The last example in the previous post illustrated the use of cubic splines to find the bending moments in a continuous beam subject to prescribed lateral displacements. A similar technique can be used to find bending moments, shear forces, reactions, and displacements in a continuous beam subject to lateral forces. The spreadsheet SplineBeam.zip contains a description of this technique and a User Defined Function (UDF) to perform the analysis.
In outline, the procedure is:
For each loaded point:
- Apply unit deflection
- Fit cubic spline and find the curvature and bending moment in the segments either side of the loaded point
- Load = change of bending moment gradient at the point load
- Scale bending moments by Applied Load / Load for unit deflection
Sum bending moments for each loaded point
Calculate shear forces and reactions from bending moment diagram
This procedure is illustrated in this example for a single point load on a three span beam (click on any image below for a full size view):
The beam support positions are specified with a Y value of zero in column B, and a deflection of -1 is entered for the load at X = 17.5. The beam slope and curvature at each node and the load location are generated by the UDF CSPLINEA() (see previous post).
The bending moment in column G is simply the curvature multiplied by the beam flexural stiffness, EI; where E is the Young’s Modulus and I is the second moment of area of the beam cross section.
The force in column H is the force required to generate the prescribed unit deflection, and is equal to the change in slope of the bending moment diagram at the load position. Dividing the specified applied load (100 kN) by the force for unit deflection gives the factor to be applied to the calculated bending moments; in this case -5.67E-04. The beam actions and deflections in Columns I to K are found by multiplying the corresponding action under unit deflection by this factor.
This procedure has been incorporated in the UDF SPLINEBEAM() which can be used to analyse continuous beams with any numbers of spans and any number of point loads:
Note that loads can only be applied at node positions, and nodes are generated by dividing each beam into a specified number of equal length segments. Nodes are numbered from 1 to number of segments + 1 from left to right for each beam.
The same analysis performed in Strand7 gives identical results.