Sometimes we are asked why the bending moment (or any other result) in combination UGT FC 1 does not correspond to the sum of the bending moment in each load group multiplied by their relevant safety factor and combination factor.
The reason: non-linear behavior. But we can explain this more easily with an example.
Model with linear behavior
- Let’s start with a linear model of a beam for which we want to calculate the bending moment in point A.
- Two load groups act on the beam.
Load case 1 Load case 2 
- We calculate the bending moment due to each load group.
Load case 1 Load case 2 
- A combination contains: 1,5 x Load case 1 + 1,0 x Load case 2
Using hand calculations we can determine the expected bending moment in the combination:
1,5 x Load case 1 + 1,0 x Load case 2 = 1,5 x 7,5 kNm – 1,0 x 15 kNm = -3,75 kNm.If we compare that to Diamonds, we get the same result.
A linear model allows an analysis based on superposition.
Model with non-linear behavior
- We take the same model as above, but this time the support on the right cannot bear tension. This is a non-linear support condition.
- We calculate the bending moment again.
Load case 1 Load case 2 The point load in this load case, doesn’t cause any tension in the support on the right. The bending moment equals the bending moment of the previous example.
The point load in this load case causes tension in the support on the right. The bending moment equals the bending moment of a cantilever beam, as if the support on the right was not present.
- We consider the same combination: 1,5 x Load case 1 + 1,0 x Load case 2
Using hand calculations we determine the expected bending moment in the combination
1,5 x Load case 1 + 1,0 x Load case 2 = 1,5 x 7,5 kNm – 1,0 x 40 kNm = -28,75 kNm.If we compare that to Diamonds, we get don’t get the same result!
That because we tried to add apples and oranges: in the graph for bending moment in load case 1 the support on the right is present, while for load case 2 that’s no longer the case.
A non-linear model doesn’t allow an analysis based on superposition. Each combination must be evaluated one by one. This model is calculated using an iterative calculation. This model will take longer to calculate than an model using superposition.
Elements causing non-linear behaviour:
- tie rods, stiffness diagrams
- supports that cannot bear tension (or compression)
- global imperfections
- 2nd order analysis
