o ePo@sdZddlmZddlmZddlmZddlmZddl m Z ddl m Z m Z ddlmZdd lmZdd l mZmZGd d 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@seZdZdZddZeddZeddZedd Zed d Z ed d Z eddZ eddZ eddZ eddZddZddZddZddZddZd d!Zd"d#Zd$d%Zd&d'Zd(d)Zd*d+Zd,S)-Trussa A Truss is an assembly of members such as beams, connected by nodes, that create a rigid structure. In engineering, a truss is a structure that consists of two-force members only. Trusses are extremely important in engineering applications and can be seen in numerous real-world applications like bridges. Examples ======== There is a Truss consisting of four nodes and five members connecting the nodes. A force P acts downward on the node D and there also exist pinned and roller joints on the nodes A and B respectively. .. image:: truss_example.png >>> from sympy.physics.continuum_mechanics.truss import Truss >>> t = Truss() >>> t.add_node("node_1", 0, 0) >>> t.add_node("node_2", 6, 0) >>> t.add_node("node_3", 2, 2) >>> t.add_node("node_4", 2, 0) >>> t.add_member("member_1", "node_1", "node_4") >>> t.add_member("member_2", "node_2", "node_4") >>> t.add_member("member_3", "node_1", "node_3") >>> t.add_member("member_4", "node_2", "node_3") >>> t.add_member("member_5", "node_3", "node_4") >>> t.apply_load("node_4", magnitude=10, direction=270) >>> t.apply_support("node_1", type="fixed") >>> t.apply_support("node_2", type="roller") cCsLg|_i|_i|_i|_g|_g|_g|_g|_i|_i|_ i|_ i|_ dS)z' Initializes the class N) _nodes_members_loads _supports _node_labels_node_positions_node_position_x_node_position_y_nodes_occupied_reaction_loads_internal_forces_node_coordinatesselfrWD:\Projects\ConvertPro\env\Lib\site-packages\sympy/physics/continuum_mechanics/truss.py__init__6s zTruss.__init__cC|jS)zL Returns the nodes of the truss along with their positions. )rrrrrnodesGz Truss.nodescCr)z7 Returns the node labels of the truss. )rrrrr node_labelsNr!zTruss.node_labelscCr)zB Returns the positions of the nodes of the truss. )rrrrrnode_positionsUr!zTruss.node_positionscCrzW Returns the members of the truss along with the start and end points. )rrrrrmembers\r!z Truss.memberscCrr$)Z_member_labelsrrrr member_labelscr!zTruss.member_labelscCr)z Returns the nodes with provided supports along with the kind of support provided i.e. pinned or roller. )rrrrrsupportsjszTruss.supportscCr)z8 Returns the loads acting on the truss. )rrrrrloadsrr!z Truss.loadscCr)z^ Returns the reaction forces for all supports which are all initialized to 0. )rrrrrreaction_loadsyr!zTruss.reaction_loadscCr)z] Returns the internal forces for all members which are all initialized to 0. )rrrrrinternal_forcesr!zTruss.internal_forcescCst|}t|}||jvrtd||jvr+||jvr+|j||j|kr+td|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)rlabelxyrrradd_nodes ,   zTruss.add_nodecCstt|jD]}|j||kr|j|}|j|}q||jvr$td|j}|D]}||j|dks?||j|dkrCtdq+|j |||f|j ||j ||f|j ||j ||t |j vrt|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 rrrr+rcopyrremoverlistrpoprr)rr.ir/r0members_duplicatememberrrr remove_nodes,    $     zTruss.remove_nodecCs||jvs||jvs||krtd|t|jvrtd|j||fr)td||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)rr+r7rrgetr)rr.startendrrr add_memberszTruss.add_membercCsz|t|jvr td|j|j|d|j|df|j|j|d|j|df|j||j|dS)a This method removes a member from the given truss. Parameters ========== label: 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) >>> 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 Trussrr2N)r7rr+rr8r)rr.rrr remove_members $$ zTruss.remove_memberc Cs||jvr td||jvrtd|jD]N}|d|krd||d|df|j|j||d|df<||j|j|d<|j||j|<|j||dt|jvrj|j|d|j|<|j|d|t|jvr|j|dkrbdt|dt|j vrdt|d t|j vr|j dt|d|j dt|d<|j dt|d |j dt|d <|j dt|d|j dt|d |j ||j |<|j |D]*}|dd kr |dt dt|d 8<|ddkr |j | |nq|j |D]+}|ddkr;|dt dt|d8<|ddkr9|j | |nq| |t dt|dd| |t dt|d d |j |nm|j|d kr|j ||j |<|j |D]+}|dd kr|dt dt|d 8<|ddkr|j | |nqw| |t dt|d d |j |n|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|fq|j|d|dkrb||j|d<d |j|j|d|f<d |j||j|df<|j|j|d|f|j||j|dfqqd 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 existsrr2pinnedR__x_yZrollerTN)rr+rr,rr8r7rstrrrrr6 apply_loadrr)rr. new_labelnodeloadr;rrrchange_node_label1s   . 4((     zTruss.change_node_labelcCs|t|jvr tdt|j}|D]+}||kr?|j|d|j|dg|j|<|j||j||j|<|j|qdS)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 Trussrr2N)r7rr+r5r8r)rr.rKr:r;rrrchange_member_labels "  zTruss.change_member_labelcCs\t|}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"r+r7rr-rlocationZ magnitude directionrrrrJs! zTruss.apply_loadcCsrt|}t|}||jvrtd||g|j|vrtd|j|||g|j|gkr7|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"r+rr6r8rPrrr remove_loads$ zTruss.remove_loadcCs||jvr td|t|jvrG|dkr3||tdt|dd||tdt|ddnI|dkrF||tdt|ddn5|j|dkrb|dkra||tdt|ddn|j|dkr||dkr|||tdt|dd||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 noderCrDrErrFrGrHN)rr+r7rrJrrIrS)rrQtyperrr apply_supports"  zTruss.apply_supportcCs||jvr td|t|jvrtd|j|dkr:||tdt|dd||tdt|ddn|j|d krP||tdt|dd|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 noderCrDrErrFrGrHN)rr+r7rrSrrIr8)rrQrrrremove_supportAs  zTruss.remove_supportc sZd}jD]&}|dtjvr+j|ddkr|d7}qj|ddkr+|d7}qdtjtj|kr>tdfddtdtjD}tdtjd}d}jD]c}|dtj vrj |dD]L}|dt d t |dd kr|dt d t |dd kr|||dt t |dd 8<||d|dtt |dd 8<qo|d7}q]d}d}jD]P}|dtjvrj|ddkr|||d7<||d|dd7<|d7}nj|ddkr||d|d7<|d7}|d7}qtjD]} j| d} j| d} tj| dj| ddj| dj| dd} j| } j| }j| dj| d| }j| dj| d| }j| dj| d| }j| dj| d| }|| d||7<|| dd||7<||d||7<||dd||7<|d7}qt|d |}i_d}t}jD]+}|dtj vrj |dD]}t|dt ttfvrt||d}qqtt|D]}t||t ttfvr7t|||dkr7d||<qjD]Z}|dtjvrj|ddkrx||jd t |dd <||djd t |dd <|d7}q<j|ddkr||jd t |dd <|d7}q= 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} rrCrBrHr2z The given truss cannot be solvedcs(g|]}ddtdtjDqS)cSsg|]}dqS)rr).0r9rrr sz*Truss.solve...rB)r3r4r)rWjrrrrXs(zTruss.solve..rDrErFg|=N)rr7rr4rr+r3r r rrrIr rr r rrr,rrrrTrrminabsr)rZcount_reaction_loadsrLZcoefficients_matrixZ load_matrixZload_matrix_rowrMcolsrowr;r>r?lengthZ start_indexZ end_indexZhorizontal_component_startZvertical_component_startZhorizontal_component_endZvertical_component_endZ forces_matrixr9Zmin_loadrYrrrsolvegs1    @(,    D         "  z Truss.solveN)__name__ __module__ __qualname____doc__rpropertyr r"r#r%r&r'r(r)r*r1r<r@rArNrOrJrSrUrVrarrrrr s@#         ,0+#[,-2* &r N)reZcmathrZsympy.core.addrZsympy.core.mulrZsympy.core.symbolrZsympy.core.sympifyrZsympyrrZ(sympy.functions.elementary.miscellaneousr Zsympy.matrices.denser r r r rrrrs