MZdPodZddlmZddlmZddlmZddlmZddl m Z ddl m Z m Z ddlmZdd lmZdd l mZmZGd d Zy )zB This module can be used to solve problems related to 2D Trusses. )inf)Add)Mul)Symbol)sympify)Matrixpi)sqrt)zeros)sincosceZdZdZdZedZedZedZedZ edZ edZ ed Z ed Z ed Zd Zd ZdZdZdZdZdZdZdZdZdZy)Trussa A Truss is an assembly of members such as beams, connected by nodes, that create a rigid structure. In engineering, a truss is a structure that consists of two-force members only. Trusses are extremely important in engineering applications and can be seen in numerous real-world applications like bridges. Examples ======== There is a Truss consisting of four nodes and five members connecting the nodes. A force P acts downward on the node D and there also exist pinned and roller joints on the nodes A and B respectively. .. image:: truss_example.png >>> from sympy.physics.continuum_mechanics.truss import Truss >>> t = Truss() >>> t.add_node("node_1", 0, 0) >>> t.add_node("node_2", 6, 0) >>> t.add_node("node_3", 2, 2) >>> t.add_node("node_4", 2, 0) >>> t.add_member("member_1", "node_1", "node_4") >>> t.add_member("member_2", "node_2", "node_4") >>> t.add_member("member_3", "node_1", "node_3") >>> t.add_member("member_4", "node_2", "node_3") >>> t.add_member("member_5", "node_3", "node_4") >>> t.apply_load("node_4", magnitude=10, direction=270) >>> t.apply_support("node_1", type="fixed") >>> t.apply_support("node_2", type="roller") cg|_i|_i|_i|_g|_g|_g|_g|_i|_i|_ i|_ i|_ y)z' Initializes the class N) _nodes_members_loads _supports _node_labels_node_positions_node_position_x_node_position_y_nodes_occupied_reaction_loads_internal_forces_node_coordinatesselfs I/usr/lib/python3/dist-packages/sympy/physics/continuum_mechanics/truss.py__init__zTruss.__init__6s`   ! " "!! "!#c|jS)zL Returns the nodes of the truss along with their positions. )rrs rnodesz Truss.nodesG {{r!c|jS)z7 Returns the node labels of the truss. )rrs r node_labelszTruss.node_labelsNs    r!c|jS)zB Returns the positions of the nodes of the truss. )rrs rnode_positionszTruss.node_positionsU ###r!c|jSzW Returns the members of the truss along with the start and end points. )rrs rmembersz Truss.members\s }}r!c|jSr+)_member_labelsrs r member_labelszTruss.member_labelscs """r!c|jS)z Returns the nodes with provided supports along with the kind of support provided i.e. pinned or roller. )rrs rsupportszTruss.supportsjs ~~r!c|jS)z8 Returns the loads acting on the truss. )rrs rloadsz Truss.loadsrr$r!c|jS)z^ Returns the reaction forces for all supports which are all initialized to 0. )rrs rreaction_loadszTruss.reaction_loadsyr)r!c|jS)z] Returns the internal forces for all members which are all initialized to 0. )rrs rinternal_forceszTruss.internal_forcess $$$r!cXt|}t|}||jvr td||jvrP||jvrB|jj ||jj |k(r td|j j|||f|jj||jj||f|jj||jj|||g|j|<y)a This method adds a node to the truss along with its name/label and its location. Parameters ========== label: String or a Symbol The label for a node. It is the only way to identify a particular node. x: Sympifyable The x-coordinate of the position of the node. y: Sympifyable The y-coordinate of the position of the node. Examples ======== >>> from sympy.physics.continuum_mechanics.truss import Truss >>> t = Truss() >>> t.add_node('A', 0, 0) >>> t.nodes [('A', 0, 0)] >>> t.add_node('B', 3, 0) >>> t.nodes [('A', 0, 0), ('B', 3, 0)] z!Node needs to have a unique labelz+A node already exists at the given positionN) rr ValueErrorrrindexrappendrr)rlabelxys radd_nodezTruss.add_nodes!6 AJ AJ D%% %@A A $'' 'A1F1F,F4K`K`KfKfghKikolAlAlGlGHIlJLJJK K KK  q!} -    $ $U +  ' 'A /  ! ! ( ( +  ! ! ( ( +-.FD " "5 )r!ctt|jD]3}|j||k(s|j|}|j |}5||jvr t d|jj}|D]7}||j|dk(s||j|dk(s.t d|jj|f|jj||jj||f|jj||j j||t|jvr|jj||t|jvr|jj||j j|y)a% This method removes a node from the truss. Parameters ========== label: String or Symbol The label of the node to be removed. Examples ======== >>> from sympy.physics.continuum_mechanics.truss import Truss >>> t = Truss() >>> t.add_node('A', 0, 0) >>> t.nodes [('A', 0, 0)] >>> t.add_node('B', 3, 0) >>> t.nodes [('A', 0, 0), ('B', 3, 0)] >>> t.remove_node('A') >>> t.nodes [('B', 3, 0)] z No such node exists in the trussrz1The node given has members already attached to itN)rangelenr#rrrr9rcopyrremoverlistrpoprr)rr<ir=r>members_duplicatemembers r remove_nodezTruss.remove_nodes0s4::' -A  #u,))!,))!, - )) )?@ @!% 2 2 4 + ZDMM&1!44vAVWXAY8Y$%XYY Z KK  q!} -    $ $U +  ' 'A /  ! ! ( ( +  ! ! ( ( +T[[)) &T^^,,""5)  " " & &u -r!ct||jvs||jvs||k(r td|t|jvr td|jj ||fr td||g|j|<d|j||f<d|j||f<d|j |<y)a This method adds a member between any two nodes in the given truss. Parameters ========== label: String or Symbol The label for a member. It is the only way to identify a particular member. start: String or Symbol The label of the starting point/node of the member. end: String or Symbol The label of the ending point/node of the member. Examples ======== >>> from sympy.physics.continuum_mechanics.truss import Truss >>> t = Truss() >>> t.add_node('A', 0, 0) >>> t.add_node('B', 3, 0) >>> t.add_node('C', 2, 2) >>> t.add_member('AB', 'A', 'B') >>> t.members {'AB': ['A', 'B']} z;The start and end points of the member must be unique nodesz9A member with the same label already exists for the trussz-A member already exists between the two nodesTrN)rr9rFrrgetr)rr<startends r add_memberzTruss.add_members8 )) )S8I8I-IUTWZZ[ [ d4==) )XY Y  ! ! % %ucl 3LM M%*3>> from sympy.physics.continuum_mechanics.truss import Truss >>> t = Truss() >>> t.add_node('A', 0, 0) >>> t.add_node('B', 3, 0) >>> t.add_node('C', 2, 2) >>> t.add_member('AB', 'A', 'B') >>> t.add_member('AC', 'A', 'C') >>> t.add_member('BC', 'B', 'C') >>> t.members {'AB': ['A', 'B'], 'AC': ['A', 'C'], 'BC': ['B', 'C']} >>> t.remove_member('AC') >>> t.members {'AB': ['A', 'B'], 'BC': ['B', 'C']} z"No such member exists in the TrussrrAN)rFrr9rrGr)rr<s r remove_memberzTruss.remove_members4 T]]+ +AB B  $ $dmmE&:1&=t}}U?STU?V%W X  $ $dmmE&:1&=t}}U?STU?V%W X MM  e $  ! ! % %e ,r!c t ||jvr td||jvr td|jD]v}|d|k(s ||d|df|j|jj||d|df<||j|jj|d<|j||j|<|jj ||dt |jvr=|j|d|j|<|jj |d|t |jvrg|j|dk(rudt|zdzt |jvrdt|zd zt |jvr|jdt|zdz|jdt|zdz<|jdt|zd z|jdt|zd z<|jj dt|zdz|jj dt|zd z|j||j|<|j|D]W}|dd k(s |dxxtdt|zd zzcc<|ddk(r|j|j|n|j|D]W}|ddk(s |dxxtdt|zdzzcc<|ddk(r|j|j|n|j|tdt|zdzd|j|tdt|zd zd |jj |n-|j|d k(r|j||j|<|j|D]W}|dd k(s |dxxtdt|zd zzcc<|ddk(r|j|j|n|j|tdt|zd zd |jj |nN|t |jvr7|j||j|<|jj |t |jD]}|j|d|dk(r||j|d<d |j||j|df<d |j|j|d|f<|jj ||j|df|jj |j|d|f|j|d|dk(s||j|d<d |j|j|d|f<d |j||j|df<|jj |j|d|f|jj ||j|dfyy )a This method changes the label of a node. Parameters ========== label: String or Symbol The label of the node for which the label has to be changed. new_label: String or Symbol The new label of the node. Examples ======== >>> from sympy.physics.continuum_mechanics.truss import Truss >>> t = Truss() >>> t.add_node('A', 0, 0) >>> t.add_node('B', 3, 0) >>> t.nodes [('A', 0, 0), ('B', 3, 0)] >>> t.change_node_label('A', 'C') >>> t.nodes [('C', 0, 0), ('B', 3, 0)] z!No such node exists for the Trussz*A node with the given label already existsrrApinnedR__x_yZrollerTN)rr9rr:rrGrFrstrrrrrE apply_loadrr)rr< new_labelnodeloadrJs rchange_node_labelzTruss.change_node_label1sN4 )) )@A A $++ +IJ J : X7e#QZ\`ab\ceijkelPmDKK 1 15$q'472K LMJSD%%d&7&7&=&=d1g&FG8<8N8Nu8UD**95**..u5Aw$t~~"6648NN474Ky1**473 D$88>>)4@#CJt3tD*#'7b=$(Gvd3u:od6J/K$KG'+Aw!|(, E(:(A(A$(G$) * )- I(>*#'7a<$(Gvd3u:od6J/K$KG'+Aw!|(, E(:(A(A$(G$) * !OOIvd3y>>QRV>V7WYZ[ OOIvd3y>>QRV>V7WY[\ KKOOE2!^^I6(B59[[5GDKK 2(, E(:*#'7b=$(Gvd3u:od6J/K$KG'+Aw!|(, E(:(A(A$(G$) * !OOIvd3y>>QRV>V7WY[\ KKOOE2 D$5559[[5GDKK 2 KKOOE2"&t}}"5 X==03tAw>7@DMM&1!4Z^D00)T]]6=RST=U1VWZ^D00$--2G2JI1VW 0044eT]]6=RST=U5VW 0044dmmF6KA6NPU5VW!]]6215a@7@DMM&1!4Z^D00$--2G2JI1VWZ^D00)T]]6=RST=U1VW 0044dmmF6KA6NPU5VW 0044eT]]6=RST=U5VW X]: Xr!c|t|jvr tdt|jj}|D]}||k(s |j|d|j|dg|j|<|jj ||j ||j |<|j j |y)aG This method changes the label of a member. Parameters ========== label: String or Symbol The label of the member for which the label has to be changed. new_label: String or Symbol The new label of the member. Examples ======== >>> from sympy.physics.continuum_mechanics.truss import Truss >>> t = Truss() >>> t.add_node('A', 0, 0) >>> t.add_node('B', 3, 0) >>> t.nodes [('A', 0, 0), ('B', 3, 0)] >>> t.change_node_label('A', 'C') >>> t.nodes [('C', 0, 0), ('B', 3, 0)] >>> t.add_member('BC', 'B', 'C') >>> t.members {'BC': ['B', 'C']} >>> t.change_member_label('BC', 'BC_new') >>> t.members {'BC_new': ['B', 'C']} z#No such member exists for the TrussrrAN)rFrr9rDrGr)rr<r]rIrJs rchange_member_labelzTruss.change_member_labels@ T]]+ +BC C!%T]] 3 8 8 : + 5U?04 f0Ea0H$--X^J_`aJb/cDMM),MM%%e,7;7L7LU7SD)))4))--e4  5r!ct|}t|}||jvr td|t|jvr!|j|j ||gy||gg|j|<y)a This method applies an external load at a particular node Parameters ========== location: String or Symbol Label of the Node at which load is applied. magnitude: Sympifyable Magnitude of the load applied. It must always be positive and any changes in the direction of the load are not reflected here. direction: Sympifyable The angle, in degrees, that the load vector makes with the horizontal in the counter-clockwise direction. It takes the values 0 to 360, inclusive. Examples ======== >>> from sympy.physics.continuum_mechanics.truss import Truss >>> from sympy import symbols >>> t = Truss() >>> t.add_node('A', 0, 0) >>> t.add_node('B', 3, 0) >>> P = symbols('P') >>> t.apply_load('A', P, 90) >>> t.apply_load('A', P/2, 45) >>> t.apply_load('A', P/4, 90) >>> t.loads {'A': [[P, 90], [P/2, 45], [P/4, 90]]} z$Load must be applied at a known nodeN)rr&r9rFrr;rlocation magnitude directions rr\zTruss.apply_loadsxBI& I& 4++ +CD D4 ,, H%,,i-CD*3Y)?(@ H%r!c:t|}t|}||jvr td||g|j|vr td|j|j ||g|j|gk(r|jj |yy)a; This method removes an already present external load at a particular node Parameters ========== location: String or Symbol Label of the Node at which load is applied and is to be removed. magnitude: Sympifyable Magnitude of the load applied. direction: Sympifyable The angle, in degrees, that the load vector makes with the horizontal in the counter-clockwise direction. It takes the values 0 to 360, inclusive. Examples ======== >>> from sympy.physics.continuum_mechanics.truss import Truss >>> from sympy import symbols >>> t = Truss() >>> t.add_node('A', 0, 0) >>> t.add_node('B', 3, 0) >>> P = symbols('P') >>> t.apply_load('A', P, 90) >>> t.apply_load('A', P/2, 45) >>> t.apply_load('A', P/4, 90) >>> t.loads {'A': [[P, 90], [P/2, 45], [P/4, 90]]} >>> t.remove_load('A', P/4, 90) >>> t.loads {'A': [[P, 90], [P/2, 45]]} z&Load must be removed from a known nodezENo load of this magnitude and direction has been applied at this nodeN)rr&r9rrErGrds r remove_loadzTruss.remove_loadsHI& I& 4++ +EF F9%T[[-BB !hii H%,,i-CD ;;x B & KKOOH % 'r!c ||jvr td|t|jvr|dk(rW|j |t dt |zdzd|j |t dt |zdzdn|dk(r|j |t dt |zdzdn|j|dk(r1|dk(rn|j|t dt |zdzdnB|j|dk(r0|dk(r+|j |t dt |zdzd||j|<y ) aH This method adds a pinned or roller support at a particular node Parameters ========== location: String or Symbol Label of the Node at which support is added. type: String Type of the support being provided at the node. Examples ======== >>> from sympy.physics.continuum_mechanics.truss import Truss >>> t = Truss() >>> t.add_node('A', 0, 0) >>> t.add_node('B', 3, 0) >>> t.apply_support('A', 'pinned') >>> t.supports {'A': 'pinned'} z%Support must be added on a known noderUrVrWrrXrYrZN)rr9rFrr\rr[ri)rretypes r apply_supportzTruss.apply_supports30 4,, ,DE EtDNN338#OOHfT#h-5G5L.MqQOOHfT#h-5G5L.MrRX%OOHfT#h-5G5L.MrR)X58#$$Xvd3x=6H6M/NPQR)X58#OOHfT#h-5G5L.MqQ'+DNN8 $r!c ||jvr td|t|jvr td|j|dk(rW|j |t dt |zdzd|j |t dt |zdzdn=|j|d k(r+|j |t dt |zdzd|jj|y ) a4 This method removes support from a particular node Parameters ========== location: String or Symbol Label of the Node at which support is to be removed. Examples ======== >>> from sympy.physics.continuum_mechanics.truss import Truss >>> t = Truss() >>> t.add_node('A', 0, 0) >>> t.add_node('B', 3, 0) >>> t.apply_support('A', 'pinned') >>> t.supports {'A': 'pinned'} >>> t.remove_support('A') >>> t.supports {} z No such node exists in the Trussz+No support has been added to the given noderUrVrWrrXrYrZN)rr9rFrrirr[rG)rres rremove_supportzTruss.remove_supportAs0 4,, ,?@ @ T$..1 1JK K~~h'83  6$s8}2DT2I+JAN  6$s8}2DT2I+JBO)X5  6$s8}2DT2I+JBO NN  x (r!c  d}|jD]S}|dt|jvs|j|ddk(r|dz }9|j|ddk(sO|dz }Udt|jzt|j|zk7r t dt dt|jzDcgc]1}t dt|jzDcgc]}dc}3}}}tdt|jzd}d}|jD]}|dt|jvr|j|dD]}|dtdt|dzdzk7s'|dtdt|dzd zk7sK||xx|dtt|dzd z zzcc<||dzxx|dtt|dzd z zzcc<|dz }d} d} |jD]}|dt|jvrn|j|ddk(r,|| | xxdz cc<|| dz| dzxxdz cc<| dz } n-|j|ddk(r|| dz| xxdz cc<| dz } | dz } t|jD]} |j| d} |j| d} t|j | d|j | dz dz|j | d|j | dz dzz}|j"j%| }|j"j%| }|j | d|j | dz |z }|j | d|j | dz |z }|j | d|j | dz |z }|j | d|j | dz |z }||dz| xx|z cc<||dzdz| xx|z cc<||dz| xx|z cc<||dzdz| xx|z cc<| dz } t'|d z|z}i|_d}t*}|jD]c}|dt|jvs|j|dD]1}t-|dtt.t0fvs#t3||d}3et t|D]<}t-||tt.t0fvs#t5|||z d ks8d||<>|jD]}|dt|jvs|j|ddk(rQ|||j(dt|dzdz<||dz|j(dt|dzd z<|dz }|j|ddk(s|||j(dt|dzd z<|dz }t|jD]} |||j6| <|dz }y cc}wcc}}w)a This method solves for all reaction forces of all supports and all internal forces of all the members in the truss, provided the Truss is solvable. A Truss is solvable if the following condition is met, 2n >= r + m Where n is the number of nodes, r is the number of reaction forces, where each pinned support has 2 reaction forces and each roller has 1, and m is the number of members. The given condition is derived from the fact that a system of equations is solvable only when the number of variables is lesser than or equal to the number of equations. Equilibrium Equations in x and y directions give two equations per node giving 2n number equations. However, the truss needs to be stable as well and may be unstable if 2n > r + m. The number of variables is simply the sum of the number of reaction forces and member forces. .. note:: The sign convention for the internal forces present in a member revolves around whether each force is compressive or tensile. While forming equations for each node, internal force due to a member on the node is assumed to be away from the node i.e. each force is assumed to be compressive by default. Hence, a positive value for an internal force implies the presence of compressive force in the member and a negative value implies a tensile force. Examples ======== >>> from sympy.physics.continuum_mechanics.truss import Truss >>> t = Truss() >>> t.add_node("node_1", 0, 0) >>> t.add_node("node_2", 6, 0) >>> t.add_node("node_3", 2, 2) >>> t.add_node("node_4", 2, 0) >>> t.add_member("member_1", "node_1", "node_4") >>> t.add_member("member_2", "node_2", "node_4") >>> t.add_member("member_3", "node_1", "node_3") >>> t.add_member("member_4", "node_2", "node_3") >>> t.add_member("member_5", "node_3", "node_4") >>> t.apply_load("node_4", magnitude=10, direction=270) >>> t.apply_support("node_1", type="pinned") >>> t.apply_support("node_2", type="roller") >>> t.solve() >>> t.reaction_loads {'R_node_1_x': 0, 'R_node_1_y': 20/3, 'R_node_2_y': 10/3} >>> t.internal_forces {'member_1': 20/3, 'member_2': 20/3, 'member_3': -20*sqrt(2)/3, 'member_4': -10*sqrt(5)/3, 'member_5': 10} rrUrTrZrAz The given truss cannot be solvedrVrWrXg|=N)rrFrrCrr9rBr r#rrr[r r r r rrr:rrrrkrrminabsr)rcount_reaction_loadsr^jrHcoefficients_matrix load_matrixload_matrix_rowr_colsrowrJrNrOlength start_index end_indexhorizontal_component_startvertical_component_starthorizontal_component_endvertical_component_end forces_matrixmin_loads rsolvez Truss.solvegs3b !KK .DAw$t~~..>>$q'*H4(A-(^^DG,h6(A-(  . S  T]]!36J!J J?@ @OTUVWZ[_[f[fWgUgOhi!53t{{3C1C+DEaEiiAc$**o-q1 KK !DAw$t{{++ KKQ0XDAwtCQL'8'= >>47FSWX[\`ab\cXdSdeiSiLjCj#O4QBtAwJsN@S8SS4#Oa$78DGC4PQ7 SVDW>$q'*H4',T2a72'A.tAv6!;6AID^^DG,h6'A.t494AID 1HC 4==) FMM&)!,E--'*C411%8;D(>s(CA(FtG]G]^cGdefGg(gio'o $(,(>(>u(Ea(HI_I_`cIdefIg(gio'o $&*&<&>#&xa#9: : s=)* )AM!$%fc3-??}Q'0158'(M!$ )KK DAw$t~~..>>$q'*H4CPQRCSD((c$q'l):4)?@CPQRSTQTCUD((c$q'l):4)?@FA^^DG,h6CPQRCSD((c$q'l):4)?@FA 4==) F,9!,rs3 $&9&M M r!