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A numerical method for determining membrane and bending stresses in photoelastic models of plates and shells

Application of the Newton-Raphson method to data analysis in three-dimensional photoelasticity leads to a new method for determination of stress components in shells with full estimation of the optical-rotation effect

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Abbreviations

fλ:

functions defined by eq (13)

h :

half thickness of a shell or plate model

i :

imaginary unit

k :

step number of the approximation procedure

n :

number of division

r jl , r ′ jl :

matrix elements ofR andR ′, respectively [eq (11)]

x,y,z :

rectangular coordinates,z axis is in the direction parallel to the wave normal

z 0 :

coordinate at the incidence of light

z k :

coordinate of a partition point, (k=1, 2,…,n)

IB :

light vector after a mathematical transformation

IB o :

incident-light vector or initial value ofIB

B x ,B y :

components of light vector

B xo B yo :

initial values ofB x andB y

C :

photoelastic constant defined by Aben

G(γ):

diagonal unitary matrix [eq (8)]

I :

unit matrix

M,M ′ :

two-by-two transformation matrices corresponding to an optical system or element

M T , M T :

transposed matrices ofM andM ′

N(z), N(Z k ) :

matrices defined by eq (2′) as a function ofz orz k

R,R ′ :

matrices representing total optical effect for reflected light [eq (11)]

S(θ):

matrix of rotation [eq (9)]

S(ω):

matrix which corresponds to a rotator

U,U ′ :

solutions of eq (4) which represent a linear transformation, respectively [eq (3)]

U k :

U(z k )

U(z), U(z k ) :

matricesU represented as a function ofz orz k

α,α′ (α*,α′*):

angles which determine characteristic directions by a model with a reflective layer

α1,α2 :

angles which determine primary and secondary characteristic directions

γ:

half characteristic phase retardation corresponding to matrixM [eqs (8) and (10)]

γ,γ (γ*,γ′*):

quarters of characteristic phase retardation for reflected light corresponding toR andR ′ by a model with a reflective layer [eq (11)]

Δ:

prefixed symbol representing increment or corrector

ζS,ζS* :

stress components re-designated in exchange for σm, θ m , σ bo , θ b , θ b and σm* etc. respectively, (s = 1,2,3,4)

ζS o :

initial values of ζS in the approximation procedure

ζS k :

approximate values of ζS* determined in thekth time approximation, (k=1, 2, 3, …)

θ:

angle betweenx axis and one of the principal-stress directions

θ m ,θ b (θ m *, θ b * :

angles betweenx axis and one of the principal directions of membrane and bending stresses, respectively

k :

number of partition point alongz axis =1, 2, …,n

σ x ,σ y ,τ xy :

stress components

σ m ,σ m * :

differences of principal membrane stresses

σ b :

difference of principal bending stresses

σ bo ,σ bo * :

maximum values of the difference of principal bending stresses at the surface of a shell model

ω:

rotational angle of principal axes of an optical system

References

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Authors and Affiliations

  1. Department of Metal Processing and Mechanical Metallurgy, Tohoku University, Sendai, Japan

    M. Chiba (Instructor) & H. Shimada (Professor)

Authors
  1. M. Chiba
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  2. H. Shimada
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Chiba, M., Shimada, H. A numerical method for determining membrane and bending stresses in photoelastic models of plates and shells. Experimental Mechanics 15, 142–147 (1975). https://doi.org/10.1007/BF02318851

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  • DOI: https://doi.org/10.1007/BF02318851

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