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Indicial Notation Appendix A Indicial Notation A.1 Indicial Notation for Vector and Matrix Operations In tensor analysis, an extensive use of indicial notation is made. Operations using Cartesian components of vectors and matrices can be expressed efficiently using this notation. A.1.1 Elements of a Matrix A set of n variables x1, x2,...,xn is denoted using indicial notation such as as xi , i = 1, 2,...,n. Let us consider now the equation that describes a plane in a three dimensional space x1, x2, x3, a1x1 + a2x2 + a3x3 = p, where ai and p are constants. This equation can be expressed alternatively as 3 ai xi = p. i=1 At this point, we can use the convention usually referred to as Einstein notation or Einstein summation convention, which implies summation over a set of indexed terms in a formula when an index appears twice in a single term along the variational rank of the indexes (1 to 3 in the above equation). Therefore, ai xi = p In the same spirit, © Springer International Publishing AG, part of Springer Nature 2018 227 E. Cueto and D. González, An Introduction to Structural Mechanics for Architects, Structural Integrity 4, https://doi.org/10.1007/978-3-319-72935-0 228 Appendix A: Indicial Notation • Vectorial scalar product: u · v = ui vi • Norm of a vector: √ √ ||u|| = u · u = ui ui • Derivative of a function: n ∂ f ∂ f df = dx = dx = f, dx ∂x i ∂x i i i i=1 i i 1 Let us define the Kronecker Delta δij as a symbol whose values can be, = δ = 1ifi j ij 0ifi = j In the same way, let us define the permutation symbol εijk as, ⎧ ⎨ 0ifi = jj= ki= k ε = , , ijk ⎩ 1ifi j k for an even number permutation −1ifi, j, k form an odd permutation A.1. Demonstrate the next properties in indicial notation: • δijδij = 3 • εijkεijk = 6 • δijv j = vi • The matrix determinant 3 × 3 can be expressed like |A| = εijkai1a j2ak3 • The three vectorial components of a vector product of two vectors are (v × w)i = εijkv j wk • The Kronecker Delta function and the permutation symbol have the next relation- ship εijkεist = δ jsδkt − δ jtδks Solution A.1. • Indexes i, j appear repeated on the equation, so we must sum them from 1 to 3. Using the definition of the Kronecker Delta function, we have δ11δ11 + δ12δ12 + δ13δ13 + δ21δ21 + δ22δ22 + δ23δ23 + δ31δ31 + +δ32δ32 + δ33δ33 = δ11δ11 + δ22δ22 + δ33δ33 = 1 + 1 + 1 = 3 • Considering i = 1 and varying j, k from 1 to 3 we have 1It is a second order tensor, actually. Appendix A: Indicial Notation 229 ε111ε111 + ε112ε112 + ε113ε113 + ε121ε121 + ε122ε122 + ε123ε123 + +ε131ε131 + ε132ε132 + ε133ε133 = ε123ε123 + ε132ε132 = = 1 · 1 + (−1) · (−1) = 2 For i = 2, ε211ε211 + ε212ε212 + ε213ε213 + ε221ε221 + ε222ε222 + ε223ε223 + +ε231ε231 + ε232ε232 + ε233ε233 = ε213ε213 + ε231ε231 = = (−1) · (−1) + 1 · 1 = 2 Finally, for i = 3, ε3 jkε3 jk = 2, we have, εijkεijk = 6 • Since δ ji vanishes for i = j then δijv j = vi , ⎤ ⎤ δ11 · v1 + δ12 · v2 + δ13 · v3 v1 ⎦ ⎦ δ21 · v1 + δ22 · v2 + δ23 · v3 = v2 = vi δ31 · v1 + δ32 · v2 + δ33 · v3 v3 • The determinant is defined by: a11 a12 a13 a21 a22 a23 = a31 a32 a33 = a11a22a33 + a21a32a13 + a31a12a23 − a11a32a23 − a12a21a33 − a13a22a31 = ε123a11a22a33 + ε231a21a32a13 + ε312a31a12a23 + ε132a11a32a23 + ε213a12a21a33 +ε321a13a22a31 = εijkai1a j2ak3, which is precisely the suggested expression (summing for i, j, k from1to3). • If, in the definition of determinant of a matrix, we change the first column by the unit vectors ei , the second component by v, and the third one by w, the vector product is given by the following indicial expression (v × w) = εijkei v j wk , (v × w)i = εijkv j wk , 230 Appendix A: Indicial Notation ˆ ˆ ˆ i j k ˆ ˆ ˆ ˆ ˆ ˆ v × w = v v v = iv2w3 + kv1w2 + jv3w1 − kv2w1 − iv3w2 − jv1w3 = 1 2 3 w1 w2 w3 εijkeˆi v j wk . To obtain the ith component it is enough to multiply by ei ,so (v × w)i = εijkv j wk . • Finally, it is easy to prove the last expression by operating component by compo- nent: –Ifj = k, both being free indexes in the expression, and taking into account that the permutation symbol vanishes when the indexes are the same, the first member of the equation is equal to zero. As per the second term, if s = j = k and t = j = k, then the second term is 1 − 1 = 0; conversely, if s = j = k and t = j = k, then the second term is 0 − 0 = 0; if s = j = k and t = j = k, then the second term is 0 − 0 = 0. –Ifs = t, following the same rationale, we obtain the same result by just changing s, t by i, j. For i = 1 for instance: ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ε123ε123 δ22δ33 − δ23δ32 1 − 0 1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ε132ε123 ⎥ ⎢ δ32δ23 − δ33δ22 ⎥ ⎢ 0 − 1 ⎥ ⎢ −1 ⎥ ε1 jkε1st ⎣ ⎦ = ⎣ ⎦ = ⎣ ⎦ = ⎣ ⎦ ε123ε132 δ23δ32 − δ22δ33 0 − 1 −1 ε132ε132 δ33δ22 − δ32δ23 1 − 0 1 ⎡ ⎤ 1 · 1 ⎢ (− ) · ⎥ = ⎢ 1 1 ⎥ = δ δ − δ δ . ⎣ 1 · (−1) ⎦ js kt jt ks (−1) · (−1) References 1. Betsch P, Steinmann P (2002) Frame-indifferent beam finite elements based upon the geomet- rically exact beam theory. Int J Numer Methods Eng 54(12):1775–1788 2. Block P, Ochsendorf J (2007) Thrust network analysis: a new methodology for three- dimensional equilibrium. J Int Assoc Shell Spatial Struct 48(3):167–173 3. Bonet J, Wood RD (2008) Nonlinear continuum mechanics for finite element analysis. Cam- bridge University Press, Cambridge 4. Calatrava S (1981) Zur Faltbarkeit von Fachwerke. PhD thesis, ETH Zurich. https://www. research-collection.ethz.ch/handle/20.500.11850/137273 5. Courant R (1943) Variational methods for the solution of problems of equilibrium and vibra- tions. Bull Am Math Assoc 42:2165–2186 6. Cross H (1930) Analysis of continuous frames by distributing fixed-end moments. In: Proceed- ings of the American society of civil engineers, ASCE, pp 919–928 7. Felippa CA (2001) A historical outline of matrix structural analysis: a play in three acts. Comput Struct 79(14):1313–1324 8. Fish J, Belytschko T (2007) A first course in finite elements. Wiley, New Jersey 9. Han SM, Benaroya H, Wei T (1999) Dynamics of transversely vibrating beams using four engineering theories. J Sound Vibr 225(5):935–988 10. Heyman J (1982) The masonry arch. Ellis Horwood 11. Huerta S (2004) Arcos, bóvedas y cúpulas. Geometría y equilibrio en el cálculo tradicional de estructuras de fábrica, Instituto Juan de Herrera 12. Huerta S (2008) The analysis of masonry architecture: a historical approach: to the memory of professor henry j. cowan. Architect Sci Rev 51(4):297–328 13. Jeleni´c G, Crisfield MA (1999) Geometrically exact 3d beam theory: implementation of a strain-invariant finite element for statics and dynamics. Comput Methods Appl Mech Eng 171(1):141–171 14. Love AEH (1927) A treatise on the mathematical theory of elasticity. Cambridge University Press, Cambridge 15. Maney GA (1915) Studies in engineering. University of Minnesota, Minneapolis 16. Muttoni A (2011) The art of structures. EPFL Press, Lausanne 17. Muttoni A, Schwartz J, Thürlimann B (1996) Design of concrete structures with stress fields. Springer Science & Business Media, Berlin 18. Reddy JN (2008) An introduction to continuum mechanics. Cambridge University Press, Cam- bridge 19. Simo JC, Vu-Quoc L (1988) On the dynamics in space of rods undergoing large motions – a geometrically exact approach. Comput Methods Appl Mech Eng 66(2):125–161 20. Turner MJ, Clough RJ, Martin HC, Topp LJ (1956) Stiffness and deflection analysis of complex structures. J Aeronaut Sci 23(9):805–823 © Springer International Publishing AG, part of Springer Nature 2018 231 E. Cueto and D. González, An Introduction to Structural Mechanics for Architects, Structural Integrity 4, https://doi.org/10.1007/978-3-319-72935-0 Index A Center of torsion, 171 Arches, 64 Chord rotation, 200, 202 Arris, 68 Collignon formula, see also Collignon– Axial internal force, 77 Jourawski formula Axial stiffness, 132 Collignon–Jourawski formula, 152 Compatibility method, see also Flexibility method B Compatibility tensor, 9 Barrel vaults, 70 Configuration, 2 Beam, 103 Reference, 2 curvature, 130 Constitutive equations, 14, 161 deep, see also Deep Beam Continuousd medium, 1 Euler–Bernoulli–Navier model, 127 Cremona diagram, 38 floor, 169 Cross method, see also Moment distribution Frenet–Serret trihedron, 105 method Gerber, 113 Cross section, 104, 105, 107, 127, 128, 149, intrinsic coordinates, 105 159 Spandrel, 169 core, 138 Timoshenko model, 157 Vierendeel, 118 Beltrami–Michell form of the elastic prob- D lem, 19 Dead loads, 46 Bending, 103 Deep beam, 157 Bending moment, 103 Deformation gradient, 4, 5, 15 Bridge Deformed configuration, 193 bowstring, 70, 77 Degree of hyperstaticity, 192 tied arch, 70 Degree of indeterminacy, 192 Buckling, 37 Direct stiffness method, 202 Displacement field, 6 Displacement gradient, 7 C Duhamel–Neumann’s law, 15 Cable beams, 47, 79 Cantilever, 71 Carryover factor, 201 E Catenary, 45 Eccentricity, 138 Cauchy strain tensor, 7 Elastic section modulus, 135 Cauchy stress tensor, 12 Envelope, 63 © Springer International Publishing AG, part of Springer Nature 2018 233 E. Cueto and D. González, An Introduction to Structural Mechanics for Architects, Structural Integrity 4, https://doi.org/10.1007/978-3-319-72935-0 234 Index Equilibrium equations, 85, 115 L Equilibrium method, see also Stiffness Limit analysis method arches, 61 External static indeterminacy, 192 Load-displacement curve, 17 Lower bound theorem, 62, 103, 158 F Field, 6 M First area moment, 132, 152 Masonry structures, 60 First Piola–Kirchhoff stress tensor, 12, 14, Mechanics of materials, 20 15 Membrane, 47 Fixed-end moments, 198 Method of consistent deformations, see also Flexibility method, 191, 194 Flexibility method Force polygon, 37
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