The stiffness matrix in matrix analysis of strutures

[chandran]

I require a detailed understanding of the stiffness matrix in the analysis of
rods,beams et

In the beginning of stiffness matrix derivation the textbooks say

force at ith degree of freedom(dof) to hold unit displacement at jth dof when all other dofs are zero.

let us analyse this statement step by step

let me take a rod and fix one end and push the other end by unit distance ,the spring will then exer a force equal to the stiffness. (step1)

This is the stiffness at ith dof due to unit displacement at ith dof.

the confusing part
Now the theory says push the jth dof by 1 unit distance and what force is required at ith dof to hold this j th displacement.(step2)

The final say by the theory is that the force reqd at ith dof is the stiffness arrived from step1+the stiffness arrived from step2

any cay explain the step 2 part.

 

[Chestermiller]

Suppose you have a square sheet of rubber and you stretch it in the length direction. To prevent it from contracting in the width direction, you need to apply a force in the width direction.