Lecture 1: STRESS IN 3D TABLE OF CONTENTS. Page

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1 1 Stress in 3D 1 1

2 Lecture 1: STRESS IN 3D TABLE OF CONTENTS Page 1.1 Introduction Mechanical Stress: Concept Mechanical Stress: Definition What Does Stress Measure? Cutting a Bod Orienting the Cut Plane Internal Forces on Elemental Area Projecting the Internal Force Resultant Defining Three Stress Components Si More Components Visualiation on Stress Cube What Happens on the Negative Faces? Notational Conventions Sign and Subscripting Conventions Matri Representation of Stress Shear Stress Reciprocit Simplifications: 2D and 1D Stress States Advanced Topics Changing Coordinate Aes A Word on Tensors Addendum: Action vs. Reaction Reminder

3 1.3 MECHANICAL STRESS: DEFINITION 1.1. Introduction This lecture introduces mechanical stresses in a 3D solid bod. It covers definitions, notational and sign conventions, stress visualiation, and reduction to two- and one-dimensional cases. Effect of coordinate aes transformation and the notion of tensors are briefl mentioned as advanced topics Mechanical Stress: Concept Mechanical stress in a solid bod is a gross abstraction of the intensit of interatomic or intermolecular forces. If we look at a solid under increasing magnification, sa through an electron microscope, we will see complicated features such as crstals, molecules and atoms appearing (and fading) at different scales. The detailed description of particle-interaction forces acting at such tin scales is not onl impractical but unnecessar for structural design and analsis. To make the idea tractable the bod is viewed as a continuum of points in the mathematical sense, and a stress state is defined at each point b a force-over-area limit process. Mechanical stress in a solid generalies the simpler concept of pressure in a fluid. A fluid in static equilibrium (the so-called hdrostatic equilibrium in the case of a liquid) can support onl a pressure state. (In a gas pressure ma be tensile or compressive whereas in a liquid it must be compressive.) A solid bod in static equilibrium can support a more general state of stress, which includes both normal and shear components. This generaliation is of major interest to structural engineers because structures, for obvious reasons, are fabricated with solid materials Mechanical Stress: Definition This section goes over the process b which the stress state at an arbitrar point of a solid bod is defined. The ke ideas are based on Newton s third law of motion. Once the phsical scenario is properl set up, mathematics takes over What Does Stress Measure? Mechanical stress measures the average intensit level of internal forces in a material (solid or fluid bod) idealied as a mathematical continuum. The phsical dimension of stress is Force per unit area, e.g. N/mm 2 (MPa) or lbs/sq-in (psi) (1.1) This measure is convenient to assess the resistance of a material to permanent deformation (ield, creep, slip) as well as rupture (fracture, cracking). Comparing working and failure stress levels allows engineers to establish strength safet factors for structures. This topic (safet factor) is further covered in Lecture 2. Stresses ma var from point to point in a bod. In the following we consider a arbitrar solid bod (which ma be a structure) in a three dimensional (3D) setting. As previoul noted, this is actuall its idealiation as a mathematical continuum. Thus the term bod actuall refers to that idealiation. 1 3

4 Lecture 1: STRESS IN 3D Applied loads cut b plane ABCD and discard blue portion Applied loads Applied loads Keep B A D ;; ; Cut plane C Discard Cutting a Bod Figure 1.1. Cutting a 3D bod through a plane passing b point. The 3D solid bod depicted on the left of Figure 1.1 is asumed to be in static equilibrium under the applied loads acting on it. We want to find the state of stress at an arbitrar point. This point will generall be located inside the bod, but it could also lie on its surface. Cut the bod b a plane ABCD that passes through (How to orient the plane is discussed in the net subsection.) The bod is divided into two. Retain one portion (red in figure) and discard the other (blue in figure), as shown on the right of Figure 1.1. To restore equilibrium, however, we must replace the discarded portion b the internal forces it had eerted on the kept portion. This is a consequence of Newton s third law: action and reaction. If ou are rust on that topic please skim the Addendum at the end of this Lecture. A B Unit vector n is the eterior normal at, meaning it points outward from the kept bod. Reference frame is a RCC sstem of aes Applied loads Kept bod D ; Cut plane C n // In the figure n has been chosen to be parallel to Figure 1.2. Orienting the cut plane ABCD b its eterior unit normal vector n Orienting the Cut Plane The cut plane ABCD is oriented b its unit normal direction vector n, or normal for short. See Figure 1.2. B convention we will draw n as emerging from and pointing outward from the kept bod, as shown in Figure 1.2. This direction identifies the so-called eterior normal. 1 4

5 1.3 MECHANICAL STRESS: DEFINITION At this point we refer the bod to a Rectangular Cartesian Coordinate (RCC) sstem of aes {,, }. This reference frame obes the right-hand orientation rule. The coordinates of point, denoted b {,, }, are called its position coordinates or position components. The position vector of is denoted b [ ] = (1.2) but we will not need to use this vector here. With respect to {,, } the normal vector has components [ ] n n = n n (1.3) in which {n, n, n } are the direction cosines of n with respect to {,, }, respectivel. Since n is a unit vector, those components must verif the unit length condition n 2 + n2 + n2 = 1. (1.4) In Figure 1.2, the cut plane ABCD has been chosen with its eterior normal parallel to the + ais. Consequentl (1.3) reduces to [ 1 n = 0. (1.5) Internal Forces on Elemental Area Recall that the action of the discarded (blue) portion of the bod on the kept (red) portion is replaced b a sstem of internal forces that restores static equilibrium. This replacement is illustrated on the right of Figure 1.3. Those internal forces generall will form a sstem of distributed forces per unit of area, which, being vectors, generall will var in magnitude and direction as we move from point to point of the cut plane, as pictured over there. Net we focus our attention on point. Pick an elemental area A around that lies on the cut plane. Call F the resultant of the internal forces that act on A. Draw that vector with origin at, as pictured on the right of Figure 1.3. Do not forget to draw also the unit normal vector n. The use of the increment smbol suggests a pass to the limit. And indeed this will be done in equations (1.6) below, to define three stress components at Projecting the Internal Force Resultant Zoom now on the elemental area about, omitting both the kept-bod and applied loads for clarit, as pictured on the left of Figure 1.4. Project the internal force resultant F on the reference aes {,, }. This produces three components: F, F and F, as shown on the right of Figure 1.4. Component F is aligned with the cut-plane normal, because n has been taken to be parallel to. This is called the normal internal force component or simpl normal force. On the other hand, components F and F lie on the cut plane. These are called tangential internal force components or simpl tangential forces. 1 5 ]

6 Lecture 1: STRESS IN 3D Applied loads ; ; Internal forces Applied loads ;; ; A Arrows are placed over and n to remind ou that the are vectors n Figure 1.3. Internal force resultant F over elemental area about point. A n Project internal force resultant on aes {,, } to get its normal and tangential components A n // Figure 1.4. Projecting the internal force resultant onto normal and tangential components Defining Three Stress Components We define the three -stress components at point b taking the limits of internal-force-overelemental-area ratios as that area shrinks to ero: σ def F = lim A 0 A normal stress component τ def = lim A 0 F A shear stress component (1.6) τ def = lim A 0 F A shear stress component σ is called a normal stress, whereas τ and τ are shear stresses. See Figure 1.5. Remark 1.1. We tacitl assumed that the limits (1.6) eist and are finite. This assumption is part of the aioms of continuum mechanics. Remark 1.2. The use of two different letters: σ and τ for normal and shear stresses, respectivel, is traditional in American undergraduate education. It follows the influential tetbooks b Timoshenko (a ke contributor to the development of Enginering Mechanics education in the US) that appeared in the s. The main reason for carring along two smbols instead of one was to emphasie their distinct phsical effects on structural materials. It has the disadvantage, however, of poor fit with the tensorial formulation used in more advanced (graduate level) courses in continuum mechanics. In those courses a more unified notation, such as σ ij or τ ij for all stress components, is used. 1 6

7 1.3 MECHANICAL STRESS: DEFINITION A n // (reproduced from previous figure for convenience) F F σ def lim τ def = = lim τ def = A 0 A A 0 A F lim A 0 A Figure 1.5. Defining stress components σ, τ and τ as force-over-area mathematical limits Si More Components Are we done? No. It turns out we need nine stress components in 3D to full characterie the stress state at a point. So far we got onl three. Si more are obtained b repeating the same procedure: find internal force resultant, project onto aes, divide b elemental area and take-the-limit, using two more cut planes. The obvious choice is to pick planes normal to the other two aes: and. Taking n parallel to and going through the motions we get three more components, one normal and two shear: σ, τ, τ. (1.7) These are called the -stress components. Finall, taking n parallel to we get three more components, one normal and two shear: σ, τ, τ. (1.8) These are called the -stress components. On grouping (1.6), (1.7) and (1.8) we arrive at a total of nine components, as required for full characteriation of the stress at a point. Now we are done Visualiation on Stress Cube The foregoing nine stress components ma be convenientl visualied on a stress cube as follows. Cut an infinitesimal cube about with sides parallel to the RCC aes {,, }, and dimensioned d, d and d, respectivel. Draw the components on the positive cube faces ( positive face is defined below) as depicted in Figure 1.6 The three positive cube faces are those with eterior (outward) normals aligned with +, + and +, respectivel. Positive (+) values for stress components on those faces are as drawn in Figure 1.6. More on sign conventions later What Happens on the Negative Faces? The stress cube has three positive (+) faces. The three opposite ones are negative ( ) faces. Outward normals at faces point along, and, respectivel. What do stresses on those faces look like? To maintain static equilibrium, stress components must be reversed. 1 7

8 Lecture 1: STRESS IN 3D Point (not pictured) is inside cube, at its center + face σ d τ τ d σ + face τ τ τ + face τ σ d Note that stresses are forces per unit area, not forces, although the look like forces in the picture. Strictl speaking, this is a "cube" onl if d=d=d, else it should be called a parallelepided; but that is difficult to pronounce. (The correct math term is ``rectangular prism'') Figure 1.6. Visualiing stresses on faces of stress cube aligned with reference frame aes. Stress components are positive as drawn. For eample, a positive σ points along the + direction on the + face, but along on the opposite face. A positive τ points along + on the + face but along on the face. To better visualie stress reversal, it is convenient to project the stress cube onto the {, } plane b looking at it from the + direction. The resulting 2D diagram, shown in Figure 1.7, clearl displas the rule given above. σ σ τ τ τ τ τ τ project onto {,} looking along - σ face σ + face τ τ σ τ τ σ + face σ face Figure 1.7. Projecting onto {, } to displa component stress reversals on going from + to faces Notational Conventions Sign and Subscripting Conventions Stress component sign conventions are as follows. For a normal stress: positive (negative) if it produces tension (compression) in the material. For a shear stress: positive if, when acting on the + face identified b the first inde, it points in the + direction identified b the second inde. Eample: τ is + if on the + face it points in the + direction. These conventions are illustrated for σ and τ on the left of Figure

9 1.4 NOTATIONAL CONVENTIONS τ + face σ Both σ and τ are + as drawn above stress acts on cut plane with n along + (the + face) τ stress points in the direction Figure 1.8. Sign and subscripting conventions for stress components. Shear stress components have two different subscript indices. The first one identifies the cut plane on which it acts as defined b the unit eterior normal to that plane. The second inde identifies component direction. That convention is illustrated for the shear stress component τ on the right of Figure 1.8. Remark 1.3. The foregoing subscripting convention plainl applies also to normal stresses, in which case the direction of the cut plane and the direction of the component merge. Because of this coalescence some authors (for instance, Beer, Johnston and DeWolf in their Mechanics of Materials book) drop one of the subscripts and denote σ, sa, simpl b σ. Remark 1.4. The sign of a normal component is phsicall meaningful since some structural materials, for eample concrete, respond differentl to tension and compression. On the other hand, the sign of a shear stress has no phsical meaning; it is entirel conventional Matri Representation of Stress The nine components of stress referred to the,, aes ma be arranged as a 3 3 matri, which is configured as [ σ τ ] τ τ σ τ (1.9) τ τ σ Note that normal stresses are placed in the diagonal of this square matri. We will call this a 3D stress matri, although in more advanced courses this is the representation of a second-order tensor called as ma be epected the stress tensor Shear Stress Reciprocit From moment equilibrium conditions on the infinitesimal stress cube it ma be shown that τ = τ, τ = τ, τ = τ, (1.10) in magnitude. In other words: switching shear stress indices does not change its value. Note, however, that inde-switched shear stresses point in different directions: τ, sa, points along whereas τ points along. For a proof of (1.10) see, for eample, pp of Beer-Johnston-DeWolf 5th ed. 1 9

10 Lecture 1: STRESS IN 3D Propert (1.10) is known as shear stress reciprocit. It follows that the stress matri is smmetric: [ ] [ ] σ τ τ σ τ τ τ = τ σ τ σ τ (1.11) τ = τ τ = τ σ smm σ Consequentl the 3D stress state depends on onl si (6) independent components: three normal stresses and three shear stresses Simplifications: 2D and 1D Stress States For certain structural configurations such as thin plates, all stress components with a subscript ma be considered negligible, and set to ero. The stress matri of (1.9) becomes [ σ τ ] 0 τ σ 0 (1.12) This two-dimensional simplification is called a plane stress state. Since τ = τ, plane stress is full characteried b just three independent stress components: the two normal stresses σ and σ, and the shear stress τ. A further simplification occurs in structures such as bars or beams, in which all stress components ecept σ ma be considered negligible and set to ero, whence the stress matri reduces to [ σ 0 ] (1.13) This is called a one-dimensional stress state. There is onl one independent stress component left: the normal stress σ Advanced Topics We mention a couple of advanced topics that either fall outside the scope of the course, or will be later covered for special cases Changing Coordinate Aes Suppose we change aes {,, } to another set {,, } that also forms a RCC sstem. The stress cube centered at is rotated to realign with {,, } as illustrated in Figure 1.9. The stress components change accordingl, as compactl shown in matri form: [ ] [ σ τ τ σ τ τ ] τ σ τ τ τ σ becomes τ σ τ τ τ σ (1.14) Can the primed components be epressed in terms of the original ones? The answer is es. All primed stress components can be epressed in terms of the unprimed ones and of the direction cosines of {,, } with respect to {,, }. This operation is called a stress transformation. 1 10

11 1.6 ADVANCED TOPICS ' ' ' σ τ d τ τ d σ τ τ τ d σ Note that is inside both cubes; the are drawn offset for clarit σ' τ' τ' d' σ' ' τ' ' τ' τ' d' τ' ' d' σ' Figure 1.9. Transforming stress components in 3D when reference aes change. For a general 3D state this operation is complicated because there are three direction cosines. In this introductor course we will cover onl transformations for the 2D plane stress state. The transformations are simpler (and more eplicit) since changing aes in 2D depends on onl one direction cosine or, equivalentl, the rotation angle about the ais. Wh bother to look at stress transformations? One important reason: material failure ma depend on the maimum normal tensile stress (for brittle materials) or the maimum absolute shear stress (for ductile materials). To find those we generall have to look at parametric rotations of the coordinate sstem, as in the skew-cut bar eample studied in the first Recitation. Once such dangerous stress maima are found for critical points of a given structure, the engineer can determine strength safet factors A Word on Tensors The state of stress at a point is not a scalar or a vector. It is a more complicated mathematical object, called a tensor (more precisel, a second-order tensor). Tensors are not covered in undergraduate engineering courses as mathematical entities. Accordingl we will deal with stresses (and strains, which are also tensors) using a phsical approach complemented with recipes. Nonetheless for those of ou interested in the deeper mathematical aspect before reaching graduate school, here is a short list that etrapolates tensors from two entities ou alread (should have) encountered in Calculus and Phsics courses. Scalars are defined b magnitude. Eamples: temperature, pressure, densit, charge. Vectors are defined b magnitude and direction. Eamples: force, displacement, velocit, acceleration, electric current. Second-order tensors are defined b magnitude and two directions. Eamples: stress, strain, electromagnetic field strength, space curvature in GRT. For mechanical stress, the two directions are: the orientation of the cut plane (as defined b its outward normal) and the orientation of the internal force component. All three kinds of objects (scalars, vectors, and second-order tensors) ma var from point to point. This means that the are epressed as functions of the position coordinates. In mathematical phsics such functions are called fields. 1 11

12 Lecture 1: STRESS IN 3D Remove the "bridge" (log) and replace its effect on the elephant b reaction forces on the legs. The elephant stas happ: nothing happens. ;; ;; ;;;;;; Strictl speaking, reaction forces are distributed over the elephant leg contact areas. The are replaced above b equivalent point forces, a.k.a. resultants, for visualiation convenience Figure How to keep a bridge-crossing elephant happ Addendum: Action vs. Reaction Reminder This is a digression about Newton s Third Law of motion, which ou should have encountered in Phsics 1. See cartoon in Figure 1.10 for a reminder of how to replace the effect of a removed phsical bod b that of its reaction forces on the kept bod. 1 12