3 Stress internal forces stress stress components Stress analysis stress transformation equations principal stresses stress invariants

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1 3 Stress orces acting at the surfaces of components were considered in the previous chapter. The task now is to eamine forces arising inside materials, internal forces. Internal forces are described using the idea of stress. There is a lot more to stress than the notion of force over area, as will become clear in this chapter. irst, the idea of surface (contact) stress distributions will be eamined, together with their relationship to resultant forces and moments. Then internal stress and traction will be discussed. The means b which internal forces are described is through the stress components, for eample z,, and this language of sigmas and subscripts needs to be mastered in order to model sensibl the internal forces in real materials. Stress analsis involves representing the actual internal forces in a real phsical component mathematicall. Some of the limitations of this are discussed in Newton s laws are used to derive the stress transformation equations, and these are then used to derive epressions for the principal stresses, stress invariants, principal directions and maimum shear stresses acting at a material particle. The practical case of two dimensional plane stress is discussed. 9

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3 Section Surface and Contact Stress The concept of the force is fundamental to mechanics and man important problems can be cast in terms of forces onl, for eample the problems considered in Chapter. However, more sophisticated problems require that the action of forces be described in terms of stress, that is, force divided b area. or eample, if one hangs an object from a rope, it is not the weight of the object which determines whether the rope will break, but the weight divided b the cross-sectional area of the rope, a fact noted b Galileo in Stress Distributions As an introduction to the idea of stress, consider the situation shown in ig. 3..a: a block of mass m and cross sectional area A sits on a bench. ollowing the methodolog of Chapter, an analsis of a free-bod of the block shows that a force equal to the weight mg acts upward on the block, ig. 3..b. Allowing for more detail now, this force will actuall be distributed over the surface of the block, as indicated in ig. 3..c. Defining the stress to be force divided b area, the stress acting on the block is mg (3..) A The unit of stress is the Pascal (Pa): Pa is equivalent to a force of Newton acting over an area of metre squared. Tpical units used in engineering applications are the kilopascal, kpa ( 0 3 Pa ), the megapascal, MPa ( 0 6 Pa ) and the gigapascal, GPa ( 0 9 Pa). mg (a) mg (b) (c) igure 3..: a block resting on a bench; (a) weight of the block, (b) reaction of the bench on the block, (c) stress distribution acting on the block The stress distribution of ig. 3..c acts on the block. B Newton s third law, an equal and opposite stress distribution is eerted b the block on the bench; one sas that the weight force of the block is transmitted to the underling bench. The stress distribution of ig. 3.. is uniform, i.e. constant everwhere over the surface. In more comple and interesting situations in which materials contact, one is more likel to obtain a non-uniform distribution of stress. or eample, consider the case of a metal ball being pushed into a similarl stiff object b a force, as 3

4 Section 3. illustrated in ig Again, an equal force acts on the underside of the ball, ig. 3..b. As with the block, the force will actuall be distributed over a contact region. It will be shown in Part II that the ball (and the large object) will deform and a circular contact region will arise where the ball and object meet, and that the stress is largest at the centre of the contact surface, ding awa to zero at the edges of contact, ig. 3..c ( in ig. 3..c). In this case, we can consider a small area of the contact region da, ig. 3..d; the force on this region is da. The total force is (3..) d da A A The stress varies from point to point over the surface but the sum (or integral) of the stresses (times areas) equals the total force applied to the ball. contact region Small region da (a) (b) (c) (d) igure 3..: a ball being forced into a large object, (a) force applied to ball, (b) reaction of object on ball, (c) a non-uniform stress distribution over the contacting surface, (d) the stress acting on a small (infinitesimal) area A given stress distribution gives rise to a resultant force, which is obtained b integration, Eqn It will also give rise to a resultant moment. This is eamined in the following eample. Eample Consider the surface shown in ig. 3..3, of length m and depth m (into the page). The stress over the surface is given b kpa, with measured in m from the lefthand side of the surface. The force acting on an element of length d at position is (see ig. 3..3b) kpa m m d da d The resultant force is then, from Eqn. 3.. the weight of the ball is neglected here the radius of which depends on the force applied and the materials in contact 3

5 Section 3. A d 0 d kpa m 4 kn The moment of the stress distribution is given b M 0 dm l da (3..3) where l is the length of the moment-arm from the chosen ais. Taking the ais to be at 0, the moment-arm is l, ig. 3..3b, and M A A 3 6 kpa m kn m 0 dm d A 0 Taking moments about the right-hand end,, one has M 3 d kpa m kn m dm A () m (a) d (b) () igure 3..3: a non-uniform stress acting over a surface; (a) the stress distribution, (b) stress acting on an element of size d 3.. Equivalent orces and Moments Sometimes it is useful to replace a stress distribution with an equivalent force, i.e. a force equal to the resultant force of the distribution and one which also gives the same moment about an ais as the distribution. ormulae for equivalent forces are derived in what follows for triangular and arbitrar linear stress distributions. Triangular Stress Distribution 33

6 Section 3. Consider the triangular stress distribution shown in ig The stress at the end is, the length of the distribution is L and the thickness into the page is t. With 0 ( ) / L, the equivalent force is, from Eqn. 3.., 0 L t 0 d 0Lt (3..4) L 0 which is just the average stress times the area of the triangle. The point of action of this force should be such that the moment of the force is equivalent to the moment of the stress distribution. Taking moments about the left hand end, for the distribution one has, from 3..3, M o L t ( ) d 0L t 3 0 Placing the force at position c, ig. 3..4, the moment of the force is M o 0 Lt / c. Equating these epressions leads to the position at which the equivalent force acts, two-thirds the wa along the triangle: c 3 L. (3..5) equivalent force () 0 o c L igure 3..4: triangular stress distribution and equivalent force Note that the moment about an ais is now the same for both the stress distribution and the equivalent force. Arbitrar Linear Stress Distribution Consider the linear stress distribution shown in ig The stress at the ends are and and this time the equivalent force is L t 0 )( / L) d Lt / ( (3..6) Taking moments about the left hand end, for the distribution one has 34

7 Section 3. M o L / 6 t ( ) d L t 0. The moment of the force is M Lt / to o L Eqn follows from 3..7 b setting 0. c. Equating these epressions leads c (3..7) 3 equivalent force () o c L igure 3..5: a non-uniform stress distribution and equivalent force The Centroid Generalising the above cases, the line of action of the equivalent force for an arbitrar stress distribution () is c t ( ) d t ( ) d d Centroid (3..8) This location is known as the centroid of the distribution. Note that most of the discussion above is for two-dimensional cases, i.e. the stress is assumed constant into the page. Three dimensional problems can be tackled in the same wa, onl now one must integrate two-dimensionall over a surface rather than one-dimensionall over a line. Also, the forces considered thus far are normal forces, where the force acts perpendicular to a surface, and the give rise to normal stresses. Normal stresses are also called pressures when the are compressive as in igs

8 Section Shear Stress Consider now the case of shear forces, that is, forces which act tangentiall to surfaces. A normal force acts on the block of ig. 3..6a. The block does not move and, to maintain equilibrium, the force is resisted b a friction force mg, where is the coefficient of friction. A free bod diagram of the block is shown in ig. 3..6b. Assuming a uniform distribution of stress, the stress and resultant force arising on the surfaces of the block and underling object are as shown. The stresses are in this case called shear stresses. (a) (b) igure 3..6: shear stress; (a) a force acting on a block, (b) shear stresses arising on the contacting surfaces 3..4 Combined Normal and Shear Stress orces acting inclined to a surface are most convenientl described b decomposing the force into components normal and tangential to the surface. Then one has both normal stress N and shear stress S, as in ig N S igure 3..7: a force giving rise to normal and shear stress over the contacting surfaces The stresses considered in this section are eamples of surface stresses or contact stresses. The arise when materials meet at a common surface. Other eamples would be sea-water pressurising a gas clinder in deep water and the stress eerted b a train wheel on a train track. 36

9 Section Problems. Consider the surface shown below, of length 4cm and unit depth (cm into the page). The stress over the surface is given b kpa, with measured in cm from the surface centre. (a) Evaluate the resultant force acting on the surface (in Newtons). (b) What is the moment about an ais (into the page) through the left-hand end of the surface? (c) What is the moment about an ais (into the page) through the centre of the surface? () 4. Consider the surface shown below, of length 4mm and unit depth (mm into the page). The stress over the surface is given b MPa, with measured from the surface centre. What is the total force acting on the surface, and the moment acting about the centre of the surface? 4 3. ind the reaction forces (per unit length) at the pin and roller for the following beam, which is subjected to a varing pressure distribution, the maimum pressure being ( ) 0kPa (all lengths are in cm give answer in N/m) [Hint: first replace the stress distribution with three equivalent forces]

10 Section A block of material of width 0cm and length m is pushed into an underling substrate b a normal force of 00 N. It is found that a uniform triangular normal stress distribution arises at the contacting surfaces, that is, the stress is maimum at the centre and dies off linearl to zero at the block edges, as sketched below right. What is the maimum pressure acting on the surface? 00 N tpical crosssection m 0cm stress distribution 38

11 Section Bod orces Surface forces act on surfaces. As discussed in the previous section, these are the forces which arise when bodies are in contact and which give rise to stress distributions. Surface forces also arise inside materials, acting on internal surfaces, ig. 3..a, as will be discussed in the following section. To complete the description of forces acting on real materials, one needs to deal with forces which arise even when bodies are not in contact; one can think of these forces as acting at a distance, for eample the force of gravit. To describe these forces, one can define the bod force, which acts on volume elements of material. ig. 3..b shows a sketch of a volume element subjected to a magnetic bod force and a gravitational bod force g. contact force internal surface e.g. air pressure volume element g (a) (b) igure 3..: forces acting on a bod; (a) surface forces acting on surfaces, (b) bod forces acting on a material volume element 3.. Weight The most important bod force is the force due to gravit, i.e. the weight force. In Chapter there were eamples involving the weight of components. In those cases it was simpl stated that the weight could be taken to be a single force acting at the component centre (for eample, Problem 3 in..3). This is true when the component is smmetrical, for eample, in the shape of a circle or a square. However, it is not true in general for a component of arbitrar shape. In what follows, the important case of a flat object of arbitrar shape will be eamined. The weight of a small volume element the total weight is V of material of densit is d gv and 39

12 Section 3. V gdv (3..) Consider the general two-dimensional case, ig. 3.., where material elements of area A i (and constant thickness t) are subjected to forces i t gai. z c, c tg i A i A i t g igure 3..: Resultant Weight on a bod The resultant, i.e. equivalent, weight force due to all elements, for a component with uniform densit, is d tg da gta, where A is the cross-sectional area. The resultant moments about the and aes, which can be positioned anwhere in the bod, are M t g da and M t g da respectivel; the moment M is shown in ig The equivalent weight force is thus positioned at, ), ig. 3.., where ( c c c da da, c Centroid of Area (3..) A A The position, ) is called the centroid of the area. The quantities da, ( c c called the first moments of area about, respectivel, the and aes. da, are 40

13 Section 3. z A i tg A i A i o M tg A i (a) (b) igure 3..3: The moment M ; (a) full view, (b) plane view 3.. Problems. Where does the resultant force due to gravit act in the triangular component shown below? (Gravit acts downward in the direction of the arrow shown, perpendicular to the component s surface.) o 90 m m 4

14 Section Internal Stress The idea of stress considered in 3. is not difficult to conceptualise since objects interacting with other objects are encountered all around us. A more difficult concept is the idea of forces and stresses acting inside a material, within the interior where neither ee nor eperiment can reach as Euler put it. It took man great minds working for centuries on this question to arrive at the concept of stress we use toda, an idea finall brought to us b Augustin Cauch, who presented a paper on the subject to the Academ of Sciences in Paris, in 8. Augustin Cauch 3.3. Cauch s Concept of Stress Uniform Internal Stress Consider first a long slender block of material subject to equilibrating forces at its ends, ig. 3.3.a. If the complete block is in equilibrium, then an sub-division of the block must be in equilibrium also. B imagining the block to be cut in two, and considering free-bod diagrams of each half, as in ig. 3.3.b, one can see that forces must be acting within the block so that each half is in equilibrium. Thus eternal loads create internal forces; internal forces represent the action of one part of a material on another part of the same material across an internal surface. We can take it that a uniform stress / A acts over this interior surface, ig. 3.3.b. imaginar cut A N S ( a) ( b) (c) igure 3.3.: a slender block of material; (a) under the action of eternal forces, (b) internal normal stress σ, (c) internal normal and shear stress 4

15 Section 3.3 Note that, if the internal forces were not acting over the internal surfaces, the two halfblocks of ig. 3.3.b would fl apart; one can thus regard the internal forces as those required to maintain material in an un-cut state. If the internal surface is at an incline, as in ig. 3.3.c, then the internal force required for equilibrium will not act normal to the surface. There will be components of the force normal and tangential to the surface, and thus both normal ( N ) and shear ( S ) stresses must arise. Thus, even though the material is subjected to a purel normal load, internal shear stresses develop. rom ig. 3.3.a, with the stress given b force divided b area, the normal and shear stresses arising on an interior surface inclined at angle to the horizontal are { Problem } N cos, S sin cos (3.3.) A A N S internal point internal surface A igure 3.3.: stress on inclined surface; (a) decomposing the force into normal and shear forces, (b) stress at an internal point Although stress is associated with surfaces, one can speak of the stress at a point. or eample, consider some point interior to the block, ig 3.3.b. The stress there evidentl depends on which surface through that point is under consideration. rom Eqn. 3.3.a, the normal stress at the point is a maimum / A when 0 and a minimum of zero o when 90. The maimum normal stress arising at a point within a material is of special significance, for eample it is this stress value which often determines whether a material will fail ( break ) there. It has a special name: the maimum principal stress. rom Eqn. 3.3.b, the maimum shear stress at the point is / A and arises on o surfaces inclined at 45. Non-Uniform Internal Stress ( a) (b) Consider a more comple geometr under a more comple loading, as in ig Again, using equilibrium arguments, there will be some stress distribution acting over an 43

16 Section 3.3 given internal surface. To evaluate these stresses is not a straightforward matter, suffice to sa here that the will invariabl be non-uniform over a surface, that is, the stress at some particle will differ from the stress at a neighbouring particle. 3 4 N S igure 3.3.3: a component subjected to a comple loading, giving rise to a nonuniform stress distribution over an internal surface Traction and the Phsical Meaning of Internal Stress All materials have a comple molecular microstructure and each molecule eerts a force on each of its neighbours. The comple interaction of countless molecular forces maintains a bod in equilibrium in its unstressed state. When the bod is disturbed and deformed into a new equilibrium position, net forces act, ig a. An imaginar plane can be drawn through the material, ig b. Unlike some of his predecessors, who attempted the etremel difficult task of accounting for all the molecular forces, Cauch discounted the molecular structure of matter and simpl replaced the imagined molecular forces acting on the plane b a single force, ig 3.3.4c. This is the force eerted b the molecules above the plane on the material below the plane and can be attractive or repulsive. Different planes can be taken through the same portion of material and, in general, a different force will act on the plane, ig 3.3.4d. ( a) ( b) ( c) (d) igure 3.3.4: a multitude of molecular forces represented b a single force; (a) molecular forces, a plane drawn through the material, replacing the molecular forces with an equivalent force, a different equivalent force acts on a different plane through the same material The definition of stress will now be made more precise. irst, define the traction at some particular point in a material as follows: take a plane of surface area S through the point, on which acts a force. Net shrink the plane as it shrinks in size both S and get smaller, and the direction in which the force acts ma change, but eventuall the ratio / S will remain constant and the force will act in a particular direction, ig The 44

17 Section 3.3 limiting value of this ratio of force over surface area is defined as the traction vector (or stress vector) t: t lim (3.3.) S0 S S a plane passing through some point in the material S S igure 3.3.5: the traction vector - the limiting value of force over area, as the surface area of the element on which the force acts is shrunk An infinite number of traction vectors act at an single point, since an infinite number of different planes pass through a point. Thus the notation lim S 0 / S is ambiguous. or this reason the plane on which the traction vector acts must be specified; this can be done b specifing the normal n to the surface on which the traction acts, ig The traction is thus a special vector associated with it is not onl the direction in which it acts but also a second direction, the normal to the plane upon which it acts. n same point with different planes passing through it (defined b different normals) n n S different forces act on different planes through the same point n S lim S S ( t n ) 0 lim S S ( t n ) 0 igure 3.3.6: two different traction vectors acting at the same point this does not mean that the force is acting on a surface of zero area the meaning of this limit is further eamined in section 5.4, in the contet of the continuum 45

18 Section 3.3 Stress Components The traction vector can be decomposed into components which act normal and parallel to the surface upon which it acts. These components are called the stress components, or simpl stresses, and are denoted b the smbol ; subscripts are added to signif the surface on which the stresses act and the directions in which the stresses act. Consider a particular traction vector acting on a surface element. Introduce a Cartesian coordinate sstem with base vectors i, j, k so that one of the base vectors is a normal to the surface, and the origin of the coordinate sstem is positioned at the point at which the traction acts. or eample, in ig , the k direction is taken to be normal to the ( plane, and t k ) t i t j t k. z (nˆ ) t z (k) t i k t j t z t igure 3.3.7: the components of the traction vector Each of these components t i is represented b ij where the first subscript denotes the direction of the normal to the plane and the second denotes the direction of the ( component. Thus, re-drawing ig as ig : t k ) zi z j zzk. The first two stresses, the components acting tangential to the surface, are shear stresses, whereas, acting normal to the plane, is a normal stress. zz z (k) t i k z j zz z igure 3.3.8: stress components the components of the traction vector this convention for the subscripts is not universall followed. Man authors, particularl in the mathematical communit, use the eact opposite convention, the first subscript to denote the direction and the second to denote the normal. It turns out that both conventions are equivalent, since, as will be shown later, ij ji 46

19 Section 3.3 The traction vector shown in igs , 3.3.8, represents the force (per unit area) eerted b the material above the surface on the material below the surface. B Newton s third law, an equal and opposite traction must be eerted b the material below the surface on (k) the material above the surface, as shown in ig (thick dotted line). If t has stress ( k) components z, z, zz, then so should t ( k) ( k) : t z( i) z( j) zz( k) t. z (k) t z zz z k i j zz ( k) t z z igure 3.3.9: equal and opposite traction vectors each with the same stress components Sign Convention for Stress Components The following convention is used: The stress is positive when the direction of the normal and the direction of the stress component are both positive or both negative The stress is negative when one of the directions is positive and the other is negative According to this convention, the three stresses in igs are all positive. Looking at the two-dimensional case for ease of visualisation, the (positive and negative) normal stresses and shear stresses on either side of a surface are as shown in ig To clarif this, consider the stress in ig. 3.30a: above the plane, the normal to the plane is in the positive direction (up) and the component acts in the positive direction (up), so this stress is positive; below the plane, the normal to the plane is in the negative direction (down) and the component acts in the negative direction (down), so this stress is positive. The simple consequence of this sign convention is that normal stresses which pull (tension) are positive and normal stresses which push (compression) are negative. Note that the shear stresses alwas go in opposite directions. 47

20 Section 3.3 ( a) (b) igure 3.3.0: stresses acting on either side of a material surface: (a) positive stresses, (b) negative stresses Eamples of negative stresses are shown in ig { Problem 4}. z e k j z i ( j) t z k j i ( j) t z t ( j) ( i j zk t j ) i j k z ( a) (b) igure 3.3.: eamples of negative stress components 3.3. Real Problems and Saint-Venant s Principle Some eamples have been given earlier of eternal forces acting on materials. In realit, an eternal force will be applied to a real material component in a comple wa. or eample, suppose that a block of material, welded to a large object at one end, is pulled at its other end b a rope attached to a metal hoop, which is itself attached to the block b a number of bolts, ig. 3.3.a. The block can be idealised as in ig 3.3.b; here, the precise details of the region in which the eternal force is applied are neglected. 48

21 Section 3.3 (a) (e) (b) (c) / ( d) A (f ) stress the same (g) stress differs here / / igure 3.3.: a block subjected to an eternal force: (a) real case, (b) ideal model, (c) stress in ideal model, (d) stress in actual material, (e) the stress in the real material, awa from the right hand end, is modelled well b either (f) or (g) According to the earlier discussion, the stress in the ideal model is as in ig. 3.3.c. One will find that, in the real material, the stress is indeed (approimatel) as predicted, but onl at an appreciable distance from the right hand end. Near where the rope is attached, the force will differ considerabl, as sketched in ig.3.3.d. Thus the ideal models of the tpe discussed in this section, and in much of this book, are useful onl in predicting the stress field in real components in regions awa from points of application of loads. This does not present too much of a problem, since the stresses internal to a structure in such regions are often of most interest. If one wants to know what happens near the bolted connection, then one will have to create a comple model incorporating all the details and the problem will be more difficult to solve. That said, it is an eperimental fact that if two different force sstems are applied to a material, but the are equivalent force sstems, as in ig. 3.3.(f,g), then the stress fields in regions awa from where the loads are applied will be the same. This is known as Saint-Venant s Principle. Tpicall, one needs to move a distance awa from where the loads are applied roughl equal to the distance over which the loads are applied. Saint-Venant s principle is etremel important in practical applications: we can replace a complicated problem b a simple model problem; the solution to this latter problem will often give us the information we require Problems. Derive Eqns

22 Section 3.3. The four sides of a square block are subjected to equal forces S, as illustrated. The length of each side is l and the block has unit depth (into the page). What normal and shear stresses act along the (dotted) diagonal? [Hint: draw a free bod diagram of the upper left hand triangle.] S S S S 3. A shaft is concreted firml into the ground. A thick steel rope is looped around the shaft and a force is applied normal to the shaft, as shown. The shaft is in static equilibrium. Draw a free bod diagram of the shaft (from the top down to ground level) showing the forces/moments acting on the shaft (including the reaction forces at the ground-level; ignore the weight of the shaft). Draw a free bod diagram of the section of shaft from the top down to the cross section at A. Draw a free bod diagram of the section of shaft from the top to the cross section at B. Roughl sketch the stresses acting over the (horizontal) internal surfaces of the shaft at A and B. B A ground 4. In ig. 3.3., which of the stress components is/are negative? 5. Label the following stress component acting on an internal material surface. Is it a positive or negative stress? z acting parallel to surface 50

23 Section Label the following shear stresses. Are the positive or negative? z 7. Label the following normal stresses. Are the positive or negative? z 8. B the definition of the traction vector t which acts on the z plane, ( j) t i j k. Sketch these three stress components on the figure below. z z 5

24 Section Equilibrium of Stress Consider two perpendicular planes passing through a point p. The stress components acting on these planes are as shown in ig. 3.4.a. These stresses are usuall shown together acting on a small material element of finite size, ig. 3.4.b. It has been seen that the stress ma var from point to point in a material but, if the element is ver small, the stresses on one side can be taken to be (more or less) equal to the stresses acting on the other side. B convention, in analses of the tpe which will follow, all stress components shown are positive. p ( a) (b) igure 3.4.: stress components acting on two perpendicular planes through a point; (a) two perpendicular surfaces at a point, (b) small material element at the point The four stresses can convenientl be written in the form of a stress matri: ij (3.4.) It will be shown below that the stress components acting on an other plane through p can be evaluated from a knowledge of onl these stress components Smmetr of the Shear Stress Consider the material element shown in ig. 3.4.b, reproduced in ig. 3.4.a below. The element has dimensions is and is subjected to uniform stresses over its sides. The resultant forces of the stresses acting on each side of the element act through the sidecentres, and are shown in ig. 3.4.b. The stresses shown are positive, but note how 5

25 Section 3.4 positive stresses can lead to negative forces, depending on the definition of the aes used. The resultant force on the complete element is seen to be zero. ( a) (b) igure 3.4.: stress components acting on a material element; (a) stresses, (b) resultant forces on each side B taking moments about an point in the block, one finds that { Problem } (3.4.) Thus the shear stresses acting on the element are all equal, and for this reason the stresses are usuall labelled, ig a, or simpl labelled, ig b. (a) (b) igure 3.4.3: shear stress acting on a material element In fact, in two-dimensional problems, the double-subscript notation is often dispensed with for simplicit, and the stress matri can be epressed as 53

26 Section 3.4, (3.4.3) to go along with the representation shown in ig igure 3.4.4: a simpler notation for D stress components (without the double subscripts) 3.4. Three Dimensional Stress The three-dimensional counterpart to the two-dimensional element of ig is shown in ig Again, all stresses shown are positive. z z z z z zz igure 3.4.5: a three dimensional material element Moment equilibrium in this case requires that,, (3.4.4) z z z z 54

27 Section 3.4 The nine stress components, si of which are independent, can be written in the matri form z ij z (3.4.5) z z zz A vector has one direction associated with it and is characterised b three components (,, z ). The stress is a quantit which has two directions associated with it (the direction of a force and the normal to the plane on which the force acts) and is characterised b the nine components of Eqn Such a mathematical object is called a tensor. Just as the three components of a vector change with a change of coordinate aes (for eample, as in ig...), so the nine components of the stress tensor change with a change of aes. This is discussed in the net section for the two-dimensional case Stress Transformation Equations Consider the case where the nine stress components acting on three perpendicular planes through a material particle are known. These components are,, etc. when using, z, aes, and can be represented b the cube shown in ig a. Rotate now the planes about the three aes these new planes can be represented b the rotated cube shown in ig b; the aes normal to the planes are now labelled,, z and the corresponding stress components with respect to these new aes are,, etc. zz z z z z z z z zz z z z (a) (b) igure 3.4.6: a three dimensional material element; (a) original element, (b) rotated element 55

28 Section 3.4 There is a relationship between the stress components,, etc. and the stress components,, etc. The relationship can be derived using Newton s Laws. The equations describing the relationship in the full three-dimensional case are ver length. Here, the relationship for the two-dimensional case will be derived this D relationship will prove ver useful in analsing man practical situations. Two-dimensional Stress Transformation Equations Assume that the stress components of ig a are known. It is required to find the stresses arising on other planes through p. Consider the perpendicular planes shown in ig b, obtained b rotating the original element through a positive (counterclockwise) angle. The new surfaces are defined b the aes. ( a) (b) igure 3.4.7: stress components acting on two different sets of perpendicular surfaces, i.e. in two different coordinate sstems; (a) original sstem, (b) rotated sstem To evaluate these new stress components, consider a triangular element of material at the point, ig Carring out force equilibrium in the direction, one has (with unit depth into the page) : AB OB cos OAsin OB sin OAcos 0 (3.4.6) Since OB AB cos, OA AB sin, and dividing through b AB, cos sin sin (3.4.7) 56

29 Section 3.4 B O A igure 3.4.8: a free bod diagram of a triangular element of material The forces can also be resolved in the direction and one obtains the relation ( )sin cos cos (3.4.8) inall, consideration of the element in ig ields two further relations, one of which is the same as Eqn o igure 3.4.9: a free bod diagram of a triangular element of material 57

30 Section 3.4 In summar, one obtains the stress transformation equations: cos sin sin cos sin cos ( sin sin ) cos D Stress Transformation Equations (3.4.9) These equations have man uses, as will be seen in the net section. In matri form, cos sin sin cos cos sin sin cos (3.4.0) These relations hold also in the case when there are bod forces, when the material is accelerating and when there are non-uniform stress fields. (This is discussed in the net section.) Problems. Derive Eqns b taking moments about the lower left corner of the block in ig Suppose that the stresses acting on two perpendicular planes through a point are ij Use the stress transformation formulae to evaluate the stresses acting on two new perpendicular planes through the point, obtained from the first set b a positive o rotation of 30. Use the conventional notation to represent the coordinate aes parallel to these new planes. 58

31 Section 3.4b 3.4b Stress Transformation: urther Aspects Here, it will be shown that the Stress Transformation Equations are valid also when (i) there are bod forces, (ii) the bod is accelerating and (iii) the stress and other quantities are not uniform. We will also eamine the full three-dimensional stress subject to the transformation. Suppose that a bod force i j acts on the material and that the material b b b is accelerating with an acceleration a ai aj. The components of bod force and acceleration are shown in ig (a reproduction of ig ). B a a b b O A igure 3.4.0: a free bod diagram of a triangular element of material, including a bod force and acceleration The bod force will var depending on the size of the material under consideration, e.g. the force of gravit b mg will be larger for larger materials; therefore consider a quantit which is independent of the amount of material: the bod force per unit mass, / m. Then, Eqn now reads b : AB OB cos OAsin OB sin OAcos / m m cos / m msin ma cos ma sin 0 b b (3.4.) where m is the mass of the triangular portion of material. The volume of the triangle is OA OB AB /sin so that, this time, when 3.4. is divided through b AB, one is left with cos sin sin AB / m / sin / m / cos a / sin a / cos b b (3.4.) 59

32 Section 3.4b where is the densit. Now, as the element is shrunk in size down to the verte O, AB 0, and Eqn is recovered. Thus the Stress Transformation Equations are valid provided the element under consideration is ver small; in the limit, the are valid at the point O. inall, consider the case where the stress is not uniform over the faces of the triangular portion of material. Intuitivel, it can be seen that, if one again shrinks the portion of material down in size to the verte O, the Stress Transformation Equations will again be valid, with the quantities,, etc. being the values at the verte. To be more precise, consider the stress acting over the face OB in ig No matter how the stress varies in the material, if the distance OB is small, the stress can be approimated b a linear stress distribution, ig. 3.4.b. This linear distribution can o itself be decomposed into two components, a uniform stress of magnitude (the value of at the verte) and a triangular distribution with maimum value. The o resultant force on the face is then OB /. This time, as the element is shrunk in size, 0 and Eqn is again recovered. The same argument can be used to show that the Stress Transformation Equations are valid for an varing stress, bod force or acceleration. B B O o O igure 3.4.: stress varing over a face; (a) stress is linear over OB if OB is small, (b) linear distribution of stress as a uniform stress and a triangular stress Three Dimensions Re-visited As the planes were rotated in the two-dimensional analsis, no consideration was given to the stresses acting in the third dimension. Considering again a three dimensional block, ig. 3.4., there is onl one traction vector acting on the plane at the material particle, t. This traction vector can be described in terms of the, z, aes as t i j k, ig 3.4.a. Alternativel, it can be described in terms of the z z zz,, z aes as t zi zj zzk, ig 3.4.b. 60

33 Section 3.4b z z z t z t z z zz zz (a) (b) igure 3.4.: a three dimensional material element; (a) original element, (b) rotated element (rotation about the z ais) With the rotation onl happening in the plane, about the z ais, one has, zz zz k k. One can thus eamine the two dimensional plane shown in ig , with i j i j. (3.4.3) z z z z Using some trigonometr, one can see that z z cos z sin. (3.4.4) sin cos z z z z t z z z cos j o i z sin z igure 3.4.: the traction vector represented using two different coordinate sstems 6

34 Section Plane Stress This section is concerned with a special two-dimensional state of stress called plane stress. It is important for two reasons: () it arises in real components (particularl in thin components loaded in certain was), and () it is a two dimensional state of stress, and thus serves as an ecellent introduction to more complicated three dimensional stress states Plane Stress The state of plane stress is defined as follows: Plane Stress: If the stress state at a material particle is such that the onl non-zero stress components act in one plane onl, the particle is said to be in plane stress. The aes are usuall chosen such that the ig plane is the plane in which the stresses act, igure 3.5.: non-zero stress components acting in the plane The stress can be epressed in the matri form Eample The thick block of uniform material shown in ig. 3.5., loaded b a constant stress o in the direction, will have 0 and all other components zero everwhere. It is therefore in a state of plane stress. 6

35 Section 3.5 o z igure 3.5.: a thick block of material in plane stress 3.5. Analsis of Plane Stress Net are discussed the stress invariants, principal stresses and maimum shear stresses for the two-dimensional plane state of stress, and tools for evaluating them. These quantities are useful because the tell us the complete state of stress at a point in simple terms. urther, these quantities are directl related to the strength and response of materials. or eample, the wa in which a material plasticall (permanentl) deforms is often related to the maimum shear stress, the directions in which flaws/cracks grow in materials are often related to the principal stresses, and the energ stored in materials is often a function of the stress invariants. Stress Invariants A stress invariant is some function of the stress components which is independent of the coordinate sstem being used; in other words, the have the same value no matter where the aes are drawn through a point. In a two dimensional space there are two stress invariants, labelled I and I. These are I I Stress Invariants (3.5.) These quantities can be proved to be invariant directl from the stress transformation equations, Eqns { Problem }. Phsicall, invariance of I and I means that the are the same for an chosen perpendicular planes through a material particle. Combinations of the stress invariants are also invariant, for eample the important quantit I I I 4 4 (3.5.) 63

36 Section 3.5 Principal Stresses Consider a material particle for which the stress, with respect to some sstem, is coordinate (3.5.3) The stress acting on different planes through the point can be evaluated using the Stress Transformation Equations, Eqns , and the results are plotted in ig The original o planes are re-visited after rotating Stress at Point Stress Degrees igure 3.5.3: stresses on different planes through a point It can be seen that there are two perpendicular planes for which the shear stress is zero, for o 58 and o. In fact it can be proved that for ever point in a material there are two (and onl two) perpendicular planes on which the shear stress is zero (see below). These planes are called the principal planes. It will also be noted from the figure that the normal stresses acting on the planes of zero shear stress are either a maimum or minimum. Again, this can be proved (see below). These normal stresses are called principal stresses. The principal stresses are labelled and, ig Principal stresses zero shear stress on these planes (principal planes) igure 3.5.4: principal stresses 64

37 Section 3.5 The principal stresses can be obtained b setting 0 in the Stress Transformation Equations, Eqns , which leads to the value of for which the planes have zero shear stress: tan Location of Principal Planes (3.5.4) or the eample stress state, Eqn , this leads to arctan and so the perpendicular planes are at and Eplicit epressions for the principal stresses can be obtained b substituting the value of from Eqn into the Stress Transformation Equations, leading to (see the Appendi to this section, 3.5.8) ( ( ) ) 4 4 ( ( ) ) Principal Stresses (3.5.5) or the eample stress state Eqn.3.5.3, one has , 0.38 Note here that one uses the smbol to represent the maimum principal stress and to represent the minimum principal stress. B maimum, it is meant the algebraicall largest stress so that, for eample, 3. rom Eqns. 3.5., 3.5.5, the principal stresses are invariant; the are intrinsic features of the stress state at a point and do not depend on the coordinate sstem used to describe the stress state. The question now arises: wh are the principal stresses so important? One part of the answer is that the maimum principal stress is the largest normal stress acting on an plane through a material particle. This can be proved b differentiating the stress transformation formulae with respect to, 65

38 Section 3.5 d d d d d d sin ( sin ( cos ( ) cos ) cos ) sin (3.5.6) The maimum/minimum values can now be obtained b setting these epressions to zero. One finds that the normal stresses are a maimum/minimum at the ver value of in Eqn the value of for which the shear stresses are zero the principal planes. Ver often the onl thing one knows about the stress state at a point are the principal stresses. In that case one can derive a ver useful formula as follows: align the coordinate aes in the principal directions, so,, 0 (3.5.7) Using the transformation formulae with the relations sin ( cos ) and cos ( cos ) then leads to ( ) ( )cos ( ) ( )cos ( )sin (3.5.8) Here, is measured from the principal directions, as illustrated in ig principal directions igure 3.5.5: principal stresses and principal directions 66

39 Section 3.5 The Third Principal Stress Although plane stress is essentiall a two-dimensional stress-state, it is important to keep in mind that an real particle is three-dimensional. The stresses acting on the plane are the normal stress zz and the shear stresses z and z, ig These are all zero (in plane stress). It was discussed above how the principal stresses occur on planes of zero shear stress. Thus the zz stress is also a principal stress. Technicall speaking, there are alwas three principal stresses in three dimensions, and (at least) one of these will be zero in plane stress. This fact will be used below in the contet of maimum shear stress. z 0 zz 0 z 0 igure 3.5.6: stresses acting on the plane Maimum Shear Stress Eqns can be used to derive an epression for the maimum shear stress. Differentiating the epression for shear stress with respect to, setting to zero and solving, shows that the maimum/minimum occurs at 45, in which case or, ma Maimum Shear Stress (3.5.9) Thus the shear stress reaches a maimum on planes which are oriented at 45 to the principal planes, and the value of the shear stress acting on these planes is as given above. Note that the formula Eqn does not let one know in which direction the shear stresses are acting but this is not usuall an important issue. Man materials respond in certain was when the maimum shear stress reaches a critical value, and the actual direction of shear 67

40 Section 3.5 stress is unimportant. The direction of the maimum principal stress is, on the other hand, important a material will in general respond differentl according to whether the normal stress is compressive or tensile. The normal stress acting on the planes of maimum shear stress can be obtained b substituting 45 back into the formulae for normal stress in Eqn , and one sees that The results of this section are summarised in ig ( ) / (3.5.0) / / ma( ) / o 45 / / igure 3.5.7: principal stresses and maimum shear stresses acting in the plane The maimum shear stress in the plane was calculated above, Eqn This is not necessaril the maimum shear stress acting at the material particle. In general, it can be shown that the maimum shear stress is the maimum of the following three terms (see the Appendi to this section, 3.5.8):, 3, 3 The first term is the maimum shear stress in the plane, i.e. the plane containing the and stresses (and given b Eqn ). The second term is the maimum shear stress in the 3 plane and the third term is the maimum shear stress in the 3 plane. These are sketched in ig below. 68

41 Section 3.5 ma ma 3 ma (a) - plane (b) -3 plane (c) -3 plane igure 3.5.8: principal stresses and maimum shear stresses In the case of plane stress, 3 zz 0, and the maimum shear stress will be ma,, (3.5.) Stress Boundar Conditions When solving problems, information is usuall available on what is happening at the boundaries of materials. This information is called the boundar conditions. Information is usuall not available on what is happening in the interior of the material information there is obtained b solving the equations of mechanics. A number of different conditions can be known at a boundar, for eample it might be known that a certain part of the boundar is fied so that the displacements there are zero. This is known as a displacement boundar condition. On the other hand the stresses over a certain part of the material boundar might be known. These are known as stress boundar conditions this case will be eamined here. General Stress Boundar Conditions It has been seen alread that, when one material contacts a second material, a force, or distribution of stress arises. This force will have arbitrar direction, ig a, and can be decomposed into the sum of a normal stress distribution N and a shear distribution S, ig b. One can introduce a coordinate sstem to describe the applied stresses, for eample the aes shown in ig c (the aes are most convenientl defined to be normal and tangential to the boundar). 69

42 Section 3.5 N S contact region (a) (b) (c) igure 3.5.9: Stress boundar conditions; (a) force acting on material due to contact with a second material, (b) the resulting normal and shear stress distributions, (c) applied stresses as stress components in a given coordinate sstem igure shows the same component as ig Shown in detail is a small material element at the boundar. rom equilibrium of the element, stresses,, equal to the applied stresses, must be acting inside the material, ig a. Note that the tangential stresses, which are the stresses in this eample, can take on an value and the element will still be in equilibrium with the applied stresses, ig b. (a) (b) igure 3.5.0: Stresses acting on a material element at the boundar, (a) normal and shear stresses, (b) tangential stresses Thus, if the applied stresses are known, then so also are the normal and shear stresses acting at the boundar of the material. Stress Boundar Conditions at a ree Surface A free surface is a surface that has nothing on one side and so there is nothing to provide reaction forces. Thus there must also be no normal or shear stress on the other side (the inside). This leads to the following, ig. 3.5.: 70

43 Section 3.5 Stress boundar conditions at a free surface: the normal and shear stress at a free surface are zero This simple fact is used again and again to solve practical problems. Again, the stresses acting normal to an other plane at the surface do not have to be zero the can be balanced as, for eample, the tangential stresses T and the stress in ig T T T S N N S 0 igure 3.5.: A free surface - the normal and shear stresses there are zero Atmospheric Pressure There is something acting on the outside free surfaces of materials the atmospheric pressure. This is a tpe of stress which is hdrostatic, that is, it acts normal at all points, as shown in ig Also, it does not var much. This pressure is present when one characterises a material, that is, when its material properties are determined from tests and so on, for eample, its Young s Modulus (see Chapter 5). The atmospheric pressure is therefore a datum stresses are reall measured relative to this value, and so the atmospheric pressure is ignored. igure 3.5.: a material subjected to atmospheric pressure 7