3.6.2: AnalyzeThII 🔷

Axial forces in beams and in-plane forces in shells influence the structural stiffness. Compressive forces decrease a structure’s stiffness, tensile forces increase it. The influence of compressive forces on displacements and cross section forces may be neglected as long as their absolute value is less than 10% of the buckling load.

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

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"	Number of load-case from which to take the normal force $N^{II}$ which cause second order theory effects. If set to −1 (the default) the minimum normal force of all load-cases is considered
"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.

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Analyze_ThII_Truss.gh