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 R2 = 0.25 in)

Mechanical springs - Extension spring terminology
Extension Spring with Twisted End

Mechanical springs - Extension spring terminology Mechanical springs - Extension spring terminology
Location of Maximum Bending and Torsion Stresses in Twisted Loops


Step 1 : Write down input parameters which are defined in the sample example.

d (in) OD (in) R2(in) Fmax(lbf) Fi(lbf) Material
0.069 1 0.25 45 10 Music wire

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.

d (in) Lo (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 (Lo < 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).

Parameter Symbol Value Unit
Wire diameter d 0.105 inch
Spring outer diameter OD 0.84 inch
Radius-1 R1 0.3675 inch
Radius-2 R2 0.25 inch
Maximum working load Fmax 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

Parameter Unit Value Symbol
Maximum working load Fmax 45 lbf
Shear stress at point B for maximum working load τB 87.26 ksi
Allowable torsional strength at point B Sall_B 111.264
Factor of safety at point B + fosB 1.28 ---
Tensile stress at point A for maximum working load σA 168.03 ksi
Allowable tensile strength at point A Sall_A 208.621
Factor of safety at point A + fosA 1.24 ---
Shear stress at spring body for maximum working load τsb 88.24 ksi
Allowable torsional strength for spring body Sall_sb 125.173
Factor of safety at spring body + fossb 1.42 ---
Ultimate tensile strength of material Sut 278.16 ksi
Material ASTM No. A228


According to calculations, the weakest point of the extension spring is point A and fosA is 1.24. If the design margin is lower than 1.24, this design is satisfactory and  can be used.