AutomotiveType Multileaf Spring: Design for Fatigue Loading
A sixleaf spring is subjected to a load at the center that varies between P_{max} and P_{min} (Figure 14.17). Estimate the total length 2L and width of each leaf.
Given: P_{min} =160 lb, P_{max} =800 lb, n =6.
Assumptions: Stress concentration at the center is such that K_f =1.2. Use a survival rate of 50% and C_f=C_e =1.
Design Decisions: We use a steel alloy spring of S_u = 200 ksi, S_e^{\prime}=78 ksi , E=30 \times 10^6 psi , \nu=0.3 , h = 0.25 in., and k = 140 lb/in. The material is shot peened. A safety factor of ns = 1.4 is applied.
From Table 7.3, C_r =1. The modified endurance limit, by Equation (7.6), S_e =(1)(1)(1)(1/1.2)78=65 ksi.
S_e=C_f C_r C_s C_t\left(1 / K_f\right) S_e^{\prime} (7.6)
Each half of a spring acts as a cantilever supporting half of the total load. The mean and the alternating loads are therefore
P_m=\frac{400+80}{2}=240 lb , \quad P_a=\frac{40080}{2}=160 lb
Inasmuch as bending stress is directly proportional to the load, we have \sigma_a / \sigma_m=P_a / P_m=2 / 3 .
The mean stress, using Equation (14.42), is
\sigma=\frac{6 P L}{n b h^2} (14.42)
\sigma_m=\frac{6 P_m L}{n b h^2}=\frac{6(240) L}{6 b(0.25)^2}=3840 \frac{L}{b} (e)
Substituting the given numerical values into Equation (7.20), we have
\sigma_m=\frac{S_u / n}{\frac{\sigma_a}{\sigma_m} \frac{S_u}{S_e}+1} (7.20)
\sigma_m=\frac{S_u / n_s}{\frac{\sigma_a}{\sigma_m} \frac{S_u}{S_e}+1}=\frac{200 / 1.4}{\frac{2}{3} \frac{200}{65}+1}=46.82 ksi
From Equation (e),
48,820=3840 \frac{L}{b} \quad \text { or } \quad b=0.082 L (f)
Because the spring is loaded at the center with 2P , Equation (14.44) becomes k=E n b h^3 / 3 L^3\left(1 \nu^2\right) . Introducing the given data results in
k=\frac{P}{\delta} \frac{E n b}{6\left(1 \nu^2\right)}\left(\frac{h}{L}\right)^3 (14.44)
140=\frac{\left(30 \times 10^6\right)(6)(0.82 L)(0.25)^3}{3 L^3(0.91)}
L = 24.56 in.
Hence, the overall length is 2L =49.12 in. The width of each of the six leaves using Equation (f) equals b=0.082(24.56)=2.014 in.
TABLE 7.3 Reliability Factors 

Survival Rate (%)  C_r 
50  1.00 
90  0.89 
95  0.87 
98  0.84 
99  0.81 
99.9  0.75 
99.99  0.70 