# Flexibility and Stiffness

Flexibility is the elastic property of a restraint and can be linear or lateral. Expansion joint flexibility values are calculated based on data provided in various catalogues, standards, or using general structural mechanics methods (for complex flexible structures). Expansion joint flexibility can be calculated using  Start-Elements: expansion joint flexibility.
• Linear Flexibility C is displacement  Δ cause by force  P of 1 kg, C=Δ/P. Linear flexibility is expressed in mm/kg.

• Rotational Flexibility C is rotation angle φ (in radians) from bending or rotating moment M of 1 kg*m, C=φ/M. Rotational flexibility is expressed in radian/(kg*m).

Stiffness is the opposite value of flexibility, K=1/C.

• Linear Stiffness K is reaction  P caused by displacement Δ of 1 mm, KC=P/Δ. Linear stiffness is expressed in kg/mm.

• Rotational Stiffness K is moment  M caused by rotation φ of 1 degree, K=M/φ. Rotational stiffness is expressed in kg*m/radian.

To input infinite flexibility (zero stiffness), the following should be input:

• rotational flexibility equal to 1 radian/(kg*m)

• linear flexibility equal to 1000 mm/kg.

Examples:

1. A rotational bellows has an allowable opening angle of φ=30=0.05236 radian; moment at allowable opening angle is М=120 kg*m. Rotational flexibility if φ/М = 0.05236/120 = 0.000436 1/kg*m.

2. An axial expansion joint has an allowable operation deformation of Δ = 25 mm. Axial force at allowable operation deformation is P = 1500 kg. Linear flexibility is Δ/P = 25/1500 = 0.0166 mm/kg.

3. Expansion joint catalogues give stiffness values for given numbers of waves. Stiffness is the opposite value of flexibility. For example, if expansion joint stiffness is K=50 kg/mm, axial flexibility is C=1/K = 1/50 = 0.02 mm/kg.

4. The number of waves is often unknown and is determined through analysis. In the case, the expansion joint should be input as completely smooth with a linear flexibility of 1000 mm/kg and rotational flexibility of 1 degree/(kg*m). After analysis deformation, the structure and elastic properties of the appropriate expansion joint are taken from a catalog. Next, a final verification piping analysis is run.

A spiral coil spring, according to OST 108.764.01-80 (revision  09), has an allowable subsidence (operation deformation) of Δ =  70 mm and allowable load (spring compression force for operation deformation) of P=3225 kg. Spring flexibility is Δ/P = 70/3225 = 0.0271 mm/kg.