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diff --git a/Mechanics_of_Materials_by_Pytel_and_Kiusalaas/Chapter02_2.ipynb b/Mechanics_of_Materials_by_Pytel_and_Kiusalaas/Chapter02_2.ipynb new file mode 100755 index 00000000..cd669b4f --- /dev/null +++ b/Mechanics_of_Materials_by_Pytel_and_Kiusalaas/Chapter02_2.ipynb @@ -0,0 +1,614 @@ +{ + "metadata": { + "name": "", + "signature": "sha256:3cb8e8b65ee50988938562d2f6cd882ccb93a3ca89c523d4423804cd1b6898ff" + }, + "nbformat": 3, + "nbformat_minor": 0, + "worksheets": [ + { + "cells": [ + { + "cell_type": "heading", + "level": 1, + "metadata": {}, + "source": [ + "Chapter 02:Strain" + ] + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Examples No:2.2.1, Page No:36" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "import math\n", + "\n", + "#Variable Decleration\n", + "#Axial Forces in lb in member AB, BC and CD\n", + "P_AB=2000 \n", + "P_BC=2000\n", + "P_CD=4000\n", + "#Other Variables\n", + "E=29*10**6 #Modulus of Elasticity in psi\n", + "#Length of each member in inches\n", + "L_AB=5*12\n", + "L_BC=4*12\n", + "L_CD=4*12\n", + "#Diameter of each member in inches\n", + "D_AB=0.5\n", + "D_BC=0.75\n", + "D_CD=0.75\n", + "\n", + "#Calculation\n", + "#Area Calculation of each member in square inches\n", + "A_AB=(pi*D_AB**2)/4\n", + "A_BC=(pi*D_BC**2)/4\n", + "A_CD=(pi*D_CD**2)/4\n", + "\n", + "#Using relation delta=(PL/AE) to compute strain\n", + "#As stress in Member CD is compression\n", + "delta=(E**-1)*((P_AB*L_AB*A_AB**-1)+(P_BC*L_BC*A_BC**-1)-(P_CD*L_CD*A_CD**-1))\n", + "\n", + "#Result\n", + "print \"The elongation in the total structure is\",round(delta,5),\"in\"\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "The elongation in the total structure is 0.01358 in\n" + ] + } + ], + "prompt_number": 4 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example 2.2.2, Page No:36" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "import math\n", + "from scipy.integrate import quad\n", + "\n", + "#Variable Decleration\n", + "E=200*10**9 #Modulus of elasticity in Pa\n", + "P=10**5 #Force acting in N\n", + "\n", + "#Calculations\n", + "#Using quad integration\n", + "#Area has been defined as a quadratic equation to integrate\n", + "def integrand(x, a, b):\n", + " return 1/(a * x + b)\n", + "a = 160\n", + "b = 800\n", + "I = quad(integrand, 0, 10, args=(a,b))\n", + "#Using delta=(P/E)*I where I is the integrand\n", + "delta=(P*E**-1)*10**6*I[0]\n", + "\n", + "#Result\n", + "print \"The elongation in the member is\",round(delta*1000,2),\"mm\"" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "The elongation in the member is 3.43 mm\n" + ] + } + ], + "prompt_number": 10 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example 2.2.3, Page No:37" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "import math\n", + "\n", + "#Variable Decelration\n", + "A_AC=0.25 #Cross Sectional Area in square inch\n", + "Load=2000 #Load at point C in lb\n", + "E=29*10**6 #Modulus of elasticity in psi\n", + "theta=(pi*40)/180 #Angle in radians\n", + "L_BC=8 #Length in ft\n", + "\n", + "#Calculations\n", + "#Using sum of forces \n", + "P_AC=Load/sin(theta) #Force in cable AC in lb\n", + "L_AC=(L_BC*12)/cos(theta) #Length of cable AC in in\n", + "\n", + "delta_AC=(P_AC*L_AC)/(E*A_AC) #elongation in inches\n", + "\n", + "delta_C=delta_AC/sin(theta) #displacement of point C in inches\n", + "\n", + "#Result\n", + "print \"The displacement of point C is\",round(delta_C,4),\"in\"" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "The displacement of point C is 0.0837 in\n" + ] + } + ], + "prompt_number": 14 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example 2.2.4, Page No:46" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "import math\n", + "\n", + "#Variable Decleration\n", + "d=0.05 #Diameter of the rod in mm\n", + "P=8000 #Load on the bar in N\n", + "E=40*10**6 #Modulus of elasticity in Pa\n", + "v=0.45 #Poisson Ratio\n", + "L=300 #Length of the rod in mm\n", + "\n", + "#Calculation\n", + "A=((pi*d**2)/4) #Area of the bar in mm^2\n", + "sigma_x=-P/A #Axial Stress in the bar in Pa\n", + "#As contact pressure resists the force\n", + "p=(v*sigma_x)/(1-v)\n", + "#Using Axial Strain formula\n", + "e_x=(sigma_x-(v*2*p))/E\n", + "#Corresponding change in length\n", + "delta=e_x*L #contraction in mm\n", + "#Without constrains of the wall\n", + "delta_w=(-P*(L*10**-3))/(E*A) #Elongation in m\n", + "\n", + "#Result\n", + "print \"The elongation in the bar is\",-round(delta,2),\"mm contraction\"" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "The elongation in the bar is 8.06 mm contraction\n" + ] + } + ], + "prompt_number": 1 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example 2.2.5, Page No:47" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "import math\n", + "\n", + "#Variable Decleration\n", + "E=500 #Modulus of elasticity in psi\n", + "v=0.48 #Poisson ratio\n", + "V=600 #Force in lb\n", + "w=5 #Width of the plate in inches\n", + "l=9 #Length of the plate in inches\n", + "t=1.75 #Thickness of the rubber layer in inches\n", + "\n", + "#Calculations\n", + "tau=V*(w*l)**-1 #Shear stress in rubber in psi\n", + "G=E/(2*(1+v)) #Bulk modulus in psi\n", + "gamma=tau/G #Shear Modulus \n", + "disp=t*gamma #Diplacement in inches\n", + "\n", + "#Result\n", + "print \"The displacement of the rubber layer is\",round(disp,4),\"in\"\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "The displacement of the rubber layer is 0.1381 in\n" + ] + } + ], + "prompt_number": 5 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example 2.2.6, Page No:52" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "import math\n", + "\n", + "#Variable Decleration\n", + "P=10**6 #Force on the member in N\n", + "Es=200 #Modulus of elasticity of steel in GPa\n", + "Ec=14 #Modulus of elasticity concrete in GPa\n", + "As=900*10**-6 #Area of steel in m^2\n", + "Ac=0.3**2 #Area of concrete block in m^2\n", + "\n", + "#Calculation\n", + "#Cross Sectional Areas\n", + "Ast=4*As #Cross Sectional Area in m^2 of Steel\n", + "Act=Ac-Ast #Cross Sectional Area of Concrete in m^2\n", + "\n", + "#Applying equilibrium to the structure\n", + "#Using the ratio of stress and modulii of elasticity we obtain the following eq\n", + "sigma_ct=P/(((Es*Ec**-1)*Ast)+Act) #Stress in Concrete in Pa\n", + "sigma_st=sigma_ct*Es*Ec**-1 #Stress in Steel in Pa\n", + "\n", + "#Result\n", + "print \"The stress in steel and concrete is as follows\",round(sigma_st*10**-6,1),\"MPa and\",round(sigma_ct*10**-6,3),\"Mpa respectively\"\n", + "\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "The stress in steel and concrete is as follows 103.6 MPa and 7.255 Mpa respectively\n" + ] + } + ], + "prompt_number": 10 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example 2.2.7, Page No:52" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "import math\n", + "\n", + "#Variable Decleration\n", + "#Say the ratio of stress in steel to concrete is R\n", + "R=14.286 \n", + "sigma_co=6*10**6 #Stress in concrete in Pa\n", + "Ast=3.6*10**-3 #Area of steel in m^2\n", + "Aco=86.4*10**-3 #Area of Concrete in m^2\n", + "\n", + "#Calculation\n", + "sigma_st=R*sigma_co #Stress in steel in Pa\n", + "#Here stress is below the allowable hence safe\n", + "P=sigma_st*Ast+sigma_co*Aco #Allowable force in N\n", + "\n", + "#Result\n", + "print \"The maximum allowable force is\",round(P*10**-3),\"kN\"" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "The maximum allowable force is 827.0 kN\n" + ] + } + ], + "prompt_number": 11 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example 2.2.8, Page No:53" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "import math\n", + "\n", + "#NOTE:The NOtation has been changed to ease coding\n", + "#Variable Decleration\n", + "d=0.005 #difference in length in inch\n", + "L=10 #Length in inch\n", + "#Area of copper and aluminium in sq.in\n", + "Ac=2 #Area of copper \n", + "Aa=3 #Area of aluminium \n", + "#Modulus of elasticity of copper and aluminium in psi\n", + "Ec=17000000 #Copper\n", + "Ea=10**7 #Aluminium\n", + "#Allowable Stress in psi\n", + "Sc=20*10**3 #Copper\n", + "Sa=10*10**3 #Aluminium\n", + "\n", + "#Calculation\n", + "#Equilibrium is Pc+Pa=P\n", + "#Hookes Law is delta_c=delta_a+0.005\n", + "#Simplfying the solution we have constants we can directly compute\n", + "A=d*Ec*(L+d)**-1\n", + "B=Ec*Ea**-1\n", + "C=L*B*(L+d)**-1\n", + "sigma_a=(Sc-A)*C**-1\n", + "\n", + "#Using equilibrium equation\n", + "P=Sc*Ac+sigma_a*Aa #Safe load in lb\n", + "\n", + "#Result\n", + "print \"The safe load on the structure is\",round(P),\"lb\"\n", + "#NOTE:Answer in the textbook has been rounded off and hence the discrepancy" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "The safe load on the structure is 60312.0 lb\n" + ] + } + ], + "prompt_number": 34 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example 2.2.9, Page No:54" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "import math\n", + "import numpy as np\n", + "\n", + "#Variable Decleration\n", + "P=50*10**3 #Load applied in N\n", + "x1=0.6 #Length in m\n", + "x2=1.6 #Length in m\n", + "L1=1 #Length of steel cable in m\n", + "L2=2 #Length of bronze cable in m\n", + "L=2.4 #Length in m\n", + "#Area in m^2\n", + "Ast=600*10**-6 #Steel\n", + "Abr=300*10**-6 #Bronze\n", + "#Modulus of elasticity in GPa\n", + "Est=200 #Steel\n", + "Ebr=83 #Bronze\n", + "\n", + "#Calculations\n", + "#Applying the equilibrium and Hookes law we solve by matrix method\n", + "a=np.array([[x1,x2],[1,-((x1*Est*Ast*L2)/(x2*Ebr*Abr))]])\n", + "b=np.array([L*P,0])\n", + "y=np.linalg.solve(a,b)\n", + "\n", + "#Stresses in Pa\n", + "sigma_st=y[0]*Ast**-1 #Stress in steel\n", + "sigma_br=y[1]/Abr #Stress in bronze\n", + "\n", + "#Result\n", + "print \"The stresses in steel and bronze are as follows\"\n", + "print round(sigma_st*10**-6,1),\"MPa and\",round(sigma_br*10**-6,1),\"MPa respectively\"" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "The stresses in steel and bronze are as follows\n", + "191.8 MPa and 106.1 MPa respectively\n" + ] + } + ], + "prompt_number": 49 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example 2.2.10, Page No:62" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "import math\n", + "\n", + "#Variable Decleration\n", + "L=2.5 #Length in m\n", + "A=1200 #Cross sectional Area in mm^2\n", + "delta_T=40 #Temperature drop in degree C\n", + "delta=0.5*10**-3 #Movement of the walls in mm\n", + "alpha=11.7*10**-6 #Coefficient of thermal expansion in /degreeC\n", + "E=200*10**9 #Modulus of elasticity in Pa\n", + "\n", + "#Calculation\n", + "#Part(1)\n", + "sigma_1=alpha*delta_T*E #Stress in the rod in Pa\n", + "\n", + "#Part(2)\n", + "#Using Hookes Law\n", + "sigma_2=E*((alpha*delta_T)-(delta*L**-1)) #Stress in the rod in Pa\n", + "\n", + "print \"The Stress in part 1 in the rod is\",round(sigma_1*10**-6,1),\"MPa\"\n", + "print \"The Stress in part 2 in the rod is\",round(sigma_2*10**-6,1),\"MPa\"" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "The Stress in part 1 in the rod is 93.6 MPa\n", + "The Stress in part 2 in the rod is 53.6 MPa\n" + ] + } + ], + "prompt_number": 53 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example 2.2.11, Page No:63" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "import math\n", + "\n", + "#Variable Decleration\n", + "delta=100 #Increase in the temperature in degreeF\n", + "Load=12000 #Load on the beam in lb\n", + "#Length in inch\n", + "Ls=2*12 #Steel\n", + "Lb=3*12 #Bronze\n", + "#Area in sq.in\n", + "As=0.75 #Steel\n", + "Ab=1.5 #Bronze\n", + "#Modulus of elasticity in psi\n", + "Es=29*10**6 #Steel\n", + "Eb=12*10**6 #Bronze\n", + "#Coefficient of thermal expansion in /degree C\n", + "alpha_s=6.5*10**-6 #Steel\n", + "alpha_b=10**-5 #Bronze\n", + "\n", + "#Calculations\n", + "#Applying the Hookes Law and equilibrium we get two equations\n", + "a=np.array([[Ls*(Es*As)**-1,-Lb*(Eb*Ab)**-1],[2,1]])\n", + "b=np.array([(alpha_b*delta*Lb-alpha_s*delta*Ls),Load])\n", + "y=np.linalg.solve(a,b)\n", + "\n", + "#Stresses\n", + "sigma_st=y[0]*As**-1 #Stress in steel in psi (T)\n", + "sigma_br=y[1]*Ab**-1 #Stress in bronze in psi (C)\n", + "\n", + "#Result\n", + "print \"The Stress in steel and bronze are as follows\"\n", + "print sigma_st,\"psi (T) and\", -sigma_br,\"psi (C)\"" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "The Stress in steel and bronze are as follows\n", + "11600.0 psi (T) and 3600.0 psi (C)\n" + ] + } + ], + "prompt_number": 58 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example 2.2.12, Page No:64" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "import math\n", + "\n", + "#Variable Decleration\n", + "P=6000 #Force in lb\n", + "Est=29*10**6 #Modulus of elasticity of steel in psi\n", + "L1=24 #Length in inches\n", + "L2=36 #Length in inches\n", + "alpha_1=6.5*10**-6 #coefficient of thermal expansion in /degree F of steel\n", + "alpha_2=10**-5 #coefficient of thermal expansion in /degree F of bronze\n", + "As=0.75 #Area os steel in sq.in\n", + "\n", + "#Calculations\n", + "delta_T=((P*L1)/(Est*As))/(alpha_2*L2-alpha_1*L1) #Change in temperature in degree F\n", + "\n", + "print \"The change in the Temperature is\",round(delta_T,1),\"F\"\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "The change in the Temperature is 32.5 F\n" + ] + } + ], + "prompt_number": 60 + } + ], + "metadata": {} + } + ] +}
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