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diff --git a/Engineering_Mechanics_of_Solids_by_Popov_E_P/chapter11.ipynb b/Engineering_Mechanics_of_Solids_by_Popov_E_P/chapter11.ipynb new file mode 100644 index 00000000..aa2f8723 --- /dev/null +++ b/Engineering_Mechanics_of_Solids_by_Popov_E_P/chapter11.ipynb @@ -0,0 +1,375 @@ +{
+ "metadata": {
+ "name": "",
+ "signature": "sha256:1a88ac8a4ae99a352f7f49d975099441ec02a55d62cfaa2e8c07de364172180d"
+ },
+ "nbformat": 3,
+ "nbformat_minor": 0,
+ "worksheets": [
+ {
+ "cells": [
+ {
+ "cell_type": "heading",
+ "level": 1,
+ "metadata": {},
+ "source": [
+ "Chapter 11:Stability of Equilibrium: columns "
+ ]
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 11.2 page number 589"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "#Given \n",
+ "h = 60 #mm - the length of the crossection \n",
+ "b = 100 #mm - the width of hte crossection \n",
+ "E = 200 #Gpa - The youngs modulus\n",
+ "stress_cr = 250 #Mpa - The proportionality limit\n",
+ "#Caliculations \n",
+ "\n",
+ "I = b*(h**3)/12 #mm3 The momentof inertia of the crossection\n",
+ "A = h*b #mm2 - The area of teh crossection \n",
+ "#From Eulier formula\n",
+ "r_min = pow((I/A),0.5) #mm - The radius of the gyration \n",
+ "#(l/r)**2= (pi**2)*E/stress_cr #From Eulier formula\n",
+ "l = (((math.pi**2)*E*(10**3)/stress_cr)**0.5)*r_min #mm - the length after which the beam starts buckling\n",
+ "print \"The length after which the beam starts buckling is \",round(l,0),\"mm\""
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "The length after which the beam starts buckling is 1539.0 mm\n"
+ ]
+ }
+ ],
+ "prompt_number": 6
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 11.6 page number 613"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "#Given\n",
+ "L = 15 #ft - The length of the each rod\n",
+ "A = 46.7 #in2 - The length of the crossection \n",
+ "r_min = 4 #in - The radius of gyration\n",
+ "stress_yp = 36 #Ksi - the yielding point stress\n",
+ "E = 29*(10**3) #ksi - The youngs modulus\n",
+ "C_c = ((2*(math.pi**2)*E/stress_yp)**0.5) #Slenderness ratio L/R\n",
+ "C_s = L*12/r_min # Slenderness ratio L/R of the present situation \n",
+ "#According to AISC formulas \n",
+ "if C_s <C_c :\n",
+ " print \"a)The following approch is solvable\"\n",
+ "else: \n",
+ " print \"The caliculation is not possible\"\n",
+ "F_S = 5.0/3 +3*C_s/(8*C_c) -(3*C_s**3)/(8*C_c**3) #Safety factor \n",
+ "Stress_all = (1 - (C_s**2)/(2*C_c**2))*stress_yp/F_S #The allowable strees \n",
+ "print \"a) The allowable stress in this case is\",round(Stress_all,2),\"Kips\" \n",
+ "#Part - B\n",
+ "#Given\n",
+ "L = 40 #ft - The length of the each rod\n",
+ "A = 46.7 #in2 - The length of the crossection \n",
+ "r_min = 4 #in - The radius of gyration\n",
+ "stress_yp = 36 #Ksi - the yielding point stress\n",
+ "E = 29*(10**3) #ksi - The youngs modulus\n",
+ "C_c = ((2*(math.pi**2)*E/stress_yp)**0.5) #Slenderness ratio L/R\n",
+ "C_s = L*12/r_min # Slenderness ratio L/R of the present situation \n",
+ "#According to AISC formulas \n",
+ "if C_s <C_c :\n",
+ " print \"b) The following approch is solvable\"\n",
+ "else: \n",
+ " print \"The caliculation is not possible\"\n",
+ "F_S = 5.0/3 +3*C_s/(8*C_c) -(3*C_s**3)/(8*C_c**3) #Safety factor \n",
+ "Stress_all = (1 - (C_s**2)/(2*C_c**2))*stress_yp/F_S #The allowable strees \n",
+ "print \"b) The allowable stress in this case is\",round(Stress_all,2),\"Kips\"\n"
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "a)The following approch is solvable\n",
+ "a) The allowable stress in this case is 18.9 Kips\n",
+ "b) The following approch is solvable\n",
+ "b) The allowable stress in this case is 11.59 Kips\n"
+ ]
+ }
+ ],
+ "prompt_number": 18
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 11.7 page number 614"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "#Given\n",
+ "L = 15 #ft - The length of the each rod \n",
+ "p = 200 #Kips The concentric load applied \n",
+ "r_min = 2.10 #in - The radius of gyration\n",
+ "stress_yp = 50 #Ksi - the yielding point stress\n",
+ "E = 29*(10**3) #ksi - The youngs modulus\n",
+ "C_c = ((2*(math.pi**2)*E/stress_yp)**0.5) #Slenderness ratio L/R\n",
+ "C_s = L*12/r_min #Slenderness ratio L/R present situation\n",
+ "if C_s <C_c :\n",
+ " print \"a)The following approch is solvable\"\n",
+ "else: \n",
+ " print \"The caliculation is not possible\"\n",
+ "F_S = 5.0/3 +3*C_s/(8*C_c) -(3*C_s**3)/(8*C_c**3) #Safety factor \n",
+ "Stress_all = (1 - (C_s**2)/(2*C_c**2))*stress_yp/F_S #The allowable strees\n",
+ "a = p/Stress_all #in2 the alloawble area of the beam \n",
+ "print \"The allowable stress in this case is\",round(Stress_all,2),\"Kips\"\n",
+ "print \"This stress requires \",round(a,2),\"in2\"\n",
+ "if a <11.5:\n",
+ " print \"This case is satisfying W8x24 section\" #From AISC Manual \n",
+ "else:\n",
+ " print \"This case is not satisfying W8x24 section\"\n",
+ " #The ans are quiet varying because of rounding\n"
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "a)The following approch is solvable\n",
+ "The allowable stress in this case is 19.14 Kips\n",
+ "This stress requires 10.45 in2\n",
+ "This case is satisfying W8x24 section\n"
+ ]
+ }
+ ],
+ "prompt_number": 31
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 11.8 pagenumber 614 "
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "#Given\n",
+ "L = 15.0 #ft - The length of the each rod\n",
+ "A = 46.7 #in2 - The length of the crossection \n",
+ "r_min = 4 #in - The radius of gyration\n",
+ "stress_yp = 36.0 #Ksi - the yielding point stress\n",
+ "E = 29*(10**3) #ksi - The youngs modulus\n",
+ "lamda = L*12*((stress_yp/E)**0.5)/(4*(math.pi)) #column slenderness ratio\n",
+ "if lamda<1.5:\n",
+ " print \"The following approach is right\"\n",
+ "else:\n",
+ " print \"The following approach is wrong\"\n",
+ "stress_cr = (0.658**(lamda**2))*stress_yp #Ksi - The critical stress \n",
+ "P_n = stress_cr*A #Kips #Nominal compressive strength \n",
+ "o = 0.85 #Resistance factor\n",
+ "p_u = o*P_n #Kips ,column design compressive strength \n",
+ "print \"column design compressive strength \",p_u,\"Kips\""
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "The following approach is right\n",
+ "column design compressive strength 1284.51846781 Kips\n"
+ ]
+ }
+ ],
+ "prompt_number": 38
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 11.9 page number 615"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "#Given \n",
+ "#FOR FLANGS\n",
+ "l = 5 #in - The length of the flang\n",
+ "b = 5 #in - Teh width of the flang\n",
+ "t = 0.312 #in - the thickness of the flang\n",
+ "L = 20 #in - Length of the beam, Extracted from AISC manuals\n",
+ "A = 4.563 #in2 - The area of crossection of the beam\n",
+ "r = 1.188 #in - radius of the gyration, Extracted from AISC manuals \n",
+ "#b/t- value of the flang \n",
+ "k = (5 -t)/(2*t) #b/t ratio \n",
+ "#AISC, lets check maximum allowable stress for slang\n",
+ "Stressf_all = 23.1 - 0.79*k #ksi The maximum allowable stress in case of flang,AISC\n",
+ "\n",
+ "#web width thickness ratio\n",
+ "k_2 = (5 -2*t)/(t)\n",
+ "if k_2<16:\n",
+ " Stressw_all = 19 #ksi - The allowable stress in case of web width\n",
+ " \n",
+ "#a) Overall buckling investment \n",
+ "k_3 = L/r #slenderness ratio\n",
+ "Stressb_all = 20.2 - 0.216*k_3#ksi The maximum allowable stress in case of Buckling,AISC\n",
+ "p_allow = A*Stressb_all #Kips The allowable concentric load \n",
+ "\n",
+ "#b) Overall buckling investment\n",
+ "L_2 = 60 #in \n",
+ "k_3 = L_2/r #slenderness ratio\n",
+ "Stressb_all_2 = 20.2 - 0.126*k_3#ksi The maximum allowable stress in case of Buckling,AISC\n",
+ "p_allow_2 = A*Stressb_all_2 #Kips The allowable concentric load \n",
+ "\n",
+ "print \"The maximum allowable stress in case of web width\",round(Stressw_all,2),\"Ksi\"\n",
+ "print \"The maximum allowable stress in case of flang\",round(Stressf_all,2),\"Ksi\"\n",
+ "print \"a) The maximum allowable load in case of Buckling\",round(p_allow,2),\"Kips\"\n",
+ "print \"b) The maximum allowable load in case of Buckling\",round(p_allow_2,2),\"Kips\"\n",
+ "\n",
+ "\n",
+ " \n"
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "The maximum allowable stress in case of web width 19.0 Ksi\n",
+ "The maximum allowable stress in case of flang 17.16 Ksi\n",
+ "a) The maximum allowable load in case of Buckling 75.58 Kips\n",
+ "b) The maximum allowable load in case of Buckling 63.14 Kips\n"
+ ]
+ }
+ ],
+ "prompt_number": 43
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 11.11 page number 620 "
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "P = 200.0 #K The force on the beam \n",
+ "L = 15 #ft - The length of the rod\n",
+ "F_y = 50.0 #Ksi \n",
+ "F_a = F_y/(5.0/3) #Ksi -AISC MANUAL ,allowable axial stress if axial force is alone\n",
+ "F_b = F_a #Allowable compressive bending stress\n",
+ "M_1 = 600.0 #k-in - The moment acting on the ends of the rod\n",
+ "M_2 = 800.0 #k-in - the moment acting on the other end of teh rod\n",
+ "B_x = 0.264 #in - Extracted from AISC manual \n",
+ "A = P/F_a + M_2*B_x/F_b #in2- The minimum area \n",
+ "print \"The minimum area is \",round(A,2),\"in2\"\n",
+ "#we will select W10x49 section \n",
+ "A_s = 14.4 #in2 - The area of the section \n",
+ "r_min = 2.54 #in The minimum radius \n",
+ "r_x = 4.35 #in \n",
+ "f_a = P/A_s #Ksi- The computed axial stress\n",
+ "f_b = M_2*B_x/A_s #Computed bending stess\n",
+ "C_c = ((2*(math.pi**2)*E/F_y)**0.5) #Slenderness ratio L/R\n",
+ "C_s = L*12/r_min # Slenderness ratio L/R of the present situation\n",
+ "if C_s <C_c :\n",
+ " print \"The following approch is solvable\"\n",
+ "else: \n",
+ " print \"The caliculation is not possible\"\n",
+ "F_a_1 = 19.3 #Ksi - AISC lets try this\n",
+ "c_m = 0.6 - 0.4*(-M_1/M_2) \n",
+ "F_e = (12*(math.pi**2)*E)/(23*(L*12/r_x)**2) \n",
+ "k = f_a/F_a_1 + c_m*f_b*(1-(f_a/F_e))/F_b #Condition mentioned in AISC\n",
+ "if k>1:\n",
+ " print \"The following W10x49 section is not satisfying our constraints since f_a/F_a_1 + c_m*f_b*(1-(f_a/F_e))/F_b\",round(k,3),\">1\"\n",
+ "else:\n",
+ " print \"The following W10x49 section is satisfying our constraints since f_a/F_a_1 + c_m*f_b*(1-(f_a/F_e))/F_b\",k,\"<1\"\n",
+ " \n",
+ "#trail - 2\n",
+ "#Lets take W10 x 60\n",
+ "A_s = 17.6 #in2 - The area of the section \n",
+ "r_min = 2.57 #in The minimum radius \n",
+ "r_x = 4.39 #in \n",
+ "f_a = P/A_s #Ksi- The computed axial stress\n",
+ "f_b = M_2*B_x/A_s #Computed bending stess\n",
+ "C_c = ((2*(math.pi**2)*E/F_y)**0.5) #Slenderness ratio L/R\n",
+ "C_s = L*12/r_min # Slenderness ratio L/R of the present situation\n",
+ "if C_s <C_c :\n",
+ " print \"The following approch is solvable\"\n",
+ "else: \n",
+ " print \"The caliculation is not possible\"\n",
+ "F_a_1 = 19.3 #Ksi - AISC lets try this\n",
+ "c_m = 0.6 - 0.4*(-M_1/M_2) \n",
+ "F_e = (12*(math.pi**2)*E)/(23*(L*12/r_x)**2) \n",
+ "k = f_a/F_a_1 + c_m*f_b*(1-(f_a/F_e))/F_b #Condition mentioned in AISC\n",
+ "if k>1:\n",
+ " print \"The following W10x49 section is not satisfying our constraints since f_a/F_a_1 + c_m*f_b*(1-(f_a/F_e))/F_b\",round(k,3),\">1\"\n",
+ "else:\n",
+ " print \"The following W10x49 section is satisfying our constraints since f_a/F_a_1 + c_m*f_b*(1-(f_a/F_e))/F_b\",round(k,2),\"<1\"\n",
+ " \n"
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "The minimum area is 13.71 in2\n",
+ "The following approch is solvable\n",
+ "The following W10x49 section is not satisfying our constraints since f_a/F_a_1 + c_m*f_b*(1-(f_a/F_e))/F_b 1.09 >1\n",
+ "The following approch is solvable\n",
+ "The following W10x49 section is satisfying our constraints since f_a/F_a_1 + c_m*f_b*(1-(f_a/F_e))/F_b 0.9 <1\n"
+ ]
+ }
+ ],
+ "prompt_number": 67
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [],
+ "language": "python",
+ "metadata": {},
+ "outputs": []
+ }
+ ],
+ "metadata": {}
+ }
+ ]
+}
\ No newline at end of file |