Rak Part II: Shell structures. Antti H. Niemi Department of Civil and Structural Engineering

Transcription

1 Rak Part II: Shell structures Antti H. Niemi Department of Civil and Structural Engineering

2 Practicalities

3 Teacher Antti H. Niemi Postdoctoral Researcher Computational Structural Engineering Office: R225b Office hours: Wed or when available 3

4 Learning outcomes Recognizing plate and shell components in load-bearing structures Understanding of the fundamentals of plate and shell theories Applying principles of virtual work and minimum potential energy for plate and shell structures Analysing plate and shell structures by using analytical and numerical methods 4

5 Content and workload Contents: I. Shell geometry (week 44, ) II. General linear theory of shells (week 45, ) III. Membrane theory of shells (week 46, ) IV. Theory of edge effects (week 47, ) V. Shell buckling test (week 48, ) VI. Project work and revision (week 49, ) Workload for part II: 2.5 ECTS ~ hours of work ~ 10 hours/week plus exam 5

6 Coursework and assignments Total maximum score is 60 points. Examinations: 48 points. Exercises, homeworks and project: 12 points (weekly average). Weekly assignments: One quiz (3 points) Two exercise problems (6 points) One turn-in homework (3 points) Exam dates: Second partial exam: Wed, 9 Dec, Final exams: Thu, Mar 3, 2016 & Thu, May 19,

7 Weekly schedule Period II: Lectures: Tue & Wed Exercises: Fri

8 Week I: Shells in engineering and shell geometry Readings: Chapters 10 & 11 from Ventsel, E., & Krauthammer, T. (2001). Thin Plates and Shells. New York (p. 658). CRC Press.

9 Learning outcomes for week I Recognizing shell components in load-bearing structures. (Must know) Ability to parametrize surfaces and visualize them. (Must know). Knowledge of the concepts of metric and curvature tensors of surfaces and ability to calculate them for given parametrizations. (Good to know) Knowledge of geometric classification of surfaces. (Good to know). 9

10 Shells in engineering

11 Haukilahti water tower Constructed in Renovated in Owned by HSY. Two nested water containers of total volume 4100 m 3. Height: 45,3 m. 11

12 Haukilahti water tower (cont.) Structural design was carried out by M.Sc. Ilmari Hyppänen. Fairly sophisticated application of plate and shell theories. A super-computer was needed to solve linear systems of algebraic equations with unknowns. 12

13 Applications of shell structures Civil and architectural engineering: Large-span roofs Liquid containers, water tanks Containment shells and cooling towers Mechanical engineering: Piping systems, pressure vessels Aircraft, rockets, missiles and submarines Biomechanics: Artery walls in human cardiovascular system 13

14 Benefits of shell structures Efficient load-carrying mechanism by combination of membrane and bending actions Very high stiffness combined with light weight High aesthetic value 14

15 Shell geometry

16 Generic definitions 3D Bodies which are bounded by two closely spaced surfaces are referred to as shells (kuorirakenne). The locus of points which lie at equal distances from these two surfaces is called the middle surface (keskipinta). The thickness (kuoren paksuus) of the shell at any point of the middle surface is defined as the length of the line segment connecting the bounding surfaces and perpendicular to the middle surface. 16

17 Generic definitions (cont.) The shell geometry is entirely determined by specifying the form of the middle surface and the thickness at each point. The thickness may vary in magnitude from point to point but, for simplicity, we assume that it is a constant denoted by. The most important geometric characteristic of shells is curvature (kaarevuus), which couples stretching and bending deformations. 17

18 Slenderness Usually, the term shell incorporates a thinness/slenderness (ohuus/hoikkuus) assumption. This means that the dimensionless or relative thickness (suhteellinen paksuus), where is some characteristic length associated to the shell middle surface, is small in comparison with unity. Typically in civil engineering we have 18

19 Geometry of surfaces Definition A parametric surface (parametrinen pinta) is a smooth map When vary over, then trace out a surface: 19

20 Geometry of surfaces (cont.) The parametric equations can be given in the form where and are unit vectors along the coordinate axes. The parameters are called curvilinear coordinates (käyräviivaiset koordinaatit) of a given surface. The curves are referred to as the coordinate curves (koordinaattikäyrät). 20

21 Example: Monge s patch 21

22 Example: hemispherical dome A hemispherical dome of radius R and centered at the origin can be parametrized as where. 22

23 Tangent vectors and arc length The vectors are tangent to the coordinate curves. The differential of the arc length can be written as so that 23

24 Tangent vectors and arc length (cont.) Assuming that the curvilinear coordinates are orthogonal, that is we have The quantities are called Lamé parameters. 24

25 Normal curvature Consider an arbitrary curve defined on the surface by where is the arc length. The tangent to the curve is The principal normal of the curve is determined by where is the radius of curvature. 25

26 Normal curvature (cont.) Projection of the curvature vector to the normal of the surface is the normal curvature of the surface: where is called the curvature tensor of the surface. The metric tensor of the surface is defined as 26

27 Principal curvatures The normal curvature can be written as where are the first and second fundamental forms of the surface. The normal curvature has two extremal values called the principal curvatures (pääkaarevuudet) and it can be shown that the corresponding directions are mutually orthogonal. 27

28 Principal curvatures (cont.) If the coordinate lines coincide with the directions of principal curvature at each point, then these coordinates are called principal curvature coordinates (pääkaarevuuskoordinaatit). In the principal curvature coordinates, we have The principal curvatures can then be determined as 28

29 Unit vectors Assuming principal curvature coordinates, we can define an orthogonal system of unit vectors on the surface: This system can be used to represent vector quantities (displacements, forces) defined on the shell middle surface. 29

30 Derivatives of the unit vectors Assuming principal curvature coordinates, the derivatives of the unit vectors can be written in terms of the geometric parameters as 30

31 Classification of shell/surface geometries

32 Surface invariants The quantities are the mean curvature and Gaussian curvature. Points on the surface as classified based on the sign of the Gaussian curvature as follows 1. Elliptic, when K>0 (1/R 1 and 1/R 2 have the same signs) 2. Parabolic, when K=0 (1/R 1 =0 or 1/R 2 =0) 3. Hyperbolic, when K<0 (1/R 1 and 1/R 2 have different signs) 32

33 Classification based on Gaussian curvature 33

34 Surfaces of translation A surface of translation (translaatiopinta) is generated by a motion of a curve which is translated such that its points describe congruent curves. Roof structures often have a shape of a translation surface The equation of such surface can be prescribed as 34

35 Surfaces of translation (cont.) Paraboloid is a quadric surface that can be represented as Elliptic paraboloid Parabolic paraboloid Hyperbolic paraboloid 35

36 Ruled surfaces A ruled surface (viivoitinpinta) is such that through every point on the surface there is a straight line that lies on the surface. Plane Hyperbolic paraboloid (doubly ruled) Cylinder 36

37 Surfaces of revolution An important class of shells in engineering practice is shells of revolution (pyörähdyskuoret). A surface of revolution (pyörähdyspinta) is created when a plane curve, called meridian, rotates about an axis. If the curve is given in the form, and revolves around the -axis, then the natural parametrization is 37

38 Surfaces of revolution (cont.) Elimination of the parameters yields which is the equation for a surface of revolution. 38

39 Example: Cooling tower A line segment with end points (2,0,0) and (0,1,3) revolves around the z-axis. What is the parametric form of the surface? 39

40 Week II: General linear theory of shells Readings: Chapter 12 from Ventsel, E., & Krauthammer, T. (2001). Thin Plates and Shells. New York (p. 658). CRC Press.

41 Learning outcomes for week II Knowledge of the basic assumptions used in linear shell theories. (Must know) Knowledge of the physical and design quantities used in linear shell theories. (Must know) Ability to distinguish theories which take into account transverse shear deformations and theories which neglect them. (Good to know) Ability to derive equilibrium equations using the principle of minimum potential energy. (Nice to know) 41

42 Goal of shell theory According to W. Koiter and J. Simmons Shell theory attempts the impossible: to provide a two-dimensional representation of an intrinsically three-dimensional phenomenon. As in the theory of thin plates, the 3D problem of shell equilibirum can be reduced to essentially 2D problem formulated on the middle surface by imposing kinematic assumptions together with assumptions on the state of the stress. The formal process is known as dimension reduction (dimensioreduktio). 42

43 Fundamentals of shell theory

44 Basic assumptions 1. Straight material fibres which are perpendicular to the middle surface before deformation remain straight after deformation and do not change their length. 2. The normal stress to the middle surface vanishes in the normal direction. Assumption 1. is sometimes referred to as the Reissner-Mindlin kinematic assumption and assumption 2. as the plane stress assumption. 44

45 Perusoletukset suomeksi 1. Kuoren keskipintaa vasten kohtisuorat materiaalisäikeet, jotka ovat suoria ennen muodonmuutosta, säilyvät muodonmuutoksessa suorina eivätkä veny tai puristu. 2. Kuoren keskipinnan normaalin suuntainen normaalijännitys oletetaan häviävän pieneksi. 45

46 All you need is Love In addition to the above assumptions, the deformation of thin elastic shell is characacterized by the Kirchhoff-Love assumption: 3. Normal fibres to the undeformed middle surface remain normal after deformation Sama suomeksi: 3. Materiaalisäikeet jotka ovat kohtisuorassa kuoren keskipintaa vasten ennen muodonmuutosta, säilyvät kohtisuorassa muodonmuutoksen jälkeen. 46

47 Derivation of shell theories Shell theories can be developed from the general 3D theory of elasticity by the following steps: Formulation of a displacement Ansatz (yrite) according to the kinematic assumption. Expansion of the strain-displacement relations. Introduction of a constitutive law (compatible with the displacement Ansatz) to write stresses in terms of strains. Minimization of the total potential energy under prescribed kinematic constraints using variational methods or equilibrium equations. 47

48 Derivation of shell theories (cont.) The displacement Ansatz is formulated with respect to the assumed curvilinear coordinate system associated to the shell middle surface. When expanding the strain-displacement relations, variation of the basis vectors from point to point must be taken into account. We outline the process assuming that are principal curvature coordinates in context of linear theory of elasticity. 48

49 Shell kinematics

50 Displacement Ansatz The 3D shell domain may be described in the global Cartesian coordinate system as where and The Reissner-Mindlin kinematic assumption can be expressed as 50

51 Displacement Ansatz (cont.) The positive directions of the displacements coincide with the directions. The dimensionless quantities stand for the rotations of the normal to the middle surface (normaalin rotaatiot/ kiertymät) about tangents to the coordinate lines and, respectively. 51

52 Strain-displacement relations If is a system of Cartesian coordinates, then the linearized strain tensor is given by Let and define a local system of Cartesian coordinates by It follows that for 52

53 Strain-displacement relations (cont.) From the formulas for the derivatives of the unit vectors, we obtain at the point P : Chain rule yields: Which can be inverted up to the accuracy of as 53

54 Strain-displacement relations (cont.) The derivatives of the displacement vector read The strains can be expanded as 54

55 Membrane strains The components of the membrane strain tensor come out as 55

56 Transverse shear strains The transverse shear strains are determined as 56

57 Bending strains Reissner-Naghdi model The components of the bending strain (or elastic curvature) tensor are 57

58 Love is all you need If the transverse shear strains are neglected by imposing the Kirchhoff-Love constraints of vanishing transverse shear strains, then the rotations become defined as The displacement field of the shell is then determined solely in terms of the displacement components 58

59 Bending strains Love-Kirchhoff-Novozhilov model The simplest expansion of the bending strains are obtained by adding to and utilizing the Codazzi relations (verify this if you are interested!): 59

60 Example: shallow shells A Shallow shell (matala kuori) is a shell which deviates only slightly from a plate. If the middle surface is descibed by the graph then the shallowness assumption reads All shells are locally shallow. 60

61 Shallow shells (cont.) For shallow shells, in terms of, we have In the simplest version of shallow shell theory, the tangential displacements are neglected in the expressions of the bending strains: 61

62 Shell statics

63 Hooke s law Hooke's law for a homogeneous and isotropic 3D body: Here is the shear modulus (liukukerroin), and and are Young s modulus (kimmokerroin) and Poisson s ratio (Poissonin luku), respectively. 63

64 Plane stress condition In the theory of thin shells it is assumed that the transverse normal stress is neglible as compared with the other stresses: Under this assumption the other direct stresses become (Exercise) In orthogonal coordinate system, the stresses are symmetric: 64

65 Strain energy The strain energy of a shell may be written by utilizing the general formula of the theory of elasticity: Here represents the 3D shell body so that 65

66 Stress resultants and couples Within the adopted accuracy, the strain energy may be expressed as It is customary to represent the general 3D stress field using stress resultants and couples (jännitysresultantit). The in-plane normal and shear forces are defined as 66

67 Stress resultants and couples (cont.) The transverse shear forces (leikkausvoimat) are defined as The bending and twisting moments (taivutus- ja vääntömomentit) are defined as The stress resultants are forces per unit length while the stress couples are moments per unit length. 67

68 Stress resultants and couples (cont.) Referring to the strain expansions derived earlier, we may write the consitutive relations in terms of the stress resultants as 68

69 Strain energy The strain energy can be written in terms of the stress resultants as 69

70 Load potential External forces consisting of surface loads applied to upper and lower surfaces of the shell and volume loads are also replaced by statically equivalent forces distributed over the middle surface. Resolving such a load into components along the directions yields the load potential 70

71 Equilibrium equations The differential equations of static equilibrium can be derived by calculus of variations: 71

72 Summary of physical quantities Kinematics Displacements of the middle surface: Dimensionless rotations of the normal to the middle surface: Membrane strains: Transverse shear strains: Bending strains or elastic curvatures: 72

73 Summary of physical quantities Statics Normal forces per unit length: Transverse shear forces per unit length: Bending and twisting moments per unit length: 73

74 Week III: Shell membrane theory Readings: Chapters 13 and 14 from Ventsel, E., & Krauthammer, T. (2001). Thin Plates and Shells. New York (p. 658). CRC Press.

75 Learning outcomes for week III Knowledge of the types of state of stress for thin shells. Knowledge of the assumptions of shell membrane theory. Ability to apply shell membrane theory in the static analysis of axisymmetrically loaded circular cylindrical shells and spherical shells. 75

76 Structure of shell equations

77 Shell equations All above discussed equations (strain-displacement, constitutive, equilibrium) are linear. For general shells, the strain-displacement relations and equilibrium equations have variable coefficients. The total order of the shell equation system is eight in the Love-Kirchhoff-Novozhilov theory and ten in the Reissner- Naghdi theory 77

78 Shell equations (cont.) It is clear, that closed-form analytic solutions to the shell equations are out of reach as a rule. As a consequence, practical stress analysis of shell structures requires simplification of the general theory or utilization of numerical methods (FEM). Becoming a confident finite element analyst takes time. Knowledge of basic analytical solution techniques for shells is useful in the verification of FE models. 78

79 Shell deformation states

80 Membrane-dominated deformations Flexural stresses are negligible compared with the membrane stresses. Governing equations can be obtained from the general equations by neglecting bending moments and transverse shear forces. 80

81 Bending-dominated deformations Membrane stresses are neglible in comparison with the flexural stresses. Even small bending moments may cause large flexural stresses. Pure bending deformations are potentially dangerous. 81

82 Intermediate or mixed state of stress Typical situation in practice where bending occurs near edges, junctions, reinforcements. A membrane-type deformation takes place at the main portion of the shell. The edge effect is localized in relatively small area. 82

83 Shell membrane theory

84 Shell membrane theory The general theory of shells can be considerably simplified if the moments are neglected. This is justified if 1. The shell has very small bending stiffness 2. The bending strains are small 84

85 Conditions for membrane state of stress Absolutely flexible shell, a membrane (cloths, sails etc.), cannot sustain any compression so that calculations remain valid only when tensile forces are obtained at all sections. A shell with finite bending stiffness, may sustain also compression in the momentless state of stress. 85

86 Governing equations of shell membrane theory Neglecting the bending moments from the general equilibrium equations for a shell element imply that also The equilibrium equations become 86

87 Governing equations of shell membrane theory (cont.) Here the number of equations matches with the number of unknowns so that these problems are internally statically determinate. Assuming that the equilibrium equations can be solved with some boundary conditions, the displacement can be determined from 87

88 Inextensional modes The general solution of the previous system may be written as where system is a particular solution of the system and is the general solution of the homegeneous Since the latter does not contribute to membrane strains, it must correspond to pure bending or rigid body displacements. 88

89 Inextensional modes (cont.) In membrane theory inextensional displacement modes (puhtaat taipumamoodit) with must be eliminated e.g. by the boundary conditions. Concerning the boundary conditions in shell membrane theory, we note that the total degree of the governing equations is four so that two boundary conditions can be specified on each edge. 89

90 Boundary conditions If the boundary conditions are specified in terms of stress resultants, then only the in-plane forces are specified. If the boundary conditions are specified, then only tangential displacements can be prescribed. Specification of transverse shear forces, moments, transverse deflections or rotations will in general conflict with the initial assumptions of the shell membrane theory. 90

91 Conditions of shell membrane theory 1. The boundaries of a shell are free from transverse shear forces and moments. 2. The transverse deflection and rotations are not constrained at the edges. 3. The shell surface is smooth. 4. The external surface and edge loads are smoothly varying functions of the coordinates. 91

92 Applications of shell membrane theory

93 Example: Cylindrical shell roof under dead load Consider a cylindrical shell roof of radius and length L. Utilization of the cylindrical coordinates to parametrize the shell middle surface gives and If is the weight per unit area of the shell surface, we have We assume that the curvilinear edges are supported by rigid diaphrams so that. 93

94 Cylindrical shell roof (cont.) Denoting, the equilibrium equations of shell membrane theory become These can be solved for the stress resultants 94

95 Example: spherical domes Consider a spherical dome of radius Parametrizing the middle surface using the meridional and circumferential angles, we get If the dome is subject to a dead load per unit surface area, then we have 95

96 Spherical dome (cont.) The governing equations become (axisymmetry!) and can be solved for 96

97 Spherical dome (cont.) The meridional force is compressive along the meridian of the shell,, regardless of the opening angle. The circumferential force changes sign when If the opening angle is greater, then the circumferential force is tensile when 97

98 Week IV: Theory of edge effects Readings: Chapter 15, 16 and 17.5 from Ventsel, E., & Krauthammer, T. (2001). Thin Plates and Shells. New York (p. 658). CRC Press.

99 Learning outcomes for week IV Knowing the importance of bending stresses for design reliability of thin shells. (Must know) Knowing the simple edge effect. (Good to know) Recognizing complex edge effects. (Nice to know) Ability to perform moment analysis of axisymmetrically loaded cylindrical and spherical shells. (Good to know) 99

100 Introduction to moment theory of shells The membrane theory is sufficiently accurate for many commonly encountered loading conditions. However, bending and shear stresses develop at locations where deflections are restricted, there is a rapid change in geometry (such as a junction), or a concentrated line/point load is acting. 100

101 Introduction to moment theory of shells (cont.) The complete governing equations of moment theory can be solved analytically only in rare special cases. However, because the bending stresses usually decay rapidly away from the disturbance which causes them, the governing equations can be simplified and solved analytically to get an approximation. Finite element method (FEM) can be used for numerical solution. Analytical solutions are useful for verifying initial designs and numerical results. 101

102 Axisymmetrically loaded circular cylindrical shells

103 Axisymmetrically loaded circular cylindrical shells Typical applications are pipelines, boilers and vessels subject to internal pressure. A closed circular cylindrical shell of radius will deform axisymmetrically if the external load and boundary conditions do not depend upon the angular coordinate. This means that and 103

104 Equations of static equilibrium The surviving equilibrium equations are The first equation yields 104

105 Strain-displacement relations The strain-displacement relations are The stress resultants and couples are and 105

106 Governing differential equation The equations can be condensed to a single differential equation or where 106

107 Solution The general solution can be represented as where is a particular solution of the non-homogeneous equation and is a complete solution of the homogeneous equation. The latter is sought in the form so that This equation has four roots 107

108 Solution (cont.) The solution has the form The constansts are determined using boundary conditions: 108

109 Long cylindrical shells For long cylindrical shells with it is possible to set near the edge For edge loads we have then 109

110 Long cylindrical shell subject to edge loads 110

111 Simplified shallow shell theory

112 Simplified shallow shell theory Simplified equilibrium equations for shallow shells in Cartesian coordinates : Here is the biharmonic operator, or bilaplacian. 112

113 Shell equation The homogeneous equation in terms of displacements: 113

114 Shell equation The shell equations can be condensed to a single equation, which in the homogeneous case reads where is the so called Vlasov operator. 114

115 Simplified theory of edge effects

116 Characterization of edge effects Membrane solution alone cannot satisfy the boundary conditions at the edges or conditions of continuity. Lines of distortion: Physical edges Lines along which external loads have discontinuities Lines along which the curvature changes abruptly Lines along which the material parameters or shell thickness change abruptly Edge effect cam be defined as a deformation state at which the displacements, stresses and strains decay rapidly normal to the line, but vary slowly along the line. 116

117 Simple edge effect Assume that the line of distortion coincides with - coordinate line and that (simple edge effect). For rapidly decaying functions in x-direction we have Integrating four times the simplified shell equation and discarding the arbitrary slowly varying functions, leads us to the governing equation of the simple edge effect: 117

118 Simple edge effect (cont.) The decaying solution has the form where In other words, the deformations and stresses associated to the simple edge effect decay in the length scale when moving away from the line of disturbance. Obviously, the edge effect encountered in axisymmetrically loaded circular cylinders is a special case of the simple edge effect. 118

119 Simple edge effect (cont.) The associated in-plane stress resultants are The moments are The transverse shear forces are 119

120 Week V: Shell buckling Readings: Chapter 19 from Ventsel, E., & Krauthammer, T. (2001). Thin Plates and Shells. New York (p. 658). CRC Press.

121 Buckling is a mysterious phenomenon Visit shellbuckling.com 121

122 Learning outcomes Knowledge of the the basic concepts of thin shell stability Knowledge of imperfection sensitivity of shells Ability to determine the critical load for circular cylindrical shells using linear stability analysis 122

123 Concepts of shell stability

124 Load-deflection diagram for columns and plates Branch OA refers to initial momentless configuration CA and AD refer to buckled configurations Transition point A is a bifurcation point and the load at that point is called the critical load The transition between stable equilibriums is smooth 124

125 Load-deflection diagram for cylindrical shell under axial loading OA and BD are stable equilibrium branches while AB is is unstable. The transition occurs by a jump. 125

126 Snap-through buckling The sudden transition from the initial configuration to the buckled configuration is termed snap-through buckling (napsahdus-lommahdus). Geometrically nonlinear theory is needed in the general case. A real shell has geometric and other imperfections which are equivalent to prebuckling deformations. Unlike columns and flat plates which are relatively insensitive to slight imperfections, shells are very sensitive prima donnas of structures. 126

127 Snap-through buckling (cont.) Three distinct critical loads characterize the buckling behavior of thin shells: 1. is the largest load up to which the initial configuration of the perfect shell remains stable with respect to infinitesimal perturbations. 2. is the largest load up to which the initial configuration of the perfect shell remains stable with respect to both infinitesimal and finite disturbances. 3. is the critical load of the real shell, or the buckling load. 127

128 Critical loads The values of can be determined using linear differential equations. Determination of requires geometrically nonlinear theory and utilization of numerical methods. The critical load is somewhere between the upper and lower critical loads. Because of the sensitivity of to imperfections, severe discrepancies between the theoretical and experimental results occur. 128

129 Critical loads (cont.) Linear stability theory is not sufficient to predict completely the buckling and postbuckling behavior of thin shells. Application of geometrically nonlinear theory is non-trivial. In practice, critical loads obtained by linear stability analysis can be reduced e.g. by so called knock-down factors (typical value is around 1/6) for determining the allowable design buckling loads. 129

130 Linear buckling analysis

131 Stability equation for shallow shells Like edge effects, buckling modes feature rapidly varying deformation components in some cases and can be analysed using shallow shell theory. Assuming that denotes the prebuckled value of the deflection and stands for the transition to adjacent configuration of equilibrium, the governing stability equation for shallow shells becomes where 131

132 Critical load for an axially compressed cylinder Consider an axially compressed circular cylinder with radius and length that is simply supported Assuming that the radial deflection after buckling is axially symmetric, we may postulate In addition, we have 132

133 Critical load for an axially compressed cylinder (cont.) Substitution of the Ansatz to the governing stability equation yields where The minimum value of occurs at 133

134 Critical load for an axially compressed cylinder (cont.) The critical value of the applied compressive load is The correspoding stress is The same critical load is obtained by considering more general modes of the form 134

135 Shell buckling test Tue & Wed Fri

136 Group work A cylindrical object is provided by the test hall staff to each group. You may also provide an object of your own (e.g. a soda can). The object is axially compressed up to failure and destroyed. One member of the group should give his/her address to the test hall staff for receiving the measument data. Each group should prepare a laboratory report (minimum 2 pages). 136

137 Laboratory report The report should include a theory section where the physical phenomenon and formulas relevant to the experiment are derived or presented. The experimental results should be compared with theoretical values (analytical/numerical) and possible discrepancies should be commented. In particular, sources of errors and uncertainties should be commented and quantified. 137

138 Laboratory report One member of each group should return the report (including names of the group members) to Mycourses by