Linear elastic spring is one of the mechanical device used in day to day life, which is capable of resisting axial loading only, and the contraction or elongation of the spring is directly proportional to the applied along the direction of the load. Proportionality constant between deformation and applied load is referred to as the spring constant(k), recoil rate, or spring stiffness k, and it has units of force per unit extent. As an elastic spring supports axial loading , we select an element coordinate system (which is also known as a local coordinate system) as an x-axis oriented along the spring length, as shown in the below figure.
a) Linear spring element with nodes, nodal displacements, and nodal forces.
b) Load-deflection curve.
Let us assume that both the nodal displacements are zero when the spring is undeformed condition, the net spring deformation is given by
δ= U2− U1
“δ” is net spring deformation
and the resultant axial force in the spring is
f = kδ= k(U2− U1)
“f” is resultant axial force
For equilibrium condition,
f1+ f2= 0 (or) f1= − f2,
Then, in terms of the applied nodal forces as
f1= −k(U2− U1)
f2= k(U2− U1)
which can be expressed in matrix form as
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