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-{
- "cells": [
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "# Chapter8-SIMPLE STRESSES AND STRAINS"
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "# example8.1 Page number243"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 1,
- "metadata": {
- "collapsed": false
- },
- "outputs": [
- {
- "name": "stdout",
- "output_type": "stream",
- "text": [
- "sectional area= 201.06 mm^2\n",
- "stress= 198.94 N/mm^2\n",
- "strain= 0.000994718394324 N/mm^2\n",
- "Elongation= 0.497 mm\n"
- ]
- }
- ],
- "source": [
- "from math import pi\n",
- "\n",
- "#variable declaration\n",
- "\n",
- "P=float(40000) #Load,N\n",
- "E=float(200000) #Modulus of elasticity for steel,N/mm^2\n",
- "L=500 #length of circular rod,mm\n",
- "d=float(16) #diameter of rod,mm\n",
- " \n",
- "A=(pi*(pow(d,2)))/4 #sectional area, mm^2\n",
- "p=P/A #stress, N/mm^2\n",
- "e=p/E #strain\n",
- "delta=(P*L)/(A*E) #Elongation,mm\n",
- "\n",
- "print \"sectional area=\",round(A,2),\"mm^2\"\n",
- "print \"stress=\",round(p,2),\"N/mm^2\"\n",
- "print \"strain=\",e,\"N/mm^2\"\n",
- "print \"Elongation=\",round(delta,3),\"mm\"\n"
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "# example8.2 Page number243"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 2,
- "metadata": {
- "collapsed": false
- },
- "outputs": [
- {
- "name": "stdout",
- "output_type": "stream",
- "text": [
- "area= 11.25 mm^2\n",
- "Elongation= 1.6 mm\n",
- "Hence, if measured length is 30.0 m.\n",
- "Actual length is 30.0016 m\n",
- "Actual length of line AB= 150.008 m.\n"
- ]
- }
- ],
- "source": [
- "#variable declaration\n",
- "\n",
- "P=float(120) # force applied during measurement,N\n",
- "E=float(200000) #Modulus of elasticity for steel,N/mm^2\n",
- "L=float(30) #length of Surveyor’s steel tape,mm\n",
- " \n",
- " \n",
- "A=15*0.75 #area, mm^2\n",
- "delta=((P*L*1000)/(A*E)) #Elongation,mm\n",
- "\n",
- "print \"area=\",round(A,2),\"mm^2\"\n",
- "print \"Elongation=\",round(delta,3),\"mm\"\n",
- "\n",
- "print \"Hence, if measured length is\", L,\"m.\"\n",
- "print \"Actual length is\" ,round((L+(delta/1000)),6),\"m\"\n",
- "\n",
- "print \"Actual length of line AB=\",round((150*(L+(delta/1000))/30),3),\"m.\""
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "# example 8.3 Page number 244\n"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 3,
- "metadata": {
- "collapsed": false
- },
- "outputs": [
- {
- "name": "stdout",
- "output_type": "stream",
- "text": [
- "Therefore, permissible stress\n",
- "p= 142.857 N/mm^2\n",
- "Load P= 160000.0 N\n",
- "A= 1120.0 mm^2\n",
- "d= 94.32 mm\n",
- "t= 3.64 mm\n",
- "Hence, use of light section is recommended.\n"
- ]
- }
- ],
- "source": [
- "from math import pi,sqrt\n",
- "\n",
- "#variable declaration\n",
- "\n",
- "Y=float(250) #Yield stress, N/mm^2\n",
- "FOS=float(1.75) #Factor of safety\n",
- "P=float(160) #Load,KN\n",
- "\n",
- "p=Y/FOS\n",
- "\n",
- "print \"Therefore, permissible stress\"\n",
- "print \"p=\",round(p,3), \"N/mm^2\"\n",
- "print \"Load P=\",P*1000,\"N\"\n",
- "\n",
- "#p=P/A\n",
- "\n",
- "A=P*1000/p #area,mm^2\n",
- "\n",
- "print \"A=\",round(A),\"mm^2\"\n",
- "\n",
- "#For hollow section of outer diameter ‘D’ and inner diameter ‘d’ A=pi*(D^2-d^2)/4\n",
- "D=float(101.6) #outer diameter,mm\n",
- "\n",
- "d=sqrt(pow(D,2)-(4*A/pi))\n",
- "\n",
- "print \"d=\",round(d,2),\"mm\"\n",
- "\n",
- "t=(D-d)/2\n",
- "print \"t=\",round(t,2),\"mm\"\n",
- "\n",
- "print \"Hence, use of light section is recommended.\"\n"
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "# example 8.4 page number 245"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 4,
- "metadata": {
- "collapsed": false
- },
- "outputs": [
- {
- "name": "stdout",
- "output_type": "stream",
- "text": [
- "Area= 314.16 mm^2\n",
- "Stress at elastic limit= 324.68 N/mm^2\n",
- "Young's modulus E= 12732.4 N/mm^22\n",
- "Percentage elongation= 28.0 %\n",
- "Percentage reduction in area= 43.75 %\n",
- "Ultimate Tensile Stress= 0.41 N/mm^2\n"
- ]
- }
- ],
- "source": [
- "from math import pi\n",
- "\n",
- "#variable declaration \n",
- "\n",
- "d=float(20) #Diameter ,mm\n",
- "Loadatelasticlimit=float(102) #Load at elastic limit,KN\n",
- "P=80 #Load for extension of o.25mm , KN\n",
- "delta=float(0.25) #extension in specimen of steel,mm\n",
- "L=200 #gauge length of specimen of steel,mm\n",
- "Finalextension=float(56) #total extension at fracture,mm\n",
- "\n",
- "\n",
- "A=(pi*pow(d,2))/4 #Area,mm^2\n",
- "print \"Area=\", round(A,2),\"mm^2\"\n",
- "\n",
- "Stressatelasticlimit=Loadatelasticlimit*1000/A #Stress at elastic limit,N/mm^2 \n",
- "print \"Stress at elastic limit=\",round(Stressatelasticlimit,2),\"N/mm^2\"\n",
- "\n",
- "E=(P*1000/A)*(delta*L) #Young’s modulus ,N/mm^2\n",
- "print \"Young's modulus E=\", round(E,2),\"N/mm^22\"\n",
- "\n",
- "Percentageelongation=Finalextension*100/L #percentage elongation,%\n",
- "print \"Percentage elongation=\", round(Percentageelongation,2),\"%\"\n",
- "\n",
- "Initialarea=(pi*pow(d,2))/4\n",
- "Finalarea=(pi*pow(15,2))/4 # total extension at fracture is 56 mm and diameter at neck is 15 mm.\n",
- "Percentagereductioninarea=(Initialarea-Finalarea)*100/Initialarea\n",
- "\n",
- "print \"Percentage reduction in area=\",round(Percentagereductioninarea,2),\"%\"\n",
- "\n",
- "UltimateLoad=130 #Maximum Load=130,kN\n",
- "UltimateTensileStress=UltimateLoad/A\n",
- "\n",
- "print\"Ultimate Tensile Stress=\",round(UltimateTensileStress,2),\"N/mm^2\"\n"
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "# example8.5 Page number247\n"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 5,
- "metadata": {
- "collapsed": false
- },
- "outputs": [
- {
- "name": "stdout",
- "output_type": "stream",
- "text": [
- "E= 56277.19 N/mm^2\n"
- ]
- }
- ],
- "source": [
- "from math import pi\n",
- "\n",
- "#variable declaration\n",
- "\n",
- "P=float(40) #Load,KN\n",
- "L1=150 #length of 1st portion,mm\n",
- "A1=pi*pow(25,2)/4 #Area of 1st portion,mm^2\n",
- "L2=250 #length of 2nd portion,mm\n",
- "A2=pi*pow(20,2)/4 #Area of 2nd portion,mm^2\n",
- "L3=150 #length of 3rd portion,mm\n",
- "A3=pi*pow(25,2)/4 #Area of 3rd portion,mm^2\n",
- "\n",
- "#E,Young's modulus ,N/mm^2\n",
- "\n",
- "#Total extension= Extension of portion 1+Extension of portion 2+Extension of portion 3\n",
- "\n",
- "#Extension=(P*1000*L)/(A*E)\n",
- "\n",
- "E=(P*1000*L1/A1)+(P*1000*L2/A2)+(P*1000*L3/A3)\n",
- "\n",
- "print \"E=\",round(E,2),\"N/mm^2\"\n",
- "\n"
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "# example8.6 Page number247"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 6,
- "metadata": {
- "collapsed": false
- },
- "outputs": [
- {
- "name": "stdout",
- "output_type": "stream",
- "text": [
- "Total extension of the bar= 0.5125 mm\n"
- ]
- }
- ],
- "source": [
- "from math import pi\n",
- "\n",
- "#variable declaration\n",
- "\n",
- "P=float(30) #Load,KN\n",
- "L1=600 #length of 1st portion,mm\n",
- "A1=40*20 #Area of 1st portion,mm^2\n",
- "\n",
- "E1=200000 # material 1 Young’s modulus,N/mm^2\n",
- " \n",
- "E2=100000 # material 2 Young’s modulus,N/mm^2\n",
- " \n",
- "\n",
- "L2=800 #length of 2nd portion,mm\n",
- "A2=30*20 #Area of 2nd portion,mm^2\n",
- "\n",
- "Extensionofportion1=(P*1000*L1)/(A1*E1) #mm\n",
- "Extensionofportion2=(P*1000*L2)/(A2*E2) #mm\n",
- "\n",
- "Totalextensionofthebar= Extensionofportion1 + Extensionofportion2\n",
- "\n",
- "print\"Total extension of the bar=\",round(Totalextensionofthebar,4),\"mm\"\n",
- "\n"
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "# example8.7 Page number248\n"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 7,
- "metadata": {
- "collapsed": false
- },
- "outputs": [
- {
- "name": "stdout",
- "output_type": "stream",
- "text": [
- "E= 200735.96 N/mm^2\n"
- ]
- }
- ],
- "source": [
- "from math import pi\n",
- "\n",
- "#variable declaration\n",
- "\n",
- "P=float(30) #Load,KN\n",
- "L1=600 #length of 1st portion,mm\n",
- "A1=pi*pow(30,2)/4 #Area of 1st portion,mm^2\n",
- "L2=400 #length of 2nd portion,mm\n",
- "A2=pi*(pow(30,2)-pow(10,2))/4 #Area of 2nd portion,mm^2\n",
- "\n",
- "#E,Young's modulus ,N/mm^2\n",
- "\n",
- "#Total extension= Extension of portion 1+Extension of portion 2\n",
- "\n",
- "#Extension=(P*1000*L)/(A*E)\n",
- "\n",
- "T=float(0.222) #Total extension of the bar,mm\n",
- "\n",
- "E=((P*1000*L1/A1)+(P*1000*L2/A2))/T \n",
- "\n",
- "print \"E=\",round(E,2),\"N/mm^2\"\n",
- "\n"
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "# example 8.10 Page number 251"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 8,
- "metadata": {
- "collapsed": false
- },
- "outputs": [
- {
- "name": "stdout",
- "output_type": "stream",
- "text": [
- "delta1= 0.2113 mm\n",
- "there is calculation mistake in book\n",
- "delta2= 0.48 mm^2\n",
- "Percentage error= 55.977 %\n",
- "there is calculation mistake in book\n"
- ]
- }
- ],
- "source": [
- "import math\n",
- "#variable declaration\n",
- "\n",
- "t=10 #steel flat thickness,mm\n",
- "b1=float(60) #tapering from b1 to b2\n",
- "b2=40\n",
- "L=600 #steel flat length\n",
- "P=float(80) #Load,KN\n",
- "E=2*100000 #Young's Modulus,N/mm^2\n",
- "\n",
- "#Extension of the tapering bar of rectangular section\n",
- "\n",
- "delta1=(P*1000*L*math.log((b1/b2),10))/(t*E*(b1-b2))\n",
- "\n",
- "print \"delta1=\",round(delta1,4),\"mm\"\n",
- "print \"there is calculation mistake in book\"\n",
- "\n",
- "#If averages cross-section is considered instead of tapering cross-section, extension is given by \n",
- "\n",
- "Aav=(b1+b2)*t/2 #mm^2\n",
- "\n",
- "delta2=(P*1000*L)/(Aav*E) #mm\n",
- "print\"delta2=\",round(delta2,3),\"mm^2\"\n",
- "\n",
- "P= (delta2-delta1)*100/delta2\n",
- "\n",
- "print\"Percentage error=\",round(P,3),\"%\"\n",
- "\n",
- "print \"there is calculation mistake in book\"\n"
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "# example8.11 page number251"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 9,
- "metadata": {
- "collapsed": false
- },
- "outputs": [
- {
- "name": "stdout",
- "output_type": "stream",
- "text": [
- "delta1= 1.194 mm\n",
- "delta2= 0.265 mm\n",
- "Total extension 1.459 mm\n"
- ]
- }
- ],
- "source": [
- "from math import pi\n",
- "\n",
- "#variable declaration\n",
- "\n",
- "P=float(200) #loading,KN\n",
- "E=200*1000\n",
- "d1=40 #Young's modulus,N/mm^2\n",
- "A= pi*pow(d1,2)/4 #Area of uniform portion,mm^2 \n",
- "L1=1500 #length of uniform portion,mm \n",
- "d2=60 #diameter of tapered section,mm\n",
- "L2=500 #length of tapered section,mm\n",
- "#Extensions of uniform portion and tapering portion are worked out separately and then added to get extension of the given bar. \n",
- "\n",
- "#Extension of uniform portion\n",
- "\n",
- "delta1=(P*1000*L1)/(A*E)\n",
- "\n",
- "print \"delta1=\",round(delta1,3),\"mm\"\n",
- "\n",
- "delta2=(P*1000*4*L2)/(E*pi*d1*d2)\n",
- "\n",
- "print \"delta2=\",round(delta2,3),\"mm\"\n",
- "\n",
- "T=delta1 + delta2 \n",
- "print \"Total extension\",round(T,3),\"mm\"\n",
- "\n",
- "\n"
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "# example 8.13 page number259"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 10,
- "metadata": {
- "collapsed": false
- },
- "outputs": [
- {
- "name": "stdout",
- "output_type": "stream",
- "text": [
- "Poisson's ratio= 0.3\n",
- "E= 203718.33 N/mm^2\n",
- "G= 78353.2 N/mm^2\n",
- "K= 169765.27 N/mm^2\n"
- ]
- }
- ],
- "source": [
- "from math import pi\n",
- "\n",
- "#variable declaration\n",
- "\n",
- "P=float(60) #load,KN\n",
- "d=float(25) #diameter,mm\n",
- "A=pi*pow(d,2)/4 #Area,mm^2\n",
- "L=float(200) #gauge length,mm\n",
- "\n",
- "delta=0.12 #extension,mm\n",
- "deltad=0.0045 #contraction in diameter,mm\n",
- "Linearstrain=delta/L\n",
- "Lateralstrain=deltad/d\n",
- "\n",
- "Pr=Lateralstrain/Linearstrain\n",
- "\n",
- "print \"Poisson's ratio=\",round(Pr,1)\n",
- "\n",
- "E=(P*1000*L)/(A*delta)\n",
- "\n",
- "print \"E=\",round(E,2),\"N/mm^2\"\n",
- "\n",
- "G=E/(2*(1+Pr)) #Rigidity modulus\n",
- "\n",
- "print \"G=\",round(G,1),\"N/mm^2\"\n",
- "\n",
- "K=E/(3*(1-(2*Pr))) #bulk modulus\n",
- "\n",
- "print \"K=\",round(K,2),\"N/mm^2\"\n"
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "# example 8.14 page number 260"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 11,
- "metadata": {
- "collapsed": false
- },
- "outputs": [
- {
- "name": "stdout",
- "output_type": "stream",
- "text": [
- "G= 76923.1 N/mm^2\n",
- "K= 166666.67 N/mm^2\n",
- "change in volume 60.0 mm^3\n"
- ]
- }
- ],
- "source": [
- "from math import pi\n",
- "\n",
- "#variable declaration\n",
- "\n",
- "E=float(2*100000) #Young's modulus,N/mm^2\n",
- "Pr=float(0.3) #poisson's ratio\n",
- "\n",
- "G=E/(2*(1+Pr)) #Rigidity modulus\n",
- "\n",
- "K=E/(3*(1-2*(Pr))) #Bulk modulus\n",
- "\n",
- "print \"G=\", round(G,1),\"N/mm^2\"\n",
- "\n",
- "print \"K=\", round(K,2), \"N/mm^2\"\n",
- "\n",
- "P=60 #Load,kN\n",
- "A=pi*pow(25,2)/4 #Area,mm^2\n",
- "\n",
- "Stress=P*1000/A #N/mm^2\n",
- "#Linear strain,ex\n",
- "\n",
- "ex=Stress/E\n",
- " \n",
- "#Lateralstrain,ey,ez\n",
- "\n",
- "ey=-1*Pr*ex\n",
- "ez=-1*Pr*ex\n",
- "\n",
- "#volumetric strain,ev=ex+ey+ez\n",
- "\n",
- "ev=ex+ey+ez\n",
- "\n",
- "v=pi*pow(25,2)*500/4\n",
- "Changeinvolume=ev*v\n",
- "\n",
- "print\"change in volume\",round(Changeinvolume,2),\"mm^3\""
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "# example 8.15 page number261"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 12,
- "metadata": {
- "collapsed": false
- },
- "outputs": [
- {
- "name": "stdout",
- "output_type": "stream",
- "text": [
- "Change in volume= 10.8 mm^3\n"
- ]
- }
- ],
- "source": [
- "#variable declaration\n",
- "# Let the x, y, z be the mutually perpendicular directions\n",
- "\n",
- "pr=float(0.3)\n",
- "PX=float(15) #Loading in x-direction,KN\n",
- "PY=float(80) #Loading in Y-direction(compressive),KN\n",
- "PZ=float(180) #Loading in Z-direction,KN\n",
- "\n",
- "#Area in X-,Y-,Z-Direction is AX,AY,AZ respectively,mm^2\n",
- "\n",
- "AX=float(10*30)\n",
- "AY=float(10*400)\n",
- "AZ=float(30*400)\n",
- "\n",
- "#stress devoloped in X-,Y-,Z- direction as px,py,pz respectively,N/mm^2\n",
- "\n",
- "px=PX*1000/AX\n",
- "py=PY*1000/AY\n",
- "pz=PZ*1000/AZ\n",
- "\n",
- "#Noting that a stress produces a strain of p/E in its own direction, the nature being same as that of stress and µ p E in lateral direction of opposite nature, and taking tensile stress as +ve, we can write expression for strains ex, ey, ez.\n",
- "E=2*100000 #young's modulus,N/mm^2\n",
- "\n",
- "ex=(px/E)+(pr*py/E)-(pr*pz/E)\n",
- "ey=(-pr*px/E)-(py/E)-(pr*pz/E)\n",
- "ez=(-pr*px/E)+(pr*py/E)+(pz/E)\n",
- "\n",
- "ev=ex+ey+ez #Volumetric strain\n",
- "\n",
- "volume=10*30*400\n",
- "\n",
- "Changeinvolume=ev*volume\n",
- "\n",
- "print \"Change in volume=\",round(Changeinvolume,2),\"mm^3\"\n"
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "# example 8.17 page number 263"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 13,
- "metadata": {
- "collapsed": false
- },
- "outputs": [
- {
- "name": "stdout",
- "output_type": "stream",
- "text": [
- "poisson's Ratio= 0.346\n",
- "Bulk modulus= 227500.0 N/mm^2\n"
- ]
- }
- ],
- "source": [
- "#variable declaration\n",
- "\n",
- "E=float(2.1*100000) #Young’s modulus of the material,N/mm^2\n",
- "G=float(0.78*100000) #modulus of rigidity,N/mm^2\n",
- "\n",
- "pr=(E/(2*G))-1\n",
- "\n",
- "print \"poisson's Ratio=\",round(pr,3)\n",
- "\n",
- "K=E/(3*(1-2*pr))\n",
- "\n",
- "print \"Bulk modulus=\",round(K,3),\"N/mm^2\" "
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "# example 8.18 page number 263"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 14,
- "metadata": {
- "collapsed": false
- },
- "outputs": [
- {
- "name": "stdout",
- "output_type": "stream",
- "text": [
- "Young's modulus= 102857.143 N\n",
- "Poisson's Ratio 0.2857\n"
- ]
- }
- ],
- "source": [
- "#variable declaration\n",
- "\n",
- "G=float(0.4*100000) #modulus of rigidity of material,N/mm^2\n",
- "K=float(0.8*100000) #bulk modulus,N/mm^2\n",
- "\n",
- "E=(9*G*K)/(3*K+G)\n",
- "\n",
- "\n",
- "print \"Young's modulus=\",round(E,3),\"N\"\n",
- "\n",
- "pr=(E/(2*G))-1\n",
- "\n",
- "print \"Poisson's Ratio\",round(pr,4)\n"
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "# example 8.19 page number 264"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 15,
- "metadata": {
- "collapsed": false
- },
- "outputs": [
- {
- "name": "stdout",
- "output_type": "stream",
- "text": [
- "Stress in aluminium strip= 23.08 N/mm^2\n",
- "Stress in steel strip= 46.15 N/mm^2\n",
- "Extension of the compound bar= 0.138 mm\n"
- ]
- }
- ],
- "source": [
- "#variable declaration\n",
- "\n",
- "L=float(600) #compound bar of length,mm\n",
- "P=float(60) #compound bar when axial tensile force ,KN\n",
- "\n",
- "Aa=float(40*20) #area of aluminium strip,mm^2\n",
- "As=float(60*15) #area of steel strip,mm^2\n",
- "\n",
- "Ea=1*100000 # elastic modulus of aluminium,N/mm^2\n",
- "Es=2*100000 # elastic modulus of steel,N/mm^2\n",
- "\n",
- "#load shared by aluminium strip be Pa and that shared by steel be Ps. Then from equilibrium condition Pa+Ps=P\n",
- "#From compatibility condition, deltaAL=deltaS\n",
- "Pa=(P*1000)/(1+((As*Es)/(Aa*Ea)))\n",
- "Ps=Pa*((As*Es)/(Aa*Ea))\n",
- "\n",
- "Sias=Pa/Aa\n",
- "print \"Stress in aluminium strip=\",round(Sias,2),\"N/mm^2\"\n",
- "Siss=Ps/As\n",
- "print \"Stress in steel strip=\",round(Siss,2),\"N/mm^2\"\n",
- "\n",
- "L=600\n",
- "#Extension of the compound bar \n",
- "deltal=(Pa*L)/(Aa*Ea)\n",
- "print\"Extension of the compound bar=\",round(deltal,3),\"mm\""
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "# example 8.20 page number 265"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 16,
- "metadata": {
- "collapsed": false
- },
- "outputs": [
- {
- "name": "stdout",
- "output_type": "stream",
- "text": [
- "stress in Copper= 75.76 N/mm^2\n",
- "stress in Steel= 126.27 N/mm^2\n"
- ]
- }
- ],
- "source": [
- "from math import pi\n",
- "\n",
- "#variable declaration\n",
- "\n",
- "Es=float(2*100000) #Young's modulus of steel rod ,N/mm^2\n",
- "Ec=float(1.2*100000) #Young's modulus of copper tube,N/mm^2\n",
- "\n",
- "di=float(25) #internal diameter,mm\n",
- "de=float(40) #external diameter,mm\n",
- "\n",
- "As=pi*pow(di,2)/4 #Area of steel rod,mm^2\n",
- "Ac=pi*(pow(de,2)-pow(di,2))/4 #Area of copper tube,mm^2\n",
- "P=120 #load, KN\n",
- "#From equation of equilibrium, Ps+Pc=P,where Ps is the load shared by steel rod and Pc is the load shared by the copper tube.\n",
- "#From compatibility condition,deltaS=deltaC\n",
- "\n",
- "Pc=(P*1000)/(1+((As*Es)/(Ac*Ec)))\n",
- "Ps=Pc*((As*Es)/(Ac*Ec))\n",
- "\n",
- "SIC=Pc/Ac #stress in copper, N/mm^2\n",
- "SIS=Ps/As #stress in steel,N/mm^2\n",
- "\n",
- "print \"stress in Copper=\",round(SIC,2),\"N/mm^2\"\n",
- "print \"stress in Steel=\",round(SIS,2),\"N/mm^2\"\n"
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "# example 8.21 page number 266"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 17,
- "metadata": {
- "collapsed": false
- },
- "outputs": [
- {
- "name": "stdout",
- "output_type": "stream",
- "text": [
- "stress in Concrete= 4.51 N/mm^2\n",
- "stress in Steel= 81.2 N/mm^2\n"
- ]
- }
- ],
- "source": [
- "from math import pi\n",
- "\n",
- "#variable declaration\n",
- "#Es/Ec=18(given)\n",
- "Er=float(18) #young modulus ratio Er=Es/Ec\n",
- "d=float(16) #steel bar diameter,mm\n",
- "#8 steel bars\n",
- "As=8*pi*pow(d,2)/4 #Area of steel bar,mm^2\n",
- "Ac=(300*500)-As #Area of concrete,mm^2\n",
- "\n",
- "P=800 #Compressive force, KN\n",
- "#From equation of equilibrium, Ps+Pc=P,where Ps is the load shared by steel bar and Pc is the load shared by the Concrete\n",
- "#From compatibility condition,deltaS=deltaC\n",
- "\n",
- "Pc=(P*1000)/(1+((As*Er)/(Ac)))\n",
- "Ps=Pc*((As*Er)/(Ac))\n",
- "\n",
- "SIC=Pc/Ac #stress in Concrete, N/mm^2\n",
- "SIS=Ps/As #stress in steel,N/mm^2\n",
- "\n",
- "print \"stress in Concrete=\",round(SIC,2),\"N/mm^2\"\n",
- "print \"stress in Steel=\",round(SIS,2),\"N/mm^2\""
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "# example 8.22 page number 267"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 18,
- "metadata": {
- "collapsed": false
- },
- "outputs": [
- {
- "name": "stdout",
- "output_type": "stream",
- "text": [
- "stress in Aluminium= 66.96 N/mm^2\n",
- "stress in Steel= 89.29 N/mm^2\n"
- ]
- }
- ],
- "source": [
- "#variable declaration\n",
- "\n",
- "Es=float(2*100000) #Young's modulus of steel ,N/mm^2\n",
- "Ea=float(1*100000) #Young's modulus of aluminium,N/mm^2\n",
- "Ls=240 #length of steel,mm\n",
- "La=160 #length of aluminium,mm\n",
- "Aa=1200 #Area of aluminium,mm^2\n",
- "As=1000 #Area of steel,mm^2\n",
- "P=250 #load, KN\n",
- "#From equation of equilibrium, Ps+2Pa=P,et force shared by each aluminium pillar be Pa and that shared by steel pillar be Ps. \n",
- "#From compatibility condition,deltaS=deltaC\n",
- "\n",
- "Pa=(P*1000)/(2+((As*Es*La)/(Aa*Ea*Ls)))\n",
- "Ps=Pa*((As*Es*La)/(Aa*Ea*Ls))\n",
- "\n",
- "SIA=Pa/Aa #stress in aluminium, N/mm^2\n",
- "SIS=Ps/As #stress in steel,N/mm^2\n",
- "\n",
- "print \"stress in Aluminium=\",round(SIA,2),\"N/mm^2\"\n",
- "print \"stress in Steel=\",round(SIS,2),\"N/mm^2\""
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "# example 8.23 page number 268\n"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 19,
- "metadata": {
- "collapsed": false
- },
- "outputs": [
- {
- "name": "stdout",
- "output_type": "stream",
- "text": [
- "ps= 91.73 N/mm^2\n",
- "pc= 44.96 N/mm^2\n"
- ]
- }
- ],
- "source": [
- "from math import pi\n",
- "\n",
- "#variable declaration\n",
- "\n",
- "# Let the force shared by bolt be Ps and that by tube be Pc. Since there is no external force, static equilibrium condition gives Ps + Pc = 0 or Ps = – Pc i.e., the two forces are equal in magnitude but opposite in nature. Obviously bolt is in tension and tube is in compression.\n",
- "#Let the magnitude of force be P. Due to quarter turn of the nut\n",
- "\n",
- "#[Note. Pitch means advancement of nut in one full turn] \n",
- "\n",
- "Ls=float(600) #length of whole assembly,mm\n",
- "Lc=float(600) #length of whole assembly,mm\n",
- "delta=float(0.5)\n",
- "ds=float(20) #diameter,mm\n",
- "di=float(28) #internal diameter,mm\n",
- "de=float(40) #external diameter,mm\n",
- "Es=float(2*100000) #Young's modulus, N/mm^2\n",
- "Ec=float(1.2*100000)\n",
- "As=pi*pow(ds,2)/4 #area of steel bolt,mm^2\n",
- "Ac=pi*(pow(de,2)-pow(di,2))/4 #area of copper tube,mm^2\n",
- "\n",
- "P= (delta*(1/Ls))/((1/(As*Es))+(1/(Ac*Ec))) #Load,N\n",
- "\n",
- "ps=P/As #stress,N/mm^2\n",
- "pc=P/Ac #copper,N/mm^2\n",
- "\n",
- "print \"ps=\",round(ps,2),\"N/mm^2\"\n",
- "print \"pc=\",round(pc,2),\"N/mm^2\""
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "# example 8.24 page number 271"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 20,
- "metadata": {
- "collapsed": false
- },
- "outputs": [
- {
- "name": "stdout",
- "output_type": "stream",
- "text": [
- "(a) p= 52.8 \tN/mm^2\n",
- "(b) p= 27.8 \tN/mm^2\n",
- " (iii) delta= 1.968 mm\n"
- ]
- }
- ],
- "source": [
- "#variable declaration\n",
- "E=float(2*100000) #Young's modulus,N/mm^2\n",
- "alpha=float(0.000012) #expansion coeffecient,/°c\n",
- "L=float(12) #length,m\n",
- "t=float(40-18) #temperature difference,°c\n",
- "\n",
- "delta=alpha*t*L*1000 #free expansion of the rails,mm \n",
- "# Provide a minimum gap of 3.168 mm between the rails, so that temperature stresses do not develop\n",
- " \n",
- "# a) If no expansion joint is provided, free expansion prevented is equal to 3.168 mm.\n",
- "\n",
- "#delta=(P*L)/(A*E) & p=P/A where p is stress, P,A is load,area \n",
- "\n",
- "p1=(delta*E)/(L*1000) #stress developed , N/mm^2\n",
- "\n",
- "print \"(a) p=\", round(p1,1),\"\tN/mm^2\"\n",
- "\n",
- "#(b) If a gap of 1.5 mm is provided, free expansion prevented delta2 = 3.168 – 1.5 = 1.668 mm.\n",
- "\n",
- "delta2=1.668 #mm\n",
- "#delta2=(P*L)/(A*E) & p=P/A where p is stress, P,A is load,area \n",
- "\n",
- "p2=(delta2*E)/(L*1000) #stress developed , N/mm^2\n",
- "\n",
- "print \"(b) p=\", round(p2,1),\"\tN/mm^2\"\n",
- "\n",
- "# If the stress developed is 20 N/mm2, then p = P/ A\n",
- "p3=20 #stress developed,N/mm^2\n",
- "delta3=delta-(p3*L*1000/E)\n",
- "\n",
- "print \" (iii) delta=\",round(delta3,3),\"mm\"\n"
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "# example 8.25 page number 272\n"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 21,
- "metadata": {
- "collapsed": false
- },
- "outputs": [
- {
- "name": "stdout",
- "output_type": "stream",
- "text": [
- "stress p= 360.0 N/mm^2\n"
- ]
- }
- ],
- "source": [
- "from math import pi\n",
- "\n",
- "#variable declaration\n",
- "\n",
- "# Let D be the diameter of ring after heating and ‘d’ be its diameter before heating\n",
- "D=float(1.2*1000) #mm\n",
- "\n",
- "#Circumference of ring after heating Ca= pi*D & Circumference of ring before heating Cb= pi*d\n",
- "\n",
- "Ca=pi*D\n",
- "Cb=pi*d\n",
- "alphas=float(0.000012) #coefficient of expansion,/°C\n",
- "t=150 #temperature change,°C\n",
- "Es=2*100000 #young's modulus,N/mm^2\n",
- "d=(Ca-Cb)/(alphas*t*pi)\n",
- "\n",
- "#when it cools expansion prevented\n",
- "#delta=pi*(D-d)\n",
- "delta=alphas*t*pi*d\n",
- "\n",
- "p=(delta*Es)/(pi*d) #stress,N/mm^2\n",
- "\n",
- "print \"stress p=\",round(p,2),\"N/mm^2\"\n"
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "# example 8.26 page number 272\n"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 22,
- "metadata": {
- "collapsed": false
- },
- "outputs": [
- {
- "name": "stdout",
- "output_type": "stream",
- "text": [
- "P= 12907.3 N\n"
- ]
- }
- ],
- "source": [
- "#variable declaration\n",
- "\n",
- "Ea=70*1000 #Young's modulus of aluminium,N/mm^2\n",
- "Es=200*1000 #Young's modulus of steel,N/mm^2\n",
- "\n",
- "alphaa=float(0.000011) #expansion coefficient,/°C\n",
- "alphas=float(0.000012) #expansion coefficient,/°C\n",
- "\n",
- "Aa=600 #Area of aluminium portion,mm^2\n",
- "As=400 #Area of steel, mm^2\n",
- "La=float(1.5) #length of aluminium portion,m\n",
- "Ls=float(3.0) #length of steel portion,m\n",
- "t=18 #temperature,°C\n",
- "\n",
- "delta=(alphaa*t*La*1000)+(alphas*t*Ls*1000) #mm\n",
- "\n",
- "P=(delta)/(((La*1000)/(Aa*Ea))+((Ls*1000)/(As*Es)))\n",
- "\n",
- "print \"P=\" ,round(P,1),\"N\"\n",
- "\n"
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "# example8.27 page number 273"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 23,
- "metadata": {
- "collapsed": false
- },
- "outputs": [
- {
- "name": "stdout",
- "output_type": "stream",
- "text": [
- "Corresponding maximum stress = 120.0 N/mm^2\n"
- ]
- }
- ],
- "source": [
- "from math import pi\n",
- "\n",
- "#variable declaration\n",
- "\n",
- "d1=float(25) # variation linearly in diameter from 25 mm to 50 mm \n",
- "d2=float(50)\n",
- "L=float(500) #length,mm\n",
- "alpha=float(0.000012) #expansion coeffecient,/°C\n",
- "t=25 #rise in temperture,°C\n",
- "E=2*100000 #Young's modulus,N/mm^2\n",
- "\n",
- "delta=alpha*t*L\n",
- "\n",
- "#If P is the force developed by supports, then it can cause a contraction of 4*P*L/(pi*d1*d2*E)\n",
- "\n",
- "P=(delta*pi*d1*d2*E)/(4*L)\n",
- "Am=pi*pow(d1,2)/4\n",
- "Ms=P/Am\n",
- "\n",
- "print \"Corresponding maximum stress = \",round(Ms,1),\"N/mm^2\"\n"
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "# example 8.28 page number 275"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 24,
- "metadata": {
- "collapsed": false
- },
- "outputs": [
- {
- "name": "stdout",
- "output_type": "stream",
- "text": [
- "stress in steel= 12.17 N/mm^2\n",
- "Stress in brass= 36.51 N/mm^2\n",
- "Shear stress in pin 18.26 N/mm^2\n"
- ]
- }
- ],
- "source": [
- "from math import pi\n",
- "\n",
- "#variable declaration\n",
- "\n",
- "Db=float(20) #diameter of brass rod,mm\n",
- "Dse=float(40) #external diameter of steel tube,mm\n",
- "Dsi=float(20) #internal diameter of steel tube,mm\n",
- "Es=float(2*100000 ) #Young's modulus steel, N/mm^2\n",
- "Eb=float(1*100000 ) #Young's modulus brass, N/mm^2\n",
- "alphas=float(0.0000116) #coeffcient of expansion of steel,/°C\n",
- "alphab=float(0.0000187) #coeffcient of expansion of brass,/°C\n",
- "t=60 #raise in temperature, °C\n",
- "As=pi*(pow(Dse,2)-pow(Dsi,2))/4 #Area of steel tube, mm^2\n",
- "Ab=pi*(pow(Db,2))/4 #Area of brass rod,mm^2\n",
- "L=1200 #length,mm\n",
- "#Since free expansion of brass is more than free expansion of steel , compressive force Pb develops in brass and tensile force Ps develops in steel to keep the final position at CC \n",
- "\n",
- "#Horizontal equilibrium condition gives Pb = Ps, say P. \n",
- "\n",
- "P=((alphab-alphas)*t*L)/((L/(As*Es))+(L/(Ab*Eb)))\n",
- "\n",
- "ps=P/As\n",
- "pb=P/Ab\n",
- "\n",
- "print \"stress in steel=\",round(ps,2),\"N/mm^2\"\n",
- "print \"Stress in brass=\",round(pb,2),\"N/mm^2\"\n",
- "\n",
- "#the pin resist the force P at the two cross- sections at junction of two bars.\n",
- "\n",
- "Shearstress=P/(2*Ab)\n",
- "print \"Shear stress in pin\",round(Shearstress,2),\"N/mm^2\"\n",
- "\n"
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "# example 8.29 page number 276"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 25,
- "metadata": {
- "collapsed": false
- },
- "outputs": [
- {
- "name": "stdout",
- "output_type": "stream",
- "text": [
- "Change in length= 1.07 mm\n",
- "Hoop stress f= 83.33 N/mm^2\n"
- ]
- }
- ],
- "source": [
- "#variable declaration\n",
- "\n",
- "L=float(1000) #length of the bar at normal temperature,mm\n",
- "As=float(50*10) #Area of steel,mm^2\n",
- "Ac=float(40*5) #Area of copper,mm^2\n",
- "#Ac = Free expansion of copper is greater than free expansion of steel . To bring them to the same position, tensile force Ps acts on steel plate and compressive force Pc acts on each copper plate. \n",
- "alphas=float(0.000012) #Expansion of coeffcient of steel,/°C\n",
- "alphac=float(0.000017 ) #Expansion of coeffcient of copper,/°C\n",
- "t=80 #raise by temperature, °C\n",
- "Es=2*100000 #Young's modulus of steel,N/mm^2\n",
- "Ec=1*100000 #Young's modulus of copper,N/mm^2\n",
- "Pc=((alphac-alphas)*t*L)/((2*L/(As*Es)) +(L/(Ac*Ec)))\n",
- "Ps=2*Pc\n",
- "\n",
- "pc=Pc/Ac #Stress in copper,N/mm^2\n",
- "ps=Ps/As #Stress in steel, N/mm^2\n",
- "\n",
- "Changeinlength=alphas*t*L+(Ps*L/(As*Es))\n",
- "\n",
- "\n",
- "print\"Change in length=\",round(Changeinlength,2),\"mm\"\n",
- "\n",
- "##example 8.30 page number 278\n",
- "\n",
- "#variable declaration\n",
- "\n",
- "p=float(2) #internal pressure, N/mm^2\n",
- "t=12 #thickness of thin cylinder,mm\n",
- "D=float(1000) #internal diameter,mm\n",
- "\n",
- "f=(p*D)/(2*t) #Hoop stress,N/mm^2\n",
- "\n",
- "print \"Hoop stress f=\",round(f,2),\"N/mm^2\"\n",
- "\n",
- "\n"
- ]
- }
- ],
- "metadata": {
- "kernelspec": {
- "display_name": "Python [Root]",
- "language": "python",
- "name": "Python [Root]"
- },
- "language_info": {
- "codemirror_mode": {
- "name": "ipython",
- "version": 2
- },
- "file_extension": ".py",
- "mimetype": "text/x-python",
- "name": "python",
- "nbconvert_exporter": "python",
- "pygments_lexer": "ipython2",
- "version": "2.7.12"
- }
- },
- "nbformat": 4,
- "nbformat_minor": 0
-}
generated by cgit (git 2.34.1) at 2024-12-19 14:57:38 +0000