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diff --git a/Mechanics_Of_Material_by_J_M_Gere/1-Tension_Comprssion_and_Shear.ipynb b/Mechanics_Of_Material_by_J_M_Gere/1-Tension_Comprssion_and_Shear.ipynb new file mode 100644 index 0000000..8332449 --- /dev/null +++ b/Mechanics_Of_Material_by_J_M_Gere/1-Tension_Comprssion_and_Shear.ipynb @@ -0,0 +1,272 @@ +{ +"cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Chapter 1: Tension Comprssion and Shear" + ] + }, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 1.1: Determine_the_compressive_stress_and_strain_in_the_post.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"d_1 = 4 ; // inner diameter (inch)\n", +"d_2 = 4.5 ; //outer diameter (inch)\n", +"P = 26000 ; // pressure in pound\n", +"L = 16; // Length of cylinder (inch)\n", +"del = 0.012 ; // shortening of post (inch)\n", +"A = (%pi/4)*((d_2^2)-(d_1^2)) //Area (inch ^2)\n", +"s = P/A; // stress\n", +"disp('psi',s,'compressive stress in the post is')\n", +"e = del / L; // strain\n", +"disp(e,'compressive strain in the post is')" + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 1.2: Calculation_of_maximum_stress.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"W = 1500; // weight (Newton)\n", +"d = 0.008 ; //diameter(meter) \n", +"g = 77000; // Weight density of steel\n", +"L = 40 ; // Length of bar (m)\n", +"A = (%pi/4)*(d^2) // Area\n", +"s_max = (1500/A) + (g*L) // maximum stress\n", +"disp('Pa',s_max,'Therefore the maximum stress in the rod is')" + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 1.3: Determination_of_various_structural_properties_of_the_pipe.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"d1 = 4.5; // diameter in inch\n", +"d2 = 6 ; // diameter in inch\n", +"A = (%pi/4)*((d2^2)-(d1^2)) // Area\n", +"P = 140 ; // pressure in K\n", +"s = -P/A ; // stress (compression)\n", +"E = 30000 ; // young's modulus in Ksi\n", +"e = s/E ; // strain\n", +"// Part (a)\n", +"del = e*4*12 // del = e*L ;\n", +"disp(del,'Change in length of the pipe is')\n", +"// Part (b)\n", +"v = 0.30; // Poissio's ratio\n", +"e_ = -(v*e)\n", +"disp(e_,'Lateral strain in the pipe is')\n", +"// Part (c)\n", +"del_d2 = e_*d2 ;\n", +"del_d1 = e_*d1;\n", +"disp('inch',del_d1,'Increase in the inner diameter is ')\n", +"// Part (d)\n", +"t = 0.75;\n", +"del_t = e_*t ;\n", +"disp('inch',del_t,'Increase in the wall thicness is')\n", +"del_t1 = (del_d2-del_d1)/2 ;\n", +"disp('del_t1 = del_t')\n", +"\n", +"\n", +"" + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 1.4: Calculation_of_average_shear_and_compressive_stress_in_a_punch.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"d = 0.02 ; // diameter in m\n", +"t = 0.008 ; // thickness in m\n", +"A = %pi*d*t ; // shear area\n", +"P = 110000 ; // prassure in Newton\n", +"A1 = (%pi/4)*(d^2); // Punch area\n", +"t_aver = P/A ; // Average shear stress \n", +"disp('Pa',t_aver,'Average shear stress in the plate is ')\n", +"s_c = P/A1 ; // compressive stress\n", +"disp('Pa',s_c,'Average compressive stress in the plate is ')" + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 1.5: Determination_of_various_structural_properties_of_the_pin.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"P = 12; // Pressure in K\n", +"t = 0.375 ; // thickness of wall in inch\n", +"theta = 40 ; // angle in degree\n", +"d_pin = 0.75 ; // diameter of pin in inch\n", +"t_G = 0.625 ; // thickness of gusset in inch\n", +"t_B = 0.375 ; //thickness of base plate in inch\n", +"d_b = 0.50 ; // diameter of bolt in inch\n", +"//Part (a)\n", +"s_b1 = P/(2*t*d_pin); // bearing stress\n", +"disp('ksi',s_b1,'Bearing stress between strut and pin')\n", +"//Part (b)\n", +"t_pin = (4*P)/(2*%pi*(d_pin^2)); // average shear stress in the \n", +"disp('ksi',t_pin,'Shear stress in pin is ')\n", +"// Part (c)\n", +"s_b2 = P/(2*t_G*d_pin); // bearing stress between pin and gusset\n", +"disp('ksi',s_b2,' Bearing stress between pin and gussets is')\n", +"// Part (d)\n", +"s_b3 = (P*cosd(40))/(4*t_B*d_b); // bearing stress between anchor bolt and base plate\n", +"disp('ksi',s_b3,'Bearing stress between anchor bolts and base plate')\n", +"// Part (e)\n", +"t_bolt = (4*cosd(40)*P)/(4*%pi*(d_b^2)); // shear stress in anchor bolt\n", +"disp('ksi',t_bolt,'Shear stress in anchor bolts is')" + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 1.7: Determination_of_allowable_tensile_load.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"b1 = 1.5 ; // width of rectangular crosssection in inch\n", +"t = 0.5 ; // thickness of rectangular crosssection in inch\n", +"b2 = 3 ; // width of enlarged rectangular crosssection in inch\n", +"d = 1 ; // diameter in inch\n", +"// Part (a)\n", +"s_1 = 16000; // maximum allowable tensile stress in Psi\n", +"P_1 = s_1*t*b1 ;\n", +"disp('lb',P_1,'The allowable load P1 is')\n", +"// Part (b)\n", +"s_2 = 11000; // maximum allowable tensile stress in Psi\n", +"P_2 = s_2*t*(b2-d) ;\n", +"disp('lb',P_2,'allowable load P2 at this section is')\n", +"//Part (c)\n", +"s_3 = 26000; // maximum allowable tensile stress in Psi\n", +"P_3 = s_3*t*d \n", +"disp('lb',P_3,'The allowable load based upon bearing between the hanger and the bolt is')\n", +"// Part (d)\n", +"s_4 = 6500; // maximum allowable tensile stress in Psi\n", +"P_4 = (%pi/4)*(d^2)*2*s_4 ;\n", +"disp('lb',P_4,'the allowable load P4 based upon shear in the bolt is')\n", +"" + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 1.8: Determination_of_required_cross_section_area_of_the_bar.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"// Horizontal component at A in N\n", +"R_ah = (2700*0.8 + 2700*2.6)/2 ;\n", +"// Horizontal component at C in N\n", +"R_ch = R_ah ;\n", +"// vertical component at C in N\n", +"R_cv = (2700*2.2 + 2700*0.4)/3 ;\n", +"// vertical component at A in N\n", +"R_av = 2700 + 2700 - R_cv ;\n", +"R_a = sqrt((R_ah^2)+(R_av^2))\n", +"R_c = sqrt((R_ch^2)+(R_cv^2))\n", +"Fab = R_a; // Tensile force in bar AB\n", +"Vc = R_c; // Shear force acting on the pin at C\n", +"s_allow = 125000000 ; // allowable stress in tension \n", +"t_allow = 45000000; // allowable stress in shear\n", +"Aab = Fab / s_allow; // required area of bar \n", +"Apin = Vc / (2*t_allow); // required area of pin\n", +"disp('m2',Apin,'Required area of bar is ')\n", +"d = sqrt((4*Apin)/%pi); // diameter in meter\n", +"disp('m',d,'Required diameter of pin is')" + ] + } +], +"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 +} diff --git a/Mechanics_Of_Material_by_J_M_Gere/11-Columns.ipynb b/Mechanics_Of_Material_by_J_M_Gere/11-Columns.ipynb new file mode 100644 index 0000000..0f47082 --- /dev/null +++ b/Mechanics_Of_Material_by_J_M_Gere/11-Columns.ipynb @@ -0,0 +1,444 @@ +{ +"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 +} diff --git a/Mechanics_Of_Material_by_J_M_Gere/12-Review_of_Centroids_and_Moments_of_Inertia.ipynb b/Mechanics_Of_Material_by_J_M_Gere/12-Review_of_Centroids_and_Moments_of_Inertia.ipynb new file mode 100644 index 0000000..6061387 --- /dev/null +++ b/Mechanics_Of_Material_by_J_M_Gere/12-Review_of_Centroids_and_Moments_of_Inertia.ipynb @@ -0,0 +1,126 @@ +{ +"cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Chapter 12: Review of Centroids and Moments of Inertia" + ] + }, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 12.2: Locating_centroid_C_of_the_cross_sectional_area.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"A1 = 6*0.5 ; // Partial Area in in2\n", +"A2 = 20.8 ; // from table E1 and E3\n", +"A3 = 8.82 ; // from table E1 and E3\n", +"y1 = (18.47/2) + (0.5/2) ; // Distance between centroid C1 and C2\n", +"y2 = 0 ; // Distance between centroid C2 and C2\n", +"y3 = (18.47/2) + 0.649 ; // Distance between centroid C3 and C2\n", +"A = A1 + A2 + A3 ; // Area of entire cross section\n", +"Qx = (y1*A1) + (y2*A2) - (y3*A3) ; // First moment of entire cross section\n", +"y_bar = Qx/A ; // Distance between x-axis and centroid of the cross section\n", +"disp('inch',-y_bar,'The distance between x-axis and centroid of the cross section is ')" + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 12.5: Determination_of_the_moment_of_inertia_Ic_with_respect_to_the_horizontal_axis.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"// Following variables are obtained from example 12.2\n", +"A1 = 6*0.5; // Partial Area in in2\n", +"A2 = 20.8; // from table E1 and E3\n", +"A3 = 8.82; // from table E1 and E3\n", +"y1 = (18.47/2) + (0.5/2); // Distance between centroid C1 and C2\n", +"y2 = 0 ; // Distance between centroid C2 and C2\n", +"y3 = (18.47/2) + 0.649; // Distance between centroid C3 and C2\n", +"A = A1 + A2 + A3; // Area of entire cross section\n", +"Qx = (y1*A1) + (y2*A2) - (y3*A3); // First moment of entire cross section\n", +"y_bar = Qx/A; // Distance between x-axis and centroid of the cross section\n", +"c_bar = -(y_bar);\n", +"//////////////////////////\n", +"I1 = (6*0.5^3)/12; // Moment of inertia of A1 \n", +"I2 = 1170; // Moment of inertia of A2 from table E1\n", +"I3 = 3.94; // Moment of inertia of A3 from table E3\n", +"Ic1 = I1 + (A1*(y1+c_bar)^2); // Moment of inertia about C-C axis of area C1\n", +"Ic2 = I2 + (A2*(y2+c_bar)^2); // Moment of inertia about C-C axis of area C2\n", +"Ic3 = I3 + (A3*(y3-c_bar)^2); // Moment of inertia about C-C axis of area C3\n", +"Ic = Ic1 + Ic2 + Ic3 ; // Moment of inertia about C-C axis of whole area\n", +"disp('in^4',Ic,'The moment of inertia of entire cross section area about its centroidal axis C-C')" + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 12.7: EX12_7.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"Ix = 29.29e06; // Moment of inertia of crosssection about x-axis\n", +"Iy = 5.667e06; // Moment of inertia of crosssection about y-axis\n", +"Ixy = -9.336e06; // Moment of inertia of crosssection \n", +"tp1 = (atand(-(2*Ixy)/(Ix-Iy)))/2 ; // Angle definig a Principle axix\n", +"tp2 = 90 + tp1 // '' \n", +"disp('degree',tp1,'The Principle axis is inclined at an angle')\n", +"disp('degree',tp2,'Second angle of inclination of Principle axis is')\n", +"Ix1 = (Ix+Iy)/2 + ((Ix-Iy)/2)*cosd(tp1) - Ixy*sind(tp1) ; // Principle Moment of inertia corresponding to tp1\n", +"Ix2 = (Ix+Iy)/2 + ((Ix-Iy)/2)*cosd(tp2) - Ixy*sind(tp2) ; // Principle Moment of inertia corresponding to tp2\n", +"disp('mm^4',Ix1,'Principle Moment of inertia corresponding to tp1')\n", +"disp('mm^4',Ix2,'Principle Moment of inertia corresponding to tp2')" + ] + } +], +"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 +} diff --git a/Mechanics_Of_Material_by_J_M_Gere/2-Axially_Loaded_Members.ipynb b/Mechanics_Of_Material_by_J_M_Gere/2-Axially_Loaded_Members.ipynb new file mode 100644 index 0000000..4f78d83 --- /dev/null +++ b/Mechanics_Of_Material_by_J_M_Gere/2-Axially_Loaded_Members.ipynb @@ -0,0 +1,339 @@ +{ +"cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Chapter 2: Axially Loaded Members" + ] + }, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 2.10: Determination_of_state_of_stress_in_a_bar.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"P = 90000; //newton\n", +"A = 1200e-6 // meter^2\n", +"s_x = -P/A ; //stress\n", +"t_1 = 25; //for the stresses on ab and cd plane\n", +"s_1 = s_x*(cosd(t_1)^2);\n", +"T_1 = -s_x*cosd(t_1)*sind(t_1) ;\n", +"t_2 = -65; //for the stresses on ad and bc plane\n", +"s_2 = s_x*(cosd(t_2)^2);\n", +"T_2 = -s_x*cosd(t_2)*sind(t_2) ;\n", +"disp('MPa respecively',s_1,T_1,' The normal and shear stresses on the plane ab and cd are')\n", +"disp('MPa respecively',s_2,T_2,' The normal and shear stresses on the plane ad and bc are')" + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 2.11: Determination_of_minimum_width_of_the_bar.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"// Value of s_x based on allowable stresses on glued joint\n", +"\n", +"s_t = -750; //psi\n", +"t = -50; //degree\n", +"T_t = -500; //psi\n", +"sg_x_1 = s_t/(cosd(t)^2)\n", +"sg_x_2 = -T_t/(cosd(t)*sind(t)) \n", +"\n", +"// Value of s_x based on allowable stresses on plastic\n", +"\n", +"sp_x_1 = -1100; //psi\n", +"T_t_p =600; //psi\n", +"t_p = 45; //degree\n", +"sp_x_2 = -T_t_p/(cosd(t_p)*sind(t_p)) \n", +"\n", +"// Minimum width of bar\n", +"\n", +"P = 8000; //lb\n", +"A = P/sg_x_2;\n", +"b_min = sqrt(A) //inch\n", +"disp('inch',b_min,'The minimum width of the bar is')" + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 2.15: Comparison_of_energy_absorbing_capacity_of_the_three_bolt.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"//Bolt with reduced shank diameter\n", +"g = 1.50; // inch\n", +"d = 0.5; //inch\n", +"t = 0.25; //inch\n", +"d_r = 0.406; //inch\n", +"L = 13.5; //inch\n", +"ratio = ( (g*(d^2)) / ( ((g-t)*(d_r^2))+(t*(d^2))) ) //U2/U1\n", +"disp(ratio,'The energy absorbing capacity of the bolts with reduced shank diameter')\n", +"// Long bolts\n", +"ratio_1 = ( (((L-t)*(d_r^2))+(t*(d^2))) / ((2*(g-t)*(d_r^2))+2*(t*(d^2))) ); //U3/2U1\n", +"disp(ratio_1,'The energy absorbing capacity of the long bolts')\n", +"" + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 2.16: Calculation_of_maximum_elongation_and_maximum_tensile_stress_in_a_bar.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"// Maximum elongation\n", +"M = 20; //kg\n", +"g = 9.81; //m/s^2\n", +"L = 2; //meter\n", +"E = 210e9; //210Gpa\n", +"h = 0.15; //meter\n", +"diameter = 0.015; //milimeter\n", +"A = (%pi/4)*(diameter^2) ; //area\n", +"D_st = ((M*g*L)/(E*A)) ;\n", +"D_max = D_st*(1+(1+(2*h/D_st))^0.5) ;\n", +"D_max_1 = sqrt(2*h*D_st) // another approach to find D_max\n", +"i = D_max / D_st // Impact factor\n", +"disp('mm',D_max,'Maximum elongation is')\n", +"// Maximum tensile stress\n", +"s_max = (E*D_max)/L ; //Maximum tensile stress\n", +"s_st = (M*g)/A ;//static stress\n", +"i_1 = s_max / s_st //Impact factor \n", +"disp('Pa',s_max,'Maximum tensile stress is')" + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 2.18: EX2_18.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"P1 = 108000; //Newton\n", +"P2 = 27000; //Newton\n", +"L = 2.2; //meter\n", +"A = 480; //mm^2\n", +"\n", +"// Displacement due to load P1 acting alone\n", +"\n", +"s = (P1/A) //stress in MPa\n", +"e = (s/70000) + (1/628.2)*((s/260)^10) //strain\n", +"D_b = e*L*1e3 //elongation in mm\n", +"disp('mm',D_b,'elongation when only P1 load acting is = ')\n", +"\n", +"// Displacement due to load P2 acting alone\n", +"\n", +"s_1 = (P2/A) //stress in MPa\n", +"e_1 = (s_1/70000) + (1/628.2)*((s_1/260)^10) //strain\n", +"D_b_1 = e_1*(L/2)*1e3 //elongation in mm (no elongation in lower half)\n", +"disp('mm',D_b_1,'elongation when only P2 load acting is = ')\n", +"\n", +"// Displacement due to both load acting simontaneously\n", +"\n", +"//upper half\n", +"s_2 = (P1/A) //stress in MPa\n", +"e_2 = (s_2/70000) + (1/628.2)*((s_2/260)^10) //strain\n", +"//lower half\n", +"s_3 = (P1+P2)/A //stress in MPa\n", +"e_3 = (s_3/70000) + (1/628.2)*((s_3/260)^10) //strain\n", +"D_b_2 = ( (e_2*L)/2 + (e_3*L)/2 ) * 1e3 // elongation in mm\n", +"disp('mm',D_b_2,'elongation when P1 and P2 both loads are acting is = ')" + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 2.1: EX2_1.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"W = 2 ; //lb\n", +"b = 10.5; //inch\n", +"c = 6.4 ; //inch\n", +"k = 4.2; //inch\n", +"p = 1/16; //inch\n", +"n = (W*b)/(c*k*p); //inch\n", +"disp(n,' No. of revolution required = ')" + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 2.2: Calculation_of_maximum_allowable_load.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"Fce_ = 2; //dummy variable\n", +"Fbd_ = 3; //dummy variable\n", +"Lbd = 480; //mm\n", +"Lce = 600; //mm\n", +"E = 205e6; //205Gpa\n", +"Abd = 1020; //mm\n", +"Ace = 520; //mm\n", +"Dbd_ = (Fbd_*Lbd)/(E*Abd); //dummy variable\n", +"Dce_ = (Fce_*Lce)/(E*Ace); //dummy variable\n", +"Da = 1; //limiting value\n", +"P = ( ( ((450+225)/225)*(Dbd_ + Dce_) - Dce_ )^(-1) ) * Da ; \n", +"Fce = 2*P; // Real value in newton\n", +"Fbd = 3*P; //real value in newton\n", +"Dbd = (Fbd*Lbd)/(E*Abd); //displacement in mm\n", +"Dce = (Fce*Lce)/(E*Ace); // displacement in mm\n", +"a = atand((Da+Dce)/675) ; //alpha in degree\n", +"disp('degree',a,'alpha = ')" + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 2.3: Calculation_of_vertical_displacement_at_point_C.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"P1 = 2100; //lb\n", +"P2 = 5600; //lb\n", +"b = 25; //inch\n", +"a = 28; //inch\n", +"A1 = 0.25; //inch^2\n", +"A2 = 0.15; //inch^2\n", +"L1 = 20; //inch\n", +"L2 = 34.8; //inch\n", +"E = 29e6; //29Gpa\n", +"P3 = (P2*b)/a ;\n", +"Ra = P3-P1;\n", +"N1 = -Ra ;\n", +"N2 = P1 ;\n", +"D = ((N1*L1)/(E*A1)) + ((N2*L2)/(E*A2)) ; //displacement\n", +"disp ('inch',D,'Downward displacement is = ')" + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 2.6: Calculation_of_the_allowable_load.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"//Numerical calculation of allowable load\n", +"d1 = 4; //mm\n", +"d2 = 3; //mm\n", +"A1 = (%pi*(d1^2))/4 ; //area\n", +"A2 = (%pi*(d2^2))/4 ; //area\n", +"L1 = 0.4; //meter\n", +"L2 = 0.3; //meter\n", +"E1 = 72e9 ; //Gpa\n", +"E2 = 45e9 ; //Gpa\n", +"f1 = L1/(E1*A1) * 1e6 ; // To cpmpensate for the mm^2\n", +"f2 = L2/(E2*A2) * 1e6 ;\n", +"s1 = 200e6; //stress\n", +"s2 = 175e6; //stress\n", +"P1 = ( (s1*A1*(4*f1 + f2))/(3*f2) ) * 1e-6 // To cpmpensate for the mm^2\n", +"P2 = ( (s2*A2*(4*f1 + f2))/(6*f1) ) * 1e-6 \n", +"disp( 'Newton',P2,'Minimum allowable stress aomong the two P1 and P2 is smaller one, therefore MAS = ')" + ] + } +], +"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 +} diff --git a/Mechanics_Of_Material_by_J_M_Gere/3-Torsion.ipynb b/Mechanics_Of_Material_by_J_M_Gere/3-Torsion.ipynb new file mode 100644 index 0000000..54bb7ee --- /dev/null +++ b/Mechanics_Of_Material_by_J_M_Gere/3-Torsion.ipynb @@ -0,0 +1,313 @@ +{ +"cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Chapter 3: Torsion" + ] + }, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 3.10: evaluation_of_the_strain_energy_for_different_cases.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"Ta = 100 ; // Torque in N-m at A\n", +"Tb = 150; // Torque in N-m at B\n", +"L = 1.6 ; // Length of shaft in meter\n", +"G = 80e09 ; // Modulus of elasticity\n", +"Ip = 79.52e-09; // polar moment of inertia in m4\n", +"Ua = ((Ta^2)*L)/(2*G*Ip) // Strain energy at A\n", +"disp('joule',Ua,'Torque acting at free end')\n", +"Ub = ((Tb^2)*L)/(4*G*Ip) // Strain energy at B\n", +"disp('joule',Ub,'Torque acting at mid point')\n", +"a = (Ta*Tb*L)/(2*G*Ip) // dummy variabble\n", +"Uc = Ua+a+Ub ; // Strain energy at C\n", +"disp('joule',Uc,'Total torque')" + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 3.11: Evaluation_of_the_strain_energy_of_a_hollow_shaft.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"t = 480 ; // Torque of constant intensity\n", +"L = 144 ; // Length of bar\n", +"G = 11.5e06; // Modulus of elasticity in Psi\n", +"Ip = 17.18 ; // Polar moment of inertia\n", +"U = ((t^2)*(L^3))/(G*Ip*6) // strain energy in in-lb\n", +"disp('in-lb',U,'The strain energu for the hollow shaft is')" + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 3.1: Calculation_of_maximum_shear_stress_and_permissible_torque_in_the_bar.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"d = 1.5; // diameter of bar in inch\n", +"L = 54 ; // Length of bar in inch\n", +"G = 11.5e06 ; // modulus of elasticity in psi \n", +"// Part (a)\n", +"T = 250 ; // torque\n", +"t_max = (16*T*12)/(%pi*(d^3)); // maximum shear stress in bar\n", +"Ip = (%pi*(d^4))/32 ; // polar miment of inertia \n", +"f = (T*12*L)/(G*Ip) ; // twist in radian\n", +"f_ = (f*180)/%pi ; // twist in degree\n", +"disp('psi',t_max,'Maximum shear stress in the bar is ')\n", +"disp('degree',f_,'Angle of twist is')\n", +"//Part (b)\n", +"t_allow = 6000 ; // allowable shear stress\n", +"T1 = (%pi*(d^3)*t_allow)/16; //allowable permissible torque in lb-in\n", +"T1_ = T1*0.0831658 ; //allowable permissible torque in lb-ft\n", +"f_allow = (2.5*%pi)/180 ; // allowable twist in radian\n", +"T2 = (G*Ip*f_allow)/L; // allowable stress via a another method\n", +"T2_ = T2*0.0831658; //allowable permissible torque in lb-ft\n", +"T_max = min(T1_,T2_); // minimum of the two\n", +"disp('lb-ft',T_max,'Maximum permissible torque in the bar is')" + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 3.2: Calculation_of_required_diameter_for_solid_and_hollow_shaft.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"T = 1200 ; // allowable torque in N-m\n", +"t = 40e06 ; // allowable shear stress in Pa\n", +"f = (0.75*%pi)/180 ; // allowable rate of twist in rad/meter\n", +"G = 78e09; // modulus of elasticity\n", +"// Part (a) : Solid shaft\n", +"d0 = ((16*T)/(%pi*t))^(1/3)\n", +"Ip = T/(G*f) ; // polar moment of inertia\n", +"d01 = ((32*Ip)/(%pi))^(1/4); // from rate of twist definition\n", +"disp('m',d0,'The required diameter of the solid shaft is ')\n", +"// Part (b) : hollow shaft\n", +"d2 = (T/(0.1159*t))^(1/3) ; // Diamater of hollow shaft in meter\n", +"// The above equation comes from solving the following four equation \n", +"// t1 = 0.1*d2 ; thickness of shaft\n", +"// d1 = d2-(2*t1) ; // diameter of inner radius\n", +"// Ip = (%pi/32)*((d2^4)-(d1^4)); // Polar moment of inertia\n", +"// r = d2/2\n", +"// t = (T*r)/Ip ; // allowable shear stress\n", +"d2_ = (T/(0.05796*G*f))^(1/4) ; // Another value of d2 by definition of theta(allow), f = T/(G*Ip)\n", +"d1 = 0.8*d2_ ; // because rate of twist governs the design\n", +"disp('m',d2,'The required diameter of the hollow shaft is ')\n", +"// Part (c) : Ratio of diameter and weight\n", +"r1 = d2_/d01 ; // diameter ratio\n", +"r2 = ((d2_^2)-(d1^2))/(d01^2) ; // Weight Ratio\n", +"disp(r1,'Ratio of the diameter of the hollow and solid shaft is')\n", +"disp(r2,'Ratio of the weight of the hollow and solid shaft is')\n", +"\n", +"\n", +"\n", +"" + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 3.4: EX3_4.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"d = 0.03 ; // diameter of the shaft in meter\n", +"T2 = 450 ; // Torque in N-m\n", +"T1 = 275 ; //\n", +"T3 = 175 ; //\n", +"Lbc = 0.5 ; // Length of shaft in meter\n", +"Lcd = 0.4 ; // Length of shaft in meter\n", +"G = 80e09 ; // Modulus of elasticity\n", +"Tcd = T2-T1 ; // torque in segment CD\n", +"Tbc = -T1 ; // torque in segment BC\n", +"tcd = (16*Tcd)/(%pi*(d^3)); // shear stress in cd segment\n", +"disp('Pa',tcd,'Shear stress in segment cd is')\n", +"tbc = (16*Tbc)/(%pi*(d^3)); // shear stress in bc segment\n", +"disp('Pa',tbc,'Shear stress in segment bc is')\n", +"Ip = (%pi/32)*(d^4); // Polar monent of inertia\n", +"fbc = (Tbc*Lbc)/(G*Ip); // angle of twist in radian\n", +"fcd = (Tcd*Lcd)/(G*Ip); // angle of twist in radian\n", +"fbd = fbc + fcd ; // angle of twist in radian\n", +"disp('radian',fbd,'Angles of twist in section BD')" + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 3.6: Calculation_of_various_stress_and_strain_in_circular_tube.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"d1 = 0.06 ; // Inner diameter in meter\n", +"d2 = 0.08 ; // Outer diameter in meter\n", +"r = d2/2; // Outer radius\n", +"G = 27e09 ; // Modulus of elasticity\n", +"T = 4000 ; // Torque in N-m\n", +"Ip = (%pi/32)*((d2^4)-(d1^4)); // Polar moment of inertia\n", +"t_max = (T*r)/Ip ; // maximum shear stress\n", +"disp('Pa',t_max,'Maximum shear stress in tube is ')\n", +"s_t = t_max ; // Maximum tensile stress\n", +"disp('Pa',s_t,'Maximum tensile stress in tube is ')\n", +"s_c = -(t_max); // Maximum compressive stress\n", +"disp('Pa',s_c,'Maximum compressive stress in tube is ')\n", +"g_max = t_max / G ; // Maximum shear strain in radian\n", +"disp('radian',g_max,'Maximum shear strain in tube is ')\n", +"e_t = g_max/2 ; // Maximum tensile strain in radian\n", +"disp('radian',e_t,'Maximum tensile strain in tube is ')\n", +"e_c = -g_max/2 ; // Maximum compressive strain in radian\n", +"disp('radian',e_c,'Maximum compressive strain in tube is ')\n", +"" + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 3.7: Calculation_of_the_required_diameter_d_of_the_shaft.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"H = 40 ; // Power in hp\n", +"s = 6000 ; // allowable shear stress in steel in psi\n", +"// Part (a)\n", +"n = 500 ; // rpm\n", +"T = ((33000*H)/(2*%pi*n))*(5042/420); // Torque in lb-in\n", +"d = ((16*T)/(%pi*s))^(1/3); // diameter in inch\n", +"disp('inch',d,'Diameter of the shaft at 500 rpm')\n", +"// Part (b)\n", +"n1 = 3000 ; // rpm\n", +"T1 = ((33000*H)/(2*%pi*n1))*(5042/420); // Torque in lb-in\n", +"d1 = ((16*T1)/(%pi*s))^(1/3); // diameter in inch\n", +"disp('inch',d1,'Diameter of the shaft at 3000 rpm')" + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 3.8: Calculation_of_maximum_shear_stress_tmax_in_the_shaft_and_the_angle_of_twist.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"d = 0.05 ; // diameter of the shaft\n", +"Lab = 1 ; // Length of shaft ab in meter\n", +"Lbc = 1.2 ; // Length of shaft bc in meter\n", +"Pa = 50000; // Power in Watt at A\n", +"Pb = 35000; // Power in Watt at B\n", +"Ip = (%pi/32)*(d^4) ; // Polar moment of inertia\n", +"Pc = 15000; // Power in Watt at C\n", +"G = 80e09; // Modulus of elasticity\n", +"f = 10 ; // frequency in Hz \n", +"Ta = Pa/(2*%pi*f) // Torque in N-m at A\n", +"Tb = Pb/(2*%pi*f) // Torque in N-m at B\n", +"Tc = Pc/(2*%pi*f) // Torque in N-m at B\n", +"Tab = Ta ; // Torque in N-m in shaft ab\n", +"Tbc = Tc ; // Torque in N-m in shaft bc\n", +"tab = (16*Tab)/(%pi*(d^3)) ; // shear stress in ab segment\n", +"fab = (Tab*Lab)/(G*Ip); // angle of twist in radian\n", +"tbc = (16*Tbc)/(%pi*(d^3)); // shear stress in ab segment\n", +"fbc = (Tbc*Lbc)/(G*Ip); // angle of twist in radian\n", +"fac = (fab+fbc)*(180/%pi); // angle of twist in degree in segment ac\n", +"tmax = Tab; // Maximum shear stress\n", +"disp('Nm',tmax,'The maximum shear stress tmax in the shaft')\n", +"disp('degree',fac,'Angle of twist in segment AC')" + ] + } +], +"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 +} diff --git a/Mechanics_Of_Material_by_J_M_Gere/4-Shear_Forces_and_Bending_Moments.ipynb b/Mechanics_Of_Material_by_J_M_Gere/4-Shear_Forces_and_Bending_Moments.ipynb new file mode 100644 index 0000000..20a910d --- /dev/null +++ b/Mechanics_Of_Material_by_J_M_Gere/4-Shear_Forces_and_Bending_Moments.ipynb @@ -0,0 +1,84 @@ +{ +"cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Chapter 4: Shear Forces and Bending Moments" + ] + }, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 4.3: Calculation_of_the_shear_force_and_the_bending_moment_of_the_cross_section.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"q = 0.2 ; // Uniform load intensity in K/ft\n", +"P = 14 ; // Concentrated load in k\n", +"Ra = 11 ; // Reaction at A from wquation of equilibrium\n", +"Rb = 9 ; // Reaction at B from wquation of equilibrium\n", +"V = 11 - 14 - (0.2*15) ; // shear force in k\n", +"disp('k',V,'Shear force at section D')\n", +"M = (11*15)-(14*6)-(0.2*15*7.5) ; // Bending moment in K-ft\n", +"disp('k-ft',M,'Bending moment at section D')\n", +"V1 = -9+(0.2*15); // Shear firce from alternative method in k\n", +"M1 = (9*9)-(0.2*7.5*15); // Bending moment from alternative method in k-ft" + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 4.7: Shear_force_nd_bending_moment_diagramme.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"q = 1 ; // Uniform load intensity in k/ft\n", +"M0 = 12 ; // Couple in k-ft\n", +"Rb = 5.25 ; // Reaction at B in k\n", +"Rc = 1.25 ; // Reaction at C in k\n", +"b = 4 ; // Length of section AB in ft\n", +"Mb = -(q*(b^2))/2 ; // Moment acting at B\n", +"disp('k-ft',Mb,'Bending moment at B')" + ] + } +], +"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 +} diff --git a/Mechanics_Of_Material_by_J_M_Gere/5-Stresses_in_Beams_Basic_Topics.ipynb b/Mechanics_Of_Material_by_J_M_Gere/5-Stresses_in_Beams_Basic_Topics.ipynb new file mode 100644 index 0000000..0ed22ea --- /dev/null +++ b/Mechanics_Of_Material_by_J_M_Gere/5-Stresses_in_Beams_Basic_Topics.ipynb @@ -0,0 +1,521 @@ +{ +"cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Chapter 5: Stresses in Beams Basic Topics" + ] + }, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 5.11: Determination_of_the_normal_stress_and_shear_stress_at_point_C.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"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", +"// Calculations from chapter 4\n", +"Mc = 17920 ; // Bending moment in ld-in\n", +"Vc = -1600 ; // Loading in lb\n", +"//\n", +"I = (b*(h^3))/12; // 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 ; // distance 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", +"disp('psi',sc,'Normal stress at C')\n", +"disp('psi',tc,'Shear stress at C')" + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 5.12: Determination_of_the_maximum_permissible_value_Pmax_of_the_loads.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"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", +"P_bending = (s*b*h^2)/(6*a); // Bending stress in N\n", +"P_shear = (2*t*b*h)/3; // shear stress in N\n", +"Pmax = P_bending; // Because bending stress governs the design\n", +"disp('N',Pmax,'the maximum permissible value Pmax of the loads')" + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 5.13: EX5_13.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"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", +"// Part (a)\n", +"t_max = ((r2^2+(r2*r1)+r1^2)*4*P)/(3*%pi*((r2^4)-(r1^4))) ; // Mximum shear stress in Psi\n", +"disp('psi',t_max,'Maximum shear stress in the pole is')\n", +"// Part (b)\n", +"d0 = sqrt((16*P)/(3*%pi*t_max)) ; // Diameter of solid circular cross section in meter\n", +"disp('m',d0,'Diameter of solid circular cross section is ')" + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 5.14: EX5_14.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"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; // Vertical force in N\n", +"I = (1/12)*((b*(h^3))-(b*(h1^3))+(t*(h1^3))) // Moment of inertia of the cros section\n", +"t_max = (V/(8*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)*(2*t_max + t_min); // Total shear force in Pa\n", +"t_avg = V/(t*h1) ; // Average shear stress in Pa\n", +"disp('Pa',t_max,'Maximum shear stress in the web is')\n", +"disp('Pa',t_min,'Minimum shear stress in the web is')\n", +"disp('Pa',T,'Total shear stress in the web is')" + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 5.15: EX5_15.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"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", +"A = b*(h-h1) + t*h1 ;// Area of cross section \n", +"Qaa = ((h+h1)/2)*b*(h-h1) + (h1/2)*(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 - ((b-t)*h1^3)/3 ; // 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)) ; // 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); // Maximum first moment of inertia below neutral axis\n", +"t_max = (V*Qmax)/(I*t); // Maximum Shear stress in Psi\n", +"disp('psi',t1,'Shear stress at the top of the web is')\n", +"disp('Psi',t_max,'Maximum Shear stress in the web is')\n", +"\n", +" " + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 5.16: determination_of_the_maximum_permissible_longitudinal_spacing_of_the_screws.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"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 ; // Distance between centroid of flange and neutral axis in mm\n", +"Q = Af*df ; // First moment of cross section of flange\n", +"I = (1/12)*(210*280^3) - (1/12)*(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", +"disp('mm',s,'The maximum permissible longitudinal spacing s of the screws is')" + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 5.17: EX5_17.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"L = 60 ; // Length of beam in inch\n", +"d = 5.5 ; // distance 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*sind(60); // Horizontal component\n", +"Pv = P*cosd(60); // Vertical component\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", +"disp('psi',sc_max,'The maximum compressive stress in the beam is')\n", +"disp('psi',st_max,'The maximum tensile stress in the beam is')" + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 5.1: Determination_of_radius_of_curvature_and_deflection_in_a_simply_supported_beam.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"L = 8 ; // length of beam in ft\n", +"h = 6 ; // Height of beam in inch\n", +"e = 0.00125 ; // elongation on the bottom surface of the beam\n", +"y = -3 ; // Distance of the bottom surface to the neutral surface of the beam in inch\n", +"r = -(y/e) ; // Radius of curvature\n", +"disp('ft',r,'radius of curvature is')\n", +"k = 1/r ; // curvature in in-1\n", +"disp('ft-1',k,'curvature')\n", +"theta = asind((L*12)/(2*r)) ; // angle in degree\n", +"disp('degree',theta,'Angle of twist')\n", +"del = r*(1-cosd(theta)); //Deflection in inch\n", +"\n", +"\n", +"" + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 5.2: EX5_2.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"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", +"M = (%pi*E*d^4)/(32*(2*R0+d)) ; // Bending moment in wire in N-m\n", +"disp('N-m',M,'Bending moment in the wire is ')\n", +"s_max = (E*d)/(2*R0+d) ; // Maximum bending stress in wire in Pa\n", +"disp('Pa',s_max,'Maximum bending stress in the wire is ')" + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 5.3: EX5_3.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"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", +"S = (b*h^2)/6 ; // Section modulus\n", +"s = (Mmax*12)/S // stress in k\n", +"st = s*1000 ; // Tensile stress\n", +"disp('psi',st,'Maximum tensile stress in the beam')\n", +"sc = -s*1000 ; // Compressive stress\n", +"disp('psi',sc,'Maximum compressive stress in the beam')" + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 5.4: EX5_4.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"q = 3200 ; // 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 ; // Reaction at A in N\n", +"Rb = 10800 ; // Reaction at B in N\n", +"Mpos = 2025 ; // Moment in Nm\n", +"Mneg = -3600 ; // Moment in Nm\n", +"y1 = t/2;\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)*(1/12);\n", +"d1 = c1-(t/2);\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", +"// Section Modulli\n", +"S1 = Iz / c1 ; // for the top surface\n", +"S2 = Iz / c2 ; // for the bottom surface\n", +"// Maximum stresses for the positive section\n", +"st = Mpos / S2 ;\n", +"disp('Pa',st,'Maximum tensile stress in the beam in positive section is')\n", +"sc = -Mpos / S1 ;\n", +"disp('Pa',sc,'Maximum compressive stress in the beam in positive section is')\n", +"// Maximum stresses for the negative section\n", +"snt = -Mneg / S1 ;\n", +"disp('Pa',snt,'Maximum tensile stress in the beam in negative section is')\n", +"snc = Mneg / S2 ;\n", +"disp('Pa',snc,'Maximum compressive stress in the beam in negative section is')\n", +"// Conclusion\n", +"st_max = st;\n", +"sc_max = snc ;" + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 5.5: Selection_of_the_suitable_size_for_the_beam.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"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", +"M = (q*L^2*12)/8 ; // Bending moment in lb-in\n", +"S = M/s ; // Section Modulli in in3\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", +"// From reference to appendix F, a beam of cross section 3*12 inch is selected\n", +"disp('Beam of crosssection 3*12 is sufficient')" + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 5.6: Calculation_of_minimum_required_diameter_in_the_wood_and_alluminium_rod.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"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", +"// Part (a) : Wood Post\n", +"s1 = 15e06 ; // Maximum allowable stress in Pa\n", +"S1 = Mmax/s1 ;// Section Modulli in m3\n", +"d1 = ((32*S1)/%pi)^(1/3); // diameter in m\n", +"disp('m',d1,'the minimum required diameter d1 of the wood post is')\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)^(1/3); // diameter in meter.....(1) \n", +"// Here equation (1) , comes from solving following three equation \n", +"// c = d2/2 (radius of tube)\n", +"// I2 = (%pi/64)*((d2^4)-((0.75*d2)^4)) (Moment of inertia)\n", +"// S2 = I2/c ;\n", +"disp('m',d2,'minimum required outer diameter d2 of the aluminum tube is')" + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 5.7: Selection_of_the_steel_beam.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"q = 2000 ; // Uniform load intensity in lb/ft\n", +"s = 18000 ; // Maximum allowable load in Psi\n", +"Ra = 18860 ; // Reaction at point A\n", +"Rb = 17140 ; // Reaction at point B\n", +"x1 = Ra/q ; // Distance in ft from left end to the point of zero shear\n", +"Mmax = (Ra*x1)-((q*(x1^2))/2) ; // Maximum bending moment in lb-ft\n", +"S = (Mmax*12)/s; // Section Modulli in in3\n", +"// Trial Beam\n", +"Ra_t = 19380 ; // Reaction at point A\n", +"Rb_t = 17670 ; // Reaction at point B\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) ; // Maximum bending moment in lb-ft\n", +"S_t = (Mmax_t*12)/s; // Section Modulli in in3\n", +"// From table E beam 12*50 is selected \n", +"disp('in3',S_t,'Beam of crosssection 12*50 is selected with section modulli')" + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 5.8: Determination_of_the_minimum_required_dimension_b_of_the_posts.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"g = 9810 ; // Specific weight of water in N/m3\n", +"h = 2; // Height of dam in m\n", +"s = 0.8 ; // Distance between square cross section in m\n", +"sa = 8e06 ; // Maximum allowable stress in Pa\n", +"b = ((g*(h^3)*s)/sa)^(1/3) ; // Dimension of croossection in m\n", +"disp('m',b,'the minimum required dimension b of the posts')" + ] + } +], +"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 +} diff --git a/Mechanics_Of_Material_by_J_M_Gere/6-Stresses_in_Beams_Advanced_Topics.ipynb b/Mechanics_Of_Material_by_J_M_Gere/6-Stresses_in_Beams_Advanced_Topics.ipynb new file mode 100644 index 0000000..6246524 --- /dev/null +++ b/Mechanics_Of_Material_by_J_M_Gere/6-Stresses_in_Beams_Advanced_Topics.ipynb @@ -0,0 +1,301 @@ +{ +"cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Chapter 6: Stresses in Beams Advanced Topics" + ] + }, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 6.1: Calculation_of_stresses_in_wood_and_steel.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"// 4*6 inch wood beam dimension\n", +"// 4*0.5 inch steel beam dimension\n", +"M = 60 ; // Moment in k-in\n", +"E1 = 1500 ; // in Ksi\n", +"E2 = 30000; // in Ksi\n", +"h1 = 5.031 ; // Distance between top surface and neutral axis of the beam in inch by solving 1500*(h1-3)*24 + 30000*(h1-6.25)*2 = 0\n", +"h2 = 6.5 - h1 ;\n", +"I1 = (1/12)*(4*6^3) + (4*6)*(h1-3)^2 ; // Momeny of inertia of the wooden cross section\n", +"I2 = (1/12)*(4*0.5^3) + (4*0.5)*(h2-0.25)^2 ; // Momeny of inertia of the steel cross section\n", +"I = I1 + I2 ; // Moment of inertia of whole cross section\n", +"// Material 1\n", +"s1a = -(M*h1*E1)/((E1*I1)+(E2*I2)) ; // Maximum compressive stress in ksi where y = h1\n", +"s1c = -(M*(-(h2-0.5))*E1)/((E1*I1)+(E2*I2)) ; // Maximum tensile stress in ksi where y = -(h2-0.5)\n", +"disp('ksi',s1a,' Maximum compressive stress in wood is')\n", +"disp('ksi',s1c,' Maximum tensile stress in wood is')\n", +"// Material 2\n", +"s2a = -(M*(-h2)*E2)/((E1*I1)+(E2*I2)); // Maximum tensile stress in ksi where y = -h2\n", +"s2c = -(M*(-(h2-0.5))*E2)/((E1*I1)+(E2*I2)); // Minimum tensile stress in ksi where y = -(h2-0.5)\n", +"disp('ksi',s2a,' Maximum tensile stress in steel is')\n", +"disp('ksi',s2c,' Minimum tensile stress in steel is')" + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 6.2: EX6_2.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"M = 3000 ; // moment in N-m\n", +"t = 0.005 ; // thickness of alluminiun in m\n", +"E1 = 72e09 ; // Modulus of elasticity of alluminium in Pa\n", +"E2 = 800e06 ; // Modulus of elasticity of Plastic core in Pa\n", +"b = 0.2 ; // Width of cross section in m\n", +"h = 0.160 ; // Height of cross section in m\n", +"hc = 0.150 ; // Height of Plastic core cross section in m\n", +"I1 = (b/12)*(h^3 - hc^3) ; // Moment of inertia of alluminium cross section\n", +"I2 = (b/12)*(hc^3) ; // Moment of inertia of Plastic core cross section\n", +"f = (E1*I1) + (E2*I2) ; // Flexural rigidity of the cross section\n", +"s1_max = (M*(h/2)*E1)/f ;\n", +"s1c = -s1_max ; // Maximum compressive stress in alluminium core in Pa\n", +"s1t = s1_max ; // Maximum tensile stress in alluminium core in Pa\n", +"disp('Pa',s1c,' Maximum compressive stress on alluminium face by the general theory for composite beams is')\n", +"disp('Pa',s1t,' Maximum tensile stress on alluminium face by the general theory for composite beams is')\n", +"s2_max = (M*(hc/2)*E2)/f ;\n", +"s2c = -s2_max ; // Maximum compressive stress in Plastic core in Pa\n", +"s2t = s2_max ; // Maximum tensile stress in Plastic core in Pa\n", +"disp('Pa',s2c,' Maximum compressive stress in plastic core by the general theory for composite beams is')\n", +"disp('Pa',s2t,' Maximum tensile stress in plastic core by the general theory for composite beams is')\n", +"// Part (b) : Calculation from approximate theory of sandwitch\n", +"s1_max1 = (M*h)/(2*I1) ;\n", +"s1c1 = -s1_max1 ; // Maximum compressive stress in alluminium core in Pa\n", +"s1t1 = s1_max1 ; // Maximum tensile stress in alluminium core in Pa\n", +"disp('Pa',s1c1,' Maximum compressive stress on alluminium core by approximate theory of sandwitch is')\n", +"disp('Pa',s1t1,' Maximum tensile stress on alluminium core by approximate theory of sandwitch is')" + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 6.3: Calculation_of_stresses_in_wood_and_steel.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"// 4*6 inch wood beam dimension\n", +"// 4*0.5 inch steel beam dimension\n", +"M = 60 ; // Moment in k-in\n", +"E1 = 1500 ; // in Ksi\n", +"E2 = 30000; // in Ksi\n", +"b = 4; // width of crosssection in inch\n", +"// Transformed Section\n", +"n = E2/E1 ; // Modular ratio\n", +"b1 = n*4 ; // Increased width of transformed cross section\n", +"// Neutral axis\n", +"h1 = ((3*4*6)+(80*0.5*6.25))/((4*6)+(80*0.5)); // Distance between top surface and neutral axis of the beam in inch\n", +"h2 = 6.5 - h1 ; // in inch\n", +"// Moment of inertia\n", +"It = (1/12)*(4*6^3) + (4*6)*(h1-3)^2 + (1/12)*(80*0.5^3) + (80*0.5)*(h2-0.25)^2 ; // Moment of inertia of transformed cross section\n", +"// Material 1\n", +"s1a = -(M*h1)/It; // Maximum tensile stress in ksi where y = h1\n", +"s1c = -(M*(-(h2-0.5)))/It; // Maximum compressive stress in ksi where y = -(h2-0.5)\n", +"disp('psi',s1a*1000,'Maximum tensile stress in wood is')\n", +"disp('psi',s1c*1000,'Maximum compressive stress in wood is')\n", +"// Material 2\n", +"s2a = -(M*(-h2)*n)/It ; // Maximum tensile stress in ksi where y = -h2\n", +"s2c = -(M*(-(h2-0.5)*n))/It ; // Minimum tensile stress in ksi where y = -(h2-0.5)\n", +"disp('psi',s2a*1000,' Maximum tensile stress in steel')\n", +"disp('psi',s2c*1000,' Minimum tensile stress in steel')" + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 6.4: EX6_4.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"q = 3000 ; // Uniform load intensity in N/m\n", +"a = 26.57 ; // tilt of the beam in degree\n", +"b = 0.1; // width of the beam\n", +"h = 0.15; // height of the beam\n", +"L = 1.6 ; // Span of the beam\n", +"qy = q*cosd(a) ; // Component of q in y direction\n", +"qz = q*sind(a) ; // Component of q in z direction\n", +"My = (qz*L^2)/8 ; // Maximum bending moment in y direction\n", +"Mz = (qy*L^2)/8 ; // Maximum bending moment in z direction\n", +"Iy = (h*b^3)/12; // Moment of inertia along y\n", +"Iz = (b*h^3)/12; // Moment of inertia alon z\n", +"s = ((3*q*L^2)/(4*b*h))*((sind(a)/b)+(cosd(a)/h));\n", +"sc = -s ; // Maximum compressive stress\n", +"st = s; // Maximum tensile stress\n", +"disp('Pa',sc,'Maximum compressive stress in the beam is')\n", +"disp('Pa',st,'Maximum tensile stress in the beam is')\n", +"// Neutral axis\n", +"l = (h/b)^2;\n", +"t = sind(a)/cosd(a);\n", +"j = l*(sind(a)/cosd(a));\n", +"be = atand(j); // Inclination of Neutral axis to z axis\n", +"disp('degree',be,'Inclination of Neutral axis to z axis is')" + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 6.5: Determination_of_the_maximum_bending_stresses_in_the_beam.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"L = 12 ; // Length of the beam in ft\n", +"P = 10 ; // Load in k acting in vertical direction\n", +"//Part (a)\n", +"h = 24 ; // Height of beam in inch\n", +"Iz = 2100 ; // Moment of inertia along z axis in in4\n", +"Iy = 42.2 ; // Moment of inertia along y axis in in4\n", +"s_max = (P*(h/2)*L*12)/Iz ; // Maximum stress in Ksi\n", +"disp('psi',s_max*1000,'Maximum tensile stress in the beam at the top of the beam')\n", +"disp('psi',-s_max*1000,'Maximum compressive stress in the beam at the bottom of the beam')\n", +"//Part (b)\n", +"a = 1 ; // Angle between y axis and the load\n", +"My = -(P*sind(a))*L*12 ; // Moment along y-axis in K-in\n", +"Mz = -(P*cosd(a))*L*12 ; // Moment along z-axis in K-in\n", +"ba = atand((My*Iz)/(Mz*Iy)); // Orientation of neutral axis\n", +"z = -3.5; y = 12 ; // Coordinates of the point A and B where maximum stress occur\n", +"s = ((My*z)/Iy)-((Mz*y)/Iz) ; // Stress in Ksi\n", +"sa = s ; // Tensile stress at A\n", +"sb = -s ; // Compressive stress in B\n", +"disp('psi',sa*1000,'The tensile stress at A is')\n", +"disp('psi',sb*1000,'The compressive stress at B is')\n", +"\n", +"\n", +"\n", +"\n", +"" + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 6.6: Calculation_of_the_bending_stresses_and_location_of_neutral_axis.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"M = 15 ; // Bending moment in k-in\n", +"t = 10 ; // Angle between line of action of moment and z-axis\n", +"// Properties of cross section\n", +"c = 0.634 ; // Location of centroid on the axis of symmetry\n", +"Iy = 2.28; // Moment of inertia in y-direction in in4\n", +"Iz = 67.4; // Moment of inertia in z-direction in in4\n", +"ya = 5 ; za = -2.6+0.634 ; // Coordinates of point A\n", +"yb = -5 ; zb = 0.634 ; // Coordinates of point B\n", +"My = M*sind(t); // Moment along y-axis\n", +"Mz = M*cosd(t); // Moment along z-axis\n", +"sa = ((My*za)/Iy)-((Mz*ya)/Iz) ; // Bending stress at point A in ksi\n", +"sb = ((My*zb)/Iy)-((Mz*yb)/Iz) ; // Bending stress at point B in ksi\n", +"disp('psi',sa*1000,'The bending stress at point A is')\n", +"disp('psi',sb*1000,'The bending stress at point B is')\n", +"// Neutral axis\n", +"j = (Iz/Iy)*(sind(t)/cosd(t)); \n", +"be = atand(j); // Inclination of neutral axis to z-axis in degree\n", +"disp('degree',be,'Inclination of neutral axis to z-axis is')\n", +"\n", +" " + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 6.9: Determination_of_the_magnitude_of_the_moment.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"b = 5 ; // in inch\n", +"b1 = 4 ; // in inch\n", +"h = 9 ; // in inch\n", +"h1 = 7.5 ; // in inch\n", +"sy = 33 ; // stress along y axis in ksi\n", +"M = (sy/12)*((3*b*h^2)-(b+(2*b1))*(h1^2)) ; // Bending moment acting in k-in\n", +"disp('k-in',M,'the magnitude of the moment M is')" + ] + } +], +"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 +} diff --git a/Mechanics_Of_Material_by_J_M_Gere/7-Analysis_of_Stress_and_Strain.ipynb b/Mechanics_Of_Material_by_J_M_Gere/7-Analysis_of_Stress_and_Strain.ipynb new file mode 100644 index 0000000..a458a37 --- /dev/null +++ b/Mechanics_Of_Material_by_J_M_Gere/7-Analysis_of_Stress_and_Strain.ipynb @@ -0,0 +1,315 @@ +{ +"cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Chapter 7: Analysis of Stress and Strain" + ] + }, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 7.1: Determination_of_the_stresses_acting_on_an_inclined_element.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"// Let x1, y1 be the transformed direction inclined at 45 deegree to the original\n", +"sx = 16000; // Direct stress in x-direction in psi\n", +"sy = 6000; // Direct stress in y-direction ''\n", +"txy = 4000; // Shear stress in y-direction ''\n", +"tyx = txy ; // Shear stress in x-direction ''\n", +"t = 45 ; // Inclination pf plane in degree \n", +"sx1 = (sx+sy)/2 + ((sx-sy)*(cosd(2*t))/2) + txy*sind(2*t); // Direct stress in x1-direction in psi\n", +"sy1 = (sx+sy)/2 - ((sx-sy)*(cosd(2*t))/2) - txy*sind(2*t); // Direct stress in y1-direction in psi\n", +"tx1y1 = - ((sx-sy)*(sind(2*t))/2) + txy*cosd(2*t) // Shear stress in psi\n", +"disp('psi',sx1,'The direct stress on the element in x1-direction is')\n", +"disp('psi',sy1,'The direct stress on the element in y1-direction is')\n", +"disp('psi',tx1y1,'The shear stress on the element')\n", +"\n", +"\n", +"" + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 7.2: Determination_of_stresses_acting_on_inclined_element.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"// Let x1, y1 be the transformed direction inclined at 15 deegree to the original\n", +"sx = -46e06; // Direct stress in x-direction in Pa\n", +"sy = 12e06; // Direct stress in y-direction ''\n", +"txy = -19e06; // Shear stress in y-direction ''\n", +"t = -15 ; // Inclination of plane in degree \n", +"sx1 = (sx+sy)/2 + ((sx-sy)*(cosd(2*t))/2) + txy*sind(2*t) // Direct stress in x1-direction in Pa\n", +"sy1 = (sx+sy)/2 - ((sx-sy)*(cosd(2*t))/2) - txy*sind(2*t) // Direct stress in y1-direction in Pa\n", +"tx1y1 = - ((sx-sy)*(sind(2*t))/2) + txy*cosd(2*t) // Shear stress in Pa\n", +"disp('Pa',sx1,'The direct stress on the element in x1-direction is')\n", +"disp('Pa',sy1,'The direct stress on the element in y1-direction is')\n", +"disp('Pa',tx1y1,'The shear stress on the element')\n", +"" + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 7.3: Determination_of_stresses_acting_on_inclined_element.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"sx = 90e06; // Direct stress in x-direction in Pa\n", +"sy = 20e06; // Direct stress in y-direction in Pa\n", +"t = 30 ; // Inclination of element in degree\n", +"savg = (sx+sy)/2 ; // Average in-plane direct stress\n", +"txy = 0 ;\n", +"R = sqrt(((sx-sy)/2)^2+(txy)^2) // Radius of mohr circle\n", +"// Point D ; at 2t = 60\n", +"sx1 = savg + R*cosd(2*t) ; // Direct stress at point D \n", +"tx1y1 = -R*sind(2*t) ; // shear stress at point D\n", +"disp('Pa',sx1,'The direct stress at point D is')\n", +"disp('Pa',tx1y1,'The shear stress at point D is')\n", +"// Point D' ; at 2t = 240\n", +"sx2 = savg + R*cosd(90 + t); // Direct stress at point D \n", +"tx2y2 = R*sind(90 + t); // shear stress at point D\n", +"disp('Pa',sx2,'The direct stress at point D_desh is')\n", +"disp('Pa',tx2y2,'The shear stress at point D_desh is')\n", +"\n", +"" + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 7.4: Determination_of_stresses_acting_on_inclined_element_using_mohrs_circle.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"sx = 90e06; // Direct stress in x-direction in Pa\n", +"sy = 20e06; // Direct stress in y-direction in Pa\n", +"t = 30 ; // Inclination of element in degree\n", +"savg = (sx+sy)/2 ; // Average in-plane direct stress\n", +"txy = 0 ;\n", +"R = sqrt(((sx-sy)/2)^2+(txy)^2) // Radius of mohr circle\n", +"// Point D ; at 2t = 60\n", +"sx1 = savg + R*cosd(2*t) ; // Direct stress at point D \n", +"tx1y1 = -R*sind(2*t) ; // shear stress at point D\n", +"// Point D ; at 2t = 240\n", +"sx2 = savg + R*cosd(90 + t); // Direct stress at point D \n", +"tx2y2 = R*sind(90 + t); // shear stress at point D\n", +"\n", +"" + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 7.5: Determination_of_stresses_acting_on_inclined_element_using_Mohrs_circle.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"sx = 15000; // Direct stress in x-direction in psi\n", +"sy = 5000; // Direct stress in y-direction ''\n", +"txy = 4000 ; // Shear stress in y-direction ''\n", +"savg = (sx+sy)/2 ; // Average in-plane direct stress\n", +"sx1 = 15000; tx1y1 = 4000; // Stress acting on face at theta = 0 degree\n", +"sx1_ = 5000; tx1y1_ = -4000; // Stress acting on face at theta = 0 degree\n", +"R = sqrt(((sx-sy)/2)^2+(txy)^2) // Radius of mohr circle\n", +"// Part (a)\n", +"t = 40 ; // Inclination of the plane in degree\n", +"f1 = atand(4000/5000) ; // Angle between line CD and x1-axis\n", +"f2 = 80 - f1 ; // Angle between line CA and x1-axis\n", +"// Point D ; \n", +"sx1 = savg + R*cosd(f2); // Direct stress at point D \n", +"tx1y1 = -R*sind(f2); // shear stress at point D\n", +"disp('psi',sx1,'The direct stres at point D')\n", +"disp('psi',tx1y1,'The shear stress at point D')\n", +"// Point D' ; \n", +"sx2 = savg - R*cosd(f2) // Direct stress at point D' \n", +"tx2y2 = R*sind(f2) // shear stress at point D'\n", +"disp('psi',sx2,'The direct stres at point D_desh')\n", +"disp('psi',tx2y2,'The shear stress at point D_desh')\n", +"//Part (b)\n", +"sp1 = savg + R ; // Maximum direct stress in mohe circle (at point P1)\n", +"tp1 = f1/2 ; // Inclination of plane of maximum direct stress\n", +"disp('degree',tp1,'with angle','psi',sp1,'The maximum direct stress at P1 is ')\n", +"sp2 = savg - R ; // Minimum direct stress in mohe circle (at point P2)\n", +"tp2 = (f1+180)/2 ; // Inclination of plane of minimum direct stress\n", +"disp('degree',tp2,'with angle','psi',sp2,'The maximum direct stress at P2 is ')\n", +"// Part (c)\n", +"tmax = R ; // Maximum shear stress in mohe circle\n", +"ts1 = -(90 - f1)/2 // Inclination of plane of maximum shear stress\n", +"disp('degree',ts1,'with plane incilation of','psi',tmax,'The Maximum shear stress is ')\n", +"\n", +"\n", +"\n", +"" + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 7.6: Determination_of_stresses_acting_on_inclined_element_using_mohrs_circle.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"sx = -50e06; // Direct stress in x-direction in psi\n", +"sy = 10e06; // Direct stress in y-direction ''\n", +"txy = -40e06 ; // Shear stress in y-direction ''\n", +"savg = (sx+sy)/2 ; // Average in-plane direct stress\n", +"sx1 = -50e06; tx1y1 = -40e06; // Stress acting on face at theta = 0 degree\n", +"sx1_ = 10e06; tx1y1_ = 40e06; // Stress acting on face at theta = 0 degree\n", +"R = sqrt(((sx-sy)/2)^2+(txy)^2); // Radius of mohr circle\n", +"// Part (a)\n", +"t = 45 ; // Inclination of the plane in degree\n", +"f1 = atand(40e06/30e06) // Angle between line CD and x1-axis\n", +"f2 = 90 - f1 ; // Angle between line CA and x1-axis\n", +"// Point D ; \n", +"sx1 = savg - R*cosd(f2); // Direct stress at point D \n", +"tx1y1 = R*sind(f2); // shear stress at point D\n", +"disp('Pa',sx1,'The direct stres at point D')\n", +"disp('Pa',tx1y1,'The shear stress at point D')\n", +"// Point D' ; \n", +"sx2 = savg + R*cosd(f2); // Direct stress at point D' \n", +"tx2y2 = -R*sind(f2); // shear stress at point D'\n", +"disp('Pa',sx2,'The direct stres at point D_desh')\n", +"disp('Pa',tx2y2,'The shear stress at point D_desh')\n", +"//Part (b)\n", +"sp1 = savg + R ; // Maximum direct stress in mohe circle (at point P1)\n", +"tp1 =(f1+180)/2 ; // Inclination of plane of maximum direct stress\n", +"disp('degree',tp1,'with angle','Pa',sp1,'The maximum direct stress at P1 is ')\n", +"sp2 = savg - R ; // Minimum direct stress in mohe circle (at point P2)\n", +"tp2 = f1/2 ; // Inclination of plane of minimum direct stress\n", +"disp('degree',tp2,'with angle','Pa',sp2,'The maximum direct stress at P2 is ')\n", +"// Part (c)\n", +"tmax = R ; // Maximum shear stress in mohe circle\n", +"ts1 = (90 + f1)/2 ;// Inclination of plane of maximum shear stress\n", +"disp('degree',ts1,'with plane incilation of','Pa',tmax,'The Maximum shear stress is ')\n", +"\n", +"\n", +"\n", +"" + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 7.7: Determination_of_various_strain_on_inclined_element.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"ex = 340e-06; // Strain in x-direction\n", +"ey = 110e-06; // Strain in y-direction\n", +"txy = 180e-06 ; // shear strain\n", +"// Part (a)\n", +"t = 30 ; // Inclination of the element in degree\n", +"ex1 = (ex+ey)/2 + ((ex-ey)/2)*cosd(2*t) + (txy/2)*(sind(2*t)); // Strain in x1 direction (located at 30 degree)\n", +"tx1y1 = 2*( -((ex-ey)/2)*sind(2*t) + (txy/2)*(cosd(2*t)) ); // Shear starin\n", +"ey1 = ex+ey-ex1 ; // Strain in y1 direction (located at 30 degree)\n", +"disp(ex1,' Strain in x1 direction (located at 30 degree) is')\n", +"disp(tx1y1,'shear strain is')\n", +"disp(ey1,' Strain in y1 direction (located at 30 degree) is')\n", +"// Part (b)\n", +"e1 = (ex+ey)/2 + sqrt(((ex-ey)/2)^2 + (txy/2)^2); // Principle stress\n", +"e2 = (ex+ey)/2 - sqrt(((ex-ey)/2)^2 + (txy/2)^2); // Principle stress\n", +"tp1 = (0.5)*atand(txy/(ex-ey)); // Angle to principle stress direction\n", +"tp2 = 90 + tp1 ; // Angle to principle stress direction\n", +"e1 = (ex+ey)/2 + ((ex-ey)/2)*cosd(2*tp1) + (txy/2)*(sind(2*tp1)); // Principle stress via another method\n", +"e2 = (ex+ey)/2 + ((ex-ey)/2)*cosd(2*tp2) + (txy/2)*(sind(2*tp2)); // Principle stress via another method\n", +"disp('degree',tp1,'with angle',e1,'The Principle stress is ')\n", +"disp('degree',tp2,'with angle',e2,'The Principle stress is ')\n", +"// Part (c)\n", +"tmax = 2*sqrt(((ex-ey)/2)^2 + (txy/2)^2); // Maxmum shear strain\n", +"ts = tp1 + 45 ; // Orientation of element having maximum shear stress \n", +"tx1y1_ = 2*( -((ex-ey)/2)*sind(2*ts) + (txy/2)*(cosd(2*ts)) ); // Shear starin assosiated with ts direction\n", +"disp('degree',ts,'with angle',tx1y1_,'The Maximum shear strain is ')\n", +"eavg = (e1+e2)/2 ; // Average atrain\n", +"disp(eavg,'The average strain is')" + ] + } +], +"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 +} diff --git a/Mechanics_Of_Material_by_J_M_Gere/8-Applications_of_Plane_Stress_Pressure_Vessels_Beams_and_Combined_Loadings.ipynb b/Mechanics_Of_Material_by_J_M_Gere/8-Applications_of_Plane_Stress_Pressure_Vessels_Beams_and_Combined_Loadings.ipynb new file mode 100644 index 0000000..2c237cb --- /dev/null +++ b/Mechanics_Of_Material_by_J_M_Gere/8-Applications_of_Plane_Stress_Pressure_Vessels_Beams_and_Combined_Loadings.ipynb @@ -0,0 +1,352 @@ +{ +"cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Chapter 8: Applications of Plane Stress Pressure Vessels Beams and Combined Loadings" + ] + }, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 8.1: Calculation_of_maximum_permissible_pressure_under_various_conditions.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"d = 18 ; // inner idameter of the hemisphere in inch\n", +"t = 1/4 ; // thickness of the hemisphere in inch\n", +"// Part (a)\n", +"sa = 14000 ; // Allowable tensile stress in Psi\n", +"Pa = (2*t*sa)/(d/2); // Maximum permissible air pressure in Psi\n", +"disp('psi',Pa,' Maximum permissible air pressure in the tank (Part(a)) is')\n", +"// Part (b)\n", +"sb = 6000 ; // Allowable shear stress in Psi\n", +"Pb = (4*t*sb)/(d/2) ; // Maximum permissible air pressure in Psi\n", +"disp('psi',Pb,' Maximum permissible air pressure in the tank (Part(b)) is')\n", +"// Part (c)\n", +"e = 0.0003 ; // Allowable Strain in Outer sufrface of the hemisphere\n", +"E = 29e06 ; // Modulus of epasticity of the steel in Psi\n", +"v = 0.28 ; // Poissions's ratio of the steel\n", +"Pc = (2*t*E*e)/((d/2)*(1-v)) ; // Maximum permissible air pressure in Psi\n", +"disp('psi',Pc,' Maximum permissible air pressure in the tank (Part(c)) is')\n", +"// Part (d)\n", +"Tf = 8100 ; // failure tensile load in lb/in \n", +"n = 2.5 ; // Required factor of safetty against failure of the weld\n", +"Ta = Tf / n ; // Allowable load in ld/in \n", +"sd = (Ta*(1))/(t*(1)); // Allowable tensile stress in Psi\n", +"Pd = (2*t*sd)/(d/2); // Maximum permissible air pressure in Psi\n", +"disp('psi',Pd,' Maximum permissible air pressure in the tank (Part(d)) is')\n", +"// Part (e)\n", +"Pallow = Pb ; // Because Shear stress in the wall governs allowable pressure inside the tank\n", +"disp('Because Shear stress in the wall governs allowable pressure inside the tank','psi',Pallow,' Maximum permissible air pressure in the tank (Part(e)) is')\n", +"" + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 8.2: Calculation_of_various_stresses_and_strain_in_cylindrical_part_of_the_vessel.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"a = 55 ; // Angle made by helix with longitudinal axis in degree\n", +"r = 1.8 ; // Inner radius of vessel in m\n", +"t = 0.02 ; // thickness of vessel in m\n", +"E = 200e09 ; // Modulus of ealsticity of steel in Pa\n", +"v = 0.3 ; // Poission's ratio of steel \n", +"P = 800e03 ; // Pressure inside the tank in Pa\n", +"// Part (a)\n", +"s1 = (P*r)/t ; // Circumferential stress in Pa\n", +"s2 = (P*r)/(2*t) ; // Longitudinal stress in Pa\n", +"// Part (b)\n", +"t_max_z = (s1-s2)/2 ; // Maximum inplane shear stress in Pa\n", +"t_max = s1/2 ; // Maximum out of plane shear stress in Pa\n", +"// Part (c)\n", +"e1 = (s1/(2*E))*(2-v) ; // Strain in circumferential direction \n", +"e2 = (s2/E)*(1-(2*v)); // Strain in longitudinal direction\n", +"// Part (d)\n", +"// x1 is the direction along the helix\n", +"theta = 90 - a ; \n", +"sx1 = ((P*r)/(4*t))*(3-cosd(2*theta)); // Stress along x1 direction\n", +"tx1y1 = ((P*r)/(4*t))*(sind(2*theta)); // Shear stress in x1y1 plane\n", +"sy1 = s1+s2-sx1 ; // Stress along y1 direction \n", +"// Mohr Circle Method\n", +"savg = (s1+s2)/2 ; // Average stress in Pa\n", +"R = (s1 - s2 )/2 ; // Radius of Mohr's Circle in Pa\n", +"sx1_ = savg - R*cosd(2*theta) ; // Stress along x1 direction\n", +"tx1y1_ = R*sind(2*theta); // Shear stress in x1y1 plane\n", +" \n", +"\n", +"\n", +"" + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 8.3: EX8_3.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"L = 6 ; // Span of the beam in ft\n", +"P = 10800 ; // Pressure acting in lb\n", +"c = 2 ; // in ft\n", +"b = 2; // Width of cross section of the beam in inch\n", +"h = 6; // Height of the cross section of the beam in inch\n", +"x = 9 ; // in inch\n", +"Ra = P/3 ; // Reaction at point at A\n", +"V = Ra ; // Shear force at section mn \n", +"M = Ra*x ; // Bending moment at the section mn\n", +"I = (b*h^3)/12 // Moment of inertia in in4\n", +"y = -3:0.1:3 ; // Variation along height\n", +"sx = -(M/I)*y; // Normal stress on crossection mn\n", +"Q = (b*(h/2-y)).*(y+((((h/2)-y)/2))) ; // First moment of rectangular cross section\n", +"txy = (V*Q)/(I*b);// Shear stress acting on x face of the stress element\n", +"s1 = (sx/2)+sqrt((sx/2).^2+(txy).^2) ; // Principal Tesile stress on the cross section\n", +"s2 = (sx/2)-sqrt((sx/2).^2+(txy).^2) ; // Principal Compressive stress on the cross section\n", +"tmax = sqrt((sx/2).^2+(txy).^2); // Maximum shear stress on the cross section\n", +"plot(sx,y,'o')\n", +"plot(txy,y,'+')\n", +"plot(s1,y,'--')\n", +"plot(s2,y,'<')\n", +"plot(tmax,y)\n", +"disp('psi',s1,'Principal Tesile stress on the cross section')\n", +"disp('psi',s2,' Principal Compressive stress on the cross section')\n", +"// Conclusions \n", +"s1_max = 14400 ; // Maximum tensile stress in Psi\n", +"txy_max = 900 ; // Maximum shear stress in Psi\n", +"t_max = 14400/2 ; // Largest shear stress at 45 degree plane" + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 8.4: Determination_of_stresses_in_the_shaft.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"d = 0.05 ; // Diameter of shaft in m\n", +"T = 2400 ; // Torque transmitted by the shaft in N-m\n", +"P = 125000; // Tensile force\n", +"s0 = (4*P)/(%pi*d^2) // Tensile stress in\n", +"t0 = (16*T)/(%pi*d^3) // Shear force \n", +"// Stresses along x and y direction\n", +"sx = 0 ;\n", +"sy = s0; \n", +"txy = -t0 ; \n", +"s1 = (sx+sy)/2 + sqrt(((sx-sy)/2)^2 + (txy)^2) ; // Maximum tensile stress \n", +"s2 = (sx+sy)/2 - sqrt(((sx-sy)/2)^2 + (txy)^2) ; // Maximum compressive stress \n", +"tmax = sqrt(((sx-sy)/2)^2 + (txy)^2) ; // Maximum in plane shear stress \n", +"disp('Pa',s1,'Maximum tensile stress')\n", +"disp('Pa',s2,'Maximum compressive stress')\n", +"disp('Pa',tmax,'Maximum in plane shear stress')" + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 8.5: Determination_of_the_maximum_allowable_internal_pressure.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"P = 12 ; // Axial load in K\n", +"r = 2.1 ; // Inner radius of the cylinder in inch\n", +"t = 0.15 ; // Thickness of the cylinder in inch\n", +"ta = 6500 ; // Allowable shear stress in Psi\n", +"// From in plane sg=hear stress\n", +"p1 = (ta - 3032)/3.5 ; // allowable internal pressure\n", +"// Above equation comes from solving the following equation\n", +"// sx = (p*r)/(2*t) - (P)/(2*%pi*r*t) ;\n", +"// sy = (p*r)/t ;\n", +"// s1 = sy\n", +"// s2 = sx \n", +"// ta = (s1-s2)/2\n", +"\n", +"// From out of the plane shear stress\n", +"// ta = s1/2\n", +"p2 = (ta + 3032)/3.5 ; // allowable internal pressure\n", +"// ta = s2/2\n", +"p3 = 6500/7 ; // allowable internal pressure\n", +"\n", +"p_allow = min(p1,p2,p3); // Minimum pressure would govern the design\n", +"disp('Becausem inimum pressure would govern the design','psi',p_allow,'Maximum allowable internal pressure ')\n", +"\n", +"" + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 8.6: Determination_of_stresses_due_to_wind_pressure.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"d1 = 0.18 ; // Inner diameter of circular pole in m\n", +"d2 = 0.22 ; // Outer diameter of circular pole in m\n", +"P = 2000; // Pressure of wind in Pa\n", +"b = 1.5 ; // Distance between centre line of pole and board in m\n", +"h = 6.6 ; // Distance between centre line of board and bottom of the ploe in m\n", +"W = P*(2*1.2) ; // Force at the midpoint of sign \n", +"V = W ; // Load\n", +"T = W*b ; // Torque acting on the pole\n", +"M = W*h ; // Moment at the bottom of the pole\n", +"I = (%pi/64)*(d2^4-d1^4) ; // Momet of inertia of cross section of the pole\n", +"sa = (M*d2)/(2*I); // Tensile stress at A \n", +"Ip = (%pi/32)*(d2^4-d1^4) ; // Polar momet of inertia of cross section of the pole\n", +"t1 = (T*d2)/(2*Ip); // Shear stress at A and B\n", +"r1 = d1/2 ; // Inner radius of circular pole in m\n", +"r2 = d2/2 ; // Outer radius of circular pole in m\n", +"A = %pi*(r2^2-r1^2); // Area of the cross section\n", +"t2 = ((4*V)/(3*A))*((r2^2 + r1*r2 +r1^2)/(r2^2+r1^2)) ; // Shear stress at point B \n", +"// Principle stresses \n", +"sxa = 0 ; sya = sa ; txya = t1;\n", +"sxb = 0 ; syb = 0 ; txyb = t1+t2 ;\n", +"// Stresses at A\n", +"s1a = (sxa+sya)/2 + sqrt(((sxa-sya)/2)^2 + (txya)^2); // Maximum tensile stress \n", +"s2a = (sxa+sya)/2 - sqrt(((sxa-sya)/2)^2 + (txya)^2) ; // Maximum compressive stress \n", +"tmaxa = sqrt(((sxa-sya)/2)^2 + (txya)^2); // Maximum in plane shear stress\n", +"disp('Pa',s1a,'Maximum tensile stress at point A is')\n", +"disp('Pa',s2a,'Maximum compressive stress at point A is')\n", +"disp('Pa',tmaxa,'Maximum in plane shear stress at point A is')\n", +"// Stress at B \n", +"s1b = (sxb+syb)/2 + sqrt(((sxb-syb)/2)^2 + (txyb)^2); // Maximum tensile stress \n", +"s2b = (sxb+syb)/2 - sqrt(((sxb-syb)/2)^2 + (txyb)^2) ; // Maximum compressive stress \n", +"tmaxb = sqrt(((sxb-syb)/2)^2 + (txyb)^2); // Maximum in plane shear stress \n", +"disp('Pa',s1b,'Maximum tensile stress at point B is')\n", +"disp('Pa',s2b,'Maximum compressive stress at point B is')\n", +"disp('Pa',tmaxb,'Maximum in plane shear stress at point B is')\n", +"\n", +"" + ] + } +, +{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 8.7: Determination_of_stresses_due_to_loads.sce" + ] + }, + { +"cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], +"source": [ +"b = 6 ; // Outer dimension of the pole in inch\n", +"t = 0.5 ; // thickness of the pole\n", +"P1 = 20*(6.75*24); // Load acting at the midpoint of the platform\n", +"d = 9 ; // Distance between longitudinal axis of the post and midpoint of platform\n", +"P2 = 800; // Load in lb\n", +"h = 52 ; // Distance between base and point of action of P2\n", +"M1 = P1*d; // Moment due to P1\n", +"M2 = P2*h; // Moment due to P2\n", +"A = b^2 - (b-2*t)^2; // Area of the cross section\n", +"sp1 = P1/A ; // Comoressive stress due to P1 at A and B\n", +"I = (1/12)*(b^4 - (b-2*t)^4); // Moment of inertia of the cross section\n", +"sm1 = (M1*b)/(2*I);// Comoressive stress due to M1 at A and B\n", +"Aweb = (2*t)*(b-(2*t)); // Area of the web\n", +"tp2 = P2/Aweb ; // Shear stress at point B by lpad P2\n", +"sm2 = (M2*b)/(2*I);// Comoressive stress due to M2 at A \n", +"sa = sp1+sm1+sm2 ; // Total Compressive stress at point A\n", +"sb = sp1+sm1; // Total compressive at point B \n", +"tb = tp2; // Shear stress at point B\n", +"// Principle stresses \n", +"sxa = 0 ; sya = -sa ; txya = 0;\n", +"sxb = 0 ; syb = -sb ; txyb = tp2 ;\n", +"// Stresses at A\n", +"s1a = (sxa+sya)/2 + sqrt(((sxa-sya)/2)^2 + (txya)^2); // Maximum tensile stress \n", +"s2a = (sxa+sya)/2 - sqrt(((sxa-sya)/2)^2 + (txya)^2); // Maximum compressive stress \n", +"tmaxa = sqrt(((sxa-sya)/2)^2 + (txya)^2); // Maximum in plane shear stress\n", +"disp('Psi',s1a,'Maximum tensile stress at point A is')\n", +"disp('Psi',s2a,'Maximum compressive stress at point A is')\n", +"disp('Psi',tmaxa,'Maximum in plane shear stress at point A is')\n", +"// Stress at B \n", +"s1b = (sxb+syb)/2 + sqrt(((sxb-syb)/2)^2 + (txyb)^2); // Maximum tensile stress \n", +"s2b = (sxb+syb)/2 - sqrt(((sxb-syb)/2)^2 + (txyb)^2); // Maximum compressive stress \n", +"tmaxb = sqrt(((sxb-syb)/2)^2 + (txyb)^2); // Maximum in plane shear stress\n", +"disp('Psi',s1b,'Maximum tensile stress at point B is')\n", +"disp('Psi',s2b,'Maximum compressive stress at point B is')\n", +"disp('Psi',tmaxb,'Maximum in plane shear stress at point B is') \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 +} |