Issue 53

I. Monetto et alii, Frattura ed Integrità Strutturale, 53 (2020) 372-393; DOI: 10.3221/IGF-ESIS.53.29 389 pertinent to this simplified model based on Euler-Bernoulli beam theory. They are given by Eqn. (25). Once again, an a priori unknown parameter EB w k enters into these expressions. Its identification requires a matching procedure similar to that presented in the third section and detailed in Appendix B also for the Timoshenko beam model. To conclude, it is worthwhile noting that for the special case of symmetric geometry (  =1) and loadings ( N 0 =0) the expressions for the compliance coefficients 0 M EB d , 0 0 V M EB EB d c  and 0 V EB c given by Eqn. (25) coincide with those reported in [10] in both isotropy and orthotropy. A PPENDIX B. M ATCHING PROCEDURE AND SUMMARY RESULTS TABLES n this Appendix the numerical results shown and discussed in the fourth section are listed in table form for different geometries (  =1,0.8,0.6,0.4). The results obtained using the model according to Timoshenko beam theory are presented in Tabs. 1 to 3 for isotropic specimens (  =  =1) with varying Poisson ratios  xz =  zx =  =0,0.3,0.5 and in Tabs. 4 to 6 for orthotropic specimens with  =1,  =0.5 and varying Poisson ratios  xz =0,0.3,0.5. Similar results through the simplified model according to Euler-Bernoulli beam theory are reported in Tabs. 7 to 9 for  =1 and  =1.   k w 0 V d 0 M d 0 N d 0 V c 0 N c 0 2.603 3.036 2.044 1.518 1 0.3 4.472 2.316 16.154 8.077 1.559 1.158 0.5 7.116 1.836 1.236 0.918 0 2.458 2.830 2.137 1.415 0.8 0.3 4.134 2.182 13.214 6.607 1.648 1.091 0.5 6.427 1.750 1.322 0.875 0 1.988 2.871 2.535 1.436 0.6 0.3 3.111 2.295 10.851 5.426 2.027 1.148 0.5 4.486 1.911 1.688 0.956 0 1.309 3.318 3.656 1.659 0.4 0.3 1.820 2.814 9.071 4.536 3.101 1.407 0.5 2.347 2.478 2.730 1.239 Table 1: Numerical results according to the Timoshenko beam based model and 0 M d based matching for isotropic materials (  =  =1). Matching the compliance coefficients using the Timoskenko beam based model In order to define the correction factor k w , a matching has been performed on the root rotations due to moment, shear and normal force presented in [3]. The numerical values used in the present paper are the difference between the root rotation compliance coefficients for the upper, say 1 K a , and lower, say 2 K a , layers given in Table 1 of [3] (the upper-script K = M 0 , V 0 , N 0 denotes the end loading moment, shear or normal force). Matching is then based on equating 1 2 K K K d a a   . For completeness, the numerical values 1 2 K K a a  are reported in Table 10. As an example, with reference to the model according to the Timoshenko beam theory, for matching based on the root rotation compliance coefficient 0 V d , under symmetrical transverse force, the use of Eqn. (18) leads to the following expression   0 0 0 4 2 1 2 3(1 ) 2 V w V V a a k      (B.1) where 0 0 0 1 2 V V V a a d   has been enforced. Substituting Eqn. (B.1) into Eqns. (19) to (24) gives the remaining compliance coefficients. These results are reported in Table 2 for isotropic specimens and in Table 5 for degenerate orthotropic specimens with  =0.5 (  =1). It is worthwhile noting that 0 M d and 0 N d for a generic  could be obtained on rescaling of   1/4 those pertinent to  =1; rescaling of the other coefficients is through more complicated expressions of the elastic I