, and hence tensile stress in addition to acceleration in each section on the left side.
(Note:The 'end' segment is found to have only a single force acting on it, but stresses on this section may be identified by splitting it into further sub-sections. It may be crudely stated that, the element of limiting thickness on the ends of this member undergo pure acceleration without deformation)
Through similar arguments, we find that sections on the right side are subjected to tensile forces of similar magnitudes. The central member (assuming that we have considered an odd number of sections) consists of the applied force and the cumulative forces "communicated" from sections on its either side, and these are distributed in such a way that this section also accelerates at 'a', immaterial of whether the other sections considered are symmetrical (in shape or number) or not. Thus, it may be broadly stated that the force application point is subjected to maximum stress in such a situation.
Observation edit
From the above discussion, it is clearly seen that, even in a simple tension/single force problem, the stress distributions in bodies in equilibrium and those that are not in equilibrium vary drastically, and can be expected to vary much more in case of complex force applications, and other material properties/interactions. However, equilibrium is a 'taken for granted' assumption even while designing members which are meant to accelerate.
It is true that the equilibrium equation is a prerequisite for solving equations in elasticity. But to what extent are such results reliable? How is it that our designs still work?
It has been my experience that digging deep into concepts leads to frustration at some stage or the other, and the relative delay in getting frustrated defines the "conceptual stamina" of the person doing it.
Yetanotherwriter (talk) 10:44, 30 April 2014 (UTC)