Sample Problem : Design of an Extension Spring which Works under Static Loading
An extension spring from music wire with full twisted end shall work in an
environment where its maximum outer diameter shall be less than 0.9 inch and the
free length of the spring must be smaller than 7.5 inch. The
initial preload shall be 10 lbf. Spring rate shall be 8 lbf/inch.
Available wire size for the production of the spring are 0.069, 0.080, 0.090, 0.105, and 0.112 in. From the given information:
(a) Determine the wire diameter to satisfy the design requirements.
(b) Find the factors of safety under a maximum static 45 lbf load. (Hook radii
will be R_{2} =
0.25 in)

Extension Spring with Twisted End 


Location of Maximum Bending and Torsion
Stresses in Twisted Loops 
Solution:
Step 1 : Write down input parameters which are
defined in the sample example.
INPUT PROPERTIES SUMMARY 
d (in) 
OD (in) 
R_{2}(in) 
F_{max}(lbf) 
F_{i}(lbf) 
Material 
0.069 
1 
0.25 
45 
10 
Music wire 
0.080 
0.090 
0.105 
0.112 
Step 2 : Visit
dimensional extension spring design and solve the problem for each of the
wire size. Known parameters are spring rate and preload so this option is
selected for the solution of the problem Physical
parameters
generated by the calculator are tabulated as follows.
INPUT PROPERTIES SUMMARY 
d (in) 
L_{o} (in) 
OD (in) 
0.069 
13.38 
0.35 
0.080 
10.2 
0.48 
0.090 
8.59 
0.61 
0.105 
7.32 
0.84 
0.112 
7.01 
0.95 
According to dimensional requirements (L
_{o}
< 7.5 in and OD < 0.9 in), the only feasible wire diameter is d=0.105 in.
Step 3 : For the
initial preload stress conditions, visit "Stress Analysis of Extension Spring for Static Loading" calculator. Select Maximum working load as the known parameter. Remember that
d and OD is found as 0.105 and 0.84 respectively in step 2 so use this number
during stress calculations. Following input parameters are used and results generated by the calculator are as follows.
Note that R1 is used as the half of mean diameter (D).
INPUT PROPERTIES SUMMARY 
Parameter 
Symbol 
Value 
Unit 
Wire diameter 
d 
0.105 
inch 
Spring outer diameter 
OD 
0.84 
inch 
Radius1 
R1 
0.3675 
inch 
Radius2 
R_{2} 
0.25 
inch 
Maximum working load 
F_{max} 
45 
lbf 
Material 
Music wire 
Allowable torsional strength for the spring body (% of Sut)
* 
45 
% 
Allowable torsional strength for the spring end (% of Sut)
* 
40 
% 
Allowable bending strength for the spring end (% of Sut)
* 
75 
% 
Note 1 : ^{*} Values are taken from the
table which is given in the supplement section of extension spring design for static loading
RESULTS 
Parameter 
Unit 
Value 
Symbol 
SPRING MATERIAL & STRESS RELEATED PARAMETERS 
Maximum working load 
F_{max} 
45 
lbf 
Shear stress at point B for maximum working load 
τ_{B} 
87.26 
ksi 
Allowable torsional strength at point B 
S_{all_B} 
111.264 
Factor of safety at point B ^{+} 
fos_{B} 
1.28 
 
Tensile stress at point A for maximum working load 
σ_{A} 
168.03 
ksi 
Allowable tensile strength at point A 
S_{all_A} 
208.621 
Factor of safety at point A ^{+} 
fos_{A} 
1.24 
 
Shear stress at spring body for maximum working load 
τ_{sb} 
88.24 
ksi 
Allowable torsional strength for spring body 
S_{all_sb} 
125.173 
Factor of safety at spring body ^{+} 
fos_{sb} 
1.42 
 
Ultimate tensile strength of material 
S_{ut} 
278.16 
ksi 
Material ASTM No. 
A228 
Conclusion:
According to calculations, the weakest point of the extension spring is point A
and fos_{A} is 1.24. If the
design margin is lower than 1.24, this design is satisfactory and can be used.
Reference: