Lecture 2 Castiglianos Energy Theorems Hans Welleman Castigliano

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Lecture 2 : Castigliano’s Energy Theorems Hans Welleman

Lecture 2 : Castigliano’s Energy Theorems Hans Welleman

Castigliano Fa ua Ir J. W. Welleman Fb ub Fc uc Fx ux Work

Castigliano Fa ua Ir J. W. Welleman Fb ub Fc uc Fx ux Work and Energy methods 2

Differentiate the Work to a specific force, e. g. Fx: Use Maxwell’s notation :

Differentiate the Work to a specific force, e. g. Fx: Use Maxwell’s notation : Ir J. W. Welleman Work and Energy methods 3

Elaborate for all … Ir J. W. Welleman Work and Energy methods 4

Elaborate for all … Ir J. W. Welleman Work and Energy methods 4

Result: Maxwell : Castigliano’s 2 nd theorem Ir J. W. Welleman Work and Energy

Result: Maxwell : Castigliano’s 2 nd theorem Ir J. W. Welleman Work and Energy methods 5

Castigliano n Work done by external loads Aext is stored in the strain energy

Castigliano n Work done by external loads Aext is stored in the strain energy Ec n Differentiate the strain energy to a force at location x to find the displacement u at x (complementary) Ir J. W. Welleman Work and Energy methods 6

Example F EI wmax 0, 5 l Ir J. W. Welleman 0, 5 l

Example F EI wmax 0, 5 l Ir J. W. Welleman 0, 5 l Work and Energy methods 7

Strain Energy in terms of the load Ir J. W. Welleman Work and Energy

Strain Energy in terms of the load Ir J. W. Welleman Work and Energy methods 8

Steps of the solution strategy n Castigliano : differentiate EC to a force. n

Steps of the solution strategy n Castigliano : differentiate EC to a force. n EC is integral of the square of the moment distribution n so …. differentiate after integration … IS THIS SMART TO DO ? ? Ir J. W. Welleman Work and Energy methods 9

Smart help Ir J. W. Welleman Work and Energy methods 10

Smart help Ir J. W. Welleman Work and Energy methods 10

Castigliano n Second n First theorem Ir J. W. Welleman Work and Energy methods

Castigliano n Second n First theorem Ir J. W. Welleman Work and Energy methods 11

Application Ir J. W. Welleman Work and Energy methods 12

Application Ir J. W. Welleman Work and Energy methods 12

Example deformation with Castigliano: • Find the displacement at the free end • Find

Example deformation with Castigliano: • Find the displacement at the free end • Find the rotation at the free end Given : Ir J. W. Welleman Work and Energy methods 13

Example Force distribution with Castigliano: • Find the support reaction at B using Castigliano

Example Force distribution with Castigliano: • Find the support reaction at B using Castigliano • Draw the moment distribution Ir J. W. Welleman Work and Energy methods 14

Application : shear stiffness Beam: n Axial loading n Bending n Shear n Torsion

Application : shear stiffness Beam: n Axial loading n Bending n Shear n Torsion Ir J. W. Welleman Work and Energy methods 15

Asumme LE, find k discrete model n n Shear stress distribution not constant over

Asumme LE, find k discrete model n n Shear stress distribution not constant over depth If LE then shear strain γ not constant! continuous model Ir J. W. Welleman Work and Energy methods 16

Equal amount of stored energy Ir J. W. Welleman Work and Energy methods 17

Equal amount of stored energy Ir J. W. Welleman Work and Energy methods 17

Analysis Software mm mm Ir J. W. Welleman Work and Energy methods 18

Analysis Software mm mm Ir J. W. Welleman Work and Energy methods 18

Beam model with shear deformation n Euler-Bernoulli – no shear deformation (one ODE) n

Beam model with shear deformation n Euler-Bernoulli – no shear deformation (one ODE) n Timoshenko beam theory – bending and shear (two coupled ODE’s) CIE 4190 Analysis of Slender Structures Ir J. W. Welleman Work and Energy methods 19