\hSrSSKJrJr SSKJr SSKJr SSKJ r SSK J r SSK J r SSKJrJr SS KJr SS KJr SS KJr SS KrSS KJr SSKJr SSKJr SSKJrJr SS/0r \"SSS/0S9r!"SS5r"g )zB This module can be used to solve problems related to 2D Trusses. )ataninf)Add)INF)Mul)Symbol)sympify)Matrixpi) import_module)sqrt)zerosN)Quantity)plot)doctest_depends_on)sincosz Truss.draw matplotlibnumpyfromlistarange) import_kwargsc$\rSrSrSrSr\S5r\S5r\S5r \S5r \S5r \S 5r \S 5r \S 5r\S 5rS rSrSrSrSrSrSrSrSrSrSr\"SS9S!Sj5rSrSrSrSr S r!g)"Trussa A Truss is an assembly of members such as beams, connected by nodes, that create a rigid structure. In engineering, a truss is a structure that consists of two-force members only. Trusses are extremely important in engineering applications and can be seen in numerous real-world applications like bridges. Examples ======== There is a Truss consisting of four nodes and five members connecting the nodes. A force P acts downward on the node D and there also exist pinned and roller joints on the nodes A and B respectively. .. image:: truss_example.png >>> from sympy.physics.continuum_mechanics.truss import Truss >>> t = Truss() >>> t.add_node(("node_1", 0, 0), ("node_2", 6, 0), ("node_3", 2, 2), ("node_4", 2, 0)) >>> t.add_member(("member_1", "node_1", "node_4"), ("member_2", "node_2", "node_4"), ("member_3", "node_1", "node_3")) >>> t.add_member(("member_4", "node_2", "node_3"), ("member_5", "node_3", "node_4")) >>> t.apply_load(("node_4", 10, 270)) >>> t.apply_support(("node_1", "pinned"), ("node_2", "roller")) c/Ul0Ul0Ul0Ul/Ul/Ul/Ul/Ul0Ul0Ul 0Ul 0Ul 0Ul g)z Initializes the class N) _nodes_members_loads _supports _node_labels_node_positions_node_position_x_node_position_y_nodes_occupied_member_lengths_reaction_loads_internal_forces_node_coordinatesselfs _/var/www/auris/envauris/lib/python3.13/site-packages/sympy/physics/continuum_mechanics/truss.py__init__Truss.__init__;sh   ! " "!!! "!#cUR$)z< Returns the nodes of the truss along with their positions. )rr*s r,nodes Truss.nodesM {{r/cUR$)z' Returns the node labels of the truss. )r!r*s r, node_labelsTruss.node_labelsTs    r/cUR$)z2 Returns the positions of the nodes of the truss. )r"r*s r,node_positionsTruss.node_positions[ ###r/cUR$)zG Returns the members of the truss along with the start and end points. )rr*s r,members Truss.membersbs }}r/cUR$)z1 Returns the length of each member of the truss. )r&r*s r,member_lengthsTruss.member_lengthsir:r/cUR$)zi Returns the nodes with provided supports along with the kind of support provided i.e. pinned or roller. )r r*s r,supportsTruss.supportsps ~~r/cUR$)z( Returns the loads acting on the truss. )rr*s r,loads Truss.loadsxr3r/cUR$)zN Returns the reaction forces for all supports which are all initialized to 0. )r'r*s r,reaction_loadsTruss.reaction_loadsr:r/cUR$)zM Returns the internal forces for all members which are all initialized to 0. )r(r*s r,internal_forcesTruss.internal_forcess $$$r/c UGHnUSnUSn[U5nUSn[U5nX0R;a [S5eXE/URR5;a [S5eURR X4U45 UR R U5 URR XE45 URR U5 URR U5 XE/URU'GM g)a This method adds a node to the truss along with its name/label and its location. Multiple nodes can be added at the same time. Parameters ========== The input(s) for this method are tuples of the form (label, x, y). label: String or a Symbol The label for a node. It is the only way to identify a particular node. x: Sympifyable The x-coordinate of the position of the node. y: Sympifyable The y-coordinate of the position of the node. Examples ======== >>> from sympy.physics.continuum_mechanics.truss import Truss >>> t = Truss() >>> t.add_node(('A', 0, 0)) >>> t.nodes [('A', 0, 0)] >>> t.add_node(('B', 3, 0), ('C', 4, 1)) >>> t.nodes [('A', 0, 0), ('B', 3, 0), ('C', 4, 1)] rz!Node needs to have a unique labelz+A node already exists at the given positionN) r r) ValueErrorvaluesrappendr!r"r#r$)r+argsilabelxys r,add_nodeTruss.add_nodes>AaDE!A AdA A... !DEE41188:: !NOO ""Ea=1!!((/$$++QF3%%,,Q/%%,,Q/12&&u-%r/cUGHn[[UR55H6nURUU:XdMURUnUR UnM8 X R ;a [S5eURR5nUH8nX RUS:XdX RUS:XdM/[S5e URRUWW45 URRU5 URRXE45 URRU5 UR RU5 X R;aURRU5 X R;aURRU5 UR RU5 GM g)a This method removes a node from the truss. Multiple nodes can be removed at the same time. Parameters ========== The input(s) for this method are the labels of the nodes to be removed. 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), ('B', 3, 0), ('C', 5, 0)) >>> t.nodes [('A', 0, 0), ('B', 3, 0), ('C', 5, 0)] >>> t.remove_node('A', 'C') >>> t.nodes [('B', 3, 0)] z No such node exists in the trussrrNz0The given node already has member attached to itN)rangelenr1r!r#r$r)rPrcopyrremover"rpopr )r+rSrUrTrVrWmembers_duplicatemembers r, remove_nodeTruss.remove_nodes|0E3tzz?+$$Q'50--a0A--a0A, 222 !CDD%)MM$6$6$8!/F f 5a 88E]]SYEZ[\E]<]()[\\0 ""E1a=1!!((/$$++QF3%%,,Q/%%,,Q/KK'KKOOE*NN*NN&&u-&&**51/r/clUGH-nUSnUSnUSnX@R;dXPR;dXE:Xa [S5eX0R;a [S5eURR XE45(a [S5eXE/URU'[ URUSURUS- S-URUSURUS- S--5UR U'SURXE4'SURXT4'SURU'GM0 g) a This method adds a member between any two nodes in the given truss. Parameters ========== The input(s) of the method are tuple(s) of the form (label, start, end). 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), ('B', 3, 0), ('C', 2, 2)) >>> t.add_member(('AB', 'A', 'B'), ('BC', 'B', 'C')) >>> t.members {'AB': ['A', 'B'], 'BC': ['B', 'C']} rrNrOz;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 nodesTN)r)rPrr%getr r&r()r+rSrTrUstartends r, add_memberTruss.add_members6AaDEaDEA$C222cAWAW6W[`[e !^__--' !\]]%%))5,77 !PQQ).| e$.2D4J4J34OPQ4RSWSiSijoSpqrSs4svw3w{|R|RSV|WXY|Z[_[q[qrw[xyz[{|{~{4/@$$U+37$$UZ037$$SZ0/0%%e,'r/cUHnX R;a [S5eURRURUSURUS45 URRURUSURUS45 URRU5 URRU5 UR RU5 M g)a This method removes members from the given truss. Parameters ========== labels: String or Symbol The label for the member to be removed. Examples ======== >>> from sympy.physics.continuum_mechanics.truss import Truss >>> t = Truss() >>> t.add_node(('A', 0, 0), ('B', 3, 0), ('C', 2, 2)) >>> t.add_member(('AB', 'A', 'B'), ('AC', 'A', 'C'), ('BC', 'B', 'C')) >>> t.members {'AB': ['A', 'B'], 'AC': ['A', 'C'], 'BC': ['B', 'C']} >>> t.remove_member('AC', 'BC') >>> t.members {'AB': ['A', 'B']} z"No such member exists in the TrussrrNN)rrPr%r_r&r()r+rSrUs r, remove_memberTruss.remove_member%s,EMM) !EFF$$(($--*>q*A4==QVCWXYCZ)[\$$(($--*>q*A4==QVCWXYCZ)[\ !!%($$((/%%))%0r/c b UGHnUSnUSnX0R;a [S5eX@R;a [S5eURGHUnUSU:XdMXESUS4URURRX5SUS45'X@RURRUS5'URUURU'URR U5 USUR ;a=UR USUR U'UR R US5 X@R ;GafUR US:XGaoS[U5-S-UR;aS[U5-S -UR;aURS[U5-S-URS[U5-S-'URS[U5-S -URS[U5-S -'URR S[U5-S-5 URR S[U5-S -5 URUURU'URUHZnUSS :XdMUS==[S[U5-S -5-ss'USS:XaURURU5 O URUHZnUSS:XdMUS==[S[U5-S-5-ss'USS:XaURURU5 O URU[S[U5-S-5S5 URU[S[U5-S -5S 5 URR U5 GO)UR US :XaURUURU'URUHZnUSS :XdMUS==[S[U5-S -5-ss'USS:XaURURU5 O URU[S[U5-S -5S 5 URR U5 OFX0R;a7URUURU'URR U5 URGHnURUSUS:XaX@RUS'S URX@RUS4'S URURUSU4'URR X0RUS45 URR URUSU45 MURUSUS:XdMX@RUS'S URURUSU4'S URX@RUS4'URR URUSU45 URR X0RUS45 GM GMX GM g )a? This method changes the label(s) of the specified node(s). Parameters ========== The input(s) of this method are tuple(s) of the form (label, new_label). 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), ('B', 3, 0)) >>> t.nodes [('A', 0, 0), ('B', 3, 0)] >>> t.change_node_label(('A', 'C'), ('B', 'D')) >>> t.nodes [('C', 0, 0), ('D', 3, 0)] rrNz!No such node exists for the Trussz*A node with the given label already existsrOpinnedR__x_yZrollerTN)r)rPrindexr!r_r strr'rrr^ apply_loadrr%)r+rSrTrU new_labelnodeloadras r,change_node_labelTruss.change_node_labelFs6AaDE!I222 !DEE444 !MNN KKDAw%'U^ef`gimnoipTq DKK$5$5u1gtAw6O$PQNW))$*;*;*A*A$q'*JK<@,E,Ed,K(- -C -1KK ,BD'+Aw!|(,Q6$s5z/$:N3O(O+/7a<,0KK,>,E,Ed,K(- -C !% 6$s9~BUVZBZ;[]^ _ $ 6$s9~BUVZBZ;[]_ ` $  6!% !:h!F9=U9K I 6,0KK,>D'+Aw"}(,Q6$s5z/$:N3O(O+/7a<,0KK,>,E,Ed,K(- -? !% 6$s9~BUVZBZ;[]_ ` $  6$ 39=U9K I 6 $  6&*mmF#}}V4Q747B;D f 5a 8^b 4 4ivAVWXAY5Z [^b 4 4dmmF6KA6NPY5Z [ $ 4 4 8 8%vAVWXAY9Z [ $ 4 4 8 8$--:OPQ:RTY9Z [!%v!6q!9T!W!D;D f 5a 8^b 4 4dmmF6KA6NPY5Z [^b 4 4ivAVWXAY5Z [ $ 4 4 8 8$--:OPQ:RTY9Z [ $ 4 4 8 8%vAVWXAY9Z ['4](r/c<UGHnUSnUSnX0R;a [S5e[UR5R5nUHnXc:XdM URUSURUS/URU'URR U5 UR UUR U'UR R U5 UR UUR U'UR R U5 M GM g)a@ This method changes the label(s) of the specified member(s). Parameters ========== The input(s) of this method are tuple(s) of the form (label, new_label) 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), ('B', 3, 0), ('D', 5, 0)) >>> t.nodes [('A', 0, 0), ('B', 3, 0), ('D', 5, 0)] >>> t.change_node_label(('A', 'C')) >>> t.nodes [('C', 0, 0), ('B', 3, 0), ('D', 5, 0)] >>> t.add_member(('BC', 'B', 'C'), ('BD', 'B', 'D')) >>> t.members {'BC': ['B', 'C'], 'BD': ['B', 'D']} >>> t.change_member_label(('BC', 'BC_new'), ('BD', 'BD_new')) >>> t.members {'BC_new': ['B', 'C'], 'BD_new': ['B', 'D']} rrNz#No such member exists for the TrussN)rrPlistr]r_r&r()r+rSrTrUrwr`ras r,change_member_labelTruss.change_member_labelsBAaDE!IMM) !FGG$($7$<$<$>!/F48MM&4I!4Ldmm\bNcdeNf3g i0 ))%0:>:N:Nu:U,,Y7,,007;?;P;PQV;W--i8--11%80r/cUHnUSnUSnUSn[U5n[U5nX0R;a [S5eX0R;a!URUR XE/5 MrXE//URU'M g)a This method applies external load(s) at the specified node(s). Parameters ========== The input(s) of the method are tuple(s) of the form (location, magnitude, direction). 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), ('B', 3, 0)) >>> P = symbols('P') >>> t.apply_load(('A', P, 90), ('A', P/2, 45), ('A', P/4, 90)) >>> t.loads {'A': [[P, 90], [P/2, 45], [P/4, 90]]} rrNrOz$Load must be applied at a known nodeN)r r)rPrrRr+rSrTlocation magnitude directions r,rvTruss.apply_loads@AtH!I!I *I *I555 !GHH{{*KK)00)1GH.7-C,DDKK)r/cpUHnUSnUSnUSn[U5n[U5nX0R;a [S5eXE/URU;a [S5eURUR XE/5 URU/:XdMURR U5 M g)a This method removes already present external load(s) at specified node(s). Parameters ========== The input(s) of this method are tuple(s) of the form (location, magnitude, direction). 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), ('B', 3, 0)) >>> P = symbols('P') >>> t.apply_load(('A', P, 90), ('A', P/2, 45), ('A', P/4, 90)) >>> t.loads {'A': [[P, 90], [P/2, 45], [P/4, 90]]} >>> t.remove_load(('A', P/4, 90), ('A', P/2, 45)) >>> t.loads {'A': [[P, 90]]} rrNrOz&Load must be removed from a known nodezENo load of this magnitude and direction has been applied at this nodeN)r r)rPrr^r_rs r, remove_loadTruss.remove_loadsFAtH!I!I *I *I555 !IJJ)X1FF$%lmmKK)00)1GH{{8$* )!r/c UGHbnUSnUSnX0R;a [S5eX0R;aUS:XaYURU[ S[ U5-S-5S45 URU[ S[ U5-S-5S45 OUS :Xa,URU[ S[ U5-S-5S45 OURUS:Xa3US :Xa,UR U[ S[ U5-S-5S45 OEURUS :Xa2US:Xa,URU[ S[ U5-S-5S45 X@RU'GMe g ) a This method adds a pinned or roller support at specified node(s). Parameters ========== The input(s) of this method are of the form (location, type). 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), ('B', 3, 0)) >>> t.apply_support(('A', 'pinned'), ('B', 'roller')) >>> t.supports {'A': 'pinned', 'B': 'roller'} rrNz%Support must be added on a known nodernrorprqrrrsN)r)rPr rvrrur)r+rSrTrtypes r, apply_supportTruss.apply_support;sY0AtHQ4D555 !HII>>1x'6$s8}:LT:Q3RTU(VW6$s8}:LT:Q3RTV(WX)6$s8}:LT:Q3RTV(WX^^H-9x'(((F4H ;Md;R4SUV)WX^^H-9x'6$s8}:LT:Q3RTU(VW+/x('r/c  UHnX R;a [S5eX R;a [S5eURUS:XaYURU[ S[ U5-S-5S45 URU[ S[ U5-S-5S45 O?URUS :Xa,URU[ S[ U5-S-5S45 URR U5 M g ) a This method removes support from specified node(s.) Parameters ========== locations: String or Symbol Label of the Node(s) at which support is to be removed. Examples ======== >>> from sympy.physics.continuum_mechanics.truss import Truss >>> t = Truss() >>> t.add_node(('A', 0, 0), ('B', 3, 0)) >>> t.apply_support(('A', 'pinned'), ('B', 'roller')) >>> t.supports {'A': 'pinned', 'B': 'roller'} >>> t.remove_support('A','B') >>> t.supports {} z No such node exists in the Trussz+No support has been added to the given nodernrorprrqrrrsN)r)rPr rrrur_)r+rSrs r,remove_supportTruss.remove_supporths.H555 !CDD/ !NOO>>(+x7$$htCM7I$7N0OQR%ST$$htCM7I$7N0OQS%TU^^H-9$$htCM7I$7N0OQS%TU""8,r/c  SnURHRnUSUR;dMURUSS:XaUS- nM5URUSS:XdMMUS- nMT S[UR5-[UR5U-:wa [ S5e[ S[UR5-5VVs/sH5n[ S[UR5-5Vs/sHnSPM snPM7 nnn[ S[UR5-S5nSnURHnUSUR;aURUSHnUS[S[US5-S-5:wdM)US[S[US5-S -5:wdMOXg==US[[US-S - 5--ss'XgS-==US[[US-S - 5--ss'M US- nM Sn Sn URHnUSUR;amURUSS:Xa*XZU ==S- ss'XZS-U S-==S- ss'U S- n O-URUSS:XaXZS-U ==S- ss'U S- n U S- n M URGHn URU Sn URU Sn [URU SURU S- S-URU SURU S- S--5nUR R#U 5nUR R#U 5nURU SURU S- U- nURU SURU S- U- nURU SURU S- U- nURU SURU S- U- nX_S-U ==U- ss'X_S-S-U ==U- ss'UUS-U ==U- ss'UUS-S-U ==U- ss'U S- n GM [%U5S -U-n0UlSn[(nURHcnUSUR;dMURUSH5n[+US5[[,[.4;dM&[1UUS5nM7 Me [ [U55HBn[+UU5[[,[.4;dM&[3UUU- 5S :dM=SUU'MD URHnUSUR;dMURUSS:XaRUUUR&S[US5-S-'UUS-UR&S[US5-S -'US- nMURUSS:XdMUUUR&S[US5-S -'US- nM URHn UUUR4U 'US- nM g s snfs snnf)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), ("node_2", 6, 0), ("node_3", 2, 2), ("node_4", 2, 0)) >>> t.add_member(("member_1", "node_1", "node_4"), ("member_2", "node_2", "node_4"), ("member_3", "node_1", "node_3")) >>> t.add_member(("member_4", "node_2", "node_3"), ("member_5", "node_3", "node_4")) >>> t.apply_load(("node_4", 10, 270)) >>> t.apply_support(("node_1", "pinned"), ("node_2", "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} rrnrOrsrNz The given truss cannot be solvedrorprqg|=N)rr r\rrPr[rr1rrrurr rr r)r!rtr r'rrrrminabsr()r+count_reaction_loadsrxjrTcoefficients_matrix load_matrixload_matrix_rowrycolsrowrarfrglength start_index end_indexhorizontal_component_startvertical_component_starthorizontal_component_endvertical_component_end forces_matrixmin_loads r,solve Truss.solvesT !KKDAw$..(>>$q'*H4(A-(^^DG,h6(A-(  S  T]]!36J!J J?@ @OTUVWZ[_[f[fWgUgOhiOh!53t{{3C1C+DE+Da+DEOhiAc$**o-q1 KKDAw$++% KKQ0DAwtCQL'8'= >>47FSWX[\`ab\cXdSdeiSiLjCj#4QBtAwJsN@S8SS4#a$78DGC4PQ7 SVDW>$q'*H4',T2a72'A.tAv6!;6AID^^DG,h6'A.t494AID 1HC mmFMM&)!,E--'*C411%8;D(>s(CA(FtG]G]^cGdefGg(gio'o $(,(>(>u(Ea(HI_I_`cIdefIg(gio'o $&*&<&>#&xa#91 s=)*AM!$%fc3-??}Q'0158'(M!$+KKDAw$..(>>$q'*H4CPQRCSD((c$q'l):4)?@CPQRSTQTCUD((c$q'l):4)?@FA^^DG,h6CPQRCSD((c$q'l):4)?@FA mmF,9!,>> from sympy.physics.continuum_mechanics.truss import Truss >>> import math >>> t = Truss() >>> t.add_node(("A", -4, 0), ("B", 0, 0), ("C", 4, 0), ("D", 8, 0)) >>> t.add_node(("E", 6, 2/math.sqrt(3))) >>> t.add_node(("F", 2, 2*math.sqrt(3))) >>> t.add_node(("G", -2, 2/math.sqrt(3))) >>> t.add_member(("AB","A","B"), ("BC","B","C"), ("CD","C","D")) >>> t.add_member(("AG","A","G"), ("GB","G","B"), ("GF","G","F")) >>> t.add_member(("BF","B","F"), ("FC","F","C"), ("CE","C","E")) >>> t.add_member(("FE","F","E"), ("DE","D","E")) >>> t.apply_support(("A","pinned"), ("D","roller")) >>> t.apply_load(("G", 3, 90), ("E", 3, 90), ("F", 2, 90)) >>> p = t.draw() >>> p # doctest: +ELLIPSIS Plot object containing: [0]: cartesian line: 1 for x over (1.0, 1.0) ... >>> p.show() z-To use this function numpy module is requiredrVrrN皙?皙?Fg?)markersshow annotationsxlimylimaxis rectangles) r ImportErrorr _draw_nodes _draw_members_draw_supports _draw_loadsrr)maxrr)r+ subs_dictrVrrr node_markersmember_rectanglessupport_markersload_annotationsxmaxxminymaxyminrxlim sing_plots r,draw Truss.drawsvduMN N 3K  '' 2  ..0' --/"++-' tt**Dt33D9!<=Dt33D9!<=Dt33D9!<=Dt33D9!<=D + $s(48#A%tCxS'8':; Sc!!# #QAq 7T_lpqtltgtvz{~v~fHLMQRUMUHUW[\_W_G`glyCDI QAq 7T_lpqtltgtvz{~v~fHLMQRUMUHUW[\_W_G`glyCDIr/c`/nURGHMn[URUS5[[4;aFURUSU;a%XRUSURUS'O[ S5e[URUS5[ :XaURUSR 5nUHjn[U5[[4;dMUS:XdXQ;a [ S5eURUS==U-ss'URUS==X-ss'Ml [URUS5[[4;aHURUSU;a'XRUSURUS'GM[ S5e[URUS5[ :XdGMURUSR 5nUHjn[U5[[4;dMUS:XdXQ;a [ S5eURUS==U-ss'URUS==X-ss'Ml GMP URH=nURURUS/URUS//SSSS.5 M? 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