{
"metadata": {
"name": ""
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"worksheets": [
{
"cells": [
{
"cell_type": "heading",
"level": 1,
"metadata": {},
"source": [
"Chapter 12:Deflection of Beams and Shaft"
]
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 12.10 Page No 610"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#Given\n",
"E = 200*10**6 #kN/m**2, stress\n",
"I = 17*10**-6 #mm**4, moment of inertia\n",
"\n",
"#The given dimension are\n",
"l_ac = 2 #m\n",
"l_cF = 4 #m\n",
"l_Fb = 2 #m\n",
"l_cb = 6 #m\n",
"l_aF = 6 #m\n",
"l_ab = 8 #m\n",
"F = 16 #kN\n",
"R_b = (F*l_cb)/l_ab\n",
"R_a = F - R_b\n",
"\n",
"#Calculation\n",
"mc = R_a*l_ac\n",
"mf = R_b*l_Fb\n",
"theta_ca = (0.5*l_ac*mc)/(E*I)\n",
"A1 = 0.5*l_aF*mf\n",
"t1_ba = (l_Fb + l_aF/3)*(A1)\n",
"A2 = 0.5*l_Fb*mf\n",
"t2_ba = (l_Fb*2*A2)/3\n",
"t_ba = (t1_ba+t2_ba)/(E*I)\n",
"theta_c = (t_ba/l_ab)-(theta_ca)\n",
"\n",
"#Display\n",
"print\"The slope at point C of the steel beam = \",round(theta_c,4),\"rad\"\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The slope at point C of the steel beam = 0.0094 rad\n"
]
}
],
"prompt_number": 3
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 12.12 Page No 612"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#Given\n",
"E = 29*10**3 #ksi, stress\n",
"I = 125 #inch**4, moment of inertia\n",
"l_ab = 12 #ft, dimension\n",
"l_bc =12 #ft\n",
"l_ac = l_ab+l_bc\n",
"R_a = -5 #kip, normal force\n",
"R_b = 10 #kip\n",
"R_c = 5 #kip\n",
"\n",
"#calculation\n",
"mb = R_a*l_ab\n",
"#Moment-Area Theorem\n",
"t_ca = (l_ab*0.5*l_ac*mb)/(E*I)\n",
"t_ba = ((1/3.0)*l_ab*0.5*l_ab*mb)/(E*I)\n",
"del_c = (t_ca - 2*t_ba)*1728\n",
"\n",
"#Display\n",
"print\"The displacement at point C for the steel overhanging beam =\",round(del_c,2),\"inch\""
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The displacement at point C for the steel overhanging beam = -2.75 inch\n"
]
}
],
"prompt_number": 1
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 12.13 Page No 620"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#Given\n",
"w = 2 #kN/m, load\n",
"L = 8 #m, length\n",
"P = 8 #kN\n",
"\n",
"#Calculations\n",
"EI_theta_A1 = (3*w*L**3)/(128) #ThetaA1 = (3wL**3)/(128EI)\n",
"EI_nu_C1 = (5*w*L**4)/(768) #NuC1 = (5wL**4)/(768EI)\n",
"EI_theta_A2 = (P*L**2)/(16) #theta_A2 = (PL**2)/(16EI)\n",
"EI_nu_C2 = (P*L**3)/(48) #nu_C2 = (PL**3)/(48EI)\n",
"theta_A = EI_theta_A1 + EI_theta_A2\n",
"nu_C = EI_nu_C1 + EI_nu_C2\n",
"\n",
"#Display\n",
"print'The slope at A in terms of EI =',theta_A,\"kNm**2\"\n",
"print'The displacement at point C in terms of EI =',nu_C,\"kNm**2\"\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The slope at A in terms of EI = 56 kNm**2\n",
"The displacement at point C in terms of EI = 138 kNm**2\n"
]
}
],
"prompt_number": 11
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 12.14 Page No 621"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#Given\n",
"w = 5 #kN/m, force per unit length\n",
"l_ab = 4 #m, length\n",
"l_bc = 2 #m\n",
"P = 10 #kN, load\n",
"M = w*l_ab #kNm\n",
"\n",
"#Calculations\n",
"EI_theta_B1 = (w*l_ab**3)/(24) #ThetaB1 = (wL**3)/(24EI)\n",
"EI_nu_C1 = l_bc*EI_theta_B1\n",
"\n",
"EI_theta_B2 = (M*l_ab)/(3) #\n",
"EI_nu_C2 = l_bc*EI_theta_B2\n",
"EI_nu_C3 = (P*l_bc**3)/(3) #nuC3 = (PL**3)/(24EI)\n",
"nu_C = -EI_nu_C1 + EI_nu_C2 + EI_nu_C3\n",
"\n",
"#Display\n",
"print'The displacement at end C of the overhanging beam, in terms of EI = ',nu_C,\"kNm**3\""
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The displacement at end C of the overhanging beam, in terms of EI = 52 kNm**3\n"
]
}
],
"prompt_number": 10
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 12.15 Page No 622"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#Given\n",
"w = 4 #kN/m, force per unit length\n",
"l = 6 #m, length\n",
"l_bc =2 #m\n",
"\n",
"#Calculations\n",
"EI_theta_B = (w*l**3)/(24.0) #ThetaB1 = (wL**3)/(24EI)\n",
"EI_nu_B = (w*l**4)/(30.0) #nuB = (wL**4)/(30EI)\n",
"nu_C = EI_nu_B + (EI_theta_B*l_bc)\n",
"\n",
"#Display\n",
"print'The displacement at end C of the cantilever beam, in terms of EI = ',nu_C,\"kNm**3\"\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The displacement at end C of the cantilever beam, in terms of EI = 244.8 kNm**3\n"
]
}
],
"prompt_number": 5
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 12.16 Page No 623"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#Given\n",
"k = 15.0 #kip/ft, force per unit length\n",
"F = 3 #kip, force\n",
"E = 29*10**3 #ksi, stress\n",
"l_ab = 3.0 #ft, length\n",
"l_ac = 1 #ft\n",
"l_cb = 2 #ft\n",
"I = 12 #in**4, moment of inertia\n",
"R_a = (F*l_cb)/(l_ab)\n",
"R_b = F-R_a\n",
"\n",
"#Calculations\n",
"mu_a = (R_a)/k\n",
"mu_b = (R_b)/k\n",
"mu_c1 = mu_b + (l_cb/l_ab)*(mu_a - mu_b)\n",
"#From fig b\n",
"a=3 #ft\n",
"b=6 #ft\n",
"L=9 #ft\n",
"mu_c2 = ((F*a*b)*(L**2 - a**2 - b**2))/(6*E*144*I*(1/20736.0)*L)\n",
"mu_c = mu_c1 + mu_c2\n",
"\n",
"#Display\n",
"print'The vertical displacement of the force at C = ',round(mu_c,3),\"ft\"\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The vertical displacement of the force at C = 0.126 ft\n"
]
}
],
"prompt_number": 11
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 12.21 Page No 643"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#Given\n",
"l = 10 #ft, length\n",
"P = 8 #kip, load\n",
"w = 2 #kip/ft, force per unit length\n",
"\n",
"#Calculation\n",
"#Compatibility Equation\n",
"EI_nu_b1 = (w*l**4)/8.0 + (5*P*l**3)/48.0 #nu_b = (wl**4)/8EI + (5Pl**3)/48EI\n",
"EI_nu_b2 = (l**3)/3.0\n",
"B_y = EI_nu_b1 / EI_nu_b2\n",
"\n",
"#Display\n",
"print\"The reactions at roller support B = \",B_y,\"kip\"\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The reactions at roller support B = 10.0 kip\n"
]
}
],
"prompt_number": 12
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 12.22 Page No 644"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#Given:\n",
"l = 8 #ft, length\n",
"l_ab = 5 #ft\n",
"l_bc = 5 #ft\n",
"l_af = 1/2.0 #inch\n",
"b = 12/1000.0 #m\n",
"w = 8 #kip, force per unit length\n",
"E = 29*10**3 #Ksi\n",
"I = 475.0 # inch**4, moment of inertia\n",
"\n",
"#Compatibility Equation:\n",
"import math\n",
"A=math.pi/4.0*(l_af**2)\n",
"muB__byFbc=l*12/(A*E)\n",
"L=l_ab+l_bc\n",
"muB=5*w*L**3*12/(48*E*I)\n",
"muB_byFbc=L**3*12/(3*E*I)\n",
"#From equation muB__=muB-muB_\n",
"Fbc=muB/(muB_byFbc+muB__byFbc)\n",
"\n",
"#Display:\n",
"print \"Force developed in the rod is\",round(Fbc,3),\"kip\"\n",
"print\"In the book: Calculation mistake\"\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Force developed in the rod is 0.042 kip\n",
"In the book: Calculation mistake\n"
]
}
],
"prompt_number": 12
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 12.23 Page No: 646"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#Given:\n",
"L = 12 #ft\n",
"#E and I are constant say\n",
"E = 29*10**3 #Ksi\n",
"I = 475 # inch**4\n",
"w = 3 #kip/ft\n",
"\n",
"#calculation\n",
"thetab=w*L**3/(48*E*I)\n",
"mub=7*w*L**4/(384*E*I)\n",
"theta_bbyBy=L**2/(2*E*I)\n",
"mu_bbyBy=L**3/(3*E*I)\n",
"theta__byMb=L/(E*I)\n",
"muB__byMb=L**2/(2*E*I)\n",
"\n",
"#From eq 1 and 2 (solving by matrix)\n",
"#72By+12Mb=-108\n",
"#576By+72Mb=-1134\n",
"M = array([[576, 72], [72, 12]])\n",
"N=([-108,-1134])\n",
"X=inv(M)*N #Inverse matrix\n",
"a=X[0,0]\n",
"b=X[1,0]*2.5\n",
"\n",
"#Display:\n",
"print\"Moment at B is\",b,\"kip-ft\""
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Moment at B is 11.25 kip-ft\n"
]
}
],
"prompt_number": 13
}
],
"metadata": {}
}
]
}