Hexahedral Mesh Refinement in Salome

     In this video I show some ways to refine the hexahedral mesh of the beam we did before. This procedure is meant to be more clever than simple making the "number of segments" parameter bigger and bigger.

Thanks again!

Cantilever beam finite element analysis using open source software

     In this video I'm modeling, meshing and computing a cantilever beam using open source software. For the model and mesh, I've used Salome7, and the finite element computations were carried out with Calculix. This is meant to be an practical introductory video to free fea software. Theres an insane amount of theoretical books about this subject out there, take a look on them before designing your own cranes (LoL). 

The code for the calculix input deck:

	*include,input=hexb.inp
	** material description
	*MATERIAL, NAME=steel
	*ELASTIC
	210000.,0.3
	*density
	7.850e-09
	*BOUNDARY
	fix,1,3,0
	** element description
	*solid section, elset=C3D20, material=steel
	*STEP
	*STATIC
	*dload
	C3D20,GRAV,9810.,0.,0.,-1.
	*el file
	S
	*node file
	U
	*end step

view raw hexbeam.inp hosted with ❤ by GitHub

Cheers!

Plotting Beam Diagrams with Python

     Today I'm sharing how to plot Beam Diagrams using python. More specifically the code is about a cantilever beam subjected to gravity load, a special case of uniformly distributed load. There's a bunch of cool python features in this post, like the very convenient use of "x>" to evaluate a function, or part of it, only after a specified value in the array.

For example, the code: (my blog font shows this symbol =, as equals, this is a minus -)

import numpy as np
x=np.linspace(1,10,10)
y=2*x*(x>5)
print x
print y

returns these two arrays.

[  1.   2.   3.   4.   5.   6.   7.     8.   9.    10.]
[  0.   0.   0.   0.   0.  12.  14.  16.  18.    20.]

     Values with "false" boolean condition returned 0, and the values with "true" condition were evaluated. This is very useful when computing singularities functions, also known as Macaulay method, due to the British mathematician William Herrick Macaulay.  Oddly this method was first proposed by the German mathematician Alfred Clebsch. 

    Other features that worth take a look is how to make very neat plots using matplotlib subplots method.

The code:

	#Program to compute and graph a cantilever beam
	#subject to Uniformly Distributed Load
	#Written by Eddie Liberato in 25/10/16
	import numpy as np
	import matplotlib.pyplot as plt
	#Beam parameters
	l=200 #mm, lenght
	h=100 #mm, height
	b=10 #mm, width
	a=0 #mm, point where load begins
	g=9810 #mm/s^2
	rho=7.850*(10**-9) #g/mm^3
	E = 210*(10**3) #N/mm^2
	I = (b*h**3)/12 #Second moment of area for square cross section
	w=rho*g*b*h #UDL by beam own weight (N/mm)
	#Reactions
	r1=w*(l-a) #Reaction at the support
	m1=(w/2)*(l**2-a**2) #Moment on the support
	#Shear, Bending and deflection arrays
	x=np.linspace(0,l)
	V=r1*(x>=0)*(x-0)**0-w*(x>=a)*(x-a)**1
	M=-m1*(x>=0)*(x-0)**0+r1*(x>=0)*(x-0)**1-(w/2)*(x>=a)*(x-a)**2
	y=(1/float (E*I))*(-(m1/2)*x**2+(r1/6)*x**3-(w/24)*(x>=a)*x**4)
	#Max Values
	maxV=max(V)
	maxM=min(M)
	maxy=min(y)
	#Diagrams plot
	textstr1 = 'max=%.2f (N)'%(maxV)
	textstr2 = 'max=%.2f (N*mm)'%(maxM)
	textstr3 = 'max=%.5f (mm)'%(maxy)
	props = dict(boxstyle='round', facecolor='white')
	fig, (ax1, ax2, ax3) = plt.subplots(3, 1, sharex=True, figsize=(10, 6))
	ax1.fill_between(x, 0, V, facecolor='#377EB8', alpha=0.8)
	ax1.set_ylabel('Shear')
	ax1.text(0.78, 0.9, textstr1, transform=ax1.transAxes, fontsize=14,
	verticalalignment='top', bbox=props)
	ax1.set_title('Cantilever Beam Subjected to Gravity Load')
	ax1.grid (True)
	ax2.fill_between(x, 0, M, facecolor='#55BA87', alpha=0.8)
	ax2.set_ylabel('Moment')
	ax2.text(0.68, 0.2, textstr2, transform=ax2.transAxes, fontsize=14,
	verticalalignment='top', bbox=props)
	ax2.grid (True)
	ax3.fill_between(x, 0, y, facecolor='#7E1137', alpha=0.8)
	ax3.set_ylabel('Deflection')
	ax3.set_xlabel('Beam lenght axis')
	ax3.text(0.02, 0.2, textstr3, transform=ax3.transAxes, fontsize=14,
	verticalalignment='top', bbox=props)
	ax3.grid (True)
	#plt.show()
	plt.savefig('beam.svg', format='svg', dpi=1200)

view raw beam.py hosted with ❤ by GitHub

The plot:

See ya!