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diff --git a/Mechanics_of_Materials_by_James_M._Gere/chapter5.ipynb b/Mechanics_of_Materials_by_James_M._Gere/chapter5.ipynb new file mode 100755 index 00000000..eded6995 --- /dev/null +++ b/Mechanics_of_Materials_by_James_M._Gere/chapter5.ipynb @@ -0,0 +1,766 @@ +{ + "metadata": { + "name": "", + "signature": "sha256:b3fea044fb61246e436e589813425563571ae1bfc62fb47f12dd9ef378e27c6d" + }, + "nbformat": 3, + "nbformat_minor": 0, + "worksheets": [ + { + "cells": [ + { + "cell_type": "heading", + "level": 1, + "metadata": {}, + "source": [ + "Chapter 5: Stresses in Beams Basic Topics" + ] + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example 5.1, page no. 307" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "\n", + "\n", + "import math\n", + "import numpy\n", + "\n", + "#initialisation\n", + "\n", + "L = 8.0 # length of beam in ft\n", + "h = 6.0 # Height of beam in inch\n", + "e = 0.00125 # elongation on the bottom surface of the beam\n", + "y = -3.0 # Dismath.tance of the bottom surface to the neutral surface of the beam in inch\n", + "\n", + "#Calculations\n", + "r = -(y/e) # Radius of curvature\n", + "print \"radius of curvature is\", round(r), \"inch\"\n", + "k = 1/r # curvature in in-1\n", + "print \"curvature\", round(k,5), \"ft-1\"\n", + "theta = numpy.degrees(numpy.arcsin(((L*12.0)/(2.0*r)))) # angle in degree\n", + "print \"Angle of twist\", round(theta,3), \"degree\"\n", + "my_del = r*(1-math.cos(math.radians(theta))) #Deflection in inch\n", + "print \"Deflection in the beam is \", round(my_del,4), \"inch\" " + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "radius of curvature is 2400.0 inch\n", + "curvature 0.00042 ft-1\n", + "Angle of twist 1.146 degree\n", + "Deflection in the beam is 0.48 inch\n" + ] + } + ], + "prompt_number": 1 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example 5.2, page no. 315" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "\n", + "\n", + "import math \n", + "\n", + "#initialisation\n", + "d = 0.004 # thickness of wire in m\n", + "R0 = 0.5 # radius of cylinder in m\n", + "E = 200e09 # Modulus of elasticity of steel\n", + "s = 1200e06 # proportional limit of steel\n", + "\n", + "#calculation\n", + "\n", + "M = (math.pi*E*d**4)/(32*(2*R0+d)) # Bending moment in wire in N-m\n", + "print \"Bending moment in the wire is \", round(M,2), \"N-m\"\n", + "s_max = (E*d)/(2*R0+d) # Maximum bending stress in wire in Pa\n", + "print \"Maximum bending stress in the wire is %e\" %(s_max), \"Pa\"" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "Bending moment in the wire is 5.01 N-m\n", + "Maximum bending stress in the wire is 7.968127e+08 Pa\n" + ] + } + ], + "prompt_number": 10 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example 5.3, page no. 316" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "\n", + "\n", + "import math \n", + "\n", + "#initialisation\n", + "L = 22 # Span of beam in ft\n", + "q = 1.5 # Uniform load intensity in k/ft\n", + "P = 12 # Concentrated in k\n", + "b = 8.75 # width of cross section of beam in inch\n", + "h = 27 # height of cross section of beam in inch\n", + "Ra = 23.59 # Reaction at point A\n", + "Rb = 21.41 # Reacyion at point B\n", + "Mmax = 151.6 # Maximum bending moment\n", + "\n", + "#calculation\n", + "\n", + "S = (b*h**2)/6 # Section modulus\n", + "s = (Mmax*12)/S # stress in k\n", + "st = s*1000 # Tensile stress\n", + "print \"Maximum tensile stress in the beam\", round(st), \"psi\"\n", + "sc = -s*1000 # Compressive stress\n", + "print \"Maximum compressive stress in the beam\", round(sc), \"psi\"" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "Maximum tensile stress in the beam 1711.0 psi\n", + "Maximum compressive stress in the beam -1711.0 psi\n" + ] + } + ], + "prompt_number": 11 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example 5.4, page no. 318" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "\n", + "\n", + "import math \n", + "\n", + "#initialisation\n", + "q = 3200.0 # Uniform load intensity in N/m\n", + "b = 0.3 # width of beam in m\n", + "h = 0.08 # Height of the beam in m\n", + "t = 0.012 # thickness of beam in m\n", + "Ra = 3600.0 # Reaction at A in N\n", + "Rb = 10800.0 # Reaction at B in N\n", + "Mpos = 2025.0 # Moment in Nm\n", + "Mneg = -3600.0 # Moment in Nm\n", + "\n", + "#calculation\n", + "y1 = t/2.0\n", + "A1 = (b-2*t)*t \n", + "y2 = h/2\n", + "A2 = h*t \n", + "A3 = A2 \n", + "c1 = ((y1*A1)+(2*y2*A2))/((A1)+(2*A2))\n", + "c2 = h - c1 \n", + "Ic1 = (b-2*t)*(t**3)*(1.0/12.0)\n", + "d1 = c1-(t/2.0)\n", + "Iz1 = (Ic1)+(A1*(d1**2))\n", + "Iz2 = 956600e-12\n", + "Iz3 = Iz2 \n", + "Iz = Iz1 + Iz2 + Iz3 # Moment of inertia of the beam cross section\n", + "\n", + "# Section Modulli\n", + "S1 = Iz / c1 # for the top surface\n", + "S2 = Iz / c2 # for the bottom surface\n", + "\n", + "# Maximum stresses for the positive section\n", + "st = Mpos / S2 \n", + "print \"Maximum tensile stress in the beam in positive section is\", st, \"Pa\"\n", + "sc = -Mpos / S1 \n", + "print \"Maximum compressive stress in the beam in positive section is\", sc, \"Pa\"\n", + "\n", + "# Maximum stresses for the negative section\n", + "snt = -Mneg / S1 \n", + "print \"Maximum tensile stress in the beam in negative section is\", snt, \"Pa\"\n", + "snc = Mneg / S2 \n", + "print \"Maximum compressive stress in the beam in negative section is\", snc, \"Pa\"\n", + "\n", + "# Conclusion\n", + "st_max = st\n", + "sc_max = snc" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "Maximum tensile stress in the beam in positive section is 50468539.6422 Pa\n", + "Maximum compressive stress in the beam in positive section is -15157118.8248 Pa\n", + "Maximum tensile stress in the beam in negative section is 26945989.0219 Pa\n", + "Maximum compressive stress in the beam in negative section is -89721848.2528 Pa\n" + ] + } + ], + "prompt_number": 5 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Exampe 5.5, page no. 325" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "\n", + "\n", + "import math \n", + "\n", + "#initialisation\n", + "L = 12 # Length of beam in ft\n", + "q = 420 # Uniform load intensity in lb/ft\n", + "s = 1800 # Allowable bending stress in psi\n", + "w = 35 # weight of wood in lb/ft3\n", + "\n", + "#calculation\n", + "M = (q*L**2*12)/8 # Bending moment in lb-in\n", + "S = M/s # Section Modulli in in3\n", + "\n", + "# From Appendix F\n", + "q1 = 426.8 # New uniform load intensity in lb/ft\n", + "S1 = S*(q1/q) # New section modulli in in3\n", + "\n", + "# From reference to appendix F, a beam of cross section 3*12 inch is selected\n", + "print (\"Beam of crosssection 3*12 is sufficient\")" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "Beam of crosssection 3*12 is sufficient\n" + ] + } + ], + "prompt_number": 6 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example 5.6, page no. 326" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "\n", + "import math\n", + "\n", + "#initialisation\n", + "P = 12000 # Lataeral load at the upper end in N\n", + "h = 2.5 # Height of post in m\n", + "Mmax = P*h # Maximum bending moment in Nm\n", + "\n", + "#calculation\n", + "# Part (a) : Wood Post\n", + "s1 = 15e06 # Maximum allowable stress in Pa\n", + "S1 = Mmax/s1 # Section Modulli in m3\n", + "d1 = ((32.0*S1)/math.pi)**(1.0/3.0) # diameter in m\n", + "print \"the minimum required diameter d1 of the wood post is\", round(d1,3), \"m\"\n", + "\n", + "# Part (b) : Alluminium tube\n", + "s2 = 50e06 # Maximum allowable stress in Pa\n", + "S2 = Mmax/s2 # Section Modulli in m3\n", + "d2 = (S2/0.06712)**(1.0/3.0) # diameter in meter.....(1) \n", + "print \"minimum required outer diameter d2 of the aluminum tube is\", round(d2,3),\"m\"" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "the minimum required diameter d1 of the wood post is 0.273 m\n", + "minimum required outer diameter d2 of the aluminum tube is 0.208 m\n" + ] + } + ], + "prompt_number": 18 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example 5.7, page no. 326" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "\n", + "\n", + "import math \n", + "\n", + "#initialisation\n", + "q = 2000.0 # Uniform load intensity in lb/ft\n", + "s = 18000.0 # Maximum allowable load in Psi\n", + "Ra = 18860.0 # Reaction at point A\n", + "Rb = 17140.0 # Reaction at point B\n", + "\n", + "#calculation\n", + "x1 = Ra/q # Distance in ft from left end to the point of zero shear\n", + "Mmax = (Ra*x1)-((q*(x1**2))/2.0) # Maximum bending moment in lb-ft\n", + "S = (Mmax*12.0)/s # Section Modulli in in3\n", + "\n", + "# Trial Beam\n", + "Ra_t = 19380.0 # Reaction at point A\n", + "Rb_t = 17670.0 # Reaction at point B\n", + "\n", + "#in Python the value for x1 differes by some points and hence the subsequent results differ\n", + "x1_t = Ra_t/q # Distance in ft from left end to the point of zero shear\n", + "Mmax_t = (Ra_t*x1_t)-((q*(x1_t**2))/2.0) # Maximum bending moment in lb-ft\n", + "S_t = (Mmax_t*12.0)/s # Section Modulli in in3\n", + "# From table E beam 12*50 is selected \n", + "print \"Beam of crosssection 12*50 is selected with section modulli\", round(S_t,2), \"in^3\"" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "Beam of crosssection 12*50 is selected with section modulli 62.6 in^3\n" + ] + } + ], + "prompt_number": 1 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example 5.8, page 329" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "\n", + "import math\n", + "\n", + "#initialisation \n", + "g = 9810 # Specific weight of water in N/m3\n", + "h = 2 # Height of dam in m\n", + "s = 0.8 # Dismath.tance between square cross section in m\n", + "sa = 8e06 # Maximum allowable stress in Pa\n", + "\n", + "#Calculations\n", + "b = ((g*(h**3)*s)/sa)**(1.0/3.0) # Dimension of croossection in m\n", + "\n", + "#Result\n", + "print \"the minimum required dimension b of the posts\", round(b,3), \"m\"" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "the minimum required dimension b of the posts 0.199 m\n" + ] + } + ], + "prompt_number": 15 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example 5.11, page no. 341" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "\n", + "\n", + "import math \n", + "\n", + "#initialisation\n", + "L = 3 # Span of beam in ft\n", + "q = 160 # Uniform load intensity in lb/in\n", + "b = 1 # Width of cross section\n", + "h = 4 # Height of cross section\n", + "\n", + "# Calculations from chapter 4\n", + "Mc = 17920 # Bending moment in ld-in\n", + "Vc = -1600 # Loading in lb\n", + "I = (b*(h**3))/12.0 # Moment of inertia in in4\n", + "sc = -(Mc*1)/I # Compressive stress at point C in psi\n", + "Ac = 1*1 # Area of section C in inch2\n", + "yc = 1.5 # dismath.tance between midlayers od section C and cross section of beam\n", + "Qc = Ac*yc # First moment of C cross section in inch3\n", + "tc = (Vc*Qc)/(I*b) # Shear stress in Psi\n", + "print \"Normal stress at C\", sc, \"psi\"\n", + "print \"Shear stress at C\", tc, \"psi\"" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "Normal stress at C -3360.0 psi\n", + "Shear stress at C -450.0 psi\n" + ] + } + ], + "prompt_number": 10 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example 5.12, page no. 342" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "\n", + "\n", + "import math\n", + "\n", + "#initialisation\n", + "s = 11e06 # allowable tensile stress in pa\n", + "t = 1.2e06 # allowable shear stress in pa\n", + "b = 0.1 # Width of cross section in m\n", + "h = 0.15 # Height of cross section in m\n", + "a = 0.5 # in m\n", + "\n", + "#Calculations\n", + "P_bending = (s*b*h**2)/(6.0*a) # Bending stress in N\n", + "P_shear = (2*t*b*h)/3.0 # shear stress in N\n", + "Pmax = P_bending # Because bending stress governs the design\n", + "\n", + "#Result\n", + "print \"the maximum permissible value Pmax of the loads\", Pmax, \"N\"" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "the maximum permissible value Pmax of the loads 8250.0 N\n" + ] + } + ], + "prompt_number": 11 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example 5.13, page no. 345" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "\n", + "\n", + "import math\n", + "\n", + "#initialisation\n", + "d2 = 4 # Outer diameter in inch\n", + "d1 = 3.2 # Inner diameter in inch\n", + "r2 = d2/2 # Outer radius in inch\n", + "r1 = d1/2 # inner radius in inch\n", + "P = 1500 # Horizontal force in lb\n", + "\n", + "#calculation\n", + "# Part (a)\n", + "t_max = ((r2**2+(r2*r1)+r1**2)*4*P)/(3*math.pi*((r2**4)-(r1**4))) # Mximum shear stress in Psi\n", + "print \"Maximum shear stress in the pole is\", round(t_max), \"psi\"\n", + "\n", + "# Part (b)\n", + "d0 = math.sqrt((16*P)/(3*math.pi*t_max)) # Diameter of solid circular cross section in meter\n", + "print \"Diameter of solid circular cross section is \", round(d0,2), \"m\"" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "Maximum shear stress in the pole is 658.0 psi\n", + "Diameter of solid circular cross section is 1.97 m\n" + ] + } + ], + "prompt_number": 20 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example 5.14, page no. 351" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "\n", + "import math\n", + "\n", + "#initialisation\n", + "b = 0.165 # in m\n", + "h = 0.320 # in m\n", + "h1 = 0.290 # in m\n", + "t = 0.0075 # in m\n", + "V = 45000.0 # Vertical force in N\n", + "\n", + "#calculation\n", + "I = (1.0/12.0)*((b*(h**3))-(b*(h1**3))+(t*(h1**3))) # Moment of inertia of the cros section\n", + "t_max = (V/(8.0*I*t))*((b*(h**2))-(b*(h1**2))+(t*(h1**2))) # Maximum shear stress in Pa\n", + "t_min = ((V*b)/(8*I*t))*(h**2-h1**2) # Minimum shear stress in Pa\n", + "T = ((t*h1)/3.0)*(2*t_max + t_min) # Total shear force in Pa\n", + "t_avg = V/(t*h1) # Average shear stress in Pa\n", + "\n", + "#Result\n", + "print \"Maximum shear stress in the web is\", round(t_max,2), \"Pa\"\n", + "print \"Minimum shear stress in the web is\", round(t_min,2), \"Pa\"\n", + "print \"Total shear stress in the web is\", round(T,2), \"N\"" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "Maximum shear stress in the web is 20985785.26 Pa\n", + "Minimum shear stress in the web is 17359517.46 Pa\n", + "Total shear stress in the web is 43015.04 N\n" + ] + } + ], + "prompt_number": 3 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example 5.15, page no. 352" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "\n", + "\n", + "import math \n", + "\n", + "#initialisation\n", + "V = 10000 # Vertical shear force in lb\n", + "b = 4 # in inch\n", + "t = 1 # in inch\n", + "h = 8 # in inch\n", + "h1 = 7 # in inch\n", + "\n", + "#calculation\n", + "A = b*(h-h1) + t*h1 # Area of cross section \n", + "Qaa = ((h+h1)/2.0)*b*(h-h1) + (h1/2.0)*(t*h1) # First moment of cross section\n", + "c2 = Qaa/A # Position of neutral axis in inch\n", + "c1 = h-c2 # Position of neutral axis in inch\n", + "Iaa = (b*h**3)/3.0 - ((b-t)*h1**3)/3.0 # Moment of inertia about the line aa\n", + "I = Iaa - A*c2**2 # Moment of inertia of crosssection\n", + "Q1 = b*(h-h1)*(c1-((h-h1)/2.0)) # First moment of area above the line nn\n", + "t1 = (V*Q1)/(I*t) # Shear stress at the top of web in Psi\n", + "Qmax = (t*c2)*(c2/2.0) # Maximum first moment of inertia below neutral axis\n", + "t_max = (V*Qmax)/(I*t) # Maximum Shear stress in Psi\n", + "\n", + "#Result\n", + "print \"Shear stress at the top of the web is\", round(t1), \"psi\"\n", + "print \"Maximum Shear stress in the web is\", round(t_max), \"Psi\"" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "Shear stress at the top of the web is 1462.0 psi\n", + "Maximum Shear stress in the web is 1762.0 Psi\n" + ] + } + ], + "prompt_number": 24 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example 5.16, page no. 357" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "\n", + "import math \n", + "\n", + "#initialisation\n", + "\n", + "Af = 40*180 # Area of flange in mm2\n", + "V = 10500 # Shear force acting on cross section\n", + "F = 800 # Allowable load in shear\n", + "df = 120 # Dismath.tance between centroid of flange and neutral axis in mm\n", + "\n", + "#calculation\n", + "Q = Af*df # First moment of cross section of flange\n", + "I = (1.0/12.0)*(210*280**3) - (1.0/12.0)*(180*200**3) # Moment of inertia of entire cross section in mm4\n", + "f = (V*Q)/I # Shear flow\n", + "s = (2*F)/f # Spacing between the screw\n", + "\n", + "#Result\n", + "print \"The maximum permissible longitudinal spacing s of the screws is\", round(s,1), \"mm\"" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "The maximum permissible longitudinal spacing s of the screws is 46.6 mm\n" + ] + } + ], + "prompt_number": 25 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example 5.17, page no. 362" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "\n", + "import math \n", + "\n", + "#initialisation\n", + "L = 60 # Length of beam in inch\n", + "d = 5.5 # dismath.tance from the point of application of the load P to the longitudinal axis of the tube in inch\n", + "b = 6 # Outer dimension of tube in inch\n", + "A = 20 # Area of cross section of tube in inch\n", + "I = 86.67 # Moment of inertia in in4\n", + "P = 1000 # in lb\n", + "theta = 60 # in degree\n", + "Ph = P*math.sin(math.radians(60)) # Horizontal component\n", + "Pv = P*math.cos(math.radians(60)) # Vertical component\n", + "\n", + "#Calculations\n", + "M0 = Ph*d # Moment in lb-in\n", + "y = -3 # Point at which maximum tensile stress occur in inch\n", + "N = Ph # Axial force\n", + "M = 9870 # Moment in lb-in\n", + "st_max = (N/A)-((M*y)/I) # Maximum tensile stress in Psi\n", + "yc = 3 # in inch\n", + "M1 = 5110 # moment in lb-in\n", + "sc_left = (N/A)-((M*yc)/I) # Stress at the left of point C in Psi\n", + "sc_right = -(M1*yc)/I # Stress at the right of point C in Psi\n", + "sc_max = min(sc_left,sc_right) # Because both are negative quantities\n", + "\n", + "#Result\n", + "print \"The maximum compressive stress in the beam is\", round(sc_max), \"psi\"\n", + "print \"The maximum tensile stress in the beam is\", round(st_max), \"psi\"" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "The maximum compressive stress in the beam is -298.0 psi\n", + "The maximum tensile stress in the beam is 385.0 psi\n" + ] + } + ], + "prompt_number": 26 + } + ], + "metadata": {} + } + ] +}
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