# 202 Index. failure, 26 field equation, 122 force, 1

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1 Index acceleration, 12, 161 admissible function, 155 admissible stress, 32 Airy's stress function, 122, 124 d'alembert's principle, 165, 167, 177 amplitude, 171 analogy, 76 anisotropic material, 20 aperiodic motion, 170 area, 38 auxiliary condition, 164 axial force, 37 axis of beam, 37 balance - of angular momentum, 12 - of energy, 14 - of linear momentum, 12 beam, 37,103 - axis, 37 - complementary energy, curved,95 - deflection, 63 - elastic axis, 53 - non-prismatic, 60 - prismatic, 37 - strain energy, thin-walled cross section, 48 beam axis - deflection, 38, 39 - slope, 38 Beltrami's equations, 23 bending - energy, non-uniform, 40 - pure, 37 - stiffness, 136, 155 Bernoulli's hypothesis, 37, 40, 96, 134 Betti's theorem, 104 bifurcation point, 151 bifurcation problem, 150 body force, 12 - field, 123 Boltzmann's theorem, 3 boundary condition, 22, 72, 73, 88, deflection, 65 - dynamical, 65, 137, geometrical, 154, 155, 157 boundary term, 155 Bredt's first formula, 79 Bredt's second formula, 79 buckling load, 154 buckling problem, 155 bulk modulus, 20 cartesian coordinates, 2 Castigliano's first theorem, 32 Castigliano's second theorem, 32, 105 centerline, 77 characteristic equation, 151, 169 circular plate, 137 collapse load, 152 column buckling, 153 compatibility equation, 11, 122, 123 complementary energy, 21, 106, 109, density, 32 - minimum, 32 - plate, specific, 22, 106 complementary virtual work, 31 compliance tensor, 20 condition - auxiliary, dynamical, 65, 137, geometrical, 154, 155, initial, kinematical, 65 - stationarity, 106 configuration, 162 conical shell, 143 conservation of mass, 12 conservative force, 162 conservative system, 162, 164, 178 constraint, 163

2 202 Index - holonomic rheonomic, scleronomic, 163 contact force, 12 continuum mechanics, 1 coordinates - cartesian, 2 - centerline, 77 - cylindrical, 10, 13 - polar, 123, 137, spherical, 11, 13 correction factor, 76 critical damping, 170 critical load, 150, cross section, 37 curvature - change of, 97, 99 - large, 96 - principal, radius of, 38, 64, 95 - small, 99 curved beam, 95 - deflection, 99 - normal stress, 98 - shear stress, 98 cylindrical coordinates, 10, 13 cylindrical shell, 143 damped oscillation, 170 damped system, 169 damping - critical, factor, force, 169 dead loading, 152 deflection, 37, 63, 99 - boundary condition, 65 - differential equation, 64 - bending, 37 - curve, mode, plate, statical, 165 deformation, 9 degree of freedom, 150, 152, 162, 165 degree of indeterminacy, 105 density, 12 deviator, 6, 11 dilatation, 11, 23 Dirichlet boundary term, 155 disk, 131, 132 displacement, 9, 22, 79 - components, 9, 86 - curved beam, 99 - differential equation, excited, method, 33 - transmitted, virtual, 15, 154, warping, 73 distortion, 11 distributed force, 56, 95 distributed load, 55 divergence theorem, 12, 14, 15,31 dyadic product, 3 dynamic absorber, 179 dynamic coupling, 178 dynamical condition, 65, 137, 157 dynamical problem, 161 effective collapse load, 152 eigenfrequency, 165, 178 eigenvalue, 6, 156, 179 Einstein's - summation convention, 2 elastic axis, 53 elastic energy, 142 elastic material, 19 elasticity, 19 elasticity tensor, 20 energy, 14,21 - balance, 15 - internal, 21 - kinetic, 14, 161, potential, 155, principle, stored, 155 Engesser's theorem, 32, 105, 111 equation of motion, 12, 161, 169, 177 equilibrium equation, 13, 15,72, 121, 134 equivalent stress, 26 - maximum principal stress, 27 - maximum shear stress, 27 - shear strain energy, 28 - total strain energy, 27 Euler buckling load, 154 Euler theory, 153 Euler-Langrange equation, 164 Eulers's equation, 124 excited displacement, 174 exciting force, 174 extensional stiffness, 155 external force, 15 external work, 14 failure, 26 field equation, 122 force, 1

3 Index axial,37 - body, 12 - conservative, contact, 12 - distributed, 56, 95 - exciting, external, 15 - gravitational, harmonic, inertia, internal, 16 - long-range, 12, 14, 15 - non-conservative, short-range, 12, 14 - transmitted, vibratory, 174 forced vibration, 171 form factor, 80 form factor of shear, 109 free vibration, 167, 168 frequency, 166 frequency equation, 178 general solution, 124, 139 generalized - coordinate, displacement, 33, force, 33, 103, 162 geometrical condition, 154, 155, 157 global force equilibrium, 12 gravitational force, 165 Hamilton's principle, 162 harmonic excitation, 171 harmonic force, 171 harmonic oscillation, 171 holonomic rheonomic constraint, 163 homogeneity, 19 homogeneous equation, 151, 156 Hooke's law, 19,24, 135 hydrostatic stress, 23 inertia force, 168 influence coefficient, 103, 104, 112, 167 initial condition, 166 internal energy, 21 - specific, 21 internal force, 16 internal work, 14 invariant - natural,6 - principal, 6 isolator, 174 isotropic material, 19 kinematical condition, 65 kinetic energy, 14, 161, 163 Kirchhoff's theory, 135 Kronecker delta, 5 Lagrange multiplier, 164 Lagrange's equation, 177 Lagrangian determinant, 178, 179 Lagrangian function, 162 Lame coefficients, 20 Lame-Navier's equation, 23 Laplace equation, 123 Laplace operator, 23, 123 large curvature, 96 linear expansion coefficient, 23 local force equilibrium, 13 logarithmic decrement, 170 long-range force, 12, 14, 15 lumped mass, 166 magnification factor, 172, 173 mass, 12, conservation of, 12 - lumped, matrix, 178 mass moment of inertia, 166 material parameter, 20 material time derivative, 12 maximum shear stress, 6 Maxwell's reciprocity theorem, 104 mean stress, 6, 23 membrane - force, theory, 143 Menabrea's theorem, 106 meridian, 143 middle surface, 131, 143 minimum complementary energy, 32 v. Mises criterion, 28 mode of vibration, 179 modulus of elasticity, 19 moment, 134, 136 moment equilibrium equation, 135 moment of inertia, 38, 40, 80 - curved beam, 97 - parallel-axis theorem, 41 - polar, 40, 73 - principal values, 40 motion, I, aperiodic, oscillatory, 170 multiplier method, 164

4 204 Index natural frequency, 179 natural invariant, 6 neutral axis, 39, 98 neutral stability, 149 Newton's second law, 12, 14, 165, 177 non-conservative force, 164 non-prismatic beam, 60 non-uniform bending, 40 non-uniform torsion, 71 normal stress, 1,37,39,97 - curved beam, 98 oscillation - damped, harmonic, steady, 171 oscillatory motion, 170 oscillatory system, 165 parallel circle, 143 parallel-axis theorem, 41 particular solution, 88, 124, 139, 171 period, 165 phase angle, 171, 178 phase response, 172 plane strain, 25, 121 plane stress, 3, 6, 24, 121 plate, Kirchhoff's theory, moment equilibrium, 135 Poisson's ratio, 19 polar coordinates, 123, 137, 143 polar moment of inertia, 40, 73 potential, 123 potential energy, 155, 163 Prandtl's stress function, 72 primary structure, 105, 112 principal - curvature, invariant, 6 - stress, 6 - values of inertia, 40 principal axis, 4, 39 - orientation, 4 principle - of Castigliano & Menabrea, 32 - of complementary virtual work, 31 - of virtual work, 15, 154, 161 prism, 121 prismatic beam, 37 product integral, 112 product of inertia, 39,40 pure bending, 37 radius of curvature, 38, 64, 95 Rayleigh quotient, 155 Rayleigh's method, 155 Rayleigh-Ritz method, 156 rectangular cross section, 44 redundant, 105, 112, 164 released structure, 105 resonance, 179 restrained warping, 85 resultant - differential equation, 55, 95 - shear force, 39, 136 rotation, 10, 82 scleronomic constraint, 163 self-equilibrating stress, 121 shear center, 53, 54 shear flow, 77, 80 shear force, 37, resultant, 39, 136 shear modulus, 20 shear strain, 9 shear stress, 2, 39, 44, 77, 82 - axial load, 56 - curved beam, 98 - maximum, 6 - non-prismatic beam, 60 - rectangular cross section, 44 - resultant, symmetric cross section, 46 - thin-walled cross section, 48 - vertical load, 57 shear-circulation theorem, 74 shell, 131 shell of revolution, 143 short-range force, 12, 14 simple beam, 37 slope, 38 - change of, 97 small curvature, 99 snap-through buckling, 152 soap-film analogy, 76 solid cross section, 71, 108 specific complementary energy, 22, 106 specific internal energy, 21 specific strain energy, 21, 110 spherical coordinates, 11, 13 spring force, 165 spring stiffness, 165 St.Venant's theory, 71 stability, 149 stable equilibrium, 149 static coupling, 178

5 Index 205 statical deflection, 165 statically detenninate, 105 statically indetenninate, structure, 32 - system, 65, 105, 112 stationarity condition, 15, 106 stationary temperature field, 122, 124 stationary value, 155 steady oscillation, 171 stiffness, 152, extensional, matrix, parallel connection, series connection, spring, 165 stored energy, 155 strain, 9, 19 - beam axis, 97, 99 - deviator, 11 - plane, 25, tensor, 10 - virtual, 15 strain energy, 21, beam, density, 32 - plate, potential, specific, 21, 110 strength criteria, 26 stress, 1, 19 - components, 2 - deviator, 6 - eigenvalue, 6 - hydrostatic, 6 - mean, 6, 23 - plane, 3, 6, 24, principal axes, 6 - principal values, self-equilibrating, tensor, 2 - transfonnation, 3 - vector, 1 - virtual, 31 - yield, 26 stress function, 72 stress resultant, 38, 96, 97, 131 stress-strain relation, 19 summation convention, 2 superposition law, 104 surface traction, 22 symmetric cross section, 45 system, conservative, 162, 164, damped, oscillatory, 165 temperature, 23 temperature field, 64, 96, 132, stationary, 122, 124 tensile strain, 9 tensor, 2 - product, 3 - second-rank, 2, 10, 11,40, 166 test function, 157 thickness, 131 thin plate, 121, theory, 142 thin-walled - 2D structure, closed section, 77 - cross section, 48, 77,109 - open section, 53, 80 torque, 73, 78, 79 torsion, 71 - non-unifonn, 71 - prismatic bar, 82 - solid cross section, 71 - thin-walled closed section, 77 - thin-walled open section, 80 - unifonn, 71 traction, 22 transfonnation, 3 - rules, 3, 11,40, 166 transmissibility, 174 transmitted - displacement, force, 174 Tresca criterion, 27 twist, 71, 79, 82, 86 unifonn torsion, 71 unit vector, 1 unstable equilibrium, 149, 151 variational principle, 164 vibration - forced, free, 167, mode of, 179 vibratory force, 174 virtual displacement, 15, 154, 161 virtual strain, 15 virtual stress, 31 virtual work, 15, 142, 154, complementary, 31 volume dilatation, 20

6 206 Index warping, 39, 71, 79 - displacement, 73 - function, 71, 72, 86 - resistance, 87, 88 - restrained, 85 warping function, 80 work, 21, external, 14 - internal, 14 yield stress, 26 Young's modulus, 19

Index A Admissible function, 163 Amplification factor, 36 Amplitude, 1, 22 Amplitude-modulated carrier, 630 Amplitude ratio, 36 Antinodes, 612 Approximate analytical methods, 647 Assumed modes method,

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