{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Examples 6 -- Structural Analysis \n", "\n", "## (Trusses, Frames & Machines)\n", "\n", "Contents:\n", "- [A. Simple 2D Truss](#A)\n", "- [B. Building modeled as 2D truss](#B)\n", "- [C. 2D Truss (method of sections)](#C)\n", "- [D. 2D Truss (sections again)](#D) \n", "- [E. 3D Truss (sections)](#E) \n", "- [F. Simple Frame](#F) \n", "- [G. Another Frame (pulley support)](#G) \n", "- [H. Yet Another Frame (pulley support)](#H) \n", "- [I. Machine (Garbage Truck)](#I) " ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "from numpy import *\n", "dtr = pi/180 # degree-to-radian conversion" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "## A. Simple 2D Truss\n", "\n", "" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "P = 1020.0 # lb\n", "X,Y = 5.7,7.6 # ft" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Analysis of whole structure.\n", "\n", "$\\sum M_A = 2XE_y-XP = 0$\n", "\n", "" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "answer:\n", " TAB= -510.0 lb\n", " TAC= 0.0 lb\n", " TBC= 637.5 lb\n", " TBD= -382.50000000000006 lb\n", " TCD= 637.5 lb\n", " TCE= 0.0 lb\n", " TDE= -510.0 lb\n" ] } ], "source": [ "# whole structure\n", "Ex = 0.\n", "Ey = P/2.\n", "Ay = P-Ey\n", "# point A\n", "TAC = 0. # zero force member\n", "TAB = -Ay\n", "# point E\n", "TCE = Ex # zero force member\n", "TDE = -Ey\n", "# point D\n", "alpha = arctan(X/Y)\n", "TCD = -TDE/cos(alpha)\n", "TBD = -TCD*sin(alpha)\n", "# point B\n", "TBC = -TBD/sin(alpha)\n", "# \n", "print ('answer:')\n", "print (' TAB=',TAB,'lb')\n", "print (' TAC=',TAC,'lb')\n", "print (' TBC=',TBC,'lb')\n", "print (' TBD=',TBD,'lb')\n", "print (' TCD=',TCD,'lb')\n", "print (' TCE=',TCE,'lb')\n", "print (' TDE=',TDE,'lb')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "## B. Building modeled as 2D truss\n", "\n", "\n", "" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [], "source": [ "Fy_top = 6.0 # kip\n", "Fy_mid = 7.0 # kip\n", "Fx_top = 5.0 # kip\n", "Fx_mid = 4.0 # kip\n", "X,Y = 32.,24. # ft" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "answer:\n", " TAB= -5.0 kip\n", " TBD= -6.0 kip\n", " TAD= 6.25 kip\n", " TAC= -9.75 kip\n", " TCD= -9.0 kip\n", " TDF= -9.25 kip\n", " TCF= 11.25 kip\n", " TCE= -23.5 kip\n" ] } ], "source": [ "F1,F2 = Fy_top,Fy_mid\n", "F3,F4 = Fx_top,Fx_mid\n", "alpha = arctan(Y/X)\n", "TCE = -F1-F2-(2*F3+F4)*tan(alpha)\n", "TCF = (F3+F4)/cos(alpha)\n", "TDF = -F1-F2+F3*tan(alpha)\n", "TCD = -F3-F4\n", "TAC = -F1-F3*tan(alpha)\n", "TAD = F3/cos(alpha)\n", "TAB = -F3\n", "TBD = -F1\n", "print ('answer:')\n", "print (' TAB=',TAB,'kip')\n", "print (' TBD=',TBD,'kip')\n", "print (' TAD=',TAD,'kip')\n", "print (' TAC=',TAC,'kip')\n", "print (' TCD=',TCD,'kip')\n", "print (' TDF=',TDF,'kip')\n", "print (' TCF=',TCF,'kip')\n", "print (' TCE=',TCE,'kip')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "## C. 2D Truss (method of sections)\n", "\n", "" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [], "source": [ "F1,F2,F3 = 4.,8.,16. # kN, top forces (left-to-right)\n", "F4 = 12. # kN, lower forces (both)\n", "X,Y = 70.,110. # cm" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [], "source": [ "# whole structure:\n", "Ay = F1+F2+F3\n", "Bx = ( X*F2 + 3*X*F3 + Y*F4 + 2*Y*F4)/Y\n", "Ax = 2*F4 - Bx" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Take the section *ABCD*:\n", "\n", "\n", "\n", "$\\sum M_D = -YA_x -2X(A_y-F_1)+XF_2-YT_{CE}=0$\n", "\n", "$\\sum F_y = A_y -F_1-F_2+T_{DE}\\tfrac{Y}{Z}=0$\n", "\n", "$\\sum F_x = A_x + T_{CE} +B_x+T_{DF}+T_{DE}\\tfrac{X}{Z}=0$" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "answer:\n", " TDE= -18.964952451498615 kN\n", " TCE= 22.181818181818183 kN\n", " TDF= -36.00000000000001 kN\n" ] } ], "source": [ "TCE = ( -Y*Ax - 2*X*(Ay-F1) + X*F2 )/Y\n", "Z = sqrt(X**2+Y**2)\n", "TDE = Z/Y*( -Ay + F1 +F2 )\n", "TDF = -Ax -TCE -Bx -TDE*X/Z\n", "print ('answer:')\n", "print (' TDE=',TDE,'kN')\n", "print (' TCE=',TCE,'kN')\n", "print (' TDF=',TDF,'kN')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "\n", "## D. 2D Truss (sections again)\n", "\n", "" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [], "source": [ "W = 420. # lb\n", "X = 4. # ft\n", "Y = 3. # ft" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "answer:\n", " TIJ= 0.0 lb\n", " TIL= -2800.0 lb\n", " TIK= -3360.0 lb\n", " THI= 3500.0 lb\n", " TGI= -8400.0 lb\n" ] } ], "source": [ "# whole structure\n", "Ay = 8*W\n", "Bx = -(8+7+6+5+4+3+2+1)*X*W/Y\n", "Ax = -Bx\n", "# section ABJI\n", "alpha = arctan(Y/X)\n", "TIL = (4*W-Ay)/sin(alpha)\n", "TJL = ( (3+2+1)*X*W - 4*X*Ay - Y*Bx )/Y\n", "TIK = -Ax - Bx - TJL - TIL*cos(alpha)\n", "TIJ = 0.\n", "# joint I\n", "THI = W/sin(alpha) - TIL\n", "TGI = TIK + (TIL-THI)*cos(alpha)\n", "print ('answer:')\n", "print (' TIJ=',TIJ,'lb')\n", "print (' TIL=',TIL,'lb')\n", "print (' TIK=',TIK,'lb')\n", "print (' THI=',THI,'lb')\n", "print (' TGI=',TGI,'lb')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "## E. 3D Truss (sections)\n", "\n", "\n", "\n", "Determine the force in member JG if $P = 16$ kN and $Q = 0$.\n", "\n", "Note this problem can be done on paper, as seen in the [solution here](http://madisoncollegephysics.net/232/ipy/images/notes06-5.pdf) (pdf)." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Shown below is another way to solve the problem, letting the code do the cross products and then using [Sympy](https://www.sympy.org/en/index.html) (a symbolic equation solver) to solve for the unknowns.\n", "\n", "If `sympy` is not installed on your system, you will have to run this:" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Defaulting to user installation because normal site-packages is not writeable\n", "Requirement already satisfied: sympy in /home/archie/.local/lib/python3.8/site-packages (1.6.2)\n", "Requirement already satisfied: mpmath>=0.19 in /home/archie/.local/lib/python3.8/site-packages (from sympy) (1.1.0)\n", "\u001b[33mWARNING: You are using pip version 20.1.1; however, version 20.2.3 is available.\n", "You should consider upgrading via the '/usr/bin/python3 -m pip install --upgrade pip' command.\u001b[0m\n", "Note: you may need to restart the kernel to use updated packages.\n" ] } ], "source": [ "pip install sympy" ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [], "source": [ "import sympy as s" ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [], "source": [ "P,Q = 16 , 0. # kN\n", "l = 2 # m" ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "moment about L=\n", "\t 8.0*Azs - 80.0 \n", "\t -1.0*Azs - 2.0*Kzs + 16.0 \n", "\t -8.0*Axs\n", "structure support reactions are \n", "\t Lz = 3.00 kN\n", "\t Kz = 3.00 kN\n", "\t Az = 10.00 kN\n" ] } ], "source": [ "# moment about L for whole structure\n", "ihat,jhat,khat = array((1.,0,0)),array((0,1.,0)),array((0,0,1.))\n", "rLK = l*ihat\n", "rLG = l/2*ihat + 2.5*l*jhat + l*khat\n", "rLA = l/2*ihat + 4*l*jhat\n", "Kzs,Azs,Axs = s.symbols('Kzs,Azs,Axs')\n", "eq = Kzs*cross(rLK,khat) + P*cross(rLG,-khat) \\\n", " + Azs*cross(rLA,khat) + Axs*cross(rLA,ihat) \n", "print ('moment about L=\\n\\t',eq[0],'\\n\\t',eq[1],'\\n\\t',eq[2])\n", "ss = s.solve( [eq[0],eq[1],eq[2]], [Kzs,Azs,Axs])\n", "Az,Kz = float(ss[Azs]),float(ss[Kzs])\n", "Lz = P-Kz-Az\n", "print('structure support reactions are ')\n", "print ('\\t Lz = {:.2f} kN'.format(Lz))\n", "print ('\\t Kz = {:.2f} kN'.format(Kz))\n", "print ('\\t Az = {:.2f} kN'.format(Az))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "See the [solution pdf](http://madisoncollegephysics.net/232/ipy/images/notes06-5.pdf) for the section diagram used below." ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "member forces are \n", "\t FJG = -12.00 kN\n", "\t FEG = -3.67 kN\n", "\t FEB = 4.50 kN\n", "(negative numbers imply compression)\n" ] } ], "source": [ "# moment about F for section\n", "rFK = array(( l, -2*l, 0 ))\n", "rFL = array(( 0, -2*l, 0 ))\n", "rFE = array(( l, 0, 0 ))\n", "rFJ = array(( l/2, -l/2, l ))\n", "rEG = array((-l/2, -l/2, l ))\n", "uEG = rEG / sqrt(sum(rEG**2))\n", "FJGs,FEGs,FEBs = s.symbols('FJGs,FEGs,FEBs')\n", "eq = cross( rFK, Kz*khat ) + cross( rFL, Lz*khat ) \\\n", " + FJGs*cross( rFJ, jhat ) + FEBs*cross( rFE, jhat ) \\\n", " + FEGs*cross( rFE, uEG )\n", "ss = s.solve( [eq[0],eq[1],eq[2]], [FJGs,FEGs,FEBs])\n", "FJG,FEG,FEB = float(ss[FJGs]), float(ss[FEGs]), float(ss[FEBs])\n", "print('member forces are ')\n", "print ('\\t FJG = {:.2f} kN'.format(FJG))\n", "print ('\\t FEG = {:.2f} kN'.format(FEG))\n", "print ('\\t FEB = {:.2f} kN'.format(FEB))\n", "print('(negative numbers imply compression)')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "\n", "## F. Simple Frame\n", "\n", "" ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [], "source": [ "F1,F2 = 74., 20. # lb\n", "X = 3.0 # in\n", "Y1,Y2,Y3=2.,4.,6. # in (bottom to top)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$\\sum M_C = -XA_y +Y_2B_x - XF_2=0$" ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "answer:\n", " Gy= 20.0 lb\n", " Ay= 74.0 lb\n", " Bx= 70.5 lb\n", " Cx= -70.5 lb\n", " Cy= -54.0 lb\n" ] } ], "source": [ "Ay,Gy = F1,F2\n", "Bx = (X*F2 + X*Ay)/Y2 # moment sum about C\n", "Cy = F2 - Ay\n", "Cx = -Bx\n", "print ('answer:')\n", "print (' Gy=',Gy,'lb')\n", "print (' Ay=',Ay,'lb')\n", "print (' Bx=',Bx,'lb')\n", "print (' Cx=',Cx,'lb')\n", "print (' Cy=',Cy,'lb')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "\n", "\n", "## G. Another Frame (pulley support)\n", "\n", "In the structure shown, cable segment *ED* and member *ECF* are horizontal. If $W=660$ lb, determine the forces that the pins *C* and *F* exert on member *ECF* and the forces that the pin at *B* exerts on member *GBF*.\n" ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [], "source": [ "W = 660. # lb\n", "LCE,LCF = 89,16. # in\n", "LCB,LAB = 26,76. # in\n", "LAG = 64. # in\n", "R = 9. # in (pulley radius)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "Whole body ($A_x=0$):\n", "\n", "$\\sum M_A= -L_{AG}Gy + (L_{CE}+R)W=0$\n", "\n", "$\\sum F_y = G_y+A_y-W=0$\n", "\n", "*ECF*: \n", "\n", "$\\sum M_C=L_{CE}W + L_{CF}F_y=0$\n", "\n", "$\\sum F_y = C_y+F_y-W=0$\n", "\n", "$\\sum F_x = C_x+F_x+W=0$\n", "\n", "*ABCD*: \n", "\n", "$\\sum M_B=(L_{CB}+R)W + L_{CB}C_x=0$\n", "\n", "$\\sum F_x = -W-C_x-B_x+A_x =0$" ] }, { "cell_type": "code", "execution_count": 19, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "answer:\n", " Fy= -3671.25 lb\n", " Cy= 4331.25 lb\n", " Fx= 228.46153846153845 lb\n", " By= -4681.875 lb\n", " Bx= 228.46153846153845 lb\n", " Cx= -888.4615384615385 lb\n" ] } ], "source": [ "# whole body:\n", "Gy = (LCE+R)*W / LAG\n", "Ay = W - Gy\n", "Ax = 0.\n", "# ECF:\n", "Fy = -LCE*W / LCF\n", "Cy = W - Fy\n", "# ABCD:\n", "Cx = -(LCB+R)*W / LCB\n", "Bx = Ax - W - Cx\n", "By = Ay - Cy\n", "Fx = -W - Cx\n", "print ('answer:')\n", "print (' Fy=',Fy,'lb')\n", "print (' Cy=',Cy,'lb')\n", "print (' Fx=',Fx,'lb')\n", "print (' By=',By,'lb')\n", "print (' Bx=',Bx,'lb')\n", "print (' Cx=',Cx,'lb')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "## H. Yet Another Frame (pulley support)\n", "\n", "" ] }, { "cell_type": "code", "execution_count": 20, "metadata": {}, "outputs": [], "source": [ "W = 420. # lb\n", "R = 5. # in (pulley radius)\n", "X1,X2,X3 = 15.,30.,25. # in (right to left)\n", "Y = 40. # in" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "For member *ABCD*:\n", "\n", "$\\sum M_A= X_1\\frac{Y}{Z}T_{BG}+(X_1+X_2)\\frac{Y}{Z}T_{CF} + (X_1+X_2+X_3)2W = 0$\n", "\n", "where $Z=\\sqrt{X_2^2+Y^2}$\n", "\n", "And for member *EFGH*:\n", "\n", "$\\sum M_E= -X_1\\frac{Y}{Z}T_{CF}-(X_1+X_2)\\frac{Y}{Z}T_{BG} - (X_1+X_2+X_3-R)W = 0$\n", "\n", "Solve these two equations simultaneously." ] }, { "cell_type": "code", "execution_count": 21, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "answer:\n", " TBG= -240.625 lb\n", " TCF= -1553.125 lb\n" ] } ], "source": [ "Z = sqrt(X2**2+Y**2)\n", "CC = array((( X1*Y/Z, (X1+X2)*Y/Z ),\n", " ( -(X1+X2)*Y/Z, -X1*Y/Z ) ))\n", "SS = array(( -(X1+X2+X3)*2*W, (X1+X2+X3-R)*W ))\n", "TBG,TCF = linalg.solve(CC,SS)\n", "print ('answer:')\n", "print (' TBG=',TBG,'lb')\n", "print (' TCF=',TCF,'lb')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "## H. Machine (Garbage Truck)\n", "\n", "\n", "\n", "The linkage shown is used on a garbage truck to lift a 2000 lb dumpster. Points A–G are pins, and member ABC is horizontal. This linkage has the feature that as the dumpster is lifted, the front edge A is lowered, allowing a more convenient height for emptying.\n", "\n", "For the position shown, when the dumpster just lifts off the ground, determine the force in hydraulic cylinder CF." ] }, { "cell_type": "code", "execution_count": 22, "metadata": {}, "outputs": [], "source": [ "W = 2000. # lb\n", "LAB,LCB,LAHx = 18.,18.,60. # in\n", "LED,LDA,LAG = 12.,24.,24. # in" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "dumpster:\n", "\n", "$\\sum M_A = L_{AG}F_G - L_{AHx}W = 0$\n", "\n", "$\\sum F_y = A_y + \\tfrac{1}{\\sqrt{2}}F_G - W =0$\n", "\n", "*EDG*:\n", "\n", "$\\sum M_E = -L_{EG}F_G - L_{ED}F_{CD} = 0$\n", "\n", "*ABC*:\n", "\n", "$\\sum M_B = -L_{AB}A_y - L_{CB}\\tfrac{1}{\\sqrt{2}}F_{CD}- L_{CB}\\tfrac{1}{\\sqrt{2}}F_{CF} = 0$" ] }, { "cell_type": "code", "execution_count": 23, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "answer:\n", " FCF = 27172 lb\n" ] } ], "source": [ "FG = LAHx*W/LAG\n", "Ay = W - FG/sqrt(2)\n", "LEG = LED + LDA + LAG\n", "FCD = -LEG*FG/LED\n", "FCF = ( -LAB*Ay*sqrt(2) - LCB*FCD )/LCB\n", "print ('answer:')\n", "print (' FCF = {:.0f} lb'.format(FCF))" ] } ], "metadata": { "anaconda-cloud": {}, "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.8.5" } }, "nbformat": 4, "nbformat_minor": 4 }