ó ~9­\c@ sedZddlmZmZddlmZmZmZmZddl m Z ddl m Z ddl mZmZmZddlmZddlmZdd lmZdd lmZdd lmZdd lmZdd lmZeddidgd6ƒZeddidgd6ƒZidgdf6Z de!fd„ƒYZ"de"fd„ƒYZ#dS(sd This module can be used to solve 2D beam bending problems with singularity functions in mechanics. iÿÿÿÿ(tprint_functiontdivision(tStSymboltdifftsymbols(tlinsolve(tsstr(tSingularityFunctiont Piecewiset factorial(tsympify(t integrate(tlimit(tplot(t import_module(tdoctest_depends_on(tlambdifyt matplotlibt__import__kwargstpyplottfromlisttnumpytlinspacesBeam.plot_loading_resultstBeamcB seZdZedƒdd„Zd„Zed„ƒZed„ƒZej d„ƒZed„ƒZ e j d „ƒZ ed „ƒZ e j d „ƒZ ed „ƒZ e j d „ƒZ ed„ƒZ ed„ƒZej d„ƒZed„ƒZej d„ƒZdd„Zdd„Zd,d„Zd,d„Zed„ƒZed„ƒZd„Zd„Zd„Zd„Zd„Zd„Zd „Zd!„Zd"„Zd#„Z d,d$„Z!d,d%„Z"d,d&„Z#d,d'„Z$e%d(d-ƒd,d+„ƒZ&RS(.sü A Beam is a structural element that is capable of withstanding load primarily by resisting against bending. Beams are characterized by their cross sectional profile(Second moment of area), their length and their material. .. note:: While solving a beam bending problem, a user should choose its own sign convention and should stick to it. The results will automatically follow the chosen sign convention. Examples ======== There is a beam of length 4 meters. A constant distributed load of 6 N/m is applied from half of the beam till the end. There are two simple supports below the beam, one at the starting point and another at the ending point of the beam. The deflection of the beam at the end is restricted. Using the sign convention of downwards forces being positive. >>> from sympy.physics.continuum_mechanics.beam import Beam >>> from sympy import symbols, Piecewise >>> E, I = symbols('E, I') >>> R1, R2 = symbols('R1, R2') >>> b = Beam(4, E, I) >>> b.apply_load(R1, 0, -1) >>> b.apply_load(6, 2, 0) >>> b.apply_load(R2, 4, -1) >>> b.bc_deflection = [(0, 0), (4, 0)] >>> b.boundary_conditions {'deflection': [(0, 0), (4, 0)], 'slope': []} >>> b.load R1*SingularityFunction(x, 0, -1) + R2*SingularityFunction(x, 4, -1) + 6*SingularityFunction(x, 2, 0) >>> b.solve_for_reaction_loads(R1, R2) >>> b.load -3*SingularityFunction(x, 0, -1) + 6*SingularityFunction(x, 2, 0) - 9*SingularityFunction(x, 4, -1) >>> b.shear_force() -3*SingularityFunction(x, 0, 0) + 6*SingularityFunction(x, 2, 1) - 9*SingularityFunction(x, 4, 0) >>> b.bending_moment() -3*SingularityFunction(x, 0, 1) + 3*SingularityFunction(x, 2, 2) - 9*SingularityFunction(x, 4, 1) >>> b.slope() (-3*SingularityFunction(x, 0, 2)/2 + SingularityFunction(x, 2, 3) - 9*SingularityFunction(x, 4, 2)/2 + 7)/(E*I) >>> b.deflection() (7*x - SingularityFunction(x, 0, 3)/2 + SingularityFunction(x, 2, 4)/4 - 3*SingularityFunction(x, 4, 3)/2)/(E*I) >>> b.deflection().rewrite(Piecewise) (7*x - Piecewise((x**3, x > 0), (0, True))/2 - 3*Piecewise(((x - 4)**3, x - 4 > 0), (0, True))/2 + Piecewise(((x - 2)**4, x - 2 > 0), (0, True))/4)/(E*I) txtCcC su||_||_||_||_||_igd6gd6|_d|_g|_i|_d|_ d|_ dS(sãInitializes the class. Parameters ========== length : Sympifyable A Symbol or value representing the Beam's length. elastic_modulus : Sympifyable A SymPy expression representing the Beam's Modulus of Elasticity. It is a measure of the stiffness of the Beam material. It can also be a continuous function of position along the beam. second_moment : Sympifyable A SymPy expression representing the Beam's Second moment of area. It is a geometrical property of an area which reflects how its points are distributed with respect to its neutral axis. It can also be a continuous function of position along the beam. variable : Symbol, optional A Symbol object that will be used as the variable along the beam while representing the load, shear, moment, slope and deflection curve. By default, it is set to ``Symbol('x')``. base_char : String, optional A String that will be used as base character to generate sequential symbols for integration constants in cases where boundary conditions are not sufficient to solve them. t deflectiontslopeiN( tlengthtelastic_modulust second_momenttvariablet _base_chart_boundary_conditionst_loadt_applied_loadst_reaction_loadstNonet_composite_typet_hinge_position(tselfRRRR t base_char((sElib/python2.7/site-packages/sympy/physics/continuum_mechanics/beam.pyt__init__Ns         cC s4djt|jƒt|jƒt|jƒƒ}|S(NsBeam({}, {}, {})(tformatRt_lengtht_elastic_modulust_second_moment(R)tstr_sol((sElib/python2.7/site-packages/sympy/physics/continuum_mechanics/beam.pyt__str__ss0cC s|jS(s- Returns the reaction forces in a dictionary.(R%(R)((sElib/python2.7/site-packages/sympy/physics/continuum_mechanics/beam.pytreaction_loadswscC s|jS(sLength of the Beam.(R-(R)((sElib/python2.7/site-packages/sympy/physics/continuum_mechanics/beam.pyR|scC st|ƒ|_dS(N(R R-(R)tl((sElib/python2.7/site-packages/sympy/physics/continuum_mechanics/beam.pyRscC s|jS(s¼ A symbol that can be used as a variable along the length of the beam while representing load distribution, shear force curve, bending moment, slope curve and the deflection curve. By default, it is set to ``Symbol('x')``, but this property is mutable. Examples ======== >>> from sympy.physics.continuum_mechanics.beam import Beam >>> from sympy import symbols >>> E, I = symbols('E, I') >>> x, y, z = symbols('x, y, z') >>> b = Beam(4, E, I) >>> b.variable x >>> b.variable = y >>> b.variable y >>> b = Beam(4, E, I, z) >>> b.variable z (t _variable(R)((sElib/python2.7/site-packages/sympy/physics/continuum_mechanics/beam.pyR …scC s+t|tƒr||_n tdƒ‚dS(Ns'The variable should be a Symbol object.(t isinstanceRR4t TypeError(R)tv((sElib/python2.7/site-packages/sympy/physics/continuum_mechanics/beam.pyR  s cC s|jS(sYoung's Modulus of the Beam. (R.(R)((sElib/python2.7/site-packages/sympy/physics/continuum_mechanics/beam.pyR§scC st|ƒ|_dS(N(R R.(R)te((sElib/python2.7/site-packages/sympy/physics/continuum_mechanics/beam.pyR¬scC s|jS(s#Second moment of area of the Beam. (R/(R)((sElib/python2.7/site-packages/sympy/physics/continuum_mechanics/beam.pyR°scC st|ƒ|_dS(N(R R/(R)ti((sElib/python2.7/site-packages/sympy/physics/continuum_mechanics/beam.pyRµscC s|jS(s Returns a dictionary of boundary conditions applied on the beam. The dictionary has three kewwords namely moment, slope and deflection. The value of each keyword is a list of tuple, where each tuple contains loaction and value of a boundary condition in the format (location, value). Examples ======== There is a beam of length 4 meters. The bending moment at 0 should be 4 and at 4 it should be 0. The slope of the beam should be 1 at 0. The deflection should be 2 at 0. >>> from sympy.physics.continuum_mechanics.beam import Beam >>> from sympy import symbols >>> E, I = symbols('E, I') >>> b = Beam(4, E, I) >>> b.bc_deflection = [(0, 2)] >>> b.bc_slope = [(0, 1)] >>> b.boundary_conditions {'deflection': [(0, 2)], 'slope': [(0, 1)]} Here the deflection of the beam should be ``2`` at ``0``. Similarly, the slope of the beam should be ``1`` at ``0``. (R"(R)((sElib/python2.7/site-packages/sympy/physics/continuum_mechanics/beam.pytboundary_conditions¹scC s |jdS(NR(R"(R)((sElib/python2.7/site-packages/sympy/physics/continuum_mechanics/beam.pytbc_slopeÖscC s||jd>> from sympy.physics.continuum_mechanics.beam import Beam >>> from sympy import symbols >>> E, I = symbols('E, I') >>> R1, R2 = symbols('R1, R2') >>> b1 = Beam(2, E, 1.5*I) >>> b2 = Beam(2, E, I) >>> b = b1.join(b2, "fixed") >>> b.apply_load(20, 4, -1) >>> b.apply_load(R1, 0, -1) >>> b.apply_load(R2, 0, -2) >>> b.bc_slope = [(0, 0)] >>> b.bc_deflection = [(0, 0)] >>> b.solve_for_reaction_loads(R1, R2) >>> b.load 80*SingularityFunction(x, 0, -2) - 20*SingularityFunction(x, 0, -1) + 20*SingularityFunction(x, 4, -1) >>> b.slope() (((80*SingularityFunction(x, 0, 1) - 10*SingularityFunction(x, 0, 2) + 10*SingularityFunction(x, 4, 2))/I - 120/I)/E + 80.0/(E*I))*SingularityFunction(x, 2, 0) + 0.666666666666667*(80*SingularityFunction(x, 0, 1) - 10*SingularityFunction(x, 0, 2) + 10*SingularityFunction(x, 4, 2))*SingularityFunction(x, 0, 0)/(E*I) - 0.666666666666667*(80*SingularityFunction(x, 0, 1) - 10*SingularityFunction(x, 0, 2) + 10*SingularityFunction(x, 4, 2))*SingularityFunction(x, 2, 0)/(E*I) R?thingeN(R RRRR RR'R((R)tbeamtviaRtEt new_lengthtnew_second_momenttnew_beam((sElib/python2.7/site-packages/sympy/physics/continuum_mechanics/beam.pytjoinæs +        cC sÜ|dks|dkrZtdt|ƒƒ}|j||dƒ|jj|dfƒn~tdt|ƒƒ}tdt|ƒƒ}|j||dƒ|j||dƒ|jj|dfƒ|jj|dfƒdS( s. This method applies support to a particular beam object. Parameters ========== loc : Sympifyable Location of point at which support is applied. type : String Determines type of Beam support applied. To apply support structure with - zero degree of freedom, type = "fixed" - one degree of freedom, type = "pin" - two degrees of freedom, type = "roller" Examples ======== There is a beam of length 30 meters. A moment of magnitude 120 Nm is applied in the clockwise direction at the end of the beam. A pointload of magnitude 8 N is applied from the top of the beam at the starting point. There are two simple supports below the beam. One at the end and another one at a distance of 10 meters from the start. The deflection is restricted at both the supports. Using the sign convention of upward forces and clockwise moment being positive. >>> from sympy.physics.continuum_mechanics.beam import Beam >>> from sympy import symbols >>> E, I = symbols('E, I') >>> b = Beam(30, E, I) >>> b.apply_support(10, 'roller') >>> b.apply_support(30, 'roller') >>> b.apply_load(-8, 0, -1) >>> b.apply_load(120, 30, -2) >>> R_10, R_30 = symbols('R_10, R_30') >>> b.solve_for_reaction_loads(R_10, R_30) >>> b.load -8*SingularityFunction(x, 0, -1) + 6*SingularityFunction(x, 10, -1) + 120*SingularityFunction(x, 30, -2) + 2*SingularityFunction(x, 30, -1) >>> b.slope() (-4*SingularityFunction(x, 0, 2) + 3*SingularityFunction(x, 10, 2) + 120*SingularityFunction(x, 30, 1) + SingularityFunction(x, 30, 2) + 4000/3)/(E*I) tpintrollertR_iÿÿÿÿitM_iþÿÿÿN(Rtstrt apply_loadR=tappendR;(R)tlocttypet reaction_loadtreaction_moment((sElib/python2.7/site-packages/sympy/physics/continuum_mechanics/beam.pyt apply_support%s,c C s|j}t|ƒ}t|ƒ}t|ƒ}|jj||||fƒ|j|t|||ƒ7_|r|jrŒd}t|ƒ‚n|||}xctd|dƒD]K}|j|j ||ƒj |||ƒt|||ƒt |ƒ8_q®WndS(s* This method adds up the loads given to a particular beam object. Parameters ========== value : Sympifyable The magnitude of an applied load. start : Sympifyable The starting point of the applied load. For point moments and point forces this is the location of application. order : Integer The order of the applied load. - For moments, order = -2 - For point loads, order =-1 - For constant distributed load, order = 0 - For ramp loads, order = 1 - For parabolic ramp loads, order = 2 - ... so on. end : Sympifyable, optional An optional argument that can be used if the load has an end point within the length of the beam. Examples ======== There is a beam of length 4 meters. A moment of magnitude 3 Nm is applied in the clockwise direction at the starting point of the beam. A point load of magnitude 4 N is applied from the top of the beam at 2 meters from the starting point and a parabolic ramp load of magnitude 2 N/m is applied below the beam starting from 2 meters to 3 meters away from the starting point of the beam. >>> from sympy.physics.continuum_mechanics.beam import Beam >>> from sympy import symbols >>> E, I = symbols('E, I') >>> b = Beam(4, E, I) >>> b.apply_load(-3, 0, -2) >>> b.apply_load(4, 2, -1) >>> b.apply_load(-2, 2, 2, end=3) >>> b.load -3*SingularityFunction(x, 0, -2) + 4*SingularityFunction(x, 2, -1) - 2*SingularityFunction(x, 2, 2) + 2*SingularityFunction(x, 3, 0) + 4*SingularityFunction(x, 3, 1) + 2*SingularityFunction(x, 3, 2) spIf 'end' is provided the 'order' of the load cannot be negative, i.e. 'end' is only valid for distributed loads.iiN( R R R$RNR#Rt is_negativet ValueErrortrangeRtsubsR ( R)tvaluetstarttordertendRtmsgtfR9((sElib/python2.7/site-packages/sympy/physics/continuum_mechanics/beam.pyRM]s-     c C s4|j}t|ƒ}t|ƒ}t|ƒ}||||f|jkr†|j|t|||ƒ8_|jj||||fƒnd}t|ƒ‚|r0|jr¼d}t|ƒ‚n|||}xctd|dƒD]K}|j|j ||ƒj |||ƒt|||ƒt |ƒ7_qÞWndS(sû This method removes a particular load present on the beam object. Returns a ValueError if the load passed as an argument is not present on the beam. Parameters ========== value : Sympifyable The magnitude of an applied load. start : Sympifyable The starting point of the applied load. For point moments and point forces this is the location of application. order : Integer The order of the applied load. - For moments, order= -2 - For point loads, order=-1 - For constant distributed load, order=0 - For ramp loads, order=1 - For parabolic ramp loads, order=2 - ... so on. end : Sympifyable, optional An optional argument that can be used if the load has an end point within the length of the beam. Examples ======== There is a beam of length 4 meters. A moment of magnitude 3 Nm is applied in the clockwise direction at the starting point of the beam. A pointload of magnitude 4 N is applied from the top of the beam at 2 meters from the starting point and a parabolic ramp load of magnitude 2 N/m is applied below the beam starting from 2 meters to 3 meters away from the starting point of the beam. >>> from sympy.physics.continuum_mechanics.beam import Beam >>> from sympy import symbols >>> E, I = symbols('E, I') >>> b = Beam(4, E, I) >>> b.apply_load(-3, 0, -2) >>> b.apply_load(4, 2, -1) >>> b.apply_load(-2, 2, 2, end=3) >>> b.load -3*SingularityFunction(x, 0, -2) + 4*SingularityFunction(x, 2, -1) - 2*SingularityFunction(x, 2, 2) + 2*SingularityFunction(x, 3, 0) + 4*SingularityFunction(x, 3, 1) + 2*SingularityFunction(x, 3, 2) >>> b.remove_load(-2, 2, 2, end = 3) >>> b.load -3*SingularityFunction(x, 0, -2) + 4*SingularityFunction(x, 2, -1) s4No such load distribution exists on the beam object.spIf 'end' is provided the 'order' of the load cannot be negative, i.e. 'end' is only valid for distributed loads.iiN( R R R$R#RtremoveRURTRVRRWR ( R)RXRYRZR[RR\R]R9((sElib/python2.7/site-packages/sympy/physics/continuum_mechanics/beam.pyt remove_load s"/      cC s|jS(sâ Returns a Singularity Function expression which represents the load distribution curve of the Beam object. Examples ======== There is a beam of length 4 meters. A moment of magnitude 3 Nm is applied in the clockwise direction at the starting point of the beam. A point load of magnitude 4 N is applied from the top of the beam at 2 meters from the starting point and a parabolic ramp load of magnitude 2 N/m is applied below the beam starting from 3 meters away from the starting point of the beam. >>> from sympy.physics.continuum_mechanics.beam import Beam >>> from sympy import symbols >>> E, I = symbols('E, I') >>> b = Beam(4, E, I) >>> b.apply_load(-3, 0, -2) >>> b.apply_load(4, 2, -1) >>> b.apply_load(-2, 3, 2) >>> b.load -3*SingularityFunction(x, 0, -2) + 4*SingularityFunction(x, 2, -1) - 2*SingularityFunction(x, 3, 2) (R#(R)((sElib/python2.7/site-packages/sympy/physics/continuum_mechanics/beam.pytloadìscC s|jS(sß Returns a list of all loads applied on the beam object. Each load in the list is a tuple of form (value, start, order, end). Examples ======== There is a beam of length 4 meters. A moment of magnitude 3 Nm is applied in the clockwise direction at the starting point of the beam. A pointload of magnitude 4 N is applied from the top of the beam at 2 meters from the starting point. Another pointload of magnitude 5 N is applied at same position. >>> from sympy.physics.continuum_mechanics.beam import Beam >>> from sympy import symbols >>> E, I = symbols('E, I') >>> b = Beam(4, E, I) >>> b.apply_load(-3, 0, -2) >>> b.apply_load(4, 2, -1) >>> b.apply_load(5, 2, -1) >>> b.load -3*SingularityFunction(x, 0, -2) + 9*SingularityFunction(x, 2, -1) >>> b.applied_loads [(-3, 0, -2, None), (4, 2, -1, None), (5, 2, -1, None)] (R$(R)((sElib/python2.7/site-packages/sympy/physics/continuum_mechanics/beam.pyt applied_loadssc! G sz|j}|j}|j}|j}t|tƒrX|jdd}|jdd}n |}}d}d} x|jD]} | d|krE|| dt|| d| dƒ7}| ddkrí|| dt|| d| dƒ8}q| ddkr|| dt|| d| dƒ| dt|| ddƒ8}qqx| d|kr¨|| dt|| d| dƒ7}| | dt|| d|| dƒ7} qx| d|krx| | dt|| d|| dƒ7} | ddkr| | dt|| d|| dƒ8} q| ddkr| | dt|| d|| dƒ| dt|| d|dƒ8} qqxqxWt dƒ} || t||dƒ7}| | t|ddƒ8} g} t ||ƒ} t | ||ƒ}| j |ƒt | |ƒ}t |||ƒ}| j |ƒt | |ƒ}t |||j |ƒ}| j |ƒt ||ƒ}t |||j |ƒ}| j |ƒt dƒ}t dƒ}t d ƒ}t d ƒ}tdƒ||t ||ƒ|}tdƒ||t ||||ƒ|||}tdƒ||t t t | |ƒ|ƒ|ƒ|}tdƒ||t ||||ƒ|}xd|jD]Y\}}||kr¿| j |j||ƒ|ƒq‡| j |j|||ƒ|ƒq‡Wxd|jD]Y\}}||kr&| j |j||ƒ|ƒqî| j |j|||ƒ|ƒqîW| j |j||ƒ|j|dƒƒtt| ||||| |Œƒ}t|dƒd } tt|| ƒƒ|_|jj|jƒ|_|ji|dd|6|dd | 6ƒj|jƒ}|ji|dd|6|dd|6|dd | 6ƒj|jƒ}|ji|||6|dd|6|dd | 6ƒj|jƒ}|ji|||6|dd|6|dd|6|dd | 6ƒj|jƒ}|t|ddƒ|t||dƒ|t||dƒ|_|t|ddƒ|t||dƒ|t||dƒ|_d S(s3 Method to find integration constants and reactional variables in a composite beam connected via hinge. This method resolves the composite Beam into its sub-beams and then equations of shear force, bending moment, slope and deflection are evaluated for both of them separately. These equations are then solved for unknown reactions and integration constants using the boundary conditions applied on the Beam. Equal deflection of both sub-beams at the hinge joint gives us another equation to solve the system. Examples ======== A combined beam, with constant fkexural rigidity E*I, is formed by joining a Beam of length 2*l to the right of another Beam of length l. The whole beam is fixed at both of its both end. A point load of magnitude P is also applied from the top at a distance of 2*l from starting point. >>> from sympy.physics.continuum_mechanics.beam import Beam >>> from sympy import symbols >>> E, I = symbols('E, I') >>> l=symbols('l', positive=True) >>> b1=Beam(l ,E,I) >>> b2=Beam(2*l ,E,I) >>> b=b1.join(b2,"hinge") >>> M1, A1, M2, A2, P = symbols('M1 A1 M2 A2 P') >>> b.apply_load(A1,0,-1) >>> b.apply_load(M1,0,-2) >>> b.apply_load(P,2*l,-1) >>> b.apply_load(A2,3*l,-1) >>> b.apply_load(M2,3*l,-2) >>> b.bc_slope=[(0,0), (3*l, 0)] >>> b.bc_deflection=[(0,0), (3*l, 0)] >>> b.solve_for_reaction_loads(M1, A1, M2, A2) >>> b.reaction_loads {A1: -5*P/18, A2: -13*P/18, M1: 5*P*l/18, M2: -4*P*l/9} >>> b.slope() (5*P*l*SingularityFunction(x, 0, 1)/18 - 5*P*SingularityFunction(x, 0, 2)/36 + 5*P*SingularityFunction(x, l, 2)/36)*SingularityFunction(x, 0, 0)/(E*I) - (5*P*l*SingularityFunction(x, 0, 1)/18 - 5*P*SingularityFunction(x, 0, 2)/36 + 5*P*SingularityFunction(x, l, 2)/36)*SingularityFunction(x, l, 0)/(E*I) + (P*l**2/18 - 4*P*l*SingularityFunction(-l + x, 2*l, 1)/9 - 5*P*SingularityFunction(-l + x, 0, 2)/36 + P*SingularityFunction(-l + x, l, 2)/2 - 13*P*SingularityFunction(-l + x, 2*l, 2)/36)*SingularityFunction(x, l, 0)/(E*I) >>> b.deflection() (5*P*l*SingularityFunction(x, 0, 2)/36 - 5*P*SingularityFunction(x, 0, 3)/108 + 5*P*SingularityFunction(x, l, 3)/108)*SingularityFunction(x, 0, 0)/(E*I) - (5*P*l*SingularityFunction(x, 0, 2)/36 - 5*P*SingularityFunction(x, 0, 3)/108 + 5*P*SingularityFunction(x, l, 3)/108)*SingularityFunction(x, l, 0)/(E*I) + (5*P*l**3/54 + P*l**2*(-l + x)/18 - 2*P*l*SingularityFunction(-l + x, 2*l, 2)/9 - 5*P*SingularityFunction(-l + x, 0, 3)/108 + P*SingularityFunction(-l + x, l, 3)/6 - 13*P*SingularityFunction(-l + x, 2*l, 3)/108)*SingularityFunction(x, l, 0)/(E*I) iiiithiÿÿÿÿtC1tC2tC3tC4iiN(R R(R.R/R5R targsRaRRR R RNRRR;RWR=tlistRtdicttzipR%R#t_hinge_beam_slopet_hinge_beam_deflection(!R)t reactionsRR3RCtItI1tI2tload_1tload_2R`Rbteqtshear_1t shear_curve_1t bending_1tmoment_curve_1tshear_2t shear_curve_2t bending_2tmoment_curve_2RcRdReRftslope_1tdef_1tslope_2tdef_2tpositionRXt constantstreaction_values((sElib/python2.7/site-packages/sympy/physics/continuum_mechanics/beam.pyt_solve_hinge_beams#sŠ.     &)H&-*-T         %57-  %  %)$9HDSAcG s¢|jdkr|j|ŒS|j}|j}tdƒ}tdƒ}t|jƒ||ƒ}t|jƒ||ƒ}g}g} t|jƒ|ƒ|} x>|j dD]/\} } | j || ƒ| } |j | ƒq©Wt| |ƒ|}x>|j dD]/\} } |j || ƒ| } | j | ƒqýWt t ||g|| ||f|ƒjdƒ}|d}tt||ƒƒ|_|jj |jƒ|_dS( s Solves for the reaction forces. Examples ======== There is a beam of length 30 meters. A moment of magnitude 120 Nm is applied in the clockwise direction at the end of the beam. A pointload of magnitude 8 N is applied from the top of the beam at the starting point. There are two simple supports below the beam. One at the end and another one at a distance of 10 meters from the start. The deflection is restricted at both the supports. Using the sign convention of upward forces and clockwise moment being positive. >>> from sympy.physics.continuum_mechanics.beam import Beam >>> from sympy import symbols, linsolve, limit >>> E, I = symbols('E, I') >>> R1, R2 = symbols('R1, R2') >>> b = Beam(30, E, I) >>> b.apply_load(-8, 0, -1) >>> b.apply_load(R1, 10, -1) # Reaction force at x = 10 >>> b.apply_load(R2, 30, -1) # Reaction force at x = 30 >>> b.apply_load(120, 30, -2) >>> b.bc_deflection = [(10, 0), (30, 0)] >>> b.load R1*SingularityFunction(x, 10, -1) + R2*SingularityFunction(x, 30, -1) - 8*SingularityFunction(x, 0, -1) + 120*SingularityFunction(x, 30, -2) >>> b.solve_for_reaction_loads(R1, R2) >>> b.reaction_loads {R1: 6, R2: 2} >>> b.load -8*SingularityFunction(x, 0, -1) + 6*SingularityFunction(x, 10, -1) + 120*SingularityFunction(x, 30, -2) + 2*SingularityFunction(x, 30, -1) R@ReRfRRiiN(R'RƒR RRR t shear_forcetbending_momentR R"RWRNRhRRgRiRjR%R#(R)RmRR3ReRft shear_curvet moment_curvet slope_eqstdeflection_eqst slope_curveR€RXteqstdeflection_curvetsolution((sElib/python2.7/site-packages/sympy/physics/continuum_mechanics/beam.pytsolve_for_reaction_loadsªs.$     . cC s|j}t|j|ƒS(s* Returns a Singularity Function expression which represents the shear force curve of the Beam object. Examples ======== There is a beam of length 30 meters. A moment of magnitude 120 Nm is applied in the clockwise direction at the end of the beam. A pointload of magnitude 8 N is applied from the top of the beam at the starting point. There are two simple supports below the beam. One at the end and another one at a distance of 10 meters from the start. The deflection is restricted at both the supports. Using the sign convention of upward forces and clockwise moment being positive. >>> from sympy.physics.continuum_mechanics.beam import Beam >>> from sympy import symbols >>> E, I = symbols('E, I') >>> R1, R2 = symbols('R1, R2') >>> b = Beam(30, E, I) >>> b.apply_load(-8, 0, -1) >>> b.apply_load(R1, 10, -1) >>> b.apply_load(R2, 30, -1) >>> b.apply_load(120, 30, -2) >>> b.bc_deflection = [(10, 0), (30, 0)] >>> b.solve_for_reaction_loads(R1, R2) >>> b.shear_force() -8*SingularityFunction(x, 0, 0) + 6*SingularityFunction(x, 10, 0) + 120*SingularityFunction(x, 30, -1) + 2*SingularityFunction(x, 30, 0) (R R R`(R)R((sElib/python2.7/site-packages/sympy/physics/continuum_mechanics/beam.pyR„ís cC s&ddlm}m}m}|jƒ}|j}|j}g}xA|D]9}t||ƒrl|jd}n|j|jdƒqGW|j ƒt t |ƒƒ}g} g} x9t |ƒD]+\} } | dkr×q¹ny/t tdƒ||| dkf|jjt ƒ|| kftdƒtfƒ} || |ƒ}g}x'|D]}|j|j||ƒƒqFW|j|| d| gƒ|jt|||| ddƒt||| dƒgƒt tt|ƒƒ}t|ƒ}| j|ƒ| j||j|ƒƒWq¹tk rãt|||| ddƒ}t||| dƒ}|j||| d| dƒ||dkrµ||krµ| j||gƒ| j|| d| gƒqä| j|ƒ| j||| d| ƒƒq¹Xq¹Wt tt| ƒƒ} t| ƒ}| | j|ƒ}||fS( sJReturns maximum Shear force and its coordinate in the Beam object.iÿÿÿÿ(tsolvetMultIntervaliitnant+t-i(tsympyRRR‘R„R RgR5RNtsortRhtsett enumerateR tfloatR#trewritetTrueRWtextendR tmaptabstmaxtindextNotImplementedError(R)RRR‘R†Rttermst singularityttermt intervalst shear_valuesR9tst shear_slopetpointstvaltpointt max_sheart initial_sheart final_sheart maximum_shear((sElib/python2.7/site-packages/sympy/physics/continuum_mechanics/beam.pytmax_shear_forcesP      P 9   < & cC s|j}t|jƒ|ƒS(s/ Returns a Singularity Function expression which represents the bending moment curve of the Beam object. Examples ======== There is a beam of length 30 meters. A moment of magnitude 120 Nm is applied in the clockwise direction at the end of the beam. A pointload of magnitude 8 N is applied from the top of the beam at the starting point. There are two simple supports below the beam. One at the end and another one at a distance of 10 meters from the start. The deflection is restricted at both the supports. Using the sign convention of upward forces and clockwise moment being positive. >>> from sympy.physics.continuum_mechanics.beam import Beam >>> from sympy import symbols >>> E, I = symbols('E, I') >>> R1, R2 = symbols('R1, R2') >>> b = Beam(30, E, I) >>> b.apply_load(-8, 0, -1) >>> b.apply_load(R1, 10, -1) >>> b.apply_load(R2, 30, -1) >>> b.apply_load(120, 30, -2) >>> b.bc_deflection = [(10, 0), (30, 0)] >>> b.solve_for_reaction_loads(R1, R2) >>> b.bending_moment() -8*SingularityFunction(x, 0, 1) + 6*SingularityFunction(x, 10, 1) + 120*SingularityFunction(x, 30, 0) + 2*SingularityFunction(x, 30, 1) (R R R„(R)R((sElib/python2.7/site-packages/sympy/physics/continuum_mechanics/beam.pyR…Cs cC s)ddlm}m}m}|jƒ}|j}|j}g}xA|D]9}t||ƒrl|jd}n|j|jdƒqGW|j ƒt t |ƒƒ}g} g} x<t |ƒD].\} } | dkr×q¹ny2t tdƒ||| dkf|jƒjt ƒ|| kftdƒtfƒ} || |ƒ}g}x'|D]}|j|j||ƒƒqIW|j|| d| gƒ|jt|||| ddƒt||| dƒgƒt tt|ƒƒ}t|ƒ}| j|ƒ| j||j|ƒƒWq¹tk ræt|||| ddƒ}t||| dƒ}|j||| d| dƒ||dkr¸||kr¸| j||gƒ| j|| d| gƒqç| j|ƒ| j||| d| ƒƒq¹Xq¹Wt tt| ƒƒ} t| ƒ}| | j|ƒ}||fS( sJReturns maximum Shear force and its coordinate in the Beam object.iÿÿÿÿ(RRR‘iiR’R“R”i(R•RRR‘R…R RgR5RNR–RhR—R˜R R™R„RšR›RWRœR RRžRŸR R¡(R)RRR‘t bending_curveRR¢R£R¤R¥t moment_valuesR9R§t moment_slopeR©RªR«t max_momenttinitial_momentt final_momenttmaximum_moment((sElib/python2.7/site-packages/sympy/physics/continuum_mechanics/beam.pyt max_bmomentesP      S 9   < & cC s‰ddlm}m}|tdƒ|jdkf|jƒ|j|jkftdƒtfƒ}||j|ƒ|jdt j ƒ}|S(s¨ Returns a Set of point(s) with zero bending moment and where bending moment curve of the beam object changes its sign from negative to positive or vice versa. Examples ======== There is is 10 meter long overhanging beam. There are two simple supports below the beam. One at the start and another one at a distance of 6 meters from the start. Point loads of magnitude 10KN and 20KN are applied at 2 meters and 4 meters from start respectively. A Uniformly distribute load of magnitude of magnitude 3KN/m is also applied on top starting from 6 meters away from starting point till end. Using the sign convention of upward forces and clockwise moment being positive. >>> from sympy.physics.continuum_mechanics.beam import Beam >>> from sympy import symbols >>> E, I = symbols('E, I') >>> b = Beam(10, E, I) >>> b.apply_load(-4, 0, -1) >>> b.apply_load(-46, 6, -1) >>> b.apply_load(10, 2, -1) >>> b.apply_load(20, 4, -1) >>> b.apply_load(3, 6, 0) >>> b.point_cflexure() [10/3] iÿÿÿÿ(RR R’itdomain( R•RR R™R R…RR›RšRtReals(R)RR R‡R©((sElib/python2.7/site-packages/sympy/physics/continuum_mechanics/beam.pytpoint_cflexure™s cC s]|j}|j}|j}|jdkr1|jS|jdsQt|jƒ|ƒSt|t ƒr¨|jdkr¨|j }d}d}d}xt t |ƒƒD]}|dkrË||ddj d}nt dƒ|t|jƒ||d|||fƒ} |t |ƒdkra||| t||dƒ|| t|||dj ddƒ7}n||| t||dƒ7}| j|||dj dƒ}qW|Stdƒ} tt dƒ|||jƒ|ƒ| } g} x>|jdD]/\} }| j|| ƒ|}| j|ƒqóWtt| | ƒƒ}| ji|dd| 6ƒ} | S(sD Returns a Singularity Function expression which represents the slope the elastic curve of the Beam object. Examples ======== There is a beam of length 30 meters. A moment of magnitude 120 Nm is applied in the clockwise direction at the end of the beam. A pointload of magnitude 8 N is applied from the top of the beam at the starting point. There are two simple supports below the beam. One at the end and another one at a distance of 10 meters from the start. The deflection is restricted at both the supports. Using the sign convention of upward forces and clockwise moment being positive. >>> from sympy.physics.continuum_mechanics.beam import Beam >>> from sympy import symbols >>> E, I = symbols('E, I') >>> R1, R2 = symbols('R1, R2') >>> b = Beam(30, E, I) >>> b.apply_load(-8, 0, -1) >>> b.apply_load(R1, 10, -1) >>> b.apply_load(R2, 30, -1) >>> b.apply_load(120, 30, -2) >>> b.bc_deflection = [(10, 0), (30, 0)] >>> b.solve_for_reaction_loads(R1, R2) >>> b.slope() (-4*SingularityFunction(x, 0, 2) + 3*SingularityFunction(x, 10, 2) + 120*SingularityFunction(x, 30, 1) + SingularityFunction(x, 30, 2) + 4000/3)/(E*I) R@RR?iiRe(R RRR'RkR"RRR5R RgRVtlenRR R…RRWRRNRhR(R)RRCRnRgRt prev_slopetprev_endR9t slope_valueReRŠtbc_eqsR€RXR‹R((sElib/python2.7/site-packages/sympy/physics/continuum_mechanics/beam.pyRÃs>      8.% +cC ss|j}|j}|j}|jdkr1|jS|jd rH|jd rHt|tƒrí|jdkrí|j}d}d}d}d}xZt t |ƒƒD]F} | dkrÍ|| ddjd}nt dƒ|t |j ƒ|| d|||fƒ} || } t | |||fƒ} | t |ƒdkr…||| t||dƒ|| t||| djddƒ7}n||| t||dƒ7}| j||| djdƒ}| j||| djdƒ}qŸW|S|j} t| dƒ}t dƒ||t t |j ƒ|ƒ|ƒ|d||dS|jds…|j} t| dƒ}t |jƒ|ƒ|S|jd r0|jdr0t|tƒr@|jdkr@|j}d}d}d}d}xZt t |ƒƒD]F} | dkr || ddjd}nt dƒ|t |j ƒ|| d|||fƒ} || } t | |||fƒ} | t |ƒdkrØ||| t||dƒ|| t||| djddƒ7}n||| t||dƒ7}| j||| djdƒ}| j||| djdƒ}qòW|S|j} t| dƒ\}}t |j ƒ|ƒ|}t ||ƒ|}g}x>|jdD]/\}}|j||ƒ|}|j|ƒqŸWtt|||fƒƒ}|ji|dd|6|dd|6ƒ}t dƒ|||St|tƒrÐ|jdkrÐ|j}d}d}d}d}xZt t |ƒƒD]F} | dkr°|| ddjd}nt dƒ|t |j ƒ|| d|||fƒ} || } t | |||fƒ} | t |ƒdkrh||| t||dƒ|| t||| djddƒ7}n||| t||dƒ7}| j||| djdƒ}| j||| djdƒ}q‚W|Std ƒ}t |jƒ|ƒ|}g}x>|jdD]/\}}|j||ƒ|}|j|ƒq Wtt||ƒƒ}|ji|dd|6ƒ}|S( sW Returns a Singularity Function expression which represents the elastic curve or deflection of the Beam object. Examples ======== There is a beam of length 30 meters. A moment of magnitude 120 Nm is applied in the clockwise direction at the end of the beam. A pointload of magnitude 8 N is applied from the top of the beam at the starting point. There are two simple supports below the beam. One at the end and another one at a distance of 10 meters from the start. The deflection is restricted at both the supports. Using the sign convention of upward forces and clockwise moment being positive. >>> from sympy.physics.continuum_mechanics.beam import Beam >>> from sympy import symbols >>> E, I = symbols('E, I') >>> R1, R2 = symbols('R1, R2') >>> b = Beam(30, E, I) >>> b.apply_load(-8, 0, -1) >>> b.apply_load(R1, 10, -1) >>> b.apply_load(R2, 30, -1) >>> b.apply_load(120, 30, -2) >>> b.bc_deflection = [(10, 0), (30, 0)] >>> b.solve_for_reaction_loads(R1, R2) >>> b.deflection() (4000*x/3 - 4*SingularityFunction(x, 0, 3)/3 + SingularityFunction(x, 10, 3) + 60*SingularityFunction(x, 30, 2) + SingularityFunction(x, 30, 3)/3 - 12000)/(E*I) R@RRR?iis3:5t4Rf(R RRR'RlR"R5R RgRVR¼RR R…RRWR!RRRNRhRR(R)RRCRnRgR½tprev_defR¾RR9R¿trecent_segment_slopetdeflection_valueR*RtconstantReRfRŠRŒRÀR€RXR‹((sElib/python2.7/site-packages/sympy/physics/continuum_mechanics/beam.pyRs¶     8 .!% B    8 .!% -  8 .!% c C s ddlm}m}|tdƒ|jdkf|jƒ|j|jkftdƒtfƒ}||j|ƒ|jdt j ƒ}|j ƒ}g|D]}|j |j|ƒ^q˜}t tt|ƒƒ}t|ƒdkrt|ƒ}||j|ƒ|fSdSdS(sq Returns point of max deflection and its coresponding deflection value in a Beam object. iÿÿÿÿ(RR R’iR¹N(R•RR R™R RRR›RšRRºRRWRhRRžR¼RŸR R&( R)RR RŠR©RŒRt deflectionstmax_def((sElib/python2.7/site-packages/sympy/physics/continuum_mechanics/beam.pytmax_deflectionŠs  ( c C sÑ|jƒ}|d kr!i}nxK|jtƒD]:}||jkrLq1n||kr1td|ƒ‚q1q1W|j|krŽ||j}n |j}t|j|ƒ|jd|fddddddd d ƒS( s Returns a plot for Shear force present in the Beam object. Parameters ========== subs : dictionary Python dictionary containing Symbols as key and their corresponding values. Examples ======== There is a beam of length 8 meters. A constant distributed load of 10 KN/m is applied from half of the beam till the end. There are two simple supports below the beam, one at the starting point and another at the ending point of the beam. A pointload of magnitude 5 KN is also applied from top of the beam, at a distance of 4 meters from the starting point. Take E = 200 GPa and I = 400*(10**-6) meter**4. Using the sign convention of downwards forces being positive. >>> from sympy.physics.continuum_mechanics.beam import Beam >>> from sympy import symbols >>> R1, R2 = symbols('R1, R2') >>> b = Beam(8, 200*(10**9), 400*(10**-6)) >>> b.apply_load(5000, 2, -1) >>> b.apply_load(R1, 0, -1) >>> b.apply_load(R2, 8, -1) >>> b.apply_load(10000, 4, 0, end=8) >>> b.bc_deflection = [(0, 0), (8, 0)] >>> b.solve_for_reaction_loads(R1, R2) >>> b.plot_shear_force() Plot object containing: [0]: cartesian line: -13750*SingularityFunction(x, 0, 0) + 5000*SingularityFunction(x, 2, 0) + 10000*SingularityFunction(x, 4, 1) - 31250*SingularityFunction(x, 8, 0) - 10000*SingularityFunction(x, 8, 1) for x over (0.0, 8.0) sValue of %s was not passed.ittitles Shear ForcetxlabelR€tylabeltValuet line_colortgN( R„R&tatomsRR RURRRW(R)RWR„tsymR((sElib/python2.7/site-packages/sympy/physics/continuum_mechanics/beam.pytplot_shear_force¡s%     'c C sÑ|jƒ}|d kr!i}nxK|jtƒD]:}||jkrLq1n||kr1td|ƒ‚q1q1W|j|krŽ||j}n |j}t|j|ƒ|jd|fddddddd d ƒS( s Returns a plot for Bending moment present in the Beam object. Parameters ========== subs : dictionary Python dictionary containing Symbols as key and their corresponding values. Examples ======== There is a beam of length 8 meters. A constant distributed load of 10 KN/m is applied from half of the beam till the end. There are two simple supports below the beam, one at the starting point and another at the ending point of the beam. A pointload of magnitude 5 KN is also applied from top of the beam, at a distance of 4 meters from the starting point. Take E = 200 GPa and I = 400*(10**-6) meter**4. Using the sign convention of downwards forces being positive. >>> from sympy.physics.continuum_mechanics.beam import Beam >>> from sympy import symbols >>> R1, R2 = symbols('R1, R2') >>> b = Beam(8, 200*(10**9), 400*(10**-6)) >>> b.apply_load(5000, 2, -1) >>> b.apply_load(R1, 0, -1) >>> b.apply_load(R2, 8, -1) >>> b.apply_load(10000, 4, 0, end=8) >>> b.bc_deflection = [(0, 0), (8, 0)] >>> b.solve_for_reaction_loads(R1, R2) >>> b.plot_bending_moment() Plot object containing: [0]: cartesian line: -13750*SingularityFunction(x, 0, 1) + 5000*SingularityFunction(x, 2, 1) + 5000*SingularityFunction(x, 4, 2) - 31250*SingularityFunction(x, 8, 1) - 5000*SingularityFunction(x, 8, 2) for x over (0.0, 8.0) sValue of %s was not passed.iRÉsBending MomentRÊR€RËRÌRÍtbN( R…R&RÏRR RURRRW(R)RWR…RÐR((sElib/python2.7/site-packages/sympy/physics/continuum_mechanics/beam.pytplot_bending_momentÕs%     'c C sÑ|jƒ}|d kr!i}nxK|jtƒD]:}||jkrLq1n||kr1td|ƒ‚q1q1W|j|krŽ||j}n |j}t|j|ƒ|jd|fddddddd d ƒS( s\ Returns a plot for slope of deflection curve of the Beam object. Parameters ========== subs : dictionary Python dictionary containing Symbols as key and their corresponding values. Examples ======== There is a beam of length 8 meters. A constant distributed load of 10 KN/m is applied from half of the beam till the end. There are two simple supports below the beam, one at the starting point and another at the ending point of the beam. A pointload of magnitude 5 KN is also applied from top of the beam, at a distance of 4 meters from the starting point. Take E = 200 GPa and I = 400*(10**-6) meter**4. Using the sign convention of downwards forces being positive. >>> from sympy.physics.continuum_mechanics.beam import Beam >>> from sympy import symbols >>> R1, R2 = symbols('R1, R2') >>> b = Beam(8, 200*(10**9), 400*(10**-6)) >>> b.apply_load(5000, 2, -1) >>> b.apply_load(R1, 0, -1) >>> b.apply_load(R2, 8, -1) >>> b.apply_load(10000, 4, 0, end=8) >>> b.bc_deflection = [(0, 0), (8, 0)] >>> b.solve_for_reaction_loads(R1, R2) >>> b.plot_slope() Plot object containing: [0]: cartesian line: -8.59375e-5*SingularityFunction(x, 0, 2) + 3.125e-5*SingularityFunction(x, 2, 2) + 2.08333333333333e-5*SingularityFunction(x, 4, 3) - 0.0001953125*SingularityFunction(x, 8, 2) - 2.08333333333333e-5*SingularityFunction(x, 8, 3) + 0.00138541666666667 for x over (0.0, 8.0) sValue of %s was not passed.iRÉtSlopeRÊR€RËRÌRÍtmN( RR&RÏRR RURRRW(R)RWRRÐR((sElib/python2.7/site-packages/sympy/physics/continuum_mechanics/beam.pyt plot_slope s%     'c C sÑ|jƒ}|d kr!i}nxK|jtƒD]:}||jkrLq1n||kr1td|ƒ‚q1q1W|j|krŽ||j}n |j}t|j|ƒ|jd|fddddddd d ƒS( s| Returns a plot for deflection curve of the Beam object. Parameters ========== subs : dictionary Python dictionary containing Symbols as key and their corresponding values. Examples ======== There is a beam of length 8 meters. A constant distributed load of 10 KN/m is applied from half of the beam till the end. There are two simple supports below the beam, one at the starting point and another at the ending point of the beam. A pointload of magnitude 5 KN is also applied from top of the beam, at a distance of 4 meters from the starting point. Take E = 200 GPa and I = 400*(10**-6) meter**4. Using the sign convention of downwards forces being positive. >>> from sympy.physics.continuum_mechanics.beam import Beam >>> from sympy import symbols >>> R1, R2 = symbols('R1, R2') >>> b = Beam(8, 200*(10**9), 400*(10**-6)) >>> b.apply_load(5000, 2, -1) >>> b.apply_load(R1, 0, -1) >>> b.apply_load(R2, 8, -1) >>> b.apply_load(10000, 4, 0, end=8) >>> b.bc_deflection = [(0, 0), (8, 0)] >>> b.solve_for_reaction_loads(R1, R2) >>> b.plot_deflection() Plot object containing: [0]: cartesian line: 0.00138541666666667*x - 2.86458333333333e-5*SingularityFunction(x, 0, 3) + 1.04166666666667e-5*SingularityFunction(x, 2, 3) + 5.20833333333333e-6*SingularityFunction(x, 4, 4) - 6.51041666666667e-5*SingularityFunction(x, 8, 3) - 5.20833333333333e-6*SingularityFunction(x, 8, 4) for x over (0.0, 8.0) sValue of %s was not passed.iRÉt DeflectionRÊR€RËRÌRÍtrN( RR&RÏRR RURRRW(R)RWRRÐR((sElib/python2.7/site-packages/sympy/physics/continuum_mechanics/beam.pytplot_deflection=s&     !tmodulesRRcC s{tdkrtdƒ‚n tj}tdkr?tdƒ‚n tj}|j}|dkrfi}nxQ|jƒjt ƒD]:}||jkr—q|n||kr|t d|ƒ‚q|q|W|j |krÙ||j }n |j }t ||j ƒj|ƒjtƒdƒ}t ||jƒj|ƒjtƒdƒ}t ||jƒj|ƒjtƒdƒ} t ||jƒj|ƒjtƒdƒ} |dt|ƒdd|ƒ} |jdd ƒ\} } | dj| || ƒƒ| djd ƒ| d j| || ƒƒ| d jd ƒ| d j| | | ƒƒ| d jd ƒ| dj| | | ƒƒ| djdƒ| jƒ| S(sÉ Returns Axes object containing subplots of Shear Force, Bending Moment, Slope and Deflection of the Beam object. Parameters ========== subs : dictionary Python dictionary containing Symbols as key and their corresponding values. .. note:: This method only works if numpy and matplotlib libraries are installed on the system. Examples ======== There is a beam of length 8 meters. A constant distributed load of 10 KN/m is applied from half of the beam till the end. There are two simple supports below the beam, one at the starting point and another at the ending point of the beam. A pointload of magnitude 5 KN is also applied from top of the beam, at a distance of 4 meters from the starting point. Take E = 200 GPa and I = 400*(10**-6) meter**4. Using the sign convention of downwards forces being positive. >>> from sympy.physics.continuum_mechanics.beam import Beam >>> from sympy import symbols >>> R1, R2 = symbols('R1, R2') >>> b = Beam(8, 200*(10**9), 400*(10**-6)) >>> b.apply_load(5000, 2, -1) >>> b.apply_load(R1, 0, -1) >>> b.apply_load(R2, 8, -1) >>> b.apply_load(10000, 4, 0, end=8) >>> b.bc_deflection = [(0, 0), (8, 0)] >>> b.solve_for_reaction_loads(R1, R2) >>> axes = b.plot_loading_results() s&Install matplotlib to use this method.s!Install numpy to use this method.sValue of %s was not passed.Ritnumidiis Shear ForcesBending MomentiRÔiR×N(RR&t ImportErrorRRRR RRÏRRURRR„RWRšR R…RR™tsubplotsRt set_titlet tight_layout(R)RWtpltRR RÐRtsheartmomentRRR©tfigtaxs((sElib/python2.7/site-packages/sympy/physics/continuum_mechanics/beam.pytplot_loading_resultsssP'           !   N(RR('t__name__t __module__t__doc__RR+R1tpropertyR2RtsetterR RRR:R;R=RGRSR&RMR_R`RaRƒRŽR„R°R…R¸R»RRRÈRÑRÓRÖRÙRRå(((sElib/python2.7/site-packages/sympy/physics/continuum_mechanics/beam.pyRsN1%  ? 8 C L ‡ C " 4 " 4 * D ƒ  4 4 4 6 tBeam3DcB seZdZedƒd„Zed„ƒZejd„ƒZed„ƒZejd„ƒZed„ƒZ e jd„ƒZ ed „ƒZ ed „ƒZ ed „ƒZ d d „Z d d„Zdd„Zd„Zd„Zd„Zd„Zd„Zd„Zd„Zd„ZRS(s This class handles loads applied in any direction of a 3D space along with unequal values of Second moment along different axes. .. note:: While solving a beam bending problem, a user should choose its own sign convention and should stick to it. The results will automatically follow the chosen sign convention. This class assumes that any kind of distributed load/moment is applied through out the span of a beam. Examples ======== There is a beam of l meters long. A constant distributed load of magnitude q is applied along y-axis from start till the end of beam. A constant distributed moment of magnitude m is also applied along z-axis from start till the end of beam. Beam is fixed at both of its end. So, deflection of the beam at the both ends is restricted. >>> from sympy.physics.continuum_mechanics.beam import Beam3D >>> from sympy import symbols, simplify >>> l, E, G, I, A = symbols('l, E, G, I, A') >>> b = Beam3D(l, E, G, I, A) >>> x, q, m = symbols('x, q, m') >>> b.apply_load(q, 0, 0, dir="y") >>> b.apply_moment_load(m, 0, -1, dir="z") >>> b.shear_force() [0, -q*x, 0] >>> b.bending_moment() [0, 0, -m*x + q*x**2/2] >>> b.bc_slope = [(0, [0, 0, 0]), (l, [0, 0, 0])] >>> b.bc_deflection = [(0, [0, 0, 0]), (l, [0, 0, 0])] >>> b.solve_slope_deflection() >>> b.slope() [0, 0, l*x*(-l*q + 3*l*(A*G*l*(l*q - 2*m) + 12*E*I*q)/(2*(A*G*l**2 + 12*E*I)) + 3*m)/(6*E*I) + q*x**3/(6*E*I) + x**2*(-l*(A*G*l*(l*q - 2*m) + 12*E*I*q)/(2*(A*G*l**2 + 12*E*I)) - m)/(2*E*I)] >>> dx, dy, dz = b.deflection() >>> dx 0 >>> dz 0 >>> expectedy = ( ... -l**2*q*x**2/(12*E*I) + l**2*x**2*(A*G*l*(l*q - 2*m) + 12*E*I*q)/(8*E*I*(A*G*l**2 + 12*E*I)) ... + l*m*x**2/(4*E*I) - l*x**3*(A*G*l*(l*q - 2*m) + 12*E*I*q)/(12*E*I*(A*G*l**2 + 12*E*I)) - m*x**3/(6*E*I) ... + q*x**4/(24*E*I) + l*x*(A*G*l*(l*q - 2*m) + 12*E*I*q)/(2*A*G*(A*G*l**2 + 12*E*I)) - q*x**2/(2*A*G) ... ) >>> simplify(dy - expectedy) 0 References ========== .. [1] http://homes.civil.aau.dk/jc/FemteSemester/Beams3D.pdf RcC stt|ƒj||||ƒ||_||_dddg|_dddg|_dddg|_dddg|_dddg|_ dS(sèInitializes the class. Parameters ========== length : Sympifyable A Symbol or value representing the Beam's length. elastic_modulus : Sympifyable A SymPy expression representing the Beam's Modulus of Elasticity. It is a measure of the stiffness of the Beam material. shear_modulus : Sympifyable A SymPy expression representing the Beam's Modulus of rigidity. It is a measure of rigidity of the Beam material. second_moment : Sympifyable or list A list of two elements having SymPy expression representing the Beam's Second moment of area. First value represent Second moment across y-axis and second across z-axis. Single SymPy expression can be passed if both values are same area : Sympifyable A SymPy expression representing the Beam's cross-sectional area in a plane prependicular to length of the Beam. variable : Symbol, optional A Symbol object that will be used as the variable along the beam while representing the load, shear, moment, slope and deflection curve. By default, it is set to ``Symbol('x')``. iN( tsuperRëR+t shear_modulustareat _load_vectort_moment_load_vectort_load_Singularityt_slopet _deflection(R)RRRíRRîR ((sElib/python2.7/site-packages/sympy/physics/continuum_mechanics/beam.pyR+ s  cC s|jS(sYoung's Modulus of the Beam. (t_shear_modulus(R)((sElib/python2.7/site-packages/sympy/physics/continuum_mechanics/beam.pyRí-scC st|ƒ|_dS(N(R Rô(R)R8((sElib/python2.7/site-packages/sympy/physics/continuum_mechanics/beam.pyRí2scC s|jS(s#Second moment of area of the Beam. (R/(R)((sElib/python2.7/site-packages/sympy/physics/continuum_mechanics/beam.pyR6scC sMt|tƒr:g|D]}t|ƒ^q}||_nt|ƒ|_dS(N(R5RhR R/(R)R9R((sElib/python2.7/site-packages/sympy/physics/continuum_mechanics/beam.pyR;s cC s|jS(s"Cross-sectional area of the Beam. (t_area(R)((sElib/python2.7/site-packages/sympy/physics/continuum_mechanics/beam.pyRîCscC st|ƒ|_dS(N(R Rõ(R)ta((sElib/python2.7/site-packages/sympy/physics/continuum_mechanics/beam.pyRîHscC s|jS(sL Returns a three element list representing the load vector. (Rï(R)((sElib/python2.7/site-packages/sympy/physics/continuum_mechanics/beam.pyt load_vectorLscC s|jS(sQ Returns a three element list representing moment loads on Beam. (Rð(R)((sElib/python2.7/site-packages/sympy/physics/continuum_mechanics/beam.pytmoment_load_vectorSscC s|jS(s  Returns a dictionary of boundary conditions applied on the beam. The dictionary has two keywords namely slope and deflection. The value of each keyword is a list of tuple, where each tuple contains loaction and value of a boundary condition in the format (location, value). Further each value is a list corresponding to slope or deflection(s) values along three axes at that location. Examples ======== There is a beam of length 4 meters. The slope at 0 should be 4 along the x-axis and 0 along others. At the other end of beam, deflection along all the three axes should be zero. >>> from sympy.physics.continuum_mechanics.beam import Beam3D >>> from sympy import symbols >>> l, E, G, I, A, x = symbols('l, E, G, I, A, x') >>> b = Beam3D(30, E, G, I, A, x) >>> b.bc_slope = [(0, (4, 0, 0))] >>> b.bc_deflection = [(4, [0, 0, 0])] >>> b.boundary_conditions {'deflection': [(4, [0, 0, 0])], 'slope': [(0, (4, 0, 0))]} Here the deflection of the beam should be ``0`` along all the three axes at ``4``. Similarly, the slope of the beam should be ``4`` along x-axis and ``0`` along y and z axis at ``0``. (R"(R)((sElib/python2.7/site-packages/sympy/physics/continuum_mechanics/beam.pyR:ZstycC s|j}t|ƒ}t|ƒ}t|ƒ}|dkr|dks[|jdc|7>> from sympy.physics.continuum_mechanics.beam import Beam3D >>> from sympy import symbols >>> l, E, G, I, A, x = symbols('l, E, G, I, A, x') >>> b = Beam3D(30, E, G, I, A, x) >>> b.apply_load(8, start=0, order=0, dir="y") >>> b.apply_load(9*x, start=0, order=0, dir="z") >>> b.bc_deflection = [(0, [0, 0, 0]), (30, [0, 0, 0])] >>> R1, R2, R3, R4 = symbols('R1, R2, R3, R4') >>> b.apply_load(R1, start=0, order=-1, dir="y") >>> b.apply_load(R2, start=30, order=-1, dir="y") >>> b.apply_load(R3, start=0, order=-1, dir="z") >>> b.apply_load(R4, start=30, order=-1, dir="z") >>> b.solve_for_reaction_loads(R1, R2, R3, R4) >>> b.reaction_loads {R1: -120, R2: -120, R3: -1350, R4: -2700} iis2Ambiguous solution for %s in different directions.N(R RRñR RVthasR¼R RhRRgRiRjR%RUtupdate(R)treactionRR3tqR`t shear_curvesRát moment_curvesR9RØtreactR†R‡tsoltsol_dictR2tkey((sElib/python2.7/site-packages/sympy/physics/continuum_mechanics/beam.pyRŽÏs$   ""?"   cC sI|j}|j}t|d |ƒt|d |ƒt|d |ƒgS(s Returns a list of three expressions which represents the shear force curve of the Beam object along all three axes. iii(R RïR (R)RRÿ((sElib/python2.7/site-packages/sympy/physics/continuum_mechanics/beam.pyR„ÿs  cC s|jƒdS(sY Returns expression of Axial shear force present inside the Beam object. i(R„(R)((sElib/python2.7/site-packages/sympy/physics/continuum_mechanics/beam.pyt axial_forcescC se|j}|j}|jƒ}t|d |ƒt|d |d|ƒt|d |d|ƒgS(s Returns a list of three expressions which represents the bending moment curve of the Beam object along all three axes. iii(R RðR„R (R)RRÕRá((sElib/python2.7/site-packages/sympy/physics/continuum_mechanics/beam.pyR…s    *cC s|jƒdS(sX Returns expression of Torsional moment present inside the Beam object. i(R…(R)((sElib/python2.7/site-packages/sympy/physics/continuum_mechanics/beam.pyttorsional_momentscC sûddlm}m}m}m}|j}|j}|j}|j}|j } t | t ƒrv| d| d} } n | } } |j } |j } |j}|dƒ}|dƒ}||| |||ƒ|ƒ|ƒ| d}|||dƒ||ƒƒjd}tdƒ}tdƒ}t t|j|dƒ|j||ƒg||ƒjdƒ}|ji|d|6|d|6ƒ}|j|ƒ}||jd<||jds,"ÿÿÿÿÿº