plugins.section2d — pyFormex 1.0.5 documentation

# Source code for plugins.section2d

    # $Id$
    ##
    ##  This file is part of pyFormex 1.0.5  (Sat Feb 16 10:40:32 CET 2019)
    ##  pyFormex is a tool for generating, manipulating and transforming 3D
    ##  geometrical models by sequences of mathematical operations.
    ##  Home page: http://pyformex.org
    ##  Project page:  http://savannah.nongnu.org/projects/pyformex/
    ##  Copyright 2004-2018 (C) Benedict Verhegghe (benedict.verhegghe@ugent.be)
    ##  Distributed under the GNU General Public License version 3 or later.
    ##
    ##  This program is free software: you can redistribute it and/or modify
    ##  it under the terms of the GNU General Public License as published by
    ##  the Free Software Foundation, either version 3 of the License, or
    ##  (at your option) any later version.
    ##
    ##  This program is distributed in the hope that it will be useful,
    ##  but WITHOUT ANY WARRANTY; without even the implied warranty of
    ##  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
    ##  GNU General Public License for more details.
    ##
    ##  You should have received a copy of the GNU General Public License
    ##  along with this program.  If not, see http://www.gnu.org/licenses/.
    ##
    #
    """Some functions operating on 2D structures.

    This is a plugin for pyFormex.
    (C) 2002 Benedict Verhegghe

    See the Section2D example for an example of its use.
    """
    from __future__ import absolute_import, division, print_function

    from pyformex.plugins import sectionize
    from numpy import *

    ## initalization should be with a closed PolyLine, after checking it is plane

    class PlaneSection(object):
        """A class describing a general 2D section.

        The 2D section is the area inside a closed curve in the (x,y) plane.
        The curve is decribed by a finite number of points and by straight
        segments connecting them.
        """

        def __init__(self, F):
            """Initialize a plane section.

            Initialization can be done either by a list of points or a set of line
            segments.

            By Points:
              Each point is connected to the following one, and (unless they are
              very close) the last one back to the first. Traversing the resulting
              path should rotate positively around the z axis to yield a positive
              surface.

            By Segments:
              It is the responsibilty of the user to ensure that the segments
              form a closed curve. If not, the calculated section data will be
              rather meaningless.
            """
            if F.nplex() == 1:
                self.F = sectionize.connectPoints(F, close=True)
            elif F.nplex() == 2:
                self.F = F
            else:
                raise ValueError("Expected a plex-1 or plex-2 Formex")

        def sectionChar(self):
            return sectionChar(self.F)

    def sectionChar(F):
        """Compute characteristics of plane sections.

        The plane sections are described by their circumference, consisting of a
        sequence of straight segments.
        The segment end point data are gathered in a plex-2 Formex.
        The segments should form a closed curve.
        The z-value of the coordinates does not have to be specified,
        and will be ignored if it is.
        The resulting path through the points should rotate positively around the
        z axis to yield a positive surface.

        The return value is a dict with the following characteristics:

        - `L`   : circumference,
        - `A`   : enclosed surface,
        - `Sx`  : first area moment around global x-axis
        - `Sy`  : first area moment around global y-axis
        - `Ixx` : second area moment around global x-axis
        - `Iyy` : second area moment around global y-axis
        - `Ixy` : product moment of area around global x,y-axes
        """
        if F.nplex() != 2:
            raise ValueError("Expected a plex-2 Formex!")
        #pf.debug("The circumference has %d segments" % F.nelems())
        x = F.x
        y = F.y
        x0 = x[:, 0]
        y0 = y[:, 0]
        x1 = x[:, 1]
        y1 = y[:, 1]
        a = (x0*y1 - x1*y0) / 2
        return {
            'L': sqrt((x1-x0)**2 + (y1-y0)**2).sum(),
            'A': a.sum(),
            'Sx': (a*(y0+y1)).sum()/3,
            'Sy': (a*(x0+x1)).sum()/3,
            'Ixx': (a*(y0*y0+y0*y1+y1*y1)).sum()/6,
            'Iyy': (a*(x0*x0+x0*x1+x1*x1)).sum()/6,
            'Ixy': -(a*(x0*y0+x1*y1+(x0*y1+x1*y0)/2)).sum()/6,
            }

    def extendedSectionChar(S):
        """Computes extended section characteristics for the given section.

        S is a dict with section basic section characteristics as returned by
        sectionChar().
        This function computes and returns a dict with the following:

        - `xG`, `yG` : coordinates of the center of gravity G of the plane section
        - `IGxx`, `IGyy`, `IGxy` : second area moments and product around axes
          through G and  parallel with the global x,y-axes
        - `alpha` : angle(in radians) between the global x,y axes and the principal
          axes (X,Y) of the section (X and Y always pass through G)
        - `IXX`, `IYY` : principal second area moments around X,Y respectively. (The
          second area product is always zero.)
        """
        xG =  S['Sy']/S['A']
        yG =  S['Sx']/S['A']
        IGxx = S['Ixx'] - S['A'] * yG**2
        IGyy = S['Iyy'] - S['A'] * xG**2
        IGxy = S['Ixy'] + S['A'] * xG*yG
        alpha, IXX, IYY = princTensor2D(IGxx, IGyy, IGxy)
        return {
            'xG': xG,
            'yG': yG,
            'IGxx': IGxx,
            'IGyy': IGyy,
            'IGxy': IGxy,
            'alpha': alpha,
            'IXX': IXX,
            'IYY': IYY,
            }

    def princTensor2D(Ixx, Iyy, Ixy):
        """Compute the principal values and directions of a 2D tensor.

        Returns a tuple with three values:

        - `alpha` : angle (in radians) from x-axis to principal X-axis
        - `IXX,IYY` : principal values of the tensor
        """
        from math import sqrt, atan2
        C = (Ixx+Iyy) * 0.5
        D = (Ixx-Iyy) * 0.5
        R = sqrt(D**2 + Ixy**2)
        IXX = C+R
        IYY = C-R
        alpha = atan2(Ixy, D) * 0.5
        return alpha, IXX, IYY

    # End