Statische Berechnungen von Stabwerken

Static in frameworks

A framework is determined by the coordinates of a number NK of nodes and a number NS of bars each connecting two nodes. The bars have a mass and load forces may drag the nodes. Two nodes are assigned as abutments. There are two cases of abutments:

1. The stand-alone framework which has a stiff connection between the two abutment nodes. With two fixed abutments this framework would be overdeterined, it is thus necessary to relax one condition. This is achieved by using a gliding abutment which is itself defined by the normal vector of the abutment plane
2. The pendular framework which requires two fixed abutments

Formal problem definition

Given a number NK of nodes and a number of slabs and two nodes for which the abutment forces are to be found. Each slab causes gravitation forces in its ending nodes. Load forces may be specified for individual nodes. All forces are noted as two component vectors or, by equal alternative, the force value and angle of direction

To be determined are:

1. Scalars for the force in every bar
2. Two components of the force vector at abutment  A
3. in abutment B
   1. Scalar for the foce value of the force parallel to the normal vector in the stand-alone case
   2. Two components of the force vector at abutment B in the other casel

corollary conditions

Since the nodes do not move, the sum of all forces at every node is equal to zero

Since the bars do not move, the sum of all forces in every bar is equal to zero

Since the framework as a whole does not move, the sum of all gravitational, load and abutment forces in A and B is equal to zero

Equation system

The equation system determines the equilibrium condition for every component of the resulting force vector at each node and hence contains 2 NK equations in that many rows

In the case of a stand-alone framework this provides the following unknowns:

1. NS scalars for the bar force amounts
2. 2 components of the abutment force in A
3. 1 scalar for the amount of abutment force B

This defines a conditions for the number of slabs in a framework with NK nodes: NS = 2NK - 3

In the case of a pendular framework the following unknown figures result:

1. NS scalars for the bar force amounds
2. 2 components of the abutment force in A
3. 2 components of the abutment force in B

In this case the number of bars is: NS = 2NK - 4

If the equiation system can be resolved, which is equal to a non-zero determinant of the Matrix A, the solution vector contains all elements sought:

Evaluated and applied on the bar vectors, we obtain the force diagram:

Analysis tool

QRS provides a light Excel tool to analyse a framework in a simple way. The figures in this post are screenshots of the worksheet for three nodes and three slabs. The current version allows for 10 nodes and a maximum number of 16 or 17 bars, depending on the static case. The angle of the gravitational force can be set arbitrarily, so the whole construction can be "inclined" with respect to gravity.

Literature:

Löwe, Harald; Statik von Fachwerken; TU Braunschweig, Institut for computational mathematics, D-38106 Braunschweig, Deutschland

Wandinger, Johannes; Einfache ebene Tragwerke; Hochschule München