Optical Measurements, Modeling, and Metrology, Volume 5

(x )i (x )ij Fj (i 1~ M, j 1~ N)  (x )ij Fj j1 N  (i 1~ M) (x )i1F1 (x )i2 F2 (x )iN FN (i 1~ M). In equation (2), Fj is the nodal forces along the boundary. Therefore, the tractions along the boundaries are determined and subsequent stress analysis can be performed if the values of Fj are determined. Linear Algorithm From the principal stress difference 1 – 2 and the principal direction  obtained by photoelasticity, the normal stress difference x – y and the shear stress xy are obtained as x y (1 2 )cos2, xy  1 2 (1 2 )sin2. (3) Therefore, the relationships between the values obtained by photoelasticity and the nodal forces Fj along the boundary can be expressed as (x y )i  (x )ij (y )ij  Fj (i 1~ M, j 1~ N), (xy )i (xy )ij Fj (i 1~ M, j 1~ N), (4) where (x – y)i and (xy)i express the normal stress difference and the shear stress at the point (xi, yi) obtained by photoelasticity. Equation (4) expresses linear equations in the unknown coefficients Fj. For numerous data points, an over-determined set of simultaneous equations is obtained. In this case, the nodal forces Fj along the boundary can be estimated using linear least-squares as F ATA1  ATS, (5) where F, A and S are the nodal force, stresses under the boundary condition of the unit force and the values obtained by pho- Fig. 1 Typical optical setup for photoelasticity Fig. 2 Finite element model with boundary condition of unit force 111

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