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+{
+"cells": [
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Chapter 11: Columns"
+ ]
+ },
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 11.1: EX11_1.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"E = 29000; // Modulus of elasticity in ksi\n",
+"spl = 42 ; // Proportional limit in ksi\n",
+"L = 25 ; // Total length of coloum in ft\n",
+"n = 2.5 ; // factor of safety\n",
+"I1 = 98 ; // Moment of inertia on horizontal axis\n",
+"I2 = 21.7 ; // Moment of inertia on vertical axis\n",
+"A = 8.25 ; // Area of the cross section\n",
+"Pcr2 = (4*%pi^2*E*I2)/((L*12)^2) ; // Criticle load if column buckles in the plane of paper\n",
+"Pcr1 = (%pi^2*E*I1)/((L*12)^2) ; // Criticle load if column buckles in the plane of paper\n",
+"Pcr = min(Pcr1,Pcr2) ; // Minimum pressure would govern the design\n",
+"scr = Pcr/A ; // Criticle stress\n",
+"Pa = Pcr/n ; // Allowable load in k\n",
+"disp('k',Pa,'The allowable load is ')\n",
+" "
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 11.2: EX11_2.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"L = 3.25 ; // Length of alluminium pipe in m\n",
+"d = 0.1 ; // Outer diameter of alluminium pipe\n",
+"P = 100000; // Allowable compressive load in N\n",
+"n =3 ; // Safety factor for eular buckling\n",
+"E = 72e09 ; // Modulus of elasticity in Pa\n",
+"l = 480e06 ; // Proportional limit\n",
+"Pcr = n*P ; // Critic;e load\n",
+"t = (0.1 - (55.6e-06)^(1/4) )/2 ; // Required thickness\n",
+"// Above formula comes from solving following equation\n",
+"// d2 = d ; d1 = d-2*t ; Pcr = n*P ; I = (%pi/64)*(d2^4-d1^4); Pcr = (2.406*%pi^2*E*I)/((L)^2) ;\n",
+"tmin = t ;\n",
+"disp('mm',tmin*1000,'The minimum required thickness of the coloumn is')\n",
+"// Supplimentry calculatios \n",
+"I = (%pi/64)*(d^4-(d-2*t)^4) ; // Moment of inertia\n",
+"A = (%pi/4)*(d^2-(d-2*t)^2) ; // Area of cross section\n",
+"r = sqrt(I/A);\n",
+"s = L/r // slenderness ratio\n",
+"scr = Pcr/A ; // Criticle stress "
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 11.3: Determination_of_longest_permissible_length_of_rod.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"P = 1500 ; // Load in lb\n",
+"e = 0.45 ; // ecentricity in inch\n",
+"h = 1.2 ; // Height of cross section in inch\n",
+"b = 0.6 ; // Width of cross section in inch\n",
+"E = 16e06 ; // Modulus of elasticity \n",
+"del = 0.12 ; // Allowable deflection in inch\n",
+"L = asec(1.2667)/0.06588 ; // Maximum allowable length possible\n",
+"// Above formula comes from solving following equation\n",
+"// Pcr = (%pi^2*E*I)/(4*(L)^2); I = (h*b^3)/12; del = e*(sec((%pi/2)*sqrt(P/Pcr))-1)\n",
+"disp('inch',L,'The longest permissible length of the bar is')\n",
+"\n",
+""
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 11.4: Calculation_of_compressive_stress_and_factor_of_safety.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"L = 25 ; // Length of coloum in ft\n",
+"P1 = 320 ; // Load in K\n",
+"P2 = 40 ; // Load in K\n",
+"E = 30000 ; // Modulus of elasticity of steel in Ksi\n",
+"P = 360 ; // Euivalent load\n",
+"e = 1.5 ; // Ecentricity of compressive load\n",
+"A = 24.1 ; // Area of the Cross section\n",
+"r = 6.05 ; // in inch\n",
+"c = 7.155 ; // in inch\n",
+"sy = 42 ;// Yeild stress of steel in Ksi\n",
+"smax = (P/A)*(1+(((e*c)/r^2)*sec((L/(2*r))*sqrt(P/(E*A))))); // Maximum compressive stress\n",
+"disp('ksi',smax,'The Maximum compressive stress in the column ')\n",
+"// Bisection method method to solve for yeilding\n",
+"function [x] = stress(a,b,f)\n",
+" N = 100;\n",
+" eps = 1e-5;\n",
+" if((f(a)*f(b))>0) then\n",
+" error('no root possible f(a)*f(b)>0');\n",
+" abort;\n",
+" end;\n",
+" if(abs(f(a))<eps) then\n",
+" error('solution at a');\n",
+" abort;\n",
+" end\n",
+" if(abs(f(b))<eps) then\n",
+" error('solution at b');\n",
+" abort;\n",
+" end\n",
+" while(N>0)\n",
+" c = (a+b)/2\n",
+" if(abs(f(c))<eps) then\n",
+" x = c ;\n",
+" x;\n",
+" return;\n",
+" end;\n",
+" if((f(a)*f(c))<0 ) then\n",
+" b = c ;\n",
+" else\n",
+" a = c ;\n",
+" end\n",
+" N = N-1;\n",
+" end\n",
+" error('no convergence');\n",
+" abort;\n",
+"endfunction\n",
+"\n",
+"deff('[y]=p(x)',['y = x + (0.2939*x*sec(0.02916*sqrt(x))) - 1012 '])\n",
+"x = stress(710,750,p);\n",
+"Py = x ; // Yeilding load in K\n",
+"n = Py/P; // Factor of safety against yeilding\n",
+"disp(n,'The factor of safety against yeilding is')\n",
+"\n",
+"\n",
+" \n",
+" \n",
+" "
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 11.5: Calculation_of_allowable_axial_load_and_maximum_permissible_length.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"E = 29000; // Modulus of elasticity in ksi\n",
+"sy = 36 ; // Yeilding stress in ksi\n",
+"L = 20 ; // Length of coloumn in ft\n",
+"r = 2.57 ; // radius of gyration of coloumn\n",
+"K = 1 ; // Effetive Length factor\n",
+"s = sqrt((2*%pi^2*E)/sy) // Criticle slenderness ratio (K*L)/r\n",
+"s_ = (L*12)/r; // Slenderness ratio\n",
+"// Part(a)\n",
+"n1 = (5/3)+((3/8)*(s_/s))-((1/8)*((s_^3)/(s^3)));// Factor of safety \n",
+"sallow = (sy/n1)*(1-((1/2)*((s_^2)/(s^2)))); // Allowable axial load\n",
+"A = 17.6; // Cross sectional area from table E1\n",
+"Pallow = sallow*A ; // Allowable axial load\n",
+"disp('k',Pallow,'Allowable axial load is')\n",
+"// Part (b)\n",
+"Pe = 200 ; // Permissible load in K\n",
+"L_ = 25 ; // Assumed length in ft\n",
+"s__ = (L_*12)/r; // Slenderness ratio\n",
+"n1_ = (5/3)+((3/8)*(s__/s))-((1/8)*((s__^3)/(s^3)));// Factor of safety \n",
+"sallow_ = (sy/n1_)*(1-((1/2)*((s__^2)/(s^2)))); // Allowable axial load\n",
+"A = 17.6 ; // Area of the cross section in^2\n",
+"Pallow = sallow_*A // Allowable load\n",
+"L1 = [24 24.4 25];\n",
+"P1 = [201 194 190];\n",
+"L_max = interpln([P1;L1],Pe); // Interpolation for getting the length correspondong to permissible load\n",
+"disp('ft',L_max,'The maximum permissible length is')\n",
+""
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 11.6: Finding_the_minimum_required_thickness_for_a_steel_pipe_column.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"L = 3.6 ; // Length of steel pipe coloumn\n",
+"d = 0.16 ; // Outer diameter in m\n",
+"P = 240e03; // Load in N\n",
+"E = 200e09; // Modulus of elasticity in Pa\n",
+"sy = 259e06 ; // yeilding stress in Pa\n",
+"Le = 2*L ; // As it in fixed-free condition\n",
+"sc = sqrt((2*%pi^2*E)/sy); // Critical slenderness ratio\n",
+"K = 2;\n",
+"// First trial\n",
+"t = 0.007; // Assumed thick ness in m\n",
+"I = (%pi/64)*(d^4-(d-2*t)^4) // Moment of inertia\n",
+"A = (%pi/4)*(d^2-(d-2*t)^2) // Area of cross section\n",
+"r = sqrt(I/A) ; // Radius of gyration\n",
+"sc_ = (K*L)/r ; // Slender ness ratio\n",
+"n2 = 1.98 ; // From equation 11.80\n",
+"sa = (sy/(2*n2))*(sc^2/sc_^2) // Allowable stress\n",
+"Pa = sa*A ; // Allowable axial load in N\n",
+"// Interpolation\n",
+"t = [7 8 9];\n",
+"Pa = [196 220 243];\n",
+"t_min = interpln([Pa;t],240) ; // Interpolation for getting the minimum length\n",
+"disp('mm',t_min,'The minimum required thickness of the steel pipe is')\n",
+""
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 11.7: Determination_of_the_minimum_required_outer_diameter_of_aluminium_tube.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"L = 16 ; // Effective length in inch\n",
+"P = 5 ; // axial load in K\n",
+"// Bisection method for solvong the quaderatic\n",
+"function [x] = stress(a,b,f)\n",
+" N = 100;\n",
+" eps = 1e-5;\n",
+" if((f(a)*f(b))>0) then\n",
+" error('no root possible f(a)*f(b)>0');\n",
+" abort;\n",
+" end;\n",
+" if(abs(f(a))<eps) then\n",
+" error('solution at a');\n",
+" abort;\n",
+" end\n",
+" if(abs(f(b))<eps) then\n",
+" error('solution at b');\n",
+" abort;\n",
+" end\n",
+" while(N>0)\n",
+" c = (a+b)/2\n",
+" if(abs(f(c))<eps) then\n",
+" x = c ;\n",
+" x;\n",
+" return;\n",
+" end;\n",
+" if((f(a)*f(c))<0 ) then\n",
+" b = c ;\n",
+" else\n",
+" a = c ;\n",
+" end\n",
+" N = N-1;\n",
+" end\n",
+" error('no convergence');\n",
+" abort;\n",
+"endfunction\n",
+"\n",
+"deff('[y]=p(x)',['y = 30.7*x^2 - 11.49*x -17.69 '])\n",
+"x = stress(0.9,1.1,p);\n",
+"d = x; // Diameter in inch\n",
+"sl = 49.97/d ; // Slenderness ration L/r\n",
+"dmin = d ; // Minimum diameter\n",
+"\n",
+"// The above equation comes from solving the following equationd for d\n",
+"// S_allow = 13.7 - 0.23*(L/r) = P/ A ;\n",
+"// A = (%pi/4)*(d^2-(d-2t)^2)\n",
+"// I = (%pi/64)*(d^4-(d-2t)^4)\n",
+"// r = sqrt(I/A)\n",
+"disp('inch',dmin,'The minimum required outer diameter of the tube is')\n",
+"\n",
+"\n",
+"\n",
+"\n",
+"\n",
+"\n",
+"\n",
+"\n",
+"\n",
+""
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 11.8: EX11_8.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"Fc = 11e06 ; // Compressive desing stress in Pa\n",
+"E = 13e09 ; // Modulus of elasticity in Pa\n",
+"// Part (a)\n",
+"Kce = 0.3 ; \n",
+"c = 0.8; \n",
+"A = 0.12*0.16 ; // Area of cross section\n",
+"Sl = 1.8/0.12 ; // Slenderness ratio\n",
+"fi = (Kce*E)/(Fc*Sl^2) ; // ratio of stresses\n",
+"Cp = ((1+fi)/(2*c)) - sqrt(((1+fi)/(2*c))^2-(fi/c)); // Coloumn stability factor \n",
+"Pa = Fc*Cp*A ; // Allowable Axial load\n",
+"disp('N',Pa,'The allowable axial load is')\n",
+"// Part (b)\n",
+"P = 100000; // Allowable Axial load\n",
+"Cp_ = P/(Fc*A) ; // Coloumn stability factor\n",
+"// Bisection method method to solve for fi\n",
+"function [x] = stress(a,b,f)\n",
+" N = 100;\n",
+" eps = 1e-5;\n",
+" if((f(a)*f(b))>0) then\n",
+" error('no root possible f(a)*f(b)>0');\n",
+" abort;\n",
+" end;\n",
+" if(abs(f(a))<eps) then\n",
+" error('solution at a');\n",
+" abort;\n",
+" end\n",
+" if(abs(f(b))<eps) then\n",
+" error('solution at b');\n",
+" abort;\n",
+" end\n",
+" while(N>0)\n",
+" c = (a+b)/2\n",
+" if(abs(f(c))<eps) then\n",
+" x = c ;\n",
+" x;\n",
+" return;\n",
+" end;\n",
+" if((f(a)*f(c))<0 ) then\n",
+" b = c ;\n",
+" else\n",
+" a = c ;\n",
+" end\n",
+" N = N-1;\n",
+" end\n",
+" error('no convergence');\n",
+" abort;\n",
+"endfunction\n",
+"deff('[y]=p(x)',['y = ((1+x)/(2*c)) - sqrt(((1+x)/(2*c))^2-(x/c)) - Cp_ '])\n",
+"x = stress(0.1,1,p); \n",
+"fi_ = x \n",
+"d_ = 0.12 ; // Diameter in m\n",
+"L_max = d_*sqrt((Kce*E)/(fi_*Fc)); // Maximum length in m\n",
+"disp('m',L_max,'The minimum allowable length is')\n",
+"// Part (c)\n",
+"b1 = [0.130 0.131 0.132]; // Two choices\n",
+"Sl1 = 2.6./b1 // slenderness ratio\n",
+"fi1 = (Kce*E)./(Fc*Sl1^2) // Ratio\n",
+"Cp1 = ((1+fi1)/(2*c)) - sqrt(((1+fi1)/(2*c)).^2-(fi1/c)); // Coloumn stability factor \n",
+"P1 = 11000.*Cp1.*b1^2 ; // Allowable atress \n",
+"Pa1 = 125; // Given allowable stress\n",
+"// Does not require display of result analysis has been shown for b = 0.131\n",
+" "
+ ]
+ }
+],
+"metadata": {
+ "kernelspec": {
+ "display_name": "Scilab",
+ "language": "scilab",
+ "name": "scilab"
+ },
+ "language_info": {
+ "file_extension": ".sce",
+ "help_links": [
+ {
+ "text": "MetaKernel Magics",
+ "url": "https://github.com/calysto/metakernel/blob/master/metakernel/magics/README.md"
+ }
+ ],
+ "mimetype": "text/x-octave",
+ "name": "scilab",
+ "version": "0.7.1"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 0
+}
generated by cgit (git 2.34.1) at 2025-03-07 19:19:49 +0000