{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Chapter 11: Energy Methods" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Example 11.01, Page number 699" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Strain energy :- \n", "0.5*L*P**2/(A*E)\n" ] } ], "source": [ "import math\n", "from sympy import symbols\n", "\n", "#Variable declaration\n", "P=symbols('P') # Force \n", "L=symbols('L') # Length\n", "A=symbols('A') # Area \n", "E=symbols('E') # Modulus of elasticity \n", "n=symbols('n') # Value\n", "\n", "#Calculation \n", "Un=((P**2)*((1/2.0)*L))/(2*A*E) + ((P**2)*((1/2.0)*L))/(2*(n**2)*A*E) # Strain energy\n", "U1=Un.subs(n,1) # Strain energy\n", "\n", "#Result\n", "print('Strain energy :- ')\n", "print(U1)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Example 11.02, Page number 700" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Strain energy :-\n", "0.364*P**2*l/AE\n" ] } ], "source": [ "import math\n", "from sympy import symbols\n", "\n", "#Variable declaration\n", "Fbc=symbols('Fbc') # Force\n", "Fbd=symbols('Fbd') # Force\n", "BC=symbols('BC') # Length\n", "BD=symbols('BD') # Length \n", "AE=symbols('AE') # Length\n", "l=symbols('l') # Length\n", "P=symbols('P') # Force\n", "n=-1\n", "\n", "#Calculation \n", "U=((Fbc**2)*(BC))/(2*AE) + ((Fbd**2)*(BD))/(2*AE) # Strain energy\n", "SE=U.subs([(BC, (0.6*l)),(BD, (0.8*l)),(Fbc, (0.6*P)),(Fbd, (0.8*P))]) # Strain energy\n", "\n", "#Result\n", "print('Strain energy :-')\n", "print(SE)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Example 11.03, Page number 701" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Strain energy :- \n", "P**2*l**3/(6*E*I)\n" ] } ], "source": [ "import math\n", "from sympy import integrate, symbols\n", "\n", "#Variable declaration\n", "P=symbols('P') # Force\n", "x=symbols('x') # Distance \n", "E=symbols('E') # Modulus of elasticity\n", "I=symbols('I') # Moment of inertia \n", "l=symbols('l') # Value \n", "n=-1\n", "\n", "#Calculation \n", "U=integrate((((P**2)*(x**2))/(2*E*I)),(x,0,l)) # Strain energy\n", "\n", "#Result\n", "print('Strain energy :- ')\n", "print(U)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Example 11.04, Page number 702" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Strain energy :- \n", "0.5*L*T**2/(G*j)\n" ] } ], "source": [ "import math\n", "from sympy import symbols\n", "\n", "#Variable declaration\n", "T=symbols('T') # Twisting couple\n", "L=symbols('L') # Length\n", "G=symbols('G') # Modulus of elasticity\n", "j=symbols('j') # Polar moment of inertia\n", "n=symbols('n') # Value \n", "\n", "#Calculation \n", "Un=((T**2)*((1/2.0)*L))/(2*G*j) + ((T**2)*((1/2.0)*L))/(2*(n**4)*G*j) # Strain energy\n", "U1=Un.subs(n,1) # Strain energy\n", "\n", "#Result\n", "print('Strain energy :- ')\n", "print(U1)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## SAMPLE PROBLEM 11.1, Page number 707" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Yield strength of steel :-\n", "36.2049720342386\n" ] } ], "source": [ "import math\n", "from sympy import symbols,solve\n", "\n", "#Variable declaration\n", "E=29*(pow(10,6)) # Elastic strain energy(psi) \n", "A=((math.pi)/4.0)*(pow(0.75,2)) # Area of cross section(in**2)\n", "L=60 # Length(in)\n", "Sy=symbols('Sy') # Stress\n", "\n", "#Calculation \n", "#Factor Of Safety\n", "U=5*120 # Strain energy(in.lb)\n", "#Strain-Energy Density \n", "V=A*L # Volume(in**3)\n", "u=(U/V) # Strain energy density(in.lb/in**3)\n", "#Yield Strength\n", "Sy=solve(Sy**2/(2.0*29.0*pow(10.0,6))-22.6,Sy) # Maximum stress(ksi)\n", "\n", "#Result\n", "print('Yield strength of steel :-')\n", "print(Sy[1]/1000) " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## SAMPLE PROBLEM 11.2, Page number 708" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Case(a): Strain energy :-\n", "0.166666666666667*P**2*a**2*b**2*(a + b)/(E*I*L**2)\n", "Case(b): Evaluated Strain energy = 0.000025 psi \n" ] } ], "source": [ "import math\n", "from sympy import integrate,symbols\n", "\n", "#Variable declaration\n", "P=40 # Force(kips)\n", "L=12 # Length(ft)\n", "a=3 # Length(ft)\n", "b=9 # Length(ft)\n", "E=29*(pow(10,6)) # Modulus of elasticity(psi) \n", "P=symbols('P') # Force\n", "b=symbols('b') # Length \n", "L=symbols('L') # Length\n", "P=symbols('P') # Force\n", "a=symbols('a') # Length \n", "x=symbols('x') # Length\n", "v=symbols('v') # Length\n", "E=symbols('E') # Modulus of elasticity\n", "I=symbols('I') # Moment of inertia\n", "\n", "#Calculation \n", "#Bending Moment\n", "Ra=(P*b)/L # Reaction\n", "Rb=(P*a)/L # Reaction\n", "\n", "M1=((P*b)/L)*x # Bending moment\n", "M2=((P*a)/L)*v # Bending moment\n", "\n", "# Case(a) Bending Moment\n", "U=(1/(2.0*E*I))*(integrate(M1**2,(x,0,a))+integrate(M2**2,(v,0,b))) # Total strain energy\n", "\n", "# Case(b) Evaluation of the Strain Energy\n", "P=40 # Central axial load(kips)\n", "a=36 # Length(in)\n", "L=144 # Length(in) \n", "b=108 # Length(in)\n", "I=248 # Moment of inertia(in**4)\n", "U=U.simplify()\n", "Usubs=(40*36*108)/(6.0*29*248*144*(pow(10,3))) # Strain energy\n", "\n", "#Result\n", "print('Case(a): Strain energy :-')\n", "print(U)\n", "print('Case(b): Evaluated Strain energy = %lf psi '%Usubs)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Example 11.06, Page number 717" ] }, { "cell_type": "code", "execution_count": 27, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Maximum value of stress in rod :-\n", "1.0*(Pm**2)**0.5/A\n" ] } ], "source": [ "import math\n", "from sympy import symbols\n", "\n", "#Variable declaration\n", "Pm=symbols('Pm') # Force\n", "L=symbols('L') # Length \n", "E=symbols('E') # Modulus of elasticity\n", "A=symbols('A') # Area\n", "m=symbols('m') # Mass \n", "v0=symbols('v0') # Velocity\n", "\n", "#Calculation \n", "Um=(5*(Pm**2)*(L))/(16.0*A*E) # Strain energy\n", "Pm=((16/5.0)*((Um*A*E)/(L)))**(1/2.0) # Force\n", "Sm=Pm/A # Stress \n", "\n", "#Result\n", "print('Maximum value of stress in rod :-')\n", "print(Sm)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Example 11.07, Page number 717" ] }, { "cell_type": "code", "execution_count": 28, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Maximum stress in the beam :-\n", "L*c*(Pm**2)**0.5/I\n" ] } ], "source": [ "import math\n", "from sympy import symbols\n", "\n", "#Variable declaration\n", "Pm=symbols('Pm') # Force\n", "L=symbols('L') # Length\n", "E=symbols('E') # Modulus of elasticity\n", "I=symbols('I') # Moment of inertia \n", "m=symbols('m') # Mass\n", "v0=symbols('v0') # Velocity\n", "Lc=symbols('Lc') # Length\n", "W=symbols('W') # Work \n", "h=symbols('h') # Height\n", "c=symbols('c') # Radius \n", "\n", "#Calculation \n", "Um=((Pm**2)*(L**3))/(6.0*E*I) # Strain energy\n", "Pm=((6)*((Um*E*I)/(L**3)))**(1/2.0) # Static force\n", "Sm=(Pm*L*c)/(I) # Maximum stress \n", "\n", "#Result\n", "print('Maximum stress in the beam :-')\n", "print(Sm)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Example 11.08, Page number 721" ] }, { "cell_type": "code", "execution_count": 29, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Case(a): Equivalent static load :-\n", "48**0.5*(E*I*m*v0**2/L**3)**0.5\n", "Case(b): Maximum stress :-\n", "3**0.5*(E*c**2*m*v0**2/(I*L))**0.5\n", "Case(c): Maximum deflection :-\n", "48**0.5*L**3*(E*I*m*v0**2/L**3)**0.5/(48*E*I)\n" ] } ], "source": [ "import math\n", "from sympy import symbols\n", "\n", "#Variable declaration\n", "Pm=symbols('Pm') # Force\n", "L=symbols('L') # Length \n", "E=symbols('E') # Modulus of elasticity\n", "I=symbols('I') # Moment of inertia\n", "m=symbols('m') # Mass\n", "v0=symbols('v0') # Velocity\n", "Lc=symbols('Lc') # Length \n", "h=symbols('h') # Height\n", "c=symbols('c') # Length\n", "xm=symbols('xm') # Distance\n", "\n", "#Calculation \n", "Um=(1/2.0)*(m)*(v0**2) # Strain energy\n", "Um=(1/2.0)*(Pm)*(xm) # Expressing Um as the work of the equivalent horizontal static load\n", "xm=((Pm)*(L**3))/(48.0*E*I) # Deflection of c corresponding to static load Pm \n", "Um=(((Pm**2)*(L**3))/(48.0*E*I))*(1/2.0) # Substituting xm in strain energy\n", "Pm=((48.0*m*(v0**2)*(E)*I)/(L**3))**(1/2.0) # Static load\n", "Sm=(Pm*Lc)/(4.0*I) # Maximum stress\n", "Sm=((3*m*(v0**2)*(E)*I)/(L*(I/c)**2))**(1/2.0) # Maximum stress after sustituting for Pm\n", "xm=Pm*((L**3)/(48.0*E*I)) # Maximum deflection\n", "\n", "#Result\n", "print('Case(a): Equivalent static load :-')\n", "print(Pm)\n", "print('Case(b): Maximum stress :-')\n", "print(Sm)\n", "print('Case(c): Maximum deflection :-')\n", "print(xm)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Example 11.09, Page number 723" ] }, { "cell_type": "code", "execution_count": 30, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Vertical deflection of B :-\n", "[0.728*L*P/(A*E)]\n" ] } ], "source": [ "import math\n", "from sympy import symbols,solve\n", "\n", "#Variable declaration\n", "Pm=symbols('Pm') # Force\n", "L=symbols('L') # Length \n", "E=symbols('E') # Modulus of elasticity \n", "A=symbols('A') # Area of crosssection\n", "m=symbols('m') # Mass \n", "v0=symbols('v0') # Velocity \n", "P=symbols('P') # Force\n", "yb=symbols('yb') # Distance\n", "\n", "#Calculation \n", "yb=solve(0.364*(((P**2.0)*L)/(A*E))-(1/2.0)*P*yb,yb)\n", "\n", "#Result\n", "print('Vertical deflection of B :-')\n", "print(yb)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Example 11.10, Page number 723" ] }, { "cell_type": "code", "execution_count": 31, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Deflection of end A taking into account of normal stress only:-\n", "[0.333333333333333*L**3*P/(E*I)]\n", "Deflection of end A taking into account of both the normal and shearing stresses.:-\n", "[0.0333333333333333*L*P**2*(3.0*E*h**2 + 10.0*G*L**2)/(E*G*I)]\n" ] } ], "source": [ "import math\n", "from sympy import symbols,solve\n", "\n", "#Variable declaration\n", "Pm=symbols('Pm') # Force \n", "L=symbols('L') # Length\n", "E=symbols('E') # Modulus of elasticity\n", "I=symbols('I') # Moment of inertia \n", "m=symbols('m') # Mass\n", "P=symbols('P') # Force\n", "yA=symbols('yA') # Distance\n", "h=symbols('h') # Height\n", "G=symbols('G') # Modulus of elasticity \n", "\n", "#Calculation \n", "yA1=solve((((P**2)*(L**3))/(6.0*E*I))-(1/2.0)*P*yA,yA) # Deflection of end A \n", "yA2=solve((((P**3)*(L**3))/(6.0*E*I))*(1+(3*E*(h**2))/(10.0*(G)*(L**2)))-(1/2.0)*(P)*(yA),yA) # Deflection of end A\n", "\n", "#Result\n", "print('Deflection of end A taking into account of normal stress only:-')\n", "print(yA1)\n", "print('Deflection of end A taking into account of both the normal and shearing stresses.:-')\n", "print(yA2)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Example 11.11, Page number 724" ] }, { "cell_type": "code", "execution_count": 32, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Angle of twist for entire shaft:-\n", "[0.53125*L*T/(G*J)]\n" ] } ], "source": [ "import math\n", "from sympy import symbols,solve\n", "\n", "#Variable declaration\n", "Pm=symbols('Pm') # Force\n", "L=symbols('L') # Length\n", "E=symbols('E') # Modulus of elasticity\n", "I=symbols('I') # Moment of inertia\n", "m=symbols('m') # Mass\n", "P=symbols('P') # Force\n", "yA=symbols('yA') # Distance \n", "h=symbols('h') # Height\n", "G=symbols('G') # Modulus of rigidity \n", "T=symbols('T') # Torque\n", "J=symbols('J') # Polar moment of inertia \n", "phyDB=symbols('phyDB') # Angle of twist\n", "\n", "#Calculation \n", "phyDB=solve((17/32.0)*(T**2)*(L)*(1/(2.0*G*J))-(1/2.0)*(T)*(phyDB),phyDB) # Angle of twist\n", "\n", "#Result\n", "print('Angle of twist for entire shaft:-')\n", "print(phyDB)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## SAMPLE PROBLEM 11.3, Page number 725 " ] }, { "cell_type": "code", "execution_count": 34, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Maximum deflection of point C:-\n", "10.2734993791682\n", "Maximum stress:-\n", "179.9296875\n" ] } ], "source": [ "import math\n", "from sympy import symbols,solve\n", "\n", "#Variable declaration\n", "ym=symbols('ym') # Distance\n", "E=symbols('E') # Modulus of elasticity\n", "I=symbols('I') # Moment of inertia\n", "L=symbols('L') # Length\n", "W=symbols('W') # Weight \n", "h=symbols('h') # Height\n", "\n", "#Calculation \n", "#Principle of Work and Energy\n", "Pm=(48*E*I)/(L**3.0) # Force \n", "U2=(1/2.0)*Pm*ym # Strain energy\n", "Eq=U2-(W*(h+ym)) # Equation\n", "\n", "#Case(a) Maximum Deflection of Point C\n", "E=73*(pow(10,9)) # Modulus of elasticity(Pa) \n", "I=(1/12.0)*((0.04)**4) # Moment of inertia(m**4) \n", "L=1 # Length(m)\n", "h=0.040 # Height(m)\n", "W=80*9.81 # Force(N) \n", "ym=solve(373.8*(pow(10,3))*(ym**2)-(784.8)*ym-31.39,ym) # Distance(mm)\n", "\n", "\n", "#Case(b) Maximum Stress\n", "Pm=48*(15.573*(pow(10,3)))*(0.01027) # Force(N)\n", "Sm=(((1/4.0)*(7677)*(0.020))/((1/12.0)*pow(0.040,4)))/(1000000.0) # Stress(MPa)\n", "\n", "#Result\n", "print('Maximum deflection of point C:-')\n", "print(ym[1]*1000)\n", "print('Maximum stress:-')\n", "print(Sm)" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.5.1" } }, "nbformat": 4, "nbformat_minor": 0 }