# Example 7.1     
 Sy   =   100 ksi
 V1  =   1 in^3
 E  =   30 Mpsi
 Ua = Sy^2 * V1 / (2 * E) ; J
 Ubl = Sy^2 * (V1/2) / (2 * E) ; J 
 Ubu = (Sy / 4)^2 * (2* V1) / (2 * E); J 
 Ubl  / Ua   
 Ubu  / Ua   

# Example 7.2     
 Se   =   207 MPa
 V1  =   1 mm^3
 E  =   207 Mpa
 rho_g  =   77 kN/m^3
 Ua = Se^2 * V1 / (2 * E) ; J 
 Ua /  rho_g / V1 ;  
 Se   =   828 MPa
 V1  =   1 mm^3
 E  =   207 Mpa
 rho_g  =   77 kN/m^3
 Ua = Se^2 * V1 / (2 * E) ; J 
 Ua /  rho_g / V1 ;  
 Se   =   2.07 MPa
 V1  =   1 mm^3
 E  =   1.034 Mpa
 rho_g  =   9.2 kN/m^3
 Ua = Se^2 * V1/ (2 * E) ; J 
 Ua /  rho_g / V1 ;  

# Example 7.3     
 P  =   100 lbf
 L1  =   60 in
 E  =   1 Mpsi
 I  =   6.46 in^4
 k  =   100 lbf/in
 h  =   12 in
 Z  =   3.56 in^3
 dst_beam = P * L1^3 / (48 * E * I) ; in 
 dst_spring = P / (2 * k) ; in 
 dst_total = dst_beam + dst_spring  ; in 
 IF = 1 + sqrt(1 + 2 * h / dst_total)  
 ddyn_beam = IF * dst_beam  ; in 
 ddyn_total = IF * dst_total  ; in 
 Pdyn = IF * P ; lbf # 756.738 lbf
 Mdyn = Pdyn * L1 / 4  ; lbf in
 sdyn = Mdyn / Z ; psi 

# Example 7.4     
 w  =   2400 rpm
 r  =   60 mm
 t  =   20 mm
 V1 = pi() * r^2 * t  ; mm^3 
 rho  =   2000 kg/m^3
 m1 = rho * V1 ; kg 
 I = (1/2) * m1 * r^2   
 U = (1/2) * I * w^2  ; J 
 G1  =   79 Gpa
 rs   =   10 mm
 Ls   =   250 mm
 Vs = pi() * rs^2 * Ls   ; mm^3 
 t = 2 * sqrt(U * G1 / Vs)  ; Mpa 
 q = t * Ls / (rs * G1)  ; deg 

# Example 7.5     
 Ki   =   1.5 
 Ki^2    

# Example 7.6     
 K0   =   1.5 
 Ki   =   3.0 
 An_over_A0   =   4.0 
 sn_over_s0 = (Ki / K0)  * An_over_A0   
 Un_over_U0 = sn_over_s0^2   

# Example 7.7     
# Original -- a     
 Ki   =   3.5 
 An_over_A0   =   0.857 
 Vn_over_V0   =   1.000 
 sn_over_s0 = An_over_A0 / Ki  
 r1 = sn_over_s0^2 * Vn_over_V0
# Redesign -- b     
 Ki   =   3.0 
 An_over_A0   =   2.000 
 Vn_over_V0   =   0.4286 
 sn_over_s0 = An_over_A0  / Ki   
 r2 = sn_over_s0^2 * Vn_over_V0
 Ub_over_Ua  = r2 / r1    

# Homework Problems:

TBD