Navigation:  Static - Beam Element > A-16 (Coupled Spring)

Objective

To verify the behavior of coupled spring which is useful in modeling bridge foundations.

 

Problem Description

In the 10 meter column [Ref 1] below, the top is subjected to the loads: Fx = 100.00 (kN), Fy = 200.00 (kN), Fz = -3000.00 (kN), Mx = 400.00 (kN-m), My = 500.00 (kN-m) and Mz = 600.00 (kN-m).  

 

Material: E = 3.25e+07 kN/ m2, ν = 0.20.

Sections: Izz = 0.0104 m4 , Iyy = 0.0417 m4 , J = 0.0286 m4 , A= 0.5 m2 , Ay = 0.4167 m2 , Az = 0.4167 m2

 

The bottom of the column is supported by a coupled spring with the following stiffness matrix terms (see “Calculation of Coupled Spring Stiffness Terms” below)

 

 

Calculation of Coupled Spring Stiffness Terms

 

The stiffness matrix terms of the coupled spring used in this example are calculated based on the following simplified bridge piers below.  On the left is the full pier (column + foundation) model A while on the right is the foundation-only model B.  In order to compute the stiffness matrix of the coupled spring, 6 loads in separate load cases (1000 kN for Px, Py and Pz; 1000 kN-m for Mx, My and Mz) are applied at the bottom of the column in Model B. We first solve the model B to obtain displacement matrix (displacements for each of these load cases).

 

 

We then invert the displacement matrix to obtain the stiffness matrix.  Note the stiffness matrix is multiplied by 1000 so the stiffness terms are in the right units as shown below.

 

 

 

Results

The nodal displacements in the coupled spring model are very close to those obtained from the full model A.  This illustrates that a coupled spring can be used to simplify the modeling of a bridge sub-structures effectively.