a khK@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)Z import_kwargsc@seZdZdZddZeddZeddZedd Zed d Z ed d Z eddZ d1ddZ ddZ d2ddZd3ddZd4ddZddZd5ddZd6dd Zd7d!d"Zd#d$Zed%d&d'd(Zd)d*Zd+d,Zd-d.Zd/d0ZdS)8Archa This class is used to solve problems related to a three hinged arch(determinate) structure. An arch is a curved vertical structure spanning an open space underneath it. Arches can be used to reduce the bending moments in long-span structures. Arches are used in structural engineering(over windows, door and even bridges) because they can support a very large mass placed on top of them. Example ======== >>> from sympy.physics.continuum_mechanics.arch import Arch >>> a = Arch((0,0),(10,0),crown_x=5,crown_y=5) >>> a.get_shape_eqn 5 - (x - 5)**2/5 >>> from sympy.physics.continuum_mechanics.arch import Arch >>> a = Arch((0,0),(10,1),crown_x=6) >>> a.get_shape_eqn 9/5 - (x - 6)**2/20 cKs@d|_t|dt|df|_t|dt|df|_d|_d|_d|vr\t|d|_d|vrrt|d|_|j|_i|_i|_|j|jd|_ i|_ ddd|_ d|_ d|_ tddtd dtd dtd di|_t|_t|_i|_i|_i|_i|_td |_td |_td |_td |_d|_d|_d|_dS) Nrcrown_xcrown_y) concentrated distributedhinge)leftrightR_A_xR_A_yR_B_xR_B_yrT) _shape_eqnr _left_support_right_support_crown_x_crown_y get_shape_eqn _conc_loads_distributed_loads_loads_loads_applied _supports_memberZ _member_forcer_reaction_forceset_points_disc_x_points_disc_y _moment_x _moment_y_load_x_load_yr_moment_x_func_moment_y_func _load_x_func _load_y_func_bending_moment _shear_force _axial_force)self left_support right_supportkwargsrCT/var/www/auris/lib/python3.9/site-packages/sympy/physics/continuum_mechanics/arch.py__init__&s> &    z Arch.__init__c Cs|jr |jStd\}}}tddd}|jr|jr|j}|j}|||d||}|||jd||jdi}t||} | d||d|}|||jdi|jdkrt dn|jr~|j}|||d||}|||jd||jdi}|||jd||jdi} t|| f||f} t | dksN| |dkrVt d | |||d| |}| ||_nt d |S) z3returns the equation of the shape of arch developedzx y caF)ZpositiverrzQprovided coordinates of crown and supports are not consistent with parabolic archzYparabolic arch cannot be constructed with the provided coordinates, try providing crown_yz(please provide crown_x to construct arch) r$rrr'r(subsr%r r& ValueErrorlenKeyError) r?xycrFx0Zy0Z parabola_eqneq1solutioneq2rCrCrDr)Hs2     zArch.get_shape_eqncCs|jS)z\ return the position of the applied load and angle (for concentrated loads) )r,r?rCrCrD get_loadsjszArch.get_loadscCs|jS)z- Returns the type of support )r.rSrCrCrDsupportsqsz Arch.supportscCs|jS)z; Returns the position of the left support. )r%rSrCrCrDr@xszArch.left_supportcCs|jS)z< Returns the position of the right support. )r&rSrCrCrDrAszArch.right_supportcCs|jS)z6 return the reaction forces generated )r0rSrCrCrDreaction_forceszArch.reaction_forceNc Cstd}td}td} t|}t|}t|}||jvrBtd|dvrRtd|dkrF|dusl||krttd |||d |j|<|j|||jvr|j||t |||| |t ||d 8<|j ||t |||7<nH| t |||| |t ||d |j|<|t ||||j |<d |j|<|d kr|durbt 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|<||jvr|j||j|d| |8<|j ||j|d7<n2|j|d | ||j|<|j|d|j |<d|j|<dS)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) rMrLrOz(load with the given label already exists)ABz1cannot use the given label, reserved for supportsrNzprovide end greater than start)startendf_yrGrz!please provide direction of force)rLrMf_xr[magangler]r[r)rrr-rIrKr+r3addr5r r7 TypeErrorr$rHrr rr*r2r4r6) r?orderlabelrYr^rZr_rMrLrOheightrCrCrD apply_loadsN2    6"0   0   .& $zArch.apply_loadc Cstd}td}td}||jvr|j||j|d}|j|d}|j|d}|j||j||t|||8<|j||t|||||t||d7<|j|}t d|d |n||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 |ntd d S) 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} rMrLrOrYrZr[rGz removed load : r]zlabel not foundN)rr+r-popr3remover7r r5printr*r2r4r6rI) r?rcrMrLrOrYrZr^valrCrCrD remove_loads2    6     $. zArch.remove_loadcCsF|dur|d|df|_|dur4|d|df|_d|_|j|_dS)a. Change position of supports. If not provided , defaults to the old value. Parameters ========== left_support: tuple (x, y) x: float x-coordinate value of the left_support y: float y-coordinate value of the left_support right_support: tuple (x, y) x: float x-coordinate value of the right_support y: float y-coordinate value of the right_support Nrr)r%r&r$r))r?r@rArCrCrDchange_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$r))r?rrrCrCrDchange_crown_positionIszArch.change_crown_positioncCsHddg}|r&||vrtd||jd<|rD||vr:td||jd<dS)a Add the type for support at each end. Can use roller or hinge support at each end. Parameters ========== left_support, right_support : string Type of support at respective end - For roller support , left_support/right_support = "roller" - For hinged support, left_support/right_support = "hinge" - defaults to hinge if value not provided Examples ======== For applying roller support at right end >>> from sympy.physics.continuum_mechanics.arch import Arch >>> a = Arch((0,0),(10,0),crown_x=5,crown_y=5) >>> a.change_support_type(right_support="roller") rollerrz%supports must only be roller or hingerrN)rIr.)r?r@rAZ support_typesrCrCrDchange_support_type]s zArch.change_support_typecCs||jks$|t|jd|jdkrLtdt|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=rLrGN) r(minr%r&rIrrr$rHr'rr/)r?rMrLrFZx_diffx1Zx2rCrCrD add_members$(  zArch.add_membercKsJ|dur|jStd}d|vr8|d}t|j|||dS|j||SdS)zl return the shear at some x-coordinates if no x value provided, returns the formula NrLdirrs)r=rrrHr?posrBrLrsrCrCrDshear_force_atszArch.shear_force_atcKsJ|dur|jStd}d|vr8|d}t|j|||dS|j||SdS)zu return the bending moment at some x-coordinates if no x value provided, returns the formula NrOrsrt)r<rrrH)r?rvrBrOrsrCrCrDbending_moment_atszArch.bending_moment_atcKsJ|dur|jStd}d|vr8|d}t|j|||dS|j||SdS)z return the axial/normal force generated at some x-coordinate if no x value provided, returns the formula NrLrsrt)r>rrrHrurCrCrDaxial_force_atszArch.axial_force_atc"CsR td}td}td}t|j}t|j}td|_td|_td|_td|_d}d}d}d} |D]T} || k} ||j | 7}||j | 7}t|| f|jdf|_t|| f|jdf|_qh|D]T} || k} ||j | 7}| |j | 7} t| | f|jdf|_t|| f|jdf|_q|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 ksH|jd d kr\|js\td d Std \}}}}}|jdd kr|jd d kr|jdt|jd|jdkrx|dkrtdnt|d}t|d}t|||d}t||jd|jd||jd|jd| d}t|||jd|j||jd|jd}t|||||f|||||f} qp|jd|jdkrt|d}t|d}t|||d}t||jd|jd||jd|jd| d}t||d}t|||||f|||||f}n|jd|jdk rpt|d}t|d}t|||d}t||jd|jd||jd|jd| d}t||d}t|||||f|||||f}n|jdd krj|jdt|jd|jdkrt|d}t||d}t|||d}t||jd|jd||jd|jd| d}t| ||jd|j||jd|jd}t|||||f|||||f} qp|jd|jdkrt|d}t|||d}t|||d}t||jd|jd||jd|jd||jd|jd| d}t| ||jd|j||jd|jd}t|||||f|||||f}n|jd|jdk rpt|d}t|||d}t|||d}t| ||jd|jd}t| ||jd|jd||jd|jd||jd|jdd}t|||||f|||||f}n|jd d k r|jdt|jd|jdkr>> 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 rMrLrOr#rTrrrnrz5member must be added if any of the supports is rollerNzR_A_x R_A_y R_B_x R_B_y TrGzDnet force in x direction not possible under the specified conditions)#rsortedr2r3rr8r9r:r;r6r4r5r7rHr&r%r'r(r.r/rirmaxrIr r r0r$r<r rrrr>r=)"r?rMrLrOZdiscontinuity_points_xZdiscontinuity_points_yZaccumulated_x_momentZaccumulated_y_momentZaccumulated_x_loadZaccumulated_y_loadpointZcondZmoment_AZmoment_hinge_leftZmoment_hinge_rightZnet_xZnet_yrr r!r"TrPrRZeq3Zeq4Zeq5rQZsymbr_ZfxZfyZ axial_forceZ shear_forcerCrCrDr s        ( "    "     "" "   "" ""      z Arch.solve)r)modulesc Cs6td}g}|}g}|}||7}|jd}|jd}t|jd|jd}|j} t|d|dd| d|dd} |}| } || 7}|j dur4|j d|jdkr| |j dd| g|j dggd d d d d |j d|jdkr4| |j dd| g|j dggd d d d d | |j g|jd| ggd dd d d | |d|ddkrt |jd| |j||jd|jdf|d|||d| |df|d| |dfddd } n\t |jd| |j||jd|jdf|d|||d| | df|d| | dfddd } | S)a7 This method returns a plot object containing the diagram of the specified arch along with the supports and forces applied to the structure. Examples ======== >>> from sympy import Symbol >>> t = Symbol('t') >>> from sympy.physics.continuum_mechanics.arch import Arch >>> a = Arch((0,0),(40,0),crown_x=20,crown_y=12) >>> a.apply_load(-1,'C',8,150,angle=270) >>> a.apply_load(0,'D',start=20,end=40,mag=-4) >>> a.apply_load(-1,'E',10,t,angle=300) >>> p = a.draw() >>> p # doctest: +ELLIPSIS Plot object containing: [0]: cartesian line: 11.325 - 3*(x - 20)**2/100 for x over (0.0, 40.0) [1]: cartesian line: 12 - 3*(x - 20)**2/100 for x over (0.0, 40.0) ... >>> p.show() rLrr皙?皙?NrG{Gzt?oZwhitenoneargsmarkerZ markersizecolorZmarkerfacecolorQ?F皙?brown)markersshow annotations rectanglesZxlimZylimZaxisZ line_color)r _draw_loads_draw_supportsr&r%rpr(r{_draw_rectangles _draw_fillerr/appendr'rr$) r?rLrrrrUxmaxxminyminymaxZlimfillerZ sing_plotrCrCrDdraws  *     z Arch.drawcCsg}|jd}|jd}t|jd|jd}|j}td|d|td|d|krnd|d|}nd|d|}|jddkr||jdg|jdd|ggdd d d d n2||jdg|jdd |ggddd d d |jddkr6||jdg|jdd|ggdd d d d n2||jdg|jdd |ggddd d d ||jdg|jdd|ggddd d d ||jdg|jdd|ggddd d d |S)Nrrrrrrng{Gz?r Zblackrrgy&1|?rg;On?_)r&r%rpr(absr.r)r?Zsupport_markersrrrrmax_diffrCrCrDrs  (            zArch._draw_supportsc Csg}|jd}|jd}t|jd|jd}|j}td|d|td|d|krnd|d|}nd|d|}|jdur|jdt|jd|jdkr||jd|jdd|f|jd|jdd|ddd n|jd|jdkrH||jd|jdd|f|jd|jdd|ddd nF||jd|jdd|ft|jd|jdd|d dd |jr|jD]L}|j|d }|j|d } |||j|d f| ||dddq|S)NrrrrrGrg{Gz?r)xywidthrdr_rrYrZ333333?orangerrrdr) r&r%rpr(rr/r{rr+) r?memberrrrrrloadsrYrZrCrCrDrRs^  (       zArch._draw_rectanglescCsg}|jd}|jd}t|jd|jd}|j}td|d|td|d|krnd|d|}nd|d|}|jD]}|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| |dfdq|j D]f}|j |d} |j |d} |j |d} t | | | | |d}t || }|D]}| dkr|d ||j|df||j|dfd ddddddn<|d ||j|df||j|dfd ddddddq| dkr||dt| dd d | | d |j|d!fdn:||dt| dd d | | d |j|d"fdqh|S)#NrrrrrLrMr_r^g{Gz? Zboldg?rZblue)rZ headlengthZ headwidthZ facecolorZ edgecolor)textrxytextfontsize fontweight arrowpropsrfz NgQ?)rrrrrYrZr[g?rrr)rrrrg?z N/mrGgffffff?g?) r&r%rpr(rr*rrr rr+rr)r?Zload_annotationsrrrrrloadrLrMr_r^rYrZx_pointsr|rCrCrDrs  (  .      zArch._draw_loadsc Cs td}g}|jd}|jd}t|jd|jd}|j}td|d|td|d|krvd|d|}nd|d|}t|jd|jd|jd|jd||}|D]J} || |j || |df|jd|jd|||dddq|S) NrLrrrrrrr) rr&r%rpr(rrrrr$rH) r?rLrrrrrrrr|rCrCrDrs&  (2 zArch._draw_filler)NN)NN)NN)NN)N)N)N)__name__ __module__ __qualname____doc__rEpropertyr)rTrUr@rArVrerkrlrmrorrrwrxryr rrrrrrrCrCrCrDrs>" !      k3   &   M j_AUrN)rZsympy.core.sympifyrZsympy.core.symbolrrZsympyrrrrr r r Zsympy.core.relationalr Zsympy.solvers.solversr Zsympy.functionsrZsympy.plottingrrZsympy.utilities.decoratorrZsympy.external.importtoolsrrrrCrCrCrDs $