Example for Springs with Boundary Conditions and its Solution


In order to determine forces and deformation of any structure or a component, one should have to give boundary conditions which constrained the structure as per customers requisite. Boundary Conditions determine how the model or structural component is externally constrained in any one of the axis. Every models must be attached to the some external point or support points. You may determine these points of support as completely controlled or as partially controlled with a Spring or spring element. Also we can define a spring support which has stiffness or rigidity in only one direction with the compression-only or tension  springs.

What is Boundary Condition

To solve the equations which is defined by the global stiffness matrix(G), So we must apply some type of constraints and/or supports or the structure will be free to move as a rigid body, this type of constraining is called Boundary condition.

Generally boundary conditions are of two types:

  • Homogeneous Boundary Conditions
  • Non-homogeneous Boundary Conditions 
  1. The most common one is the homogeneous boundary conditions, which are occur at locations of the model, that are completely prevented from structural movement.
  2. The second one is non-homogeneous boundary conditions, which are occur where finite an extremely small (none zero) values of displacement is specified, that is, such as the settlement of a support.

Example for Springs with Boundary Conditions

Let us think about the two element system as described before where the Node 1 is attached to a fixed support as shown in below figure, and yielding the displacement constraint U1= 0, k1= 50 lb/in, k2= 75 lb/in, for these conditions the forces is given as F2=F3=75 lb/in, and here we need to determine nodal displacements U2 and U3?

Free Body Diagram of Example:

example BC

By substituting the above specified values into the below equation which is derived earlier:

final form

We have

after substituting

As we see nodal force F1 becomes an unknown reaction force, it is just because of the constraint of zero displacement at node 1. Formally, first algebraic equation which is represented in this matrix equation becomes:

-50U2  = F1

This represents the constrained equation, which it illustrates the equilibrium condition of the node at which the displacement is constrained.

The second and third equation become,

we have eq

By solving the above equation, we obtain  the displacements

U2 = 3 inch


U3 = 4 inch.

Note that, the matrix equations will governing the unknown displacements which are obtained by the simply striking out the first row and column of the 3 * 3 matrix system, and since the constrained displacement value is zero, and this value is homogeneous.

The displacement boundary condition not equal to the zero (which is non-homogeneous), then this is not possible and the matrices need to be control differently.

Hence, This is the example for springs with boundary conditions and its Solution.

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Hello, I am an aircraft structural analyst with industrial experience and a master degree on aerospace structures. Currently working for an aerospace company as a stress
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