Notes on Elasticity and Finite Element Analysis. Paul S. Steif Department of Mechanical Engineering Carnegie Mellon University

Transcription

1 Notes on Elasticit and Finite Element Analsis Paul S. Steif Department of Mechanical Engineering Carnegie Mellon Universit All mechanical and civil engineering students take at least one course in mechanics of materials. For man students though, this is their last formal course in the mechanics of deforming solid bodies. Yet, it is increasingl likel for practicing engineers to be called upon to do finite element analsis (FEA) as part of their engineering activities. A thorough grounding in the theor of linear elasticit which underlies FEA, while advantageous, is less and less common. These notes attempt to enable students with onl a background in mechanics of materials to be more effective users of FEA. These notes are divided into the following Chapters: 1. Essential Variables in Elasticit The ke quantities of elasticit - displacements, strains and stresses - are defined so as to give the reader a phsical intuitive feel for them. 2. Formulation of an elasticit problem The essential elements in using elasticit are covered, including the stress-strain law, boundar conditions, and the field equations of equilibrium and compatibilit. 3. Finite Element Method The major steps in carring out a finite element analsis are reviewed and their relations to elements of elasticit are pointed out. 4. Stress Concentrations and Singularities Two major pitfalls in finite element analsis stress concentrations and singularities are discussed. Since, displacement constraints so often lead to singularities, the means of appling minimal displacements to constrain a bod which is otherwise under applied forces is also covered. 5. Eamining the Results of FEA Various methods used b eperienced analsts to test the accurac of finite element results are reviewed, including the use approimate analses based on mechanics of materials. Copright 2005 Paul S. Steif 1

2 INTRODUCTION 1. ESSENTIAL VARIABLES IN ELASTICITY Consider a plate with elastic modulus E and Poisson ratio ν, which is subjected equal and opposite tensile forces P (not shown) in the horizontal direction acting at the ends. t w Figure of plate under tension You should alread know from the stud of aial loading how to calculate the tensile stress L the strain the stretch σ = P A σ ε = E δ = εl = PL EA and the transverse strain σ ε t = νε = ν E In these formulas, the cross-sectional area A is equal to wt. The aial loading could have actuall been applied to the plate in a number of different was. Two ver different tpes of loading uniforml distributed and concentrated - are shown in the net Figure. The different resulting shapes of the plate are also shown with the dotted lines. Now, in either case we would have used the simple equations above (note that P would equal to the net force of the distribution on one end). In both cases, the simple equations give some correct information about what the plate eperiences. Copright 2005 Paul S. Steif 2

3 However, the two loadings do produce somewhat different results in the two plates. The simple equations of aial loading do not give the whole stor. To distinguish between the two tpes of loading and deformation, we need a more comple theor: elasticit theor. Load applied uniforml each end Load applied at a single point each end In both cases the plates are longer (δ > 0) and narrower (ε t < 0), but the final shape is slightl different. However, even these variables cannot be uniquel interpreted for the case of the concentrated applied load. Is δ the stretch of the center-line joining the points of the force application or the top or bottom boundar of the plate? Likewise, the plate width (w) narrows b different amounts depending on where ou are in the plate. Clearl, the aial theor, which ields single values for stress, strain, stretch and transverse strain, fails to capture the spatial variation that is evident in the concentrated force loading. In elasticit, we acknowledge that each part of the plate ma be eperiencing a different mechanical state. Surel, three small elements, one at the center of the plate, one just where the load is applied and one in the upper right corner are undergoing ver different mechanical states. In elasticit, we define the quantities that enable us to describe a mechanical state that varies at ever point in the bod. The essential quantities of elasticit are: displacement (motion of a point), strain (stretching of a small line segment through a point) and stress (force per unit area transmitted through an internal surface at a point). Since these quantities var from point to point, the are able to capture the spatial variation that distinguishes the two loadings. DISPLACEMENT The most fundamental and simplest aspect of the mechanical state to describe is the displacement or motion. Under the action of the loading, each point of the bod moves to a slightl different position. The displacement or change in position along the -ais is denoted b u. Displacement has units of length (e.g., m, cm, in.). The displacement or change in position along the -ais is denoted b u. Each of the displacements has a magnitude and a sign. If u is positive, then the point moves in the positive -direction (relative to its initial position) and similarl for u. Copright 2005 Paul S. Steif 3

4 We have identified si points in the bod. Their positions prior to the application of the load are labeled as A, B, C, D, E and F. When the load is applied (and kept acting on the bod), each of these points moves. The positions of these points while the plate is stretched are A, B, C, D, E and F (of course, A moves to A and so forth). C B A C A B F F E E D D For each the points, here are the signs of the displacement. Check each one to make sure that it makes sense to ou. A: u < 0, u > 0 B: u < 0, u = 0 C: u < 0, u < 0 D: u > 0, u > 0 E: u > 0, u = 0 F: u > 0, u < 0 You might also notice that u is the same at points A, B and C. B contrast, for the case of loads being applied at a single point at each end, the displacement u at B is more negative than at A. It might worth pointing out that the same aial loading applied uniforml at the end to an identical plate could have resulted in the motion shown in the following figure. Here, to avoid cluttering the diagram, we onl label the final positions D, E and F. In this case the point A does not move, points B and C move in the negative direction to B and C respectivel (the move down b different amounts). C A B E F D F E D Now the displacements have the following signs: A: u = 0, u = 0 B: u = 0, u < 0 C: u = 0, u < 0 D: u > 0, u = 0 E: u > 0, u < 0 F: u > 0, u < 0 Copright 2005 Paul S. Steif 4

5 You can imagine that the same loading produces the same final shape in the two plates: the final widths and heights would be identical. The differing displacements occur because the plates are restrained differentl against movement as the load was applied. Strain The change in shape of a bod is full described b the displacements at ever point. However, displacements do not directl describe the deformation. We now consider the strain, which is a local measure of deformation. Consider a bod, such as the plate above, prior to being loaded. Etch a straight segment in the bod. When forces are applied to the bod the segment gets longer, shorter or stas the same length. (The segment also moves and rotates.) The change in length is related to the displacements at the two ends (just as the stretch δ of a bar under aial loading is related to displacement u at the ends). Original configuration of plate with segment etched Plate deformed with etched segment shown as stretched The strain of the segment is defined b change in length strain = original length The strain at this point, in this direction, is defined as the above ratio in the limit of segment becoming ver (infinitesimall small). It is important to recognize that for the same bod, same loading and same point, segments of different initial orientations will have different strains. In principle, we might be interested in the strain at an initial orientation. Fortunatel, it turns out that the strains of segments of different orientations at the same point are related to one another. In fact, if ou know the strains of segments with three distinct orientations, then ou can find the strain for an other orientation. In practice, we set up a set of - aes. Then, the strain for an orientation relative to these aes can be found if the following three strains are known: Copright 2005 Paul S. Steif 5

6 The strain of a segment originall along (denoted b ε ) The strain of a segment originall along (denoted b ε ) The shear strain of a pair of initiall perpendicular segments, one originall along and one originall along (denoted b γ ). The shear strain is the change in this initial right angle, as measured in radians. We illustrate the definitions b considering a pair of lines that are perpendicular before the load is applied. Change of this segment s length controls ε Change of this angle controls γ Change of this segment s length controls ε Pair of segments before deformation Configuration of pair of segments as deformed The signs of these strains are important. If the segment originall along gets longer, then ε > 0; if it gets shorter, then ε < 0, and if it remains the same length, then ε = 0. The same convention for signs applies to ε. If the initial angle becomes < 90, then γ > 0; if this angle becomes > 90, then γ < 0; if this remains a right angle, then γ = 0. In summar, the three strains ε, ε, and γ full describe the state of strain at a point. The describe how a tin square located at the point in question becomes distorted. These three strains also allow one to calculate the strain of a segment of an other orientation (although we have not shown how this is done). Here is an eample that illustrates the ideas of displacement and strain. Consider a bod onto which a 1 mm b 1 mm square is etched with lower left corner initiall located at the point (,) = (50,60). All coordinates are in millimeters. Under the action of the loads on the bod, the four corners move to the points shown. C: (50,61) D: (51,61) C : (54,62.998) D (?) A: (50,60) B: (51,60) A : (54,62) B : (55.001,61.997) Positions of element corners when bod is undeformed Positions of element corners when bod is deformed Copright 2005 Paul S. Steif 6

7 Here are the displacements at the four corners: A: u = 4, u = 2 B: u = 4.001, u = C: u = 4, u = D: u = 4.001, u = Here is the final shape again, this time with the dimensions (in mm) indicated. Recall this was originall a 1 mm b 1 mm square The strain ε is found from the change in length of the segment AB. One can use the Pthagorean theorem to find the final length of AB; the sides of the triangle are.003 and Using a calculator, ou will find that this final length is The small rotation of AB contributes insignificantl. The motion of B along the direction of AB contributes to the change in length while the motion of B perpendicular to AB does not contribute. So slight rotations of a segment do not affect the normal strain along the segment. ε A' B' AB = = = AB 1 The strain ε is found from the change in length of the segment AC. (In this case, the - positions of A and C are unchanged, so there is no rotation to consider). ε A' C' AC = = = AC 1 You ma recall that the change in length of a rod in aial loading could be related to the difference in displacement at its ends. This idea applies here as well. That is, we could find the strains of AB and AC in the following alternative wa: ε B A' B' AB u u = = AB AB A = = ε C A A' C' AC u u = = AC AC = 1 = Copright 2005 Paul S. Steif 7

8 The segment AB rotates clockwise b The segment AC does not rotate, so the angle BAC in total increases b.003 radians. Therefore, the shear strain γ = If we are reall looking at ver small segments coming off the point A, then the stretch and rotation of the segment CD is identical to that of segment AB. This is because the are in the same initial orientation, and the are, essentiall, at the same point (the segments are so small). Likewise, the segment BD strains and rotates the same as segment AC. Therefore, the point D is located at (55.001,62.995). Since the segment ends alwas sta connected (e.g., A B and A C remain connected at A ), the square ABCD turns into a parallelogram A B C D. This idea of a parallelogram is ver general. If, before deforming a bod, one etches a tin square at an point and at an initial orientation, then that element will alwas deform into a parallelogram. The particular shape and orientation of parallelogram will depend on: the deformation of the bod, the point where the square is located, and the initial orientation of the square. Seeing a square deforming into a parallelogram is a ver useful wa of visualizing strain at a point. Stresses Your eperience so far with stress is likel to be based on the formulas ou have used before, such as σ = P/A in aial loading or σ = -M/I in bending. But, as ou start to eplore elasticit, ou need a much better understanding of the concept of stress. It is useful to be able to think about stress in two was: as it relates to the deformation, and as it relates to the applied forces. Stresses as related to deformation arctan = We have seen that a tin square etched into a bod deforms into a parallelogram when the bod is loaded. Imagine ou were given the tin square and told to deform it into such a parallelogram. You would need to appl forces to the surface of the square. Such forces also act on the small element when it is part of the bod. Internal forces must be acting on the square from the material surrounding the square to keep it in the deformed shape (the parallelogram). The internal forces are shown in the following figure. Stresses are these internal forces per unit area. Copright 2005 Paul S. Steif 8

9 Element as undeformed Element as deformed, with forces on its faces that maintain deformed shape Stresses as related to applied forces When forces are applied to a bod, these forces are transmitted through the bod. These forces locall squeeze the material together, pull it apart, or shear it. To be specific, we need to focus on two clumps of material separated b a plane. The edge of the plane is the line between the clumps. The plane runs into the bod perpendicularl to the drawing. These two clumps are in contact and when the bod is loaded, the two clumps can eert forces on each other. To reveal these forces, one needs to separate the clumps and then draw the equal and opposite forces. The material on each side of the segment ma pull or push on the other side: this is normal stress tension or compression. The material on each side of the segment ma tend to cause the material on the other side to slide one wa or the other: this is shear stress. Unloaded bod with neighboring clump identified Eternal forces on bod Eploded view of clumps with forces between them We sa that these forces are transmitted through the plane between the clumps. The size of the clumps is irrelevant. The force is associated with the plane between them. To visualize these forces in another wa, imagine perforating sheet with small holes over a short length. Then, when the loads are applied, if the tension across the perforation is too large, the perforated segment will pop open. Of course, if the stresses are ver large, this tear will propagate across the bod.) Whether the perforated segment pops open is a barometer for the tensile force acting across that segment. Copright 2005 Paul S. Steif 9

10 Bod unloaded with segment perforated (weakened b holes) Loading of bod causes perforated segment to pop open We now define stress. Consider a given point in a plate that is about to be loaded. Now load the plate. Consider an infinitesimall small surface at some orientation running through the point in question (which divides two clumps of material). As shown above, there are normal and shear forces acting across this surface. The normal stress is equal to the normal force divided b the area of the surface. (Even though the surface area is infinitesimal, so is the force; their ratio is a finite number.) Likewise, the shear stress is equal to the shear force divided b the area of the surface. The stresses will have units for force per area, such as N/m 2. Even though we are considering a single bod, with fied loading and a fied point in the bod, it is important to emphasize that the normal and shear stresses depend on the orientation of the surface considered. This can be seen from the case of a plate under simple tension. Two surfaces are shown in the following figure; these surfaces are at the same point. Across one surface, which is perpendicular to the -ais (parallel to the - ais), there is clearl a tension. If we were to make a cut in the plate at that point parallel to that surface, the cut would pop open. B contrast, if we were to make a cut along the second surface, which is perpendicular to the -ais (parallel to the -ais), nothing would happen. Under this loading, there is no force acting across the second surface. There is force transmitted through this surface No force transmitted through this surface Bod loaded in tension, showing two surfaces of different orientation through same point have different internal forces. The above description does not help us actuall find the traction in practice, since we can rarel measure such a force directl. The description just gives the definition for the stress across a plane. Copright 2005 Paul S. Steif 10

11 Since the stresses acting across different planes through a point are different, do we have go to the effort of re-calculating the stresses for all such orientations? The answer is no: It is turns out that we onl need to know stress across two planes to find all the others. This situation is somewhat similar to what we discussed for the strain. Above, we stated that the strain of a segment of an orientation at a point can be found from just three strains (that is, strains of various orientations are not completel independent). Well, in a similar manner, the stresses across an surface through a point can similarl be found from the stresses across just two perpendicular surfaces. In particular, consider a bod under a given loading. And focus on a surface that is parallel to the -ais (or perpendicular to the -ais). Across this surface there can be equal and opposite forces in the -direction and equal and opposite forces in the - direction. The equal and opposite forces in the -direction acting across this surface are normal forces. The normal stress, or force per unit area, in the -direction is denoted b σ. If the stress is tensile, then σ is positive (compressive corresponds to σ negative). The equal and opposite forces in the -direction acting across this surface are shear forces. The shear stress, or force per unit area, in the -direction is denoted b τ. This shear stress acts on a plane with normal in the -direction, and acts in the -direction. When is τ is positive, the forces act in the directions shown. σ Surface of interest τ Bod loaded with internal surface perpendicular to -ais highlighted. Stress components along - and -aes are defined in terms of the transmitted forces. Consider the same bod, same loading and same point, but now a second surface through this point, that is parallel to the -ais (or perpendicular to the -ais). The equal and opposite forces in the -direction acting across this surface are normal forces. The normal stress in the -direction is denoted b σ. Positive σ corresponds to tension and negative to compression. The equal and opposite forces in the -direction acting across this surface are shear forces; corresponding to these forces is the shear stress τ. This shear stress acts on a plane with normal in the -direction, in the -direction. When is τ is positive, the forces act in the directions shown. Copright 2005 Paul S. Steif 11

12 Surface of interest σ τ Bod loaded with internal surface perpendicular to -ais highlighted. Stress components along - and -aes are defined in terms of the transmitted forces. If ou know the quantities σ, σ, τ and τ at a point, then ou can find the traction acting on a surface of an orientation (ou will learn how to do this later). In fact, even these four quantities are not independent of each other. Sa, we zero in again on a ver small square located at the point of interest, with sides aligned with the - and -aes. This is the bold hollow square in the diagram below. Sa someone gives us the quantities σ, σ, τ and τ at the point. This means that we know the forces per unit area that act across the surfaces of the small element. We can then isolate the square and draw a free bod diagram of it. On the surface of the element, we draw the forces (or actuall forces per unit area) that are eerted b the surrounding material (the four shaded squares in the diagram below). If we assume that all the stresses σ, σ, τ and τ are positive, then the forces are in the directions shown. Notice that this square is so small that the forces acting across the right side are equal and opposite to the forces acting across the left side. That is, the left and right side are identical surfaces: the have the same orientation and the are essentiall at the same point (since the square is ver small). The stresses are likewise the same for the top and bottom surfaces of the square. σ Element to be focused on is in center. Stresses are due to surrounding elements σ τ τ σ τ τ σ Stresses due to forces eerted b surrounding material on element of focus Consider now this square with all the forces acting on it. The forces are actuall equal to the stresses times the area of each surface, but all four surfaces have equal area (which might as well be one). Is this element in equilibrium? On the left face and right face σ acts equal and oppositel so the produce no net force. Similarl, τ on the left face and Copright 2005 Paul S. Steif 12

13 right face is equal and opposite. On the top and bottom, σ and τ are equal and opposite and so produce no net force. So the sum of forces on this element is zero. Consider, however, the moments. τ produces a positive moment about the z-ais, and τ produces a negative moment about the z-ais. These moments onl balance if τ and τ are equal, which the must be. Therefore, since we have established that τ = τ, we onl need to be considered about three stresses at the point, which we denote b σ, σ, and τ. From these one can, with methods to be developed later, determine the stress acting upon an other surface through the point. Again, the stresses σ, σ, and τ, are reall just stresses on two particular planes. Sometimes we can find σ, σ, and τ from the loads using methods of mechanics of materials. Other times, the loads ma be so oriented, or the aes have been so chosen, that the stresses σ, σ, and τ are not readil found. However, there is nothing special about these three stresses. If we find the stresses acting on two other planes, then we can alwas find the stresses acting on an additional planes. When we do an elasticit analsis or use the finite element method, we will define the bod relative to - and -aes. The results of the analsis will be two displacements u and u, three strains ε, ε, and γ, and three stresses σ, σ, and τ at ever point. In the case of the finite element method, ou will find these same 8 quantities (u, u, ε, ε, γ, σ, σ, and τ ) at each of the discrete points (nodes) that ou or the program defines. This entire discussion presumes a planar (2-D) with forces in the plane. If, instead, the problem is three dimensional, then there are 3 displacements, 6 strains and 6 stresses. Copright 2005 Paul S. Steif 13

14 2. FORMULATION OF AN ELASTICITY PROBLEM INTRODUCTION We have alread introduced the variables - displacements, strains and stresses which elasticit theor uses to describe a deforming bod. Converting a phsical situation in which stresses or deflections are of interest into an elasticit problem involves the same tpes of steps as for mechanics of materials. One needs to describe: the shape(s) of the deforming bod or bodies. the material properties of the deforming bod or bodies. the loads that are acting on the bodies and/or the constraints on the motion of the deforming bodies. Shape To describe the shape, one usuall needs to specif the bounding lines or surfaces of the bod. While this is sometimes time consuming and tedious to do, it is conceptuall straightforward. Often, the challenging part is deciding just how much of the original bodies one should analze. This skill, that of modeling to reduce a comple phsical problem into one suitable for analsis, is developed over much time. We will not delve into this issue here. Materials When we are considering an elasticit problem, the material properties are those which relate stress and strain. We describe this now in some detail. You studied aial loading in Mechanics of Materials. This loading is depicted below. t P w P L You learned that the stress is the aial force divided b the cross-sectional area (wt). The strain is the change in length divided b the initial length L. A laman s terms, stress is Copright 2005 Paul S. Steif 14

15 how hard ou pull on something and strain is how much it stretches. The intrinsic stiffness of the material controls how hard ou must pull on something to get it to stretch a given amount. The intrinsic stiffness of the material is captured b the elastic modulus or Young s modulus E. E is dependent on the material; materials such as steel, polethlene, and natural rubber all have ver different elastic moduli. However, the elastic modulus is independent of the size of the material or the forces acting upon it. In studing mechanics of materials, ou probabl onl considered stress acting in one direction (sa, uniaial tension). The aial and the transverse strain were related to that stress through the Young s modulus E, and the Poisson ratio ν. Here we present the relation between stress and strain assuming normal stresses acts in both the - and - directions. ε ε σ = E σ = E σ ν E σ ν E To see that this makes sense, imagine onl the stress σ is non-zero. Substitute σ = 0 in the above equations. Then, the strain, ε, which is parallel to σ (the aial strain), is related to σ b E. Likewise, the transverse strain ε, is related to the aial stress σ b ν and E. These equations also make sense when σ is non-zero (σ = 0). In that case the aial strain is ε and the transverse strain is ε. This relation states that the strains due to σ and σ acting simultaneousl are just equal to the sum of the strains when σ and σ act individuall. It is important to recognize that ou can have stress in one direction, but no strain in that direction. Or, ou can strain in one direction, but no stress in that direction. This is because both stresses can produce both strains and hence the can counteract one another. Here is an eample of the stresses counteracting each other. Sa that E = 200 GPa and ν = 0.3. σ = 100 MPa σ = 30 MPa σ = 30 MPa σ = 100 MPa Copright 2005 Paul S. Steif 15

16 In this case, the strains are ε σ = E σ ν E = (0.3) = 0 ε σ = E σ ν E = (0.3) = So the strain ε due to the two stresses cancel. In Mechanics of Materials ou have studied shear stress and shear strain. For most materials, there is no intrinsic direction in the material. Materials with no intrinsic direction are called isotropic. (Wood with its grain and some fiber-reinforced composites are eamples of materials that are not isotropic, but anisotropic.) For isotropic materials, normal stresses are related to normal strains and shear stresses are related to shear strains. Normal stress σ produces no shear strain γ, and the shear stress τ produces no normal strain ε (or ε ). Therefore, the relationship ou derived in Mechanics of Materials, namel τ = G γ, still holds. We use this same relationship ecept we use the notation τ for τ and γ for γ. Therefore, the stress-strain relationship for shear stress and strain is γ = τ /G. This is illustrated below. τ γ τ τ τ You ma have learned before in mechanics of materials that E, ν and G are related for isotropic materials. This relation is: E = 2G(1+ν) Therefore, ou reall need to specif onl two material properties. In most finite element programs ou are to specif E and ν. It is probabl appropriate to mention here also that deciding on the values to use for E and ν is sometimes challenging, depending on the material. Copright 2005 Paul S. Steif 16

17 Forces and Displacements on the Boundar As we have seen, the leap from mechanics of materials to elasticit involves a much finer grain description of what is happening in the bod. In elasticit, we look at displacements, strains and stresses at each point of the bod. Elasticit likewise involves a finer grain description of eternal influences acting upon the bod through its boundar. Sa a bod is loaded and that the loads keep the bod in a deformed configuration. At each point of the boundar, there is some force per unit area acting upon the bod. This force could be zero, but ma not be. Also, each point of the boundar has undergone some displacement in going from the undeformed shape (before loading) to the deformed shape (with loads acting). This displacement could be zero. One important idea in elasticit is that ou cannot specif the displacement and the force at a point on the boundar. To see this idea, consider a spring, with constant k = 20 N/cm, which is fied at the left end. u = 3 cm P = 40 N What can we sa about the right end? We can specif the force to have some value, sa P = 40 N. In that case, we would know the displacement to be u = 2 cm. Or we could specif the displacement to have some value, sa u = 3 cm. In that case, we would know the force to be P = 60 N. But, we cannot sa P = 40 N and u = 3 cm, because the would obviousl be inconsistent. While for this simple problem we would know how to pick P and u to be consistent, in the case of an elasticit problem in which we are loading a complicated bod all around its bod, we can almost never know how to make a force and a displacement consistent. For this reason, we cannot specif the force and the displacement in some direction at the same point of the boundar. On the boundar of a (two-dimensional) elastic bod we actuall are concerned about displacements in two directions and about forces in two directions. Thus, we must specif the force or the displacement in each direction of the two directions. However, the idea just discussed still holds: we cannot specif both and the force and the displacement in same direction. To summarize, we must prescribe: 1. displacement in the -direction OR force in the -direction at each point along the boundar of the bod Copright 2005 Paul S. Steif 17

18 and 2. displacement in the -direction OR force in the -direction at each point along the boundar of the bod Here is an eample that illustrates several possible boundar conditions. The bod is a rectangular plate. At the left, the bod is fied along two segments, but is otherwise free on that face. There is uniform pressure acting on the right face. Above and below the bod the surface has a distribution of rollers. Rollers make the normal displacement zero, but the tangential force zero. bod 100 MPa Here is how we translate these phsical conditions into boundar conditions that elasticit theor understands. At the right: We are not specifing about the motion of this boundar or its displacements. We are saing that there is a force per unit area of magnitude 100 MPa acting in the negative -direction. We are saing that there is a force in the -direction on this face is zero. So on this boundar we are prescribing force (or equivalentl force per area) in both the - and -directions, but not displacements. At the top and bottom: Awa from the rollers, the surface is free: the force in and directions is zero (the displacement is free to be whatever it wants to be). At the rollers the vertical displacement u = 0, and the force in the -direction is zero. Notice that there will be a force in the -direction and there will be displacements in the -direction, but we don t know them in advance. At the left: Where the boundar is fied, the displacements are zero. Elsewhere on the boundar, the forces are zero. Copright 2005 Paul S. Steif 18

19 The bod is redrawn with the boundar conditions labeled. Notice that the condition of specifing the force on the right surface is equivalent to specifing two of the stress components, σ and τ. Likewise, to the right of the rollers on the top and bottom surfaces, where the forces are zero, this is equivalent to saing that two of the stress components σ and τ are zero. To remind ou, the stress components are drawn on an elementar square. Also, a small square is drawn at the top and right boundaries so ou can see which components are acting on those faces. σ τ σ u = 0, τ = 0 σ = 0, τ = 0 u = 0 u = 0 { { σ = 0 τ = 0 σ = -100 MPa τ = 0 u = 0, τ = 0 σ = 0, τ = 0 Copright 2005 Paul S. Steif 19

20 St. Venant s Principle One of the most powerful ideas in the mechanics of deformable bodies is a principle which is ascribed to St. Venant. It is of great help in setting boundar conditions. When we introduced elasticit, we said that mechanics of materials differs from elasticit in that elasticit takes a much more fine-grained approach. Elasticit considers displacements, strains and stresses in all directions at all points of the bod. Also elasticit involves a much more detailed treatment of the loading of the boundar. As we have said, either the displacement or the force must be specified in each direction at each point of the boundar. This is a lot of information, information which we often don t have. St. Venant s principle is helpful in reducing the amount of information that we need to specif. Here is the principle: If ou appl a force and/or a moment to some region of the boundar, then the stresses far from that region of the boundar don t care eactl how ou appl that force and/or moment. Here is an eample. Onl part of the boundar loading of the bod is shown. Sa ou think that the real loading is best represented b a uniform distribution q shown on the left bod. St. Venant s principle sas that the stresses far from this part of the boundar will be essentiall the same if the distributed force is replaced b a concentrated force P = qh. This force must act at the point where the distribution was centered so that the concentrated force produces the same net moment about an point as the original force. The stresses are unaffected b the change in loading at points that are approimatel 3 h or farther awa from the region of loading, where h describes the region of the original loading. Stresses the same beond region of radius 3h from load h q Stresses the same beond region of radius 3h from load 3h P = qh Provided ou are prepared to forgo knowledge of the stresses right where the loads are applied, this principle allows ou to be ver fleible in specifing force boundar conditions. The stresses are unaffected b the change in loading at points that are approimatel 3 h or farther awa from the region of loading, where h describes the region of the original loading. Copright 2005 Paul S. Steif 20

21 EQUATIONS OF EQUILIBRIUM AND COMPATIBILITY When ou solved problems using mechanics of materials ou used three principles. One principle was the material law or stress-strain relation. For the case of elasticit, we have discussed that above. The second principle is mechanical equilibrium, that is the sum of forces and moments both equal zero. The third principle is geometric compatibilit, that is the geometric relations between the motions and the strains. Elasticit also uses these principles, FEA which is based on elasticit uses these principles. Unless ou solve equations of elasticit directl, ou probabl don t need to know how equilibrium and geometric compatibilit are implemented in elasticit. For completeness though, we present these equations below: Equilibrium When the stresses var from point to point, the are in equilibrium with each other and with an forces applied on the boundar. For three dimensional stress state, this results in 3 partial differential equations involving the 6 components of stress. When there is a planar state of stress (onl σ, σ, and τ ), then there are onl two such equations and the are as follows: F σ τ = 0 implies + = 0 F τ σ = 0 implies + = 0 Geometric Compatibilit The strains are related to variations in the displacements. These relations generalize the basic relation from aial loading ε = δ/l and a similar one for shear strain. For displacements in three-dimensions, this results in 6 partial differential equations involving the strains and the displacements. For a planar deformation, there are 3 relations between strain and displacement ε u = ε = u γ = u u + In total, for three-dimensional problems there are 15 equations for the 6 stresses, 6 strains and 3 displacements. For a plane stress problem there are 8 equations for the 3 stresses (σ, σ, and τ ), 3 strains (ε, ε, and γ ) and 2 displacements (u and u ). In addition, there are boundar conditions, which we have described above for the case of plane stress problems. Copright 2005 Paul S. Steif 21

22 3. FINITE ELEMENT METHOD The equations of elasticit are partial differential equations that are to be satisfied at ever point of the bod. Furthermore the displacements or stresses must satisf specified values on the boundar. Because the equations of elasticit are so difficult to solve, the finite element method has been developed to solve these equations on the computer. The results of this method are stresses or displacements that are approimate relative to what elasticit would give. Ver briefl, the finite element method solves these equations b breaking up the bod into small, but finite regions. The method presumes that the ke quantities, displacements, strains and stresses, have a simple spatial distribution within each element. For eample, the stresses might var linearl with position in each element. These ke quantities are then chosen to satisf the equations of elasticit approimatel. Breaking up the bod appropriatel into a mesh of elements is important to getting good estimates of the stresses. Most finite element programs have a variet of methods to help ou do meshing. In the end, though, the mesh will consist of a set of node points distributed around the bod. The node points will be connected b lines and the lines form the boundaries of the elements. This is illustrated in the figure below. Boundar nodes elements nodes Below are the major steps in a finite element analsis. Define the Region This is describing the shape and position of the bod. Define Material Properties Usuall this includes the elastic modulus (E) and the Poisson ratio (ν), although other parameters could be of interest (such as thermal epansion coefficient). Define Tpe(s) of Element One must choose the tpe of elements that are to be used, which depends on the tpe of analsis. One tpe of element differs from another insofar as the variables that it deals Copright 2005 Paul S. Steif 22

23 with and how each variable varies spatiall in the element. For eample, for doing a twodimensional elasticit analsis, in an particular finite element program there will onl be one or two element tpes that is appropriate. This step is unnecessar in elasticit and is peculiar to doing FEA. Define Mesh The bod must be broken up into elements and nodes. This step is unnecessar in elasticit and is peculiar to doing FEA. Define Loading Usuall this involves describing the eternal influences across the boundar of the bod (If gravit is important, this force acts at all points of the bod, not just the boundar). Eternal influences acting across the boundar are usuall represented b specifing the displacement or the force over each node point on the boundar. Sometimes a distributed force can be defined over a part of the boundar. FEA has a default assumption on the boundar conditions. If ou specif nothing about the displacement u at a boundar point, then that is equivalent to specifing the eternal force F applied to that point to be zero. The point is free to move in the -direction. If ou specif nothing about the displacement u at a boundar point, then that is equivalent to specifing the eternal force F applied to that point to be zero. The point is free to move in the -direction. While finite element programs allow ou to enter distributed loads (force per length or per area), it is sometimes useful to be able to convert a distributed load to forces at the nodes. One wa to do that is to associate a portion of the boundar with each node. The net force from the distribution which acts over that portion of the boundar is applied to the node. This process is illustrated with the following eample. A distributed force of 100 MPa acts on the right boundar of the bod shown. Sa the bod is 1 cm thick. Sa the bod is broken into elements, with four elements along the right boundar. We associate with each of the boundar nodes a certain area of the right boundar. The middle three nodes each has 4 cm of area, and the nodes at the top and the bottom have 2 cm of area. The areas associated with two of the nodes are highlighted in gra. For each node, the force is the force per area multiplied b the area associated with that node. This means that the corner nodes (at the top and bottom) have ( )(.02)(.01) or 20,000 N of force. The three other nodes have ( )(.04)(.01) or 40,000 N of force. Notice that the total force on the nodes is 3(40,000)+2(20,000) = 160,000 N. This agrees with the total applied force of ( )(.16)(.01). Copright 2005 Paul S. Steif 23

24 16 cm 100 MPa 4 cm ( )(.02)(.01) = 20,000 N ( )(.04)(.01) = 40,000 N Request Solution When doing FEA, one usuall just pushes a button which solves the problem that has been specified. This solves the equations of elasticit approimatel. Inspect Results When doing FEA, one can view the results in a variet was. More on Meshing As mentioned above, the finite element method solves the equations of elasticit approimatel. How good is that approimation? It depends on the elasticit problem, and it depends on the mesh. There are some elasticit problems which can never be solved with FEA; this is discussed later in the chapter on Stress Concentrations and Singularities. For problems with no singularities, FEA is capable of offering estimates of stress that are comparable to those given b elasticit. As we said above, FEA works b assuming an approimate variation of stress over each element, for eample a linear variation. FEA gives the most accurate answers if the actual stress varies over the element in the same wa as assumed b the element. So if the stress actuall varies linearl with position (as it does in pure bending), then FEA with elements assuming a linear variation in stress will approimate the actual stress field eactl. Under most loadings, the stresses will var in a more comple wa. So how can FEA approimate this accuratel? Consider a general function () that is continuous and smooth. If ou look at () over a small enough region, () can be approimated b a linear function (using the derivative). This means that if the elements are taken to be small enough, so the stresses Copright 2005 Paul S. Steif 24

25 locall within the element are nearl linear, then FEA can approimate the stress distribution. Now, finite element analsts take a much simpler approach. The often sense where the stresses will be varing rapidl with position, and the put more elements there. Stress concentrations are just such regions where the stress varies rapidl. Where the stress varies slowl with position, fewer elements are used. Depending on the need, finite element analsts will do the calculations on the same problem using more than one mesh. A coarse mesh would have fewer elements and a fine mesh has more elements; the might also use a medium mesh. Sometimes the mesh is onl refined in the region where stresses are known to be varing rapidl. Tpicall, as the mesh is refined, the estimate of stress gets better. But, as the mesh gets ver fine, the additional improvement becomes less. Below we show the ideal variation in one stress component (σ ) at one point as a function of the number of elements. The finite element results would be the dots. The curve is meant to represent some fit through the points. It can be seen the stress changes with the number of elements, but that it appears to approach a value which would not change much more as the number of elements gets an large. The stress at this point is said to converge. σ Number of Elements Copright 2005 Paul S. Steif 25

26 4. Stress Concentrations and Singularities As has been pointed out earlier, a comprehensive theor commonl used to calculate stresses in bodies is the theor of linear elasticit. This theor is confined to circumstances in which the material remains linear elastic and changes in the shape and orientation of the bod are small. Nevertheless, these assumptions are relevant to a vast arra of situations commonl encountered in engineering. When there is a situation in which the assumptions of linear elasticit are valid, then linear elasticit provides the best predictions of stresses and deflection. However, there are circumstances in which simpler theories, those of mechanics of materials (aial loading, torsion and bending), provide predictions of stresses and deflection that are as good as, or nearl as good as, those of elasticit. But there are also situations in which the stresses or deflections cannot be estimated b such simple theories. Moreover, we usuall cannot solve the equations of linear elasticit either (even though the would give the best answers). That is because the equations of elasticit are comple partial differential equations. This was the motivation for developing finite element analsis (FEA). In finite element analsis we solve the equations of elasticit approimatel. Sometimes that approimation is ver good and other times it is not. It is useful for ou to have a sense of the relative accurac of these different methods. Increasing accurac Mechanics of Materials FEA Elasticit It is rare on a dail basis that engineers doing stress analsis will do an elasticit analsis. The will either do mechanics of materials or FEA. Nevertheless, it is useful to bear in mind that the stresses given b elasticit theor would be ideal, if the equations of elasticit could be solved. Further, the FEA results ma be as good as the elasticit solution (as the ideal). The are never better than the ideal, and sometimes, if ou are not careful, the are much worse than the idea. For some problems, mechanics of materials, FEA and elasticit give answers that are essentiall equivalent. If ou know this in advance, and as ou gain eperience ou will be able to detect such problems, then ou can just use mechanics of materials since it is the easiest to use. There are some situations in which mechanics of materials, FEA and Elasticit give different answers, although if ou are eperienced and thoughtful in doing FEA, ou can get the FEA to give answers that are close to those of elasticit. In some of those situations, ou can also use mechanics of materials in conjunction with Tables to get good estimates of the stresses. Finall, there are other situations in which neither FEA nor mechanics of materials can give the results of elasticit. In those situations, ou need to be etremel careful in interpreting the FEA or mechanics of materials results. Copright 2005 Paul S. Steif 26

27 In the net Chapter, we outline situations in which mechanics of materials leads to reasonabl accurate predictions of stress and deflection. (We also show there how mechanics of materials can be used to give ver rough estimates of these quantities, just to make sure ou have not made some gross error in FEA.) Here we consider situations in which FEA or elasticit is necessar to arrive at estimates of stress. Tpicall, these involve situations in which the stresses are locall concentrated, beond the normal variations of stress found in mechanics materials. It will be important to distinguish between situations in which FEA, if used appropriatel, can give good estimates of stresses, and those situations in FEA cannot. FEA is a ver powerful tool if used properl, but it can be misleading or dangerous in the hands of someone who is not full prepared to use it. In considering situations in which stresses are concentrated, we distinguish between two tpes of situations: Stress Concentrations and Singularities associated with Boundar Shape These concentrations are real and of genuine concern in design Stress Concentrations and Singularities induced b loading or supports These are due to the idealized, unrealistic was in which loads or supports are commonl applied in FEA Stress Concentrations and Singularities associated with Boundar Shape The influence of boundar shape on stresses is most easil discussed b reference to the problem of a bar with a transition from one width to another. In particular, we call attention to portions of the boundar at which there is a transition from a straight segment of one orientation to a straight segment of another orientation. Straight portions Radiused transition H re-entrant non-re-entrant h Copright 2005 Paul S. Steif 27

28 Near the transition, or corner, the bod occupies a sector with some angle. The sectors are labeled in the figure. When the sector occupies angle which is greater than 180, the transition is considered a re-entrant corner. When the sector occupies angle which is less than 180, the transition is considered a non-re-entrant corner. At a re-entrant corner (angle > 180 ), the stresses will be greater than if this feature were absent. In this situation the stress would be greater than the simple aial loading prediction of P/ht (where t is the thickness into the paper). B contrast, for non-re-entrant corners, the stresses are low (even lower than P/Ht here). The ratio of the actual maimum stress to the stress P/A is termed the stress concentration. The stress concentration depends on the radius of the transition. The smaller the radius, the more rapid the transition from one boundar to another, the higher is the stress concentrations. It is widel known in mechanical design that one needs to consider reducing stress concentrations, particularl when the loads are alternating, that is, when fatigue is of concern. Ver sharp corners are to be avoided. One can talk theoreticall about the radius of the transition going to zero, that is, an immediate transition from one orientation to the other or a trul sharp corner. In this case the theor of elasticit sas that the stresses are infinite. Of course, no corner is trul sharp; moreover, the assumptions of linear elasticit break down in such regions. Thus, the predictions of infinite stresses are not reall correct. Nevertheless, there are advanced theories (fracture mechanics) that enable one to interpret the infinite stress in a meaningful wa. Consider now the use of FEA to analze such a transition. When the radius is known, and one wants an accurate estimate of the stresses, then the right radius has to be included in the finite element model. Also, a sufficient number of mesh points must be placed around the transition. On the other hand, it is sometimes ver convenient to ignore the radius and set up a finite element model with the two boundaries meeting at a point. The FEA results near the transition will be completel unreliable. Elasticit theor, which FEA is tring to approimate, sas that the stresses are infinite at the corner. But, FEA alwas gives finite stresses. Rather than converge, the FEA stresses at the corner will simpl continue to increase as more mesh points are placed near the transition. Therefore, one has to recognize such situations in which the FEA results cannot offer good estimates of stresses. Concentration/singularit due to Loading We have introduced St. Venant s principle earlier in the chapter on formulation of elasticit problems. It states that the stresses far from region of load application are dependent onl on the load, not on the precise wa in which the loads are applied. However, the stresses in the immediate vicinit of load application are dependent on how the loads are applied. If the force is applied over a smaller area then the stresses locall Copright 2005 Paul S. Steif 28