4 Stress We have talked about internal forces, distributed them uniformly over an area and they became a normal stress acting perpendicular to some internal surface at a point, or a shear stress acting tangentially, in plane, at the point. Up to now, the choice of planes upon which these stress components act, their orientation within a solid, was dictated by the geometry of the solid and the nature of the loading. We have said nothing about how these stress components might change if we looked at a set of planes of another orientation at the point. And up to now, we have said little about how these normal and shear stresses might vary with posi-tion throughout a solid.1 Now we consider a more general situation, namely an arbitrarily shaped solid which may be subjected to all sorts of externally applied loads - distributed or concentrated forces and moments. We are going to lift our gaze up from the world of crude structural elements such as truss bars in tension, shafts in torsion, or beams in bending to view these “solids” from a more abstract perspective. They all become special cases of a more general stuff we call a solid continuum. Likewise, we develop a more general and more abstract representation of inter-nal forces, moving beyond the notions of shear force, internal torque, uni-axial tension or compression and internal bending moment. Indeed, we have already done so in our representation of the internal force in a truss element as a normal stress, in our representation of torque in a thin-walled, circular shaft as the result-ant of a uniformly distributed shear stress, in our representation of internal forces in a cylinder under internal pressure as a hoop stress (and as an axial stress). We want to develop our vocabulary and vision in order to speak intelligently about stress in its most general form. We address two questions: • How do the normal and shear components of stress acting on a plane at a given point change as we change the orientation of the plane at the point. • How might stresses vary from one point to another throughout a contin-uum; The first bullet concerns the transformation of components of stress at a point; the second introduces the notion of stress field. We take them in turn. 1. The beam is the one exception. There we explored how different normal stress distributions over a rectangu-lar cross-section could be equivalent to a bending moment and zero resultant force.106 Chapter 4 4.1 Stress: The Creature and its Components We first address what we need to know to fully define “stress at a point” in a solid continuum. We will see that the stress at a point in a solid continuum is defined by its scalar components. Just as a vector quantity, say the velocity of a projectile, is defined by its three scalar components, we will see that the stress at a point in a solid continuum is defined by its nine scalar components. Now you are probably quite familiar with vector quantities - quantities that have three scalar components. But you probably have not encountered a quantity like stress that require more than three scalar values to fix its value at a point. This is a new kind of animal in our menagerie of variables; think of it as a new species, a new creature in our zoo. But don’t let the number nine trouble you. It will lead to some algebraic complexity, compared to what we know how to do with vectors, but we will find that stress, a second order tensor, behaves as well as any vector we are familiar with. The figure below is meant to illustrate the more general, indeed, the most gen-eral state of stress at a point. It requires some explanation: y z σx σz σxz σzy σzx σyxyxP σy σyz x y z x y z x y z σxy The odd looking structural element, fixed to the ground at bottom and to the left, and carrying what appears to be a uniformly distributed load over a portion of its bottom and a concentrated load on its top, is meant to symbolize an arbitrarily loaded, arbitrarily constrained, arbitrarily shaped solid continuum. It could be a beam, a truss, a thin-walled cylinder though it looks more like a potato — which too is a solid continuum. At any arbitrarily chosen point inside this object we can ask about the value of the stress at the point, say the point P. But what do we mean by “value”; value of what at that point? Think about the same question applied to a vector quantity: What do we mean when we say we know the value of the velocity of a projectile at a point in its tra-jectory? We mean we know its magnitude - its “speed” a scalar - and its direction. Direction is fully specified if we know two more scalar quantities, e.g., the direc-tion the vector makes with respect to the axes as measured by the cosine of the angles it makes with each axis. More simply, we have fully defined the velocity atStress 107 a point if we specify its three scalar components with respect to some reference coordinate frame - say its x, y and z components. Now how do we know this fully defines the vector quantity? We take as our criterion that anyone, anyone in the world (of mathematical physicists and engi-neers), would agree that they have in hand the same thing, no matter what coordi-nate frame they favor, no matter how they viewed the motion of the projectile. (We do insist that they are not displacing or rotating relative to one another, i.e., they all reside in the same inertial frame). This is assured if, after transforming the scalar components defined with respect to one reference frame to another, we obtain values for the components any other observer sees. It is then the equations which transform the values of the components of the vector from one frame to another which define what a vector is. This is like defin-ing a thing by what it does, e.g., “you are what you eat”, a behaviorists perspec-tive - which is really all that matters in mathematical physics and in engineering. Reid: Hey Katie: what do you think of all this talk about components? Isn’t he going off the deep end here? Katie: What do you mean, “...off the deep end”? Reid: I mean why don’t we stick with the stuff we were doing about beams and trusses? I mean that is the useful stuff. This general, abstract continuum business does nothing for me. Katie: There must be a reason, Reid, why he is