3.6.2: AnalyzeThII

The "Analyze ThII"-component exist for convenience. Second order theory analysis can also be specified per load-case-combination via the "Load-Case-Combination Settings"-component and calculated by an "Analyze"-component.

Section 3.2.5.3 contains details regarding second order theory calculations.

In Karamba3D distinction is made between normal forces $N$ which cause stresses in the members and normal forces $N^{II}$ which result in second order effects (see also [10]). At first sight this concept seems weird. How can there be two kinds of normal forces in the same beam? Well, in reality there can’t. In a computer program it is no problem: stresses get calculated as $\sigma = N/A$ and $N^{II}$ is used for determining second order effects only. The advantage is, that in the presence of several load-cases one can chose for each element the largest compressive force as $N^{II}$. This gives a lower limit for the structure's stiffness. A re-evaluation of the load-cases using these $N^{II}$ values leads to a structural response which is too soft. However the different load-cases may then be safely superimposed.

Use the “AnalyzeThII”-component for automatically determining the normal forces $N^{II}$ from cross section forces $N_{x} \cdot N ^{II}$influences a structure's stiffness which in turn impacts the distribution of cross section forces $N_x$. Thus an iterative procedure with repeated updates of $N^{II}$-forces needs to be applied.

Fig 3.6.2: Deflection of simply supported beam under single load in mid-span and axial compressive load

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AnalysisComponent_ThII.gh

Fig. 3.6.2 shows the same system as in fig. 3.5.1. This time with results according to first and second order theory. When comparing the transverse deflections in load-case two one can see that the maximum deflection increased from $0.24[m]$ to $0.28[m]$ due to the effect of the axial compressive load.

The “AnalyzeThII”-component features the following input-plugs:

"Model"	Model to be considered
"LC"	Name of load-case or load-case-combination from which to take the normal force $N^{II}$ which cause second order theory effects. By default all the minimum normal force of all load-cases is considered. The hint in section Load-Case-Combinations explains how load-cases and load-case-combinations can be excluded from the default list.
"RTol"	The determination of $N^{II}$ is an iterative process. The value of “RTol” is the upper limit of displacement increments from one iteration to the next.
"MaxIter"	Supply here the maximum number of iterations for determining $N^{II}$. The default is 50. In case “RTol” can not be reached within the preset number of iterations the component turns orange.
"NoTenNII"	Tension forces increase the stiffness of a structure. Setting “NoTenNII” to “True” limits $N^{II}$ to negative values.

The normal forces $N^{II}$ get attached to the model and will be considered in all further analysis steps. They impact the results of the “Analyze”-, “Buckling Modes”-, “Natural Vibrations”- and “Optimize Cross Sections”-components. For imperfection loads $N^{II}$-forces have a direct impact on the applied loads.

Use the “NII” button in submenu “Tags” of the “ModelView”-component to display $N^{II}$-forces.