3.2.5.3 Load-Case-Combination Settings

The "Load Case Combination Settings"-component lets you specify the way Karamba3D analyzes load-case combinations.

The name of the load-case combination to which the settings apply can be supplied via the "LCase" input-plug and defaults to "LC0". Regular expressions starting with '&' and simplified regular expressions ending with '$' allow to select multiple load-case combinations at once.

Fig. 3.2.5.3.1: Options for small displacement calculation with and without update of $N^{II}$-forces.

Type of Calculation

The component's drop-down menu serves to select the way how a load-case combination shall be processed by components further down the data-stream.

With nothing specified, the "Analyze"-component performs for load-combinations small displacement calculations without iteratively updating second order theory forces (called $N^{II}$).

Here some explanations in case you are not familiar with the concept of first and second theory calculations in structural computations:

• Small displacement calculations imply that the impact of transverse displacements on the elongation of elements and thus axial force can be neglected. This assumption normally holds if a structure's maximum displacement is roughly less than halve the cross-section height.

• Under the small displacement assumption equilibrium of forces in a structure can be calculated formulated for the undeformed or deformed structure. The former approach is called first order theory (Th. I) the latter second order theory (Th. II). Second order theory covers effects like buckling of beams and shells or stiffening of ropes via pre-tension. Compressive axial (think of beams) or in-plane normal forces in shells soften a system and increase existing bending moments, tensile forces stiffen a system and reduce bending moments (see also [10]). The contribution of second order theory effects to the system stiffness is called the geometric stiffness. 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.

• Karamba3D differentiates between normal forces $N$which cause stresses in the cross-section and $N^{II}$-forces which cause second order theory effects and impact a structure's stiffness. 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.

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

The Small Displacements Option

Selecting the 'Small Disp.' option in the drop-down menu (see fig. 3.2.5.3.1) tells the "Analyze"-component to perform a single calculation step for the load-case-combination without updating $N^{II}$.

The input-plug 'InitialNII' determines whether user-defined $N^{II}$-values shall be considered or not. It defaults to true. Use 'Modify Element'-components (see section 3.1.8: Modify Element) for specifying initial $N^{II}$-values and thus setting an initial geometric stiffnesses.

The Small Displacements Th. II Option

This option lets the 'Analyze'-component apply an iterative procedure for updating the $N^{II}$-values based on the normal forces $N$. In case of statically indeterminate systems these two quantities influence each other. However, in case of reasonably stable structures their values rapidly converge. These input-values can be provided:

"InitialNII"	True if initial value of NII from element definitions shall be used. Same as above.
"CombiNII"	If "true", $N^{II}$for each element is taken as the smallest value resulting from the load-cases of the load-case-combination. For details see below.
"NoTenNII"	Tension forces increase the stiffness of a structure. Setting “NoTenNII” to “True” limits $N^{II}$to negative values.
"RTol"	The determination of $N^{II}$ is an iterative process. The value of “RTol” is the upper limit of resultant displacement-, force- or $N^{II}$-increments divided by the corresponding absolute value.
"MaxIter"	Supply here the maximum number of iterations for determining $N^{II}$. The default is 50. In case “RTol” cannot be reached within the preset number of iterations the component turns orange.

Setting "CombiNII" to true (the default) or false impacts the character of results reached and speed of calculation. These are the advantages and liabilities of enabling "CombiNII":

• Using the minimum $N^{II}$of all load-cases results in a lower limit of the geometric stiffness. Thus, the structure behaves too soft. The resulting cross section forces and displacements will be overestimated.
• As compared to the more precise procedure available with 'CombiNII' set to 'false', convergence of $N^{II}$ will be faster: The stiffnessmatrix needs to be assembled and solved only once per iteration step for the load-case-combination as a hole. The more precise procedure with individual $N^{II}$-values necessitates a separate stiffness matrix for each load-case.
• With 'CombiNII' enabled, the load-case results may be linearly superimposed since they are based on the same system stiffness.