RiePodZddlmZddlmZddlmZddlmZddl m Z ddl m Z m Z ddlmZdd lmZdd l mZmZGd d Zd S)zB This module can be used to solve problems related to 2D Trusses. )inf)Add)Mul)Symbol)sympify)Matrixpi)sqrt)zeros)sincosc eZdZdZdZedZedZedZedZ edZ edZ ed Z ed Z ed Zd Zd ZdZdZdZdZdZdZdZdZdZdS)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|_ dS)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 Glib/python3.11/site-packages/sympy/physics/continuum_mechanics/truss.py__init__zTruss.__init__6sf   ! " "!! "!#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!cFt|}t|}||jvrtd||jvrN||jvrE|j||j|krtd|j|||f|j||j||f|j||j|||g|j |<dS)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_nodesE6 AJJ AJJ D% % %@AA A $' ' 'A1F,F,F4K`KfKfghKiKikolAlGlGHIlJlJLJLJJKK K K  q!} - - -   $ $U + + +  ' 'A / / /  ! ( ( + + +  ! ( ( + + +-.FD "5 ) ) )r!ctt|jD]-}|j||kr|j|}|j|}.||jvrt d|j}|D]?}||j|dks||j|dkrt d@|j |||f|j ||j ||f|j ||j ||t|j vr|j ||t|jvr|j||j|dS)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 4 + Z ZDM&1!444vAVWXAY8Y8Y$%XYYY9Z K  q!} - - -   $ $U + + +  ' 'A / / /  ! ( ( + + +  ! ( ( + + +T[)))) &&&T^,,,,""5)))  " & &u - - - - -r!cP||jvs||jvs||krtd|t|jvrtd|j||frtd||g|j|<d|j||f<d|j||f<d|j|<dS)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 ) ) )S8I-I-IUTWZZZ[[ [ d4=)) ) )XYY Y  ! % %ucl 3 3 -LMM 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]++ + +ABB B  $ $dmE&:1&=t}U?STU?V%W X X X  $ $dmE&:1&=t}U?STU?V%W X X X M  e $ $ $  ! % %e , , , , ,r!c  ||jvrtd||jvrtd|jD]}|d|kr||d|df|j|j||d|df<||j|j|d<|j||j|<|j||dt |jvr;|j|d|j|<|j|d|t |jvr|j|dkrdt|zdzt |j vrdt|zd zt |j vr|j dt|zdz|j dt|zdz<|j dt|zd z|j dt|zd z<|j dt|zdz|j dt|zd z|j ||j |<|j |D]l}|dd kr^|dxxtdt|zd zzcc<|ddkr |j | |nm|j |D]l}|ddkr^|dxxtdt|zdzzcc<|ddkr |j | |nm| |tdt|zdzd| |tdt|zd zd |j |n7|j|d kr|j ||j |<|j |D]l}|dd kr^|dxxtdt|zd zzcc<|ddkr |j | |nm| |tdt|zd zd |j |nE|t |j vr/|j ||j |<|j |t |jD]}|j|d|dkr||j|d<d |j||j|df<d |j|j|d|f<|j||j|df|j|j|d|f|j|d|dkr||j|d<d |j|j|d|f<d |j||j|df<|j|j|d|f|j||j|dfd S)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_label1s4 ) ) )@AA A $+ + +IJJ J : X: X7e##QZ\`ab\ceijkelPmDK 1 15$q'472K L LMJSD%d&7&=&=d1g&F&FG8<8Nu8UD*95*..u555Aw$t~"6"66648N474Ky1**47333 D$8$888>)4@@#CJJt3tD**#'7b==$(GGGvd3u::od6J/K/K$KGGG'+Aw!||(, E(:(A(A$(G(G(G$)E $1 )- I(>**#'7a<<$(GGGvd3u::od6J/K/K$KGGG'+Aw!||(, E(:(A(A$(G(G(G$)E $0 !OOIvd3y>>>QRV>V7W7WYZ[[[ OOIvd3y>>>QRV>V7W7WY[\\\ KOOE2222!^I6(BB59[5GDK 2(, E(:**#'7b==$(GGGvd3u::od6J/K/K$KGGG'+Aw!||(, E(:(A(A$(G(G(G$)E $1 !OOIvd3y>>>QRV>V7W7WY[\\\ KOOE222 D$5$55559[5GDK 2 KOOE222"&t}"5"5 X X=03tAw>>7@DM&1!4Z^D0)T]6=RST=U1VWZ^D0$-2G2JI1VW 044eT]6=RST=U5VWWW 044dmF6KA6NPU5VWWWW!]6215a@@7@DM&1!4Z^D0$-2G2JI1VWZ^D0)T]6=RST=U1VW 044dmF6KA6NPU5VWWW 044eT]6=RST=U5VWWWu: X: Xr!c|t|jvrtdt|j}|D]}||krw|j|d|j|dg|j|<|j||j||j|<|j|dS)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]++ + +BCC C!%T] 3 3 8 8 : : + 5 5U??04 f0Ea0H$-X^J_`aJb/cDM),M%%e,,,7;7LU7SD))4)--e444  5 5r!ct|}t|}||jvrtd|t|jvr$|j|||gdS||gg|j|<dS)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_loadsBI&& I&& 4+ + +CDD D4 ,,,, H%,,i-CDDDDD*3Y)?(@ H%%%r!cPt|}t|}||jvrtd||g|j|vrtd|j|||g|j|gkr|j|dSdS)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+ + +EFF F9%T[-BBB !hiii H%,,i-CDDD ;x B & & KOOH % % % % % ' &r!c ||jvrtd|t|jvr|dkro||t dt |zdzd||t dt |zdzdn|dkr7||t dt |zdzdn|j|dkr>|dkr7||t dt |zdzdnN|j|dkr=|dkr7||t dt |zdzd||j|<d S) 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_supports0 4, , ,DEE EtDN33338##OOHfT#h--5G5L.M.MqQQQOOHfT#h--5G5L.M.MrRRRRX%%OOHfT#h--5G5L.M.MrRRR)X558##$$Xvd3x==6H6M/N/NPQRRR)X558##OOHfT#h--5G5L.M.MqQQQ'+DN8 $ $ $r!c D||jvrtd|t|jvrtd|j|dkro||t dt |zdzd||t dt |zdzdnH|j|d kr7||t dt |zdzd|j|d S) a4 This method removes support from a particular node Parameters ========== location: String or Symbol Label of the Node at which support is to be removed. Examples ======== >>> from sympy.physics.continuum_mechanics.truss import Truss >>> t = Truss() >>> t.add_node('A', 0, 0) >>> t.add_node('B', 3, 0) >>> t.apply_support('A', 'pinned') >>> t.supports {'A': 'pinned'} >>> t.remove_support('A') >>> t.supports {} z No such node exists in the Trussz+No support has been added to the given noderUrVrWrrXrYrZN)rr9rFrrirr[rG)rres rremove_supportzTruss.remove_supportAs0 4, , ,?@@ @ T$.11 1 1JKK K~h'833  6$s8}}2DT2I+J+JANNN  6$s8}}2DT2I+J+JBOOOO)X55  6$s8}}2DT2I+J+JBOOO N  x ( ( ( ( (r!c  d}jD]W}|dtjvr9j|ddkr|dz };j|ddkr|dz }Xdtjztj|zkrt dfdt dtjzD}tdtjzd}d}jD]}|dtj vrj |dD]}|dtdt|dzd zkr|dtdt|dzd zkrq||xx|dtt|dzd z zzcc<||dzxx|dtt|dzd z zzcc<|dz }d}d}jD]}|dtjvrj|ddkr8|||xxdz cc<||dz|dzxxdz cc<|dz }n5j|ddkr||dz|xxdz cc<|dz }|dz }tjD]} j| d} j| d} tj| dj| dz dzj| dj| dz dzz} j| } j| }j| dj| dz | z }j| dj| dz | z }j| dj| dz | z }j| dj| 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]s}|dtj vrUj |dD]A}t-|dtt.t0fvrt3||d}Btt t|D]L}t-||tt.t0fvr!t5|||z d krd||<MjD]}|dtjvrj|ddkr[||jdt|dzd z<||dzjdt|dzd z<|dz }j|ddkr.||jdt|dzd z<|dz }tjD]} ||j| <|dz }dS)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 solvedc lg|]0}dtdtjzD1S)cg|]}dS)r).0rHs r z*Truss.solve...sEEEaEEEr!rT)rBrCr)rsjrs rrtzTruss.solve..s?iii!EE53t{3C3C1C+D+DEEEiiir!rVrWrXg|=N)rrFrrCrr9rBr r#rrr[r r r r rrr:rrrrkrrminabsr)rcount_reaction_loadsr^coefficients_matrix load_matrixload_matrix_rowr_colsrowrJrNrOlength start_index end_indexhorizontal_component_startvertical_component_starthorizontal_component_endvertical_component_end forces_matrixrHmin_loadrus` rsolvez Truss.solvegsb !K . .DAw$t~....>$q'*H44(A-((^DG,h66(A-( S   T]!3!36J!J J J?@@ @iiiiuUVWZ[_[fWgWgUgOhOhiiiAc$*oo-q11 K ! !DAw$t{++++ KQ0XXDAwtCQLL'8'= > >>>47FSWX[\`ab\cXdXdSdeiSiLjLjCjCj#O444QBtAwJsN@S@S8SS444#Oa$7888DGC4PQ7 SVDWDW$q'*H44',T222a7222'A.tAv666!;666AIDD^DG,h66'A.t4449444AID 1HCC4=))  FM&)!,E-'*C41%8;Ds(CA(FtG]^cGdefGg(gio'o $(,(>u(Ea(HI_`cIdefIg(gio'o $&*&>>#&xa#9#9s=))** ) )AM!$%%fc3-???}Q'011588'(M!$K  DAw$t~....>$q'*H44CPQRCSD(c$q'll):4)?@CPQRSTQTCUD(c$q'll):4)?@FAA^DG,h66CPQRCSD(c$q'll):4)?@FA4=))  F,9!,rs $$$$$$&&&&&&999999&&&&&&M M M M M M M M M M r!