Beams - structural beams#

This notebook shows how to use Beam elements model cantilever beams, how to get the results and how those results compare to theory.

A “Beam” in DAVE consists of a number of discrete segments and has a different formulation than the beams typically found in FEM packages.

from DAVE import *
s = Scene();
Equilibrium-core version = 2.48 from c:\python\miniconda3\envs\book\lib\site-packages\pyo3d.cp39-win_amd64.pyd
default resource folders:
c:\python\miniconda3\envs\book\lib\site-packages\DAVE\resources
C:\Users\beneden\DAVE_models
C:\data\Dave\Book\DAVE-book\DAVE-notebooks
Blender found at: C:\Program Files\Blender Foundation\Blender 3.3\blender.exe

Beams are elastic nodes that can be created between axis systems.

left = s.new_axis("left")
right = s.new_axis("right")
beam = s.new_beam("beam", nodeA=left, nodeB = right,
                        EA = 1000,
                        EIy = 1e5, EIz = 1e5, GIp = 1e3,
                        L=10,
                        mass = 0,
                        n_segments=20)
right.set_free()
s.solve_statics()
c:\python\miniconda3\envs\book\lib\site-packages\DAVE\scene.py:1668: UserWarning: new_axis is deprecated, use new_frame instead
  warnings.warn("new_axis is deprecated, use new_frame instead")
True
from DAVE.jupyter import *;
show(s, camera_pos=(5,-10,6), lookat = (5,0,2), width=1000, height = 200)
 Warning! Please use "settings.use_parallel_projection" instead!
../_images/beams_5_1.png
c:\python\miniconda3\envs\book\lib\site-packages\DAVE\jupyter\jupyter.py:194: UserWarning: VTK/Vedo issue: plotter is None
  warnings.warn("VTK/Vedo issue: plotter is None")

A Beam node is rigidly fixed to the axis systems at its ends. It is fixed in the positive X-direction, so it departs from node A along the X-axis and arrives at ndoe B from the negative X-direction.

At the momement we have a massless beam with its left side fixed (axis system “left” is fixed) and its right size free (the right end of the beam is fixed to axis system “right”, but “right” is not connected to anything and is free to move).

This is one of those cases that frequently occur in textbooks and for which we can easily check the results.

cantilever.png

First place a point on the right side so that we can apply a force:

s.new_point("point", parent = right)
force = s.new_force("force", parent = "point")

Cantilever beam with a force#

force.force = (0,0,-100)
s.solve_statics()
True
show(s, camera_pos=(5,-10,0), lookat = (5,0,0), width=1000, height = 300)
c:\python\miniconda3\envs\book\lib\site-packages\DAVE\visual_helpers\vtkHelpers.py:533: RuntimeWarning: invalid value encountered in divide
  axis = axis / length
c:\python\miniconda3\envs\book\lib\site-packages\DAVE\visual_helpers\vtkHelpers.py:594: RuntimeWarning: invalid value encountered in divide
  axis = axis / length
 Warning! Please use "settings.use_parallel_projection" instead!
../_images/beams_15_2.png

Theory tells us that the deflection and rotation of the right end should be:

\(\delta = -PL^3 / 3EI\)

and

\(\Theta = PL^2 / 2EI\)

Get the values from the model:

L = beam.L
EI = beam.EIy
P = -force.force[2]

And calculate their values

The deflection is the vertical displacement of the axis system on the right side of the beam:

right.z
-0.33604653085104813
-P*L**3 / (3*EI)
-0.3333333333333333

Theta is the slope. The slope can be calculated from the rotation of the axis system, but it is also available as “tilt”, which is a percentage.

right.tilt_y / 100
0.05014161613258149
P*L**2 / (2*EI)
0.05

It is also possible to obtain the forces in the beam.

The moments are available at the nodes.

import matplotlib.pyplot as plt
plt.plot(beam.X_nodes, beam.bending[:,1])
plt.xlabel('Distance along the beam [m]')
plt.ylabel('Moment about y-axis [kN*m]');
../_images/beams_27_0.png

Cantilever with moment#

We can do the same with a moment

force.force = (0,0,0)
force.moment = (0,1000,0)
s.solve_statics()
True
show(s, camera_pos=(5,-10,3), lookat = (5,0,-1), width=1000, height = 300)
c:\python\miniconda3\envs\book\lib\site-packages\DAVE\visual_helpers\vtkHelpers.py:533: RuntimeWarning: invalid value encountered in divide
  axis = axis / length
 Warning! Please use "settings.use_parallel_projection" instead!
../_images/beams_31_2.png
c:\python\miniconda3\envs\book\lib\site-packages\DAVE\jupyter\jupyter.py:194: UserWarning: VTK/Vedo issue: plotter is None
  warnings.warn("VTK/Vedo issue: plotter is None")

Theory tells us that the deflection and rotation of the right end should be:

\(\delta = ML^2 / 2EI\)

and

\(\Theta = ML / EI\)

Get the values from the model:

M = force.moment[1]  # moment about Y-axis
M*L**2 / (2*EI)
0.5
-right.z
0.4995839925972531
M*L / EI
0.1
right.tilt_y / 100
0.09983341664682813

Tension and torsion#

Beam can also take tension and torsion.

force.force = (100,0,0)
force.moment = (100,0,0)
s.solve_statics()
True
show(s, camera_pos=(5,-10,3), lookat = (5,0,-1), width=1000, height = 300)
 Warning! Please use "settings.use_parallel_projection" instead!
../_images/beams_41_1.png

Tension and torsion forces in the beam are available not at the nodes but at the centers for the beam segments.

plt.plot(beam.X_midpoints, beam.torsion);
../_images/beams_43_0.png
plt.plot(beam.X_midpoints, beam.tension);
../_images/beams_44_0.png

Now this looks horrible!

Fortunately this is just the way matplotlib displays data which is almost constant.

Looking at the data directly it appears to be comfortingly constant:

beam.torsion
[99.99999999999781,
 99.99999999999814,
 99.99999999999709,
 99.99999999999842,
 99.99999999999537,
 99.99999999999955,
 99.9999999999967,
 99.99999999999459,
 100.00000000000048,
 99.99999999999915,
 99.99999999999892,
 100.00000000000448,
 100.00000000000048,
 99.9999999999956,
 100.00000000000381,
 99.99999999998425,
 100.00000000000159,
 100.00000000000026,
 100.00000000000136,
 99.99999999998536]
import matplotlib
matplotlib.rcParams['axes.formatter.useoffset'] = False
plt.plot(beam.X_midpoints, beam.torsion);
plt.ylim((99.9, 100.1));
../_images/beams_48_0.png
plt.plot(beam.X_midpoints, beam.tension);
plt.ylim((99.9, 100.1));
../_images/beams_49_0.png

Distributed load#

Modelling a distributed load is not possible in this way. Loads can only be added at points.

To model a distributed load we would have to manually discretise the model and add discrete loads at each node.

But these is a shortcut that we can take: The beams can have a mass. We can use the mass to model a distributed load as gravity.

s.delete('force')
q = 10 # kN/m
beam.mass = beam.L * q  / 9.81
s.solve_statics()
True
show(s, camera_pos=(5,-10,0), lookat = (5,0,0), width=1000, height = 300)
 Warning! Please use "settings.use_parallel_projection" instead!
../_images/beams_56_1.png

Theory tells us that the deflection and rotation of the right end should be:

\(\delta = qL^4 / 8EI\)

and

\(\Theta = qL^3 / 6EI\)

-q*L**4 / (8*EI)
-0.125
right.z
-0.12539340701469195
q*L**3 / (6*EI)
0.016666666666666666
right.tilt_y / 100
0.01668937486888241

Large deflections#

All the tests done so far were performed on small deflections. That is for a good reason. The Euler/Bernouilli beam theory is only applicable to small displacements. This is because in this theory the moment is derived from the deflection:

\(d^2/dx^2(EI d^2w / dx^2) = q\)

so the moment in the beam is proprtional to the change in slope. This is only valid for small changes.

The implementation in DAVE is valid for large deflections, it is just that the analytical formula used above are not.

Number of segments#

Beams use a discrete model (see theory https://davedocs.online/beams.html).

When a higher number of segments is used:

  • The model become more accurate

  • The solver becomes slower

  • (The numerical errors build up)

So the number of nodes should be selected with some caution. In general 20 seems to be a good number for beams although in some cases (for example pure tension or torsion) a single segment is just as good.

It is easy to use the “plot_effect” function in scene to check the effect of the number of segments.

Here we calculate the effect of the number of segments on the vertical position of the second end of the beam:

s.plot_effect(evaluate="s['point'].gz",
   change_property="n_segments",
    change_node="beam",
    start=1,
    to=25.0,
    steps=25);
setting 1.0 results in -0.25043539538010456
setting 2.0 results in -0.15639089359229383
setting 3.0 results in -0.13899103714473474
setting 4.0 results in -0.1329040421012639
setting 5.0 results in -0.1300872247358817
setting 6.0 results in -0.1285572544156294
setting 7.0 results in -0.12763482040356924
setting 8.0 results in -0.1270361175408637
setting 9.0 results in -0.12662568342623784
setting 10.0 results in -0.12633209478116098
setting 11.0 results in -0.12611496195310684
setting 12.0 results in -0.12594964178054247
setting 13.0 results in -0.12582112540100712
setting 14.0 results in -0.12571911178709774
setting 15.0 results in -0.12563692195711026
setting 16.0 results in -0.12556940415251053
setting 17.0 results in -0.12551358019856465
setting 18.0 results in -0.12546679931017593
setting 19.0 results in -0.12542720873744273
setting 20.0 results in -0.1253934070146907
setting 21.0 results in -0.12536431820383623
setting 22.0 results in -0.12533910520139316
setting 23.0 results in -0.12531710883253766
setting 24.0 results in -0.1252978043245916
setting 25.0 results in -0.12528076971477245
../_images/beams_66_21.png

– end –