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{
"metadata": {
"name": "",
"signature": "sha256:c3ab3016805321c8a28748b1937c77b2125357dfc661b20013df0624feaa950e"
},
"nbformat": 3,
"nbformat_minor": 0,
"worksheets": [
{
"cells": [
{
"cell_type": "heading",
"level": 1,
"metadata": {},
"source": [
"Chapter 16: Bending of open and closed thin-walled beams \n"
]
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 16.1 Pg. No.456"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"# Fig 16.6 Dimensions\n",
"from __future__ import division\n",
"import math\n",
"\n",
"#variable declaration\n",
"l=300 #length of I section (mm)\n",
"b=200 #width of I section beam (mm)\n",
"w1=25 #width of center section of I beam (mm)\n",
"w2=20 #width of upper and lower section of I beam (mm)\n",
"M=100*10**6 #moment applied in vertical plane (N*mm)\n",
"\n",
"#second moment of area Ixx\n",
"# Ixx=b*l^3/12\n",
"Ixx=b*l**3/12-(b-w1)*(l-w2-w2)**3/12\n",
"print \"\\nSecond moment of area of I section beam = %5.3e mm^4\"%(Ixx)\n",
"\n",
"#sigma_z=My/I reference 16.9\n",
"# @ \n",
"y=150\n",
"sigma_z=M*y/Ixx\n",
"print \"\\ndirect stress at the top of the I section (y=150) = %3.2f N/mm^2 (compression)\"%(sigma_z)\n",
"\n",
"# @\n",
"y=-150\n",
"sigma_z=M*y/Ixx\n",
"print \"\\ndirect stress at the bottom of the I section (y=-150) = %3.2f N/mm^2 (tension)\"%(sigma_z)\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"\n",
"Second moment of area of I section beam = 1.937e+08 mm^4\n",
"\n",
"direct stress at the top of the I section (y=150) = 77.45 N/mm^2 (compression)\n",
"\n",
"direct stress at the bottom of the I section (y=-150) = -77.45 N/mm^2 (tension)\n"
]
}
],
"prompt_number": 13
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 16.2 Pg. No.457"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"# Fig 16.6 reference\n",
"from __future__ import division\n",
"import math\n",
"\n",
"#variable declaration\n",
"l=300 #length of I section (mm)\n",
"b=200 #width of I section beam (mm)\n",
"w1=25 #width of center section of I beam (mm)\n",
"w2=20 #width of upper and lower section of I beam (mm)\n",
"M=100*10**6 #moment applied in vertical plane (N*mm)\n",
"\n",
"# second moment of area\n",
"# Iyy=wb^3/12\n",
"Iyy=2*20*200**3/12+260*25**3/12\n",
"print \"\\nSecond moment of area of I section beam = %5.3e mm^4\"%(Iyy)\n",
"\n",
"#sigma_z=Mx/I \n",
"# @ \n",
"x=100\n",
"sigma_z=M*x/Iyy\n",
"print \"\\ndirect stress at the top of the I section (y=150) = %3.2f N/mm^2 (compression)\"%(sigma_z)\n",
"\n",
"# @\n",
"x=-100\n",
"sigma_z=M*x/Iyy\n",
"print \"\\ndirect stress at the bottom of the I section (y=-150) = %3.2f N/mm^2 (tension)\"%(sigma_z)\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"\n",
"Second moment of area of I section beam = 2.701e+07 mm^4\n",
"\n",
"direct stress at the top of the I section (y=150) = 370.30 N/mm^2 (compression)\n",
"\n",
"direct stress at the bottom of the I section (y=-150) = -370.30 N/mm^2 (tension)\n"
]
}
],
"prompt_number": 17
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 16.3 Pg.No.458 "
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"from __future__ import division\n",
"import math\n",
"\n",
"#variable declaration\n",
"l=300 #length of I section (mm)\n",
"b=200 #width of I section beam (mm)\n",
"w1=25 #width of center section of I beam (mm)\n",
"w2=20 #width of upper and lower section of I beam (mm)\n",
"M=100*10**6 #moment applied in vertical plane (N*mm)\n",
"theta=30 #angle at which bending moment is applied(degree)\n",
"\n",
"Mx=M*math.cos(math.radians(30))\n",
"My=M*math.sin(math.radians(30))\n",
"\n",
"# sigma_z=Mx/Ixx*y+My/Iyy*x\n",
"# @top left hand corner\n",
"y=150\n",
"x=-100\n",
"sigma_z=Mx/Ixx*y-My/Iyy*x \n",
"print \"\\ndirect stress at the top left hand corner = %3.1f N/mm^2 (tension)\"%(sigma_z)\n",
"\n",
"# @ top right hand corner\n",
"x=100\n",
"y=150\n",
"sigma_z=Mx/Ixx*y-My/Iyy*x \n",
"print \"\\ndirect stress at the top right hand corner = %3.1f N/mm^2 (compression)\"%(sigma_z)\n",
"\n",
"alpha=math.atan(My*Ixx/Mx/Iyy)\n",
"print \"\\ninclination = %3.1f degree\\n\"%(alpha*180/math.pi)"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"\n",
"direct stress at the top left hand corner = 252.2 N/mm^2 (tension)\n",
"\n",
"direct stress at the top right hand corner = -118.1 N/mm^2 (compression)\n",
"\n",
"inclination = 76.4 degree\n",
"\n"
]
}
],
"prompt_number": 38
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 16.4 Pg.No.466"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"# reference Fig 16.13\n",
"import math\n",
"from __future__ import division\n",
"#variable declaration\n",
"l=80 #length of one section (mm)\n",
"w=8 #thickness of each section (mm)\n",
"b1=40 #width of one section (mm)\n",
"b2=80 #width of other section (mm)\n",
"Mx=1500*10*3 #bending moment (N.mm)\n",
"My=0\n",
"\n",
"#centroid\n",
"# in this example C is taken at top surface x axis aligned to AB and y axis is \n",
"# aligned to E surface \n",
"y_bar=((b1+b2)*w*w/2+l*w*(w+l/2))/(l*w+(b1+b2)*w)\n",
"print \"\\ny coordinate of centroid = %3.1f mm\"%(y_bar)\n",
"\n",
"x_bar=((b1+b2)*w*((b1+b2)/2-b1+w/2)+l*w*(w/2))/(l*w+(b1+b2)*w)\n",
"print \"\\nx coordinate of centroid = %3.1f mm\"%(x_bar)\n",
"\n",
"#second area of moment\n",
"#IB=IC+Ab^2 IB=moment about any point,IC=moment about centroid, b=distace between both points\n",
"Ixx=(b1+b2)*w**3/12+(b1+b2)*w*(y_bar-w/2)**2+w*l**3/12+l*w*(w+l/2-y_bar)**2\n",
"print \"\\nsecond moment of area about x axis = %3.2e mm^4\"%(Ixx)\n",
"\n",
"Iyy=w*(b1+b2)**3/12+(b1+b2)*w*((b1+b2)/2-b1+w/2-x_bar)**2+l*w**3/12+l*w*(x_bar-w/2)**2\n",
"print \"\\nsecond moment of area about y axis = %3.2e mm^4\"%(Iyy)\n",
"\n",
"Ixy=(b1+b2)*w*(y_bar-w/2)*((b1+b2)/2-b1+w/2-x_bar)+l*w*(w+l/2-y_bar)*(x_bar-w/2)\n",
"print \"\\nsecond moment of area about x and y axis = %3.2e mm^4\"%(Ixy)\n",
"\n",
"Mx=15*10**5\n",
"def f(x,y):\n",
" return Mx*(Iyy*y-Ixy*x)/(Ixx*Iyy-Ixy**2)\n",
"sigma_z_max= f(-8,-66.4)\n",
"print \"\\nmaximum direct shear stress = %3.0f N/mm^2 (compressive)\\n\"%(sigma_z_max)"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"\n",
"y coordinate of centroid = 21.6 mm\n",
"\n",
"x coordinate of centroid = 16.0 mm\n",
"\n",
"second moment of area about x axis = 1.09e+06 mm^4\n",
"\n",
"second moment of area about y axis = 1.31e+06 mm^4\n",
"\n",
"second moment of area about x and y axis = 3.38e+05 mm^4\n",
"\n",
"maximum direct shear stress = -96 N/mm^2 (compressive)\n",
"\n"
]
}
],
"prompt_number": 62
},
{
"cell_type": "code",
"collapsed": false,
"input": [],
"language": "python",
"metadata": {},
"outputs": []
}
],
"metadata": {}
}
]
}
|
