{
"metadata": {
"name": "",
"signature": "sha256:8adf459ea4014e5f8ac2990e8d05feaccf30880e758648b173bb1fd18b74b81a"
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"worksheets": [
{
"cells": [
{
"cell_type": "heading",
"level": 1,
"metadata": {},
"source": [
"Appendix A: Review of Properties of Plane Area"
]
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example A.1, Page No:486"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import math\n",
"\n",
"#Variable Decleration\n",
"A=2000 #Area of the plane in mm^2\n",
"Ix=40*10**6 #Momnet of Inertia in mm^4\n",
"d1=90 #Distance in mm\n",
"d2=70 #Distance in mm\n",
"\n",
"#Calculations\n",
"Ix_bar=Ix-(A*d1**2) #Moment of Inertia along x_bar axis in mm^4\n",
"Iu=Ix_bar+A*d2**2 #Moment of Inertia along U-axis in mm^4\n",
"\n",
"#Result\n",
"print Ix_bar\n",
"print \"The moment of inertia along u-axis is\",round(Iu,1),\"mm^4\""
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"23800000\n",
"The moment of inertia along u-axis is 33600000.0 mm^4\n"
]
}
],
"prompt_number": 4
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example A.2, Page No:486"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import math\n",
"\n",
"#Variable Decleration\n",
"R=45 #Radius of the circle in mm\n",
"r=20 #Radius of the smaller circle in mm\n",
"h=100 #Depth of the straight section in mm\n",
"\n",
"#Calculations\n",
"#Part 1\n",
"\n",
"#Triangle\n",
"b=2*R #Breadth in mm\n",
"A_t=b*h*0.5 #Area in mm^2\n",
"Ix_bar_t=b*h**3*36**-1 #Moment of inertia in mm^4\n",
"y_bar1=2*3**-1*h #centroidal axis in mm\n",
"Ix_t=Ix_bar_t+A_t*y_bar1**2 #moment of inertia in mm^4\n",
"\n",
"#Semi-circle\n",
"A_sc=pi*R**2*0.5 #Area of the semi-circle in mm^2\n",
"Ix_bar_sc=0.1098*R**4 #Moment of inertia in mm^4\n",
"y_bar2=h+(4*R*(3*pi)**-1) #Distance of centroid in mm\n",
"Ix_sc=Ix_bar_sc+A_sc*y_bar2**2 #Moment of inertia in mm^4\n",
"\n",
"#Circle\n",
"A_c=pi*r**2 #Area of the circle in mm^2\n",
"Ix_bar_c=pi*r**4*4**-1 #Moment of inertia in mm^4\n",
"y_bar3=h #Distance of centroid in mm\n",
"Ix_c=Ix_bar_c+A_c*y_bar3**2 #Moment of inertia in mm^4\n",
"\n",
"#Composite Area\n",
"A=A_t+A_sc-A_c #Total area in mm^2\n",
"Ix=Ix_t+Ix_sc-Ix_c #Moment of inertia in mm^4\n",
"\n",
"#Part 2\n",
"y_bar=(A_t*y_bar1+A_sc*y_bar2-A_c*y_bar3)/(A) #Location of centroid in mm\n",
"Ix_bar=Ix-A*y_bar**2 #Moment of inertia in mm^4\n",
"\n",
"#Result\n",
"print \"Moment of inertia about x-axis is\",round(Ix),\"mm^4\"\n",
"print \"Moment of inertia about the centroidal axis is\",round(Ix_bar),\"mm^4\""
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Moment of inertia about x-axis is 55377079.0 mm^4\n",
"Moment of inertia about the centroidal axis is 7744899.0 mm^4\n"
]
}
],
"prompt_number": 14
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example A.3, Page No:488"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import math\n",
"\n",
"#Variable Decleration\n",
"t=20 #Thickness in mm\n",
"h=140 #Depth in mm\n",
"w=180 #Width in mm\n",
"\n",
"#Calculations\n",
"Ixy_1=0+(h*t*t*0.5*h*0.5) #product of inertia in mm^4\n",
"Ixy_2=0+((w-t)*t*(w+t)*0.5*t*0.5) #Product of inertia in mm^4\n",
"Ixy=Ixy_1+Ixy_2 #Product of inertia in mm^4\n",
"\n",
"#Result\n",
"print \"The Product of inertia is\",round(Ixy),\"mm^4\""
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The Product of inertia is 5160000.0 mm^4\n"
]
}
],
"prompt_number": 16
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example A.4, Page No:495"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import math\n",
"\n",
"#Variable Decleration\n",
"t=30 #Thickness in mm\n",
"h=200 #Depth of the section in mm\n",
"w=160 #Width in mm\n",
"the=50 #Angle in degrees\n",
"\n",
"\n",
"#Calculations\n",
"A1=t*h #Area of the web portion in mm^2\n",
"A2=(w-t)*t #Area of the flange portion in mm^2\n",
"x_bar=(A1*t*0.5+A2*(t+(w-t)*0.5))/(A1+A2) #Location of x_bar in mm\n",
"y_bar=(A1*h*0.5+A2*t*0.5)/(A1+A2) #Location of y_bar in mm\n",
"\n",
"#Simplfying the computation\n",
"a=t*h**3*12**-1\n",
"b=A1*(200*0.5-y_bar)**2\n",
"c=(w-t)*t**3*12**-1\n",
"d=A2*(t*0.5-y_bar)**2\n",
"Ix_bar=a+b+c+d #Moment of inertia about x-axis in mm^4\n",
"\n",
"#Simplifying the computation\n",
"p=h*t**3*12**-1\n",
"q=A1*(t*0.5-x_bar)**2\n",
"r=t*(w-t)**3*12**-1\n",
"s=A2*((w-t)*0.5+t-x_bar)**2\n",
"Iy_bar=p+q+r+s #Moment of inertia about y-axis in mm^4\n",
"\n",
"#Simplfying the computation\n",
"a1=(t*0.5-x_bar)*(h*0.5-y_bar)\n",
"a2=(t*0.5-y_bar)*((w-t)*0.5+t-x_bar)\n",
"Ixy_bar=A1*a1+A2*a2 #Moment of inertia in mm^4\n",
"\n",
"#Part 1\n",
"#Simplfying the computation\n",
"a3=(Ix_bar+Iy_bar)*0.5\n",
"a4=(0.5*(Ix_bar-Iy_bar))**2\n",
"a5=Ixy_bar**2\n",
"I1=a3+np.sqrt(a4+a5) #Moment of inertia in mm^4\n",
"I2=a3-np.sqrt(a4+a5) #Moment of inertia in mm^4\n",
"\n",
"ThetaRHS=-(2*Ixy_bar)/(Ix_bar-Iy_bar) #RHS of the tan term\n",
"theta1=arctan(ThetaRHS)*0.5*180*pi**-1 #Angle in degrees\n",
"theta2=theta1+90 #Angle in degrees\n",
"\n",
"#Part 2\n",
"Iu=a3+np.sqrt(a4)*np.cos(2*the*pi*180**-1)-(Ixy_bar)\\\n",
" *np.sin(2*the*pi*180**-1) #Moment of inertia in mm^4\n",
"Iv=a3-np.sqrt(a4)*np.cos(2*the*pi*180**-1)+(Ixy_bar)\\\n",
" *np.sin(2*the*pi*180**-1) #Moment of inertia in mm^4\n",
"Iuv=np.sqrt(a4)*np.sin(2*the*pi*180**-1)+(Ixy_bar)\\\n",
" *np.cos(2*the*pi*180**-1) #Moment of inertia in mm^4\n",
" \n",
" \n",
"#Result\n",
"print \"The Principal Moment of inertias are as follows\"\n",
"print \"I1=\",round(I1),\"mm^4 and I2=\",round(I2),\"mm^4\"\n",
"print \"Princial direction are theta1=\",round(theta1,1), \"degrees\"\\\n",
" \" theta2=\",round(theta2,1),\"degrees\"\n",
"print \"The moment of inertia along the uv-axis is\",round(Iuv),\"mm^4\" "
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The Principal Moment of inertias are as follows\n",
"I1= 47240734.0 mm^4 and I2= 11198811.0 mm^4\n",
"Princial direction are theta1= 31.6 degrees theta2= 121.6 degrees\n",
"The moment of inertia along the uv-axis is 10817183.0 mm^4\n"
]
}
],
"prompt_number": 51
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example A.5, Page No:497"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import math\n",
"\n",
"#Variable Decleration\n",
"Ix_bar=37.37*10**6 #Moment of inertia in mm^4\n",
"Iy_bar=21.07*10**6 #Moment of inertia in mm^4\n",
"Ixy_bar=-16.073*10**6 #Moment of inertia in mm^4\n",
"\n",
"#Calculations\n",
"b=(Ix_bar+Iy_bar)*0.5 #Parameter for the circle in mm^4\n",
"R=sqrt(((Ix_bar-Iy_bar)*0.5)**2+Ixy_bar**2) #Radius of the Mohr's Circle in mm^4\n",
"\n",
"#Part 1\n",
"I1=b+R #MI in mm^4\n",
"I2=b-R #MI in mm^4\n",
"theta1=arcsin(abs(Ixy_bar)/R)*180*pi**-1*0.5 #Angle in degrees\n",
"theta2=theta1+90 #Angle in degrees\n",
"\n",
"#Part 2\n",
"alpha=(100-theta1*2)*0.5 #Angle in degrees\n",
"Iu=round(b,2)+round(R,3)*round(np.cos(alpha*pi*180**-1),2) #MI in mm^4\n",
"Iv=round(b,2)-round(R,3)*round(np.cos(alpha*pi*180**-1),2) #MI in mm^4\n",
"Iuv=R*np.sin(2*alpha*pi*180**-1) #MI in mm^4\n",
"\n",
"#Result\n",
"print \"The Principal Moment of inertias are as follows\"\n",
"print \"I1=\",round(I1),\"mm^4 and I2=\",round(I2),\"mm^4\"\n",
"print \"Princial direction are theta1=\",round(theta1,1), \"degrees\"\\\n",
" \" theta2=\",round(theta2,1),\"degrees\"\n",
"print \"The moment of inertia along the uv-axis is\",round(Iuv),\"mm^4\" \n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The Principal Moment of inertias are as follows\n",
"I1= 47241205.0 mm^4 and I2= 11198795.0 mm^4\n",
"Princial direction are theta1= 31.6 degrees theta2= 121.6 degrees\n",
"The moment of inertia along the uv-axis is 10817230.0 mm^4\n"
]
}
],
"prompt_number": 73
}
],
"metadata": {}
}
]
}