U 9%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_coordinatesselfrf/var/www/html/Darija-Ai-API/env/lib/python3.8/site-packages/sympy/physics/continuum_mechanics/truss.py__init__6szTruss.__init__cCs|jS)zL Returns the nodes of the truss along with their positions. )rrrrrnodesGsz Truss.nodescCs|jS)z7 Returns the node labels of the truss. )rrrrr node_labelsNszTruss.node_labelscCs|jS)zB Returns the positions of the nodes of the truss. )rrrrrnode_positionsUszTruss.node_positionscCs|jSzW Returns the members of the truss along with the start and end points. )rrrrrmembers\sz Truss.memberscCs|jSr")Z_member_labelsrrrr member_labelscszTruss.member_labelscCs|jS)z Returns the nodes with provided supports along with the kind of support provided i.e. pinned or roller. )rrrrrsupportsjszTruss.supportscCs|jS)z8 Returns the loads acting on the truss. )rrrrrloadsrsz Truss.loadscCs|jS)z^ Returns the reaction forces for all supports which are all initialized to 0. )rrrrrreaction_loadsyszTruss.reaction_loadscCs|jS)z] Returns the internal forces for all members which are all initialized to 0. )rrrrrinternal_forcesszTruss.internal_forcescCst|}t|}||jkr$tdn||jkrZ||jkrZ|j||j|krZtdnT|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||jkrJtdn|j}|D]0}||j|dks||j|dkrXtdqX|j |||f|j ||j ||f|j ||j ||t |j kr|j ||t |jkr|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)rangelenrrrrr)rcopyrremoverlistrpoprr)rr,ir-r.members_duplicatememberrrr remove_nodes(     $      zTruss.remove_nodecCs||jks||jks||kr&tdnf|t|jkr>tdnN|j||frXtdn4||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)r5rrgetr)rr,startendrrr add_members   zTruss.add_membercCs||t|jkrtdn`|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 Trussrr0N)r5rr)rr6r)rr,rrr remove_members  $$ zTruss.remove_memberc Cs||jkrtdn||jkr,tdn|jD]}|d|kr2||d|df|j|j||d|df<||j|j|d<|j||j|<|j||dt|jkr|j|d|j|<|j|d|t|jkr|j|dkrdt|dt|j krdt|d t|j kr|j dt|d|j dt|d<|j dt|d |j dt|d <|j dt|d|j dt|d |j ||j |<|j |D]X}|dd kr|dt dt|d 8<|ddkr|j | |q&q|j |D]X}|ddkr0|dt dt|d8<|ddkr|j | |qq0| |t dt|dd| |t dt|d d |j |n|j|d kr|j ||j |<|j |D]X}|dd kr|dt dt|d 8<|ddkrN|j | |qXq| |t dt|d d |j |n,|t|j kr|j ||j |<|j |t|jD]}|j|d|dkrL||j|d<d |j||j|df<d |j|j|d|f<|j||j|df|j|j|d|fn|j|d|dkr||j|d<d |j|j|d|f<d |j||j|df<|j|j|d|f|j||j|dfqq2d 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 existsrr0pinnedR__x_yZrollerTN)rr)rr*rr6r5rstrrrrr4 apply_loadrr)rr, new_labelnodeloadr9rrrchange_node_label1s|      . 8((       zTruss.change_node_labelcCs|t|jkrtdnjt|j}|D]V}||kr*|j|d|j|dg|j|<|j||j||j|<|j|q*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 Trussrr0N)r5rr)r3r6r)rr,rIr8r9rrrchange_member_labels  " zTruss.change_member_labelcCs\t|}t|}||jkr$tdn4|t|jkrH|j|||gn||gg|j|<dS)a This method applies an external load at a particular node Parameters ========== location: String or Symbol Label of the Node at which load is applied. magnitude: Sympifyable Magnitude of the load applied. It must always be positive and any changes in the direction of the load are not reflected here. direction: Sympifyable The angle, in degrees, that the load vector makes with the horizontal in the counter-clockwise direction. It takes the values 0 to 360, inclusive. Examples ======== >>> from sympy.physics.continuum_mechanics.truss import Truss >>> from sympy import symbols >>> t = Truss() >>> t.add_node('A', 0, 0) >>> t.add_node('B', 3, 0) >>> P = symbols('P') >>> t.apply_load('A', P, 90) >>> t.apply_load('A', P/2, 45) >>> t.apply_load('A', P/4, 90) >>> t.loads {'A': [[P, 90], [P/2, 45], [P/4, 90]]} z$Load must be applied at a known nodeN)rr r)r5rr+rlocationZ magnitude directionrrrrHs!  zTruss.apply_loadcCsrt|}t|}||jkr$tdn0||g|j|kr@tdn|j|||g|j|gkrn|j|dS)a; This method removes an already present external load at a particular node Parameters ========== location: String or Symbol Label of the Node at which load is applied and is to be removed. magnitude: Sympifyable Magnitude of the load applied. direction: Sympifyable The angle, in degrees, that the load vector makes with the horizontal in the counter-clockwise direction. It takes the values 0 to 360, inclusive. Examples ======== >>> from sympy.physics.continuum_mechanics.truss import Truss >>> from sympy import symbols >>> t = Truss() >>> t.add_node('A', 0, 0) >>> t.add_node('B', 3, 0) >>> P = symbols('P') >>> t.apply_load('A', P, 90) >>> t.apply_load('A', P/2, 45) >>> t.apply_load('A', P/4, 90) >>> t.loads {'A': [[P, 90], [P/2, 45], [P/4, 90]]} >>> t.remove_load('A', P/4, 90) >>> t.loads {'A': [[P, 90], [P/2, 45]]} z&Load must be removed from a known nodezENo load of this magnitude and direction has been applied at this nodeN)rr r)rr4r6rNrrr remove_loads$   zTruss.remove_loadcCs||jkrtdn|t|jkr|dkrh||tdt|dd||tdt|ddq|dkr||tdt|ddnj|j|dkr|dkr||tdt|ddn4|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 noderArBrCrrDrErFN)rr)r5rrHrrGrQ)rrOtyperrr apply_supports     zTruss.apply_supportcCs||jkrtdn|t|jkr,tdn|j|dkrx||tdt|dd||tdt|ddn,|j|d kr||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 noderArBrCrrDrErFN)rr)r5rrQrrGr6)rrOrrrremove_supportAs    zTruss.remove_supportc s`d}jD]L}|dtjkr j|ddkr<|d7}q j|ddkr |d7}q dtjtj|kr|tdfddtdtjD}tdtjd}d}jD]}|dtj krzj |dD]}|dt d t |dd kr|dt d t |dd kr|||dt t |dd 8<||d|dtt |dd 8<q|d7}qd}d}jD]}|dtjkr,j|ddkr|||d7<||d|dd7<|d7}n4j|ddkr,||d|d7<|d7}|d7}qtjD]n} 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}qBt|d |}i_d}t}jD]V}|dtj kr؈j |dD],}t|dt ttfkrt||d}qqtt|D]<}t||t ttfkrdS)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} rrAr@rFr0z The given truss cannot be solvedcs(g|] }ddtdtjDqS)cSsg|]}dqS)rr).0r7rrr sz*Truss.solve...r@)r1r2r)rUjrrrrVszTruss.solve..rBrCrDg|=N)rr5rr2rr)r1r rrrrGr rr r rrr*rrrrRrrminabsr)rZcount_reaction_loadsrJZcoefficients_matrixZ load_matrixZload_matrix_rowrKcolsrowr9r<r=length start_indexZ end_indexZhorizontal_component_startZvertical_component_startZhorizontal_component_endZvertical_component_endZ forces_matrixr7Zmin_loadrWrrrsolvegs1     @(.    D          "   z Truss.solveN)__name__ __module__ __qualname____doc__rpropertyrr r!r#r$r%r&r'r(r/r:r>r?rLrMrHrQrSrTr`rrrrr s>#         ,0+#[,-2*&r N)rdZcmathrZsympy.core.addrZsympy.core.mulrZsympy.core.symbolrZsympy.core.sympifyrZsympyrrZ(sympy.functions.elementary.miscellaneousr Zsympy.matrices.denser r r r rrrrs