R m = radius to midsurface of shell (in.) 



H = Poisson's ratio 



K = buckling coefficient 



Therefore, the buckling coefficient K = 1.15 derived by Zoelly applied only to 

 small deflections. Because of the impossibility of obtaining perfectly spherical 

 shells, the classical small deflection theory for elastic instability has been found 

 to predict critical pressures that are actually 20 to 300% higher than experimen- 

 tal collapse pressures of spherical shells. Research 16, 17, 18 has shown that the 

 deviation of experimental collapse pressure from the calculated critical pressure 

 for perfect spheres is a function of deviations from nominal sphericity and nom- 

 inal wall thickness. Therefore, attempts were made to modify the classical 

 formula and to specify tighter dimensional tolerances for the spheres. The 

 resulting semiempirical expression 16 of 



3E(J- 



^ 



(2) 



where R is the external radius (in.), has been found to show good agreement 

 with experimental data so long as the deviation in sphericity in shells is less 

 than 0.03t and the minimum wall thickness value is used for the term t in 

 Equation 2. To make this equation applicable to acrylic plastic material, one 

 further modification is necessary. In place of Young's Modulus E, based on 

 linear relationship between stress and strain, new terms must be substituted 

 that reflect the nonlinear relationship between stress and strain in acrylic 

 plastic material. The new term for E may be tangent modulus of elas ticity, 

 E t , secant modulus of elasticity, E s , or a hybrid expression of -v/E s E t that 

 takes both moduli into consideration. The new term for ju may be p v , a 

 variable that also reflects the nonlinear relationship between stress and 

 Poisson's ratio in acrylic plastic. 



The secant modulus, tangent modulus, and variable Poisson's ratio 

 are generally derived from typical uniaxial compression stress— strain curves 

 of acrylic plastic test specimens. In order to arrive at the stress level in the 

 sphere so that the proper E s , E t , and m v can be substituted into Equation 2, 

 the following simplified equation was used 



P(R ° |2 

 s ■ 2fi-r < 3 > 



where s denotes average membrane stress, which for the purposes intended is 

 of sufficient accuracy. 



31 



