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

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© 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 Method of sections, 86, 96, 103, 108, 116 Foundation, 138 Moment distribution method, 200 Frame, 114 Movement, 2 three-hinged, 114 invertible, 2 Free body diagram, 37, 79 regular, 2 Funicular curve, 41, 43 Funicular polygon, 41, 103 N G Navier form of the elastic problem, 19 Girder, 169 Neutral axis, 137 Green-Lagrange strain tensor, 16 Nominal stress tensor, 12 Groin vaults, 68

O H Ordinary differential equation, 179 Hollow profile, 156 Hooke’s law, 17 Hypar, see also Hyperbolic paraboloids P Hyperbolic paraboloids, 68 Pavilion vaults, 66, 68 Hyperstatic structures, 47, 61, 192 Permanent loads, 45 Hypothesis of small displacements, 2 Perturbations Hypothesis of small strains, 7 arches, 61, 64 Polar decomposition, 7 Prandtl function, 176 I Principle of Virtual Displacements, 89 Internal force, 33, 106 Principle of Virtual Forces, 89 axial, 34 Principle of Virtual Work, 87, 140 diagrams, 108, 111, 112 shear, 106 Internal moment bending, 107 R diagrams, 108 Radius of gyration, 139 torsion, 107 Rise, 33 twisting, 107 Rotation tensor, 7 Internal static indeterminacy, 192 Invariants of tensor, 12 Isostatic structures, 36, 192, 194 S Saint Venant’s principle, 1 Saint Venant tensor, see also Compatibility K tensor Kronecker delta tensor, 7 Second Piola–Kirchhoff stress tensor, 12, 16 Index 235

Shape function, 207 Prandtl model, 169 Shear center, 155 rate of twist, 169 Shear flow, 154 Saint Venant model, 169 Slenderness ratio, 46 Torsional stiffness, 175, 185 Slope deflection method, 199 Torsion flow, 184 Small displacements, 2, 34 Traction, 37 Small strains, 7 Truss, 71, 75 Span, 33 Howe, 93 Specific weight, 13 lattice, 95 Spindle, 68 long, 94 Springer (of an arch), 61 Poncelet, 94 Stable structure, 192 Queen-post, 76 Statically determinate structure, 62, 80, 114, three-dimensional, 95 115 Statically indeterminate structure, 80, 84 Stiffness U axial, 88 Unstable structure, 80, 192 Stiffness method, 191, 194, 195 Strength of materials, 20 Stress vector, 10 V Strong form, 204 Vector of internal forces, 106 Strut, 71 Vector of internal moments, 106 Support, 37 Vierendeel beam, 118 pinned, 78 Virtual displacement, 89 roller, 78 Von Mises stress, 18, 153 Voussoir, 61

T Thin-walled profiles, 153 W Thrust Network Analysis (TNA), 68 Warping function, 177 Tie, 71 Weak form, 204 Timoshenko shear coefficient, 161 Torsion, 169 center of torsion, 171 Y constant, 175 Yield stress, 45