o 3IhK@sdZddlmZddlmZmZddlmZmZm Z m Z m Z m Z m Z ddlmZddlmZddlmZddlmZdd lmZdd lmZdd lmZed d dgidZGdddZdS)zJ This module can be used to solve probelsm related to 2D parabolic arches )sympify)Symbolsymbols)diffsqrtcossinatanradMin)Eq)solve) Piecewise)plot)limit)doctest_depends_on) import_modulenumpyfromlistarange) import_kwargsc@seZdZdZddZeddZeddZedd Zed d Z ed d Z eddZ d1ddZ ddZ d1ddZd1ddZd1ddZddZd2ddZd2dd Zd2d!d"Zd#d$Zed%d&d'd(Zd)d*Zd+d,Zd-d.Zd/d0ZdS)3Archa This class is used to solve problems related to a three hinged arch(determinate) structure. An arch is a curved vertical structure spanning an open space underneath it. Arches can be used to reduce the bending moments in long-span structures. Arches are used in structural engineering(over windows, door and even bridges) because they can support a very large mass placed on top of them. Example ======== >>> from sympy.physics.continuum_mechanics.arch import Arch >>> a = Arch((0,0),(10,0),crown_x=5,crown_y=5) >>> a.get_shape_eqn 5 - (x - 5)**2/5 >>> from sympy.physics.continuum_mechanics.arch import Arch >>> a = Arch((0,0),(10,1),crown_x=6) >>> a.get_shape_eqn 9/5 - (x - 6)**2/20 cKs@d|_t|dt|df|_t|dt|df|_d|_d|_d|vr.t|d|_d|vr9t|d|_|j|_i|_i|_|j|jd|_ i|_ ddd|_ d|_ d|_ tddtd dtd dtd di|_t|_t|_i|_i|_i|_i|_td |_td |_td |_td |_d|_d|_d|_dS) Nrcrown_xcrown_y) concentrated distributedhinge)leftrightR_A_xR_A_yR_B_xR_B_yrT) _shape_eqnr _left_support_right_support_crown_x_crown_y get_shape_eqn _conc_loads_distributed_loads_loads_loads_applied _supports_member _member_forcer_reaction_forceset_points_disc_x_points_disc_y _moment_x _moment_y_load_x_load_yr_moment_x_func_moment_y_func _load_x_func _load_y_func_bending_moment _shear_force _axial_force)self left_support right_supportkwargsrEf/home/air/sanwanet/gpt-api/venv/lib/python3.10/site-packages/sympy/physics/continuum_mechanics/arch.py__init__&s> &     z Arch.__init__c Cs|jr|jStd\}}}tddd}|jr`|jr`|j}|j}|||d||}|||jd||jdi}t||} | d||d|}|||jdi|jdkr^t d|S|jr|j}|||d||}|||jd||jdi}|||jd||jdi} t|| f||f} t | dks| |dkrt d | |||d| |}| ||_|St d ) z3returns the equation of the shape of arch developedzx y caF)positiverrzQprovided coordinates of crown and supports are not consistent with parabolic archzYparabolic arch cannot be constructed with the provided coordinates, try providing crown_yz(please provide crown_x to construct arch) r%rrr(r)subsr&r r' ValueErrorlenKeyError) rAxycrHx0y0 parabola_eqneq1solutioneq2rErErFr*Hs4    zArch.get_shape_eqncC|jS)z\ return the position of the applied load and angle (for concentrated loads) )r-rArErErF get_loadsjzArch.get_loadscCrX)z- Returns the type of support )r/rYrErErFsupportsqr[z Arch.supportscCrX)z; Returns the position of the left support. )r&rYrErErFrBxr[zArch.left_supportcCrX)z< Returns the position of the right support. )r'rYrErErFrCr[zArch.right_supportcCrX)z6 return the reaction forces generated )r2rYrErErFreaction_forcer[zArch.reaction_forceNc Cstd}td}td} t|}t|}t|}||jvr!td|dvr)td|dkr|dus5||kr9td |||d |j|<|j|||jvry|j||t |||| |t ||d 8<|j ||t |||7<n$| t |||| |t ||d |j|<|t ||||j |<d |j|<|d krk|durt d|j d|i} || |tt||tt|||d|j|<|j||j|||jvr|j||j|d||j|d7<|j||j|d7<n|j|d||j|d|j|<|j|d|j|<||jvrK|j||j|d| |8<|j ||j|d7<n|j|d | ||j|<|j|d|j |<d|j|<dSdS)aN This method adds load to the Arch. Parameters ========== order : Integer Order of the applied load. - For point/concentrated loads, order = -1 - For distributed load, order = 0 label : String or Symbol The label of the load - should not use 'A' or 'B' as it is used for supports. start : Float - For concentrated/point loads, start is the x coordinate - For distributed loads, start is the starting position of distributed load mag : Sympifyable Magnitude of the applied load. Must be positive end : Float Required for distributed loads - For concentrated/point load , end is None(may not be given) - For distributed loads, end is the end position of distributed load angle: Sympifyable The angle in degrees, the load vector makes with the horizontal in the counter-clockwise direction. Examples ======== For applying distributed load >>> from sympy.physics.continuum_mechanics.arch import Arch >>> a = Arch((0,0),(10,0),crown_x=5,crown_y=5) >>> a.apply_load(0,'C',start=3,end=5,mag=-10) For applying point/concentrated_loads >>> from sympy.physics.continuum_mechanics.arch import Arch >>> a = Arch((0,0),(10,0),crown_x=5,crown_y=5) >>> a.apply_load(-1,'C',start=2,mag=15,angle=45) rPrOrRz(load with the given label already exists)ABz1cannot use the given label, reserved for supportsrNzprovide end greater than start)startendf_yrJrz!please provide direction of force)rOrPf_xrbmaganglerdrbr)rrr.rLrNr,r5addr7r r9 TypeErrorr%rKrr rr+r4r6r8) rAorderlabelr`rerarfrPrOrRheightrErErF apply_loadsP2   6"0  0   .& $zArch.apply_loadc Cstd}td}td}||jvro|j||j|d}|j|d}|j|d}|j||j||t|||8<|j||t|||||t||d7<|j|}t d|d |d S||j vr|j||j |d}|j ||j||j||j |d||7<|j ||j |d ||j |d8<|j ||j |d 8<|j||j |d8<|j |}t d|d |d Std ) a This methods removes the load applied to the arch Parameters ========== label : String or Symbol The label of the applied load Examples ======== >>> from sympy.physics.continuum_mechanics.arch import Arch >>> a = Arch((0,0),(10,0),crown_x=5,crown_y=5) >>> a.apply_load(0,'C',start=3,end=5,mag=-10) >>> a.remove_load('C') removed load C: {'start': 3, 'end': 5, 'f_y': -10} rPrOrRr`rarbrJz removed load : rdzlabel not foundN)rr,r.popr5remover9r r7printr+r4r6r8rL) rArjrPrOrRr`rarevalrErErF remove_loads2    6     $. zArch.remove_loadcCsF|dur |d|df|_|dur|d|df|_d|_|j|_dS)a. Change position of supports. If not provided , defaults to the old value. Parameters ========== left_support: tuple (x, y) x: float x-coordinate value of the left_support y: float y-coordinate value of the left_support right_support: tuple (x, y) x: float x-coordinate value of the right_support y: float y-coordinate value of the right_support Nrr)r&r'r%r*)rArBrCrErErFchange_support_position+s  zArch.change_support_positioncCs||_||_d|_|j|_dS)a Change the position of the crown/hinge of the arch Parameters ========== crown_x: Float The x coordinate of the position of the hinge - if not provided, defaults to old value crown_y: Float The y coordinate of the position of the hinge - if not provided defaults to None N)r(r)r%r*)rArrrErErFchange_crown_positionIs zArch.change_crown_positioncCsLddg}|r||vrtd||jd<|r$||vrtd||jd<dSdS)a Add the type for support at each end. Can use roller or hinge support at each end. Parameters ========== left_support, right_support : string Type of support at respective end - For roller support , left_support/right_support = "roller" - For hinged support, left_support/right_support = "hinge" - defaults to hinge if value not provided Examples ======== For applying roller support at right end >>> from sympy.physics.continuum_mechanics.arch import Arch >>> a = Arch((0,0),(10,0),crown_x=5,crown_y=5) >>> a.change_support_type(right_support="roller") rollerrz%supports must only be roller or hingerrN)rLr/)rArBrC support_typesrErErFchange_support_type]s zArch.change_support_typecCs||jks|t|jd|jdkr&tdt|jd|jdd|jtd}t|j|||j dd}t ||j|}|j |}|j |}|||f|_ dS)z This method adds a member/rod at a particular height y. A rod is used for stability of the structure in case of a roller support. rz&position of support must be between y=z and y=rOrJN) r)minr&r'rLrrr%rKr(rr0)rArPrOrHx_diffx1x2rErErF add_members$(  zArch.add_membercKF|dur|jStd}d|vr|d}t|j|||dS|j||S)zl return the shear at some x-coordinates if no x value provided, returns the formula NrOdirr~)r?rrrKrAposrDrOr~rErErFshear_force_atzArch.shear_force_atcKr})zu return the bending moment at some x-coordinates if no x value provided, returns the formula NrRr~r)r>rrrK)rArrDrRr~rErErFbending_moment_atrzArch.bending_moment_atcKr})z return the axial/normal force generated at some x-coordinate if no x value provided, returns the formula NrOr~r)r@rrrKrrErErFaxial_force_atrzArch.axial_force_atc"CsT td}td}td}t|j}t|j}td|_td|_td|_td|_d}d}d}d} |D]*} || k} ||j | 7}||j | 7}t|| f|jdf|_t|| f|jdf|_q4|D]*} || k} ||j | 7}| |j | 7} t| | f|jdf|_t|| f|jdf|_qa|j ||jd ||jd|j ||jd ||jd} |j ||j ||j|j ||j ||j} |j ||jd ||j|j ||j ||j|j ||jd ||j|j ||j ||j}|j ||jd}|j ||jd}|jdd ks$|jd d kr.|js.td d Std \}}}}}|jdd kra|jd d kra|jdt|jd|jdkr|dkratdt|d}t|d}t|||d}t||jd|jd||jd|jd| d}t|||jd|j||jd|jd}t|||||f|||||f}n~|jd|jdkrt|d}t|d}t|||d}t||jd|jd||jd|jd| d}t||d}t|||||f|||||f}n+|jd|jdkr_t|d}t|d}t|||d}t||jd|jd||jd|jd| d}t||d}t|||||f|||||f}n|jdd kr|jdt|jd|jdkrt|d}t||d}t|||d}t||jd|jd||jd|jd| d}t| ||jd|j||jd|jd}t|||||f|||||f}nc|jd|jdkrKt|d}t|||d}t|||d}t||jd|jd||jd|jd||jd|jd| d}t| ||jd|j||jd|jd}t|||||f|||||f}n|jd|jdkrt|d}t|||d}t|||d}t| ||jd|jd}t| ||jd|jd||jd|jd||jd|jdd}t|||||f|||||f}n|jd d kr|jdt|jd|jdkrt|d}t||d}t|||d}t|||jd|j||jd|jd}t| ||jd|jdd}t|||||f|||||f}n|jd|jdkr}t|d}t|||d}t|||d}t|||jd|jd}t| ||jd|jd||jd|jdd}t|||||f|||||f}n|jd|jdkrt|d}t|||d}t|||d}t|||jd|j||jd|jd}t| ||jd|jd||jd|jd}t|||||f|||||f}nUt|||d}t|||d}t||jd|jd||jd|jd| d}t|||jd|j||jd|jd}t||||f||||f}|jD] }|||j|<q<|j |||j ||||||jd|||j ||i|jd |_tt|j|}|j||}|j||}|t||t |} | t ||t|}!| |_!|!|_"d S)a This method solves for the reaction forces generated at the supports, and bending moment and generated in the arch and tension produced in the member if used. Examples ======== >>> from sympy.physics.continuum_mechanics.arch import Arch >>> a = Arch((0,0),(10,0),crown_x=5,crown_y=5) >>> a.apply_load(0,'C',start=3,end=5,mag=-10) >>> a.solve() >>> a.reaction_force {R_A_x: 8, R_A_y: 12, R_B_x: -8, R_B_y: 8} >>> from sympy import Symbol >>> t = Symbol('t') >>> from sympy.physics.continuum_mechanics.arch import Arch >>> a = Arch((0,0),(16,0),crown_x=8,crown_y=5) >>> a.apply_load(0,'C',start=3,end=5,mag=t) >>> a.solve() >>> a.reaction_force {R_A_x: -4*t/5, R_A_y: -3*t/2, R_B_x: 4*t/5, R_B_y: -t/2} >>> a.bending_moment_at(4) -5*t/2 rPrOrRr$rTrrrurz5member must be added if any of the supports is rollerNzR_A_x R_A_y R_B_x R_B_y TrJzDnet force in x direction not possible under the specified conditions)#rsortedr4r5rr:r;r<r=r8r6r7r9rKr'r&r(r)r/r0rprmaxrLr r r2r%r>r rrrr@r?)"rArPrOrRdiscontinuity_points_xdiscontinuity_points_yaccumulated_x_momentaccumulated_y_momentaccumulated_x_loadaccumulated_y_loadpointcondmoment_Amoment_hinge_leftmoment_hinge_rightnet_xnet_yr r!r"r#TrUrWeq3eq4eq5rVsymbrffxfy axial_force shear_forcerErErFr s        ( "   "  "  " " " " ""      z Arch.solve)r)modulesc Cs2td}g}|}g}|}||7}|jd}|jd}t|jd|jd}|j} t|d|dd| d|dd} |}| } || 7}|j dur|j d|jdkru| |j dd| g|j dggd d d d d |j d|jdkr| |j dd| g|j dggd d d d d | |j g|jd| ggd dd d d | |d|ddkrt |jd| |j||jd|jdf|d|||d| |df|d| |dfddd } | St |jd| |j||jd|jdf|d|||d| | df|d| | dfddd } | S)a7 This method returns a plot object containing the diagram of the specified arch along with the supports and forces applied to the structure. Examples ======== >>> from sympy import Symbol >>> t = Symbol('t') >>> from sympy.physics.continuum_mechanics.arch import Arch >>> a = Arch((0,0),(40,0),crown_x=20,crown_y=12) >>> a.apply_load(-1,'C',8,150,angle=270) >>> a.apply_load(0,'D',start=20,end=40,mag=-4) >>> a.apply_load(-1,'E',10,t,angle=300) >>> p = a.draw() >>> p # doctest: +ELLIPSIS Plot object containing: [0]: cartesian line: 11.325 - 3*(x - 20)**2/100 for x over (0.0, 40.0) [1]: cartesian line: 12 - 3*(x - 20)**2/100 for x over (0.0, 40.0) ... >>> p.show() rOrr皙?皙?NrJ{Gzt?owhitenoneargsmarker markersizecolormarkerfacecolorQ?F皙?brown)markersshow annotations rectanglesxlimylimaxis line_color)r _draw_loads_draw_supportsr'r&rxr)r_draw_rectangles _draw_fillerr0appendr(rr%) rArOrrrr\xmaxxminyminymaxlimfiller sing_plotrErErFdraws  *    z Arch.drawcCsg}|jd}|jd}t|jd|jd}|j}td|d|td|d|kr7d|d|}nd|d|}|jddkr`||jdg|jdd|ggdd d d d n||jdg|jdd |ggddd d d |jddkr||jdg|jdd|ggdd d d d n||jdg|jdd |ggddd d d ||jdg|jdd|ggddd d d ||jdg|jdd|ggddd d d |S)Nrrrrrrug{Gz?r blackrrgy&1|?rg;On?_)r'r&rxr)absr/r)rAsupport_markersrrrrmax_diffrErErFrs  (        zArch._draw_supportsc Csg}|jd}|jd}t|jd|jd}|j}td|d|td|d|kr7d|d|}nd|d|}|jdur|jdt|jd|jdkrv||jd|jdd|f|jd|jdd|ddd nO|jd|jdkr||jd|jdd|f|jd|jdd|ddd n#||jd|jdd|ft|jd|jdd|d dd |jr|jD]%}|j|d }|j|d } |||j|d f| ||dddq|S)NrrrrrJrg{Gz?r)xywidthrkrfrr`ra333333?orangerrrkr) r'r&rxr)rr0rrr,) rAmemberrrrrrloadsr`rarErErFrRs^  (      zArch._draw_rectanglescCsg}|jd}|jd}t|jd|jd}|j}td|d|td|d|kr7d|d|}nd|d|}|jD]n}|j|d}|j|d} |j|d} |j|d} |d |tt| |d | t t| |d f|| fd d d dddddd||d| dd d |tt| |d| t t| |dfdqB|j D]}|j |d} |j |d} |j |d} t | | | | |d}t || }|D]D}| dkr|d ||j|df||j|dfd ddddddq|d ||j|df||j|dfd ddddddq| dkrH||dt| dd d | | d |j|d!fdq||dt| dd d | | d |j|d"fdq|S)#NrrrrrOrPrfreg{Gz? boldg?rblue)r headlength headwidth facecolor edgecolor)textrxytextfontsize fontweight arrowpropsrmz NgQ?)rrrrr`rarbg?rrr)rrrrg?z N/mrJgffffff?g?) r'r&rxr)rr+rrr rr,rr)rAload_annotationsrrrrrloadrOrPrfrer`rax_pointsrrErErFrs  (  .      zArch._draw_loadsc Cs td}g}|jd}|jd}t|jd|jd}|j}td|d|td|d|kr;d|d|}nd|d|}t|jd|jd|jd|jd||}|D]%} || |j || |df|jd|jd|||dddq^|S) NrOrrrrrrr) rr'r&rxr)rrrrr%rK) rArOrrrrrrrrrErErFrs&  (2 zArch._draw_filler)NN)N)__name__ __module__ __qualname____doc__rGpropertyr*rZr\rBrCr]rlrrrsrtrwr|rrrr rrrrrrrErErErFrs@" !      k 3  &   M j_A UrN)rsympy.core.sympifyrsympy.core.symbolrrsympyrrrrr r r sympy.core.relationalr sympy.solvers.solversr sympy.functionsrsympy.plottingrrsympy.utilities.decoratorrsympy.external.importtoolsrrrrErErErFs $