# The Linear Algorithm Strikes Again

This post on the OpenSees message board reminded me of another reason not to use the Linear algorithm, even when you have a linear model. Some elements need that second iteration in order to record all of their response.

Not only `shellMITC4` mentioned on the message board, but also the beloved `forceBeamColumn`. If you define member loads on a `forceBeamColumn` element and use the Linear algorithm, the member loads are not applied correctly. I’m not exactly sure why.

Consider a simply-supported beam with a uniform distributed load. The midspan moment is $wL^2/8$, the support reactions are $wL/2$, and the support rotations are $wL^3/(24EI)$.

``````import openseespy.opensees as ops

L = 240.0
E = 29000.0
A = 20.0
I = 1400.0
w = 1.5

ops.wipe()
ops.model('basic','-ndm',2,'-ndf',3)

ops.node(1,0,0); ops.fix(1,1,1,0)
ops.node(2,L,0); ops.fix(2,1,1,0)

ops.geomTransf('Linear',12)

ops.section('Elastic',1,E,A,I)
ops.beamIntegration('Lobatto',1,1,5)

ops.element('forceBeamColumn',1,1,2,12,1)

ops.timeSeries('Linear',23)
ops.pattern('Plain',1,23)

Nsteps = 2
ops.algorithm('Linear')
ops.analysis('Static')

ops.analyze(Nsteps)
ops.reactions()``````

The analysis results are shown below. Although the reactions are correct, the midspan moment is off by a factor of 2/3 and the rotations by 1/2.

Switch to the Newton algorithm and all is good.

You may ask “Why would you analyze an elastic beam using a force-based element in the first place?” Let’s say you had a nonlinear model, then backpedaled to an elastic model by switching from fiber sections to elastic sections. Then you had doubts about the analysis, so you give the Linear algorithm a try to see if everything makes sense. I do this all the time.

## 3 thoughts on “The Linear Algorithm Strikes Again”

1. Ask says:

Hi sir,
In my model when I am using linear algorithm with LoadControl and ‘sp’ pattern with ‘Path’ timeSeries, my plot is matching with experimental value.
But when I am using Newton algorithm I am getting a wrong result ?
Of course my model is non-linear – still it is giving correct answer with linear algorithm. What could be the reason ?

Like

1. Positive Definite says:

2. Positive Definite says: