CivilEngineeringBible.com CivilEngineeringBible.com

Introduction Concepts & Formulas VideosSolved problemsDownload Files
 Introduction  Concepts & Formulas  Watch! (Videos) Solved problems Download Files

Plane Trusses

Courses > Finite Elements Method > Basics of Finite Elements > Plane Trusses


Introduction on Plane Trusses :

A truss is a structure in which members are arranged in such a way that they are subjected to axial loads only. The joints in trusses are considered to be pinned. In plane trusses, all members are assumed to be in the x-y plane.



Concepts and Formulas of Plane Trusses:

 

A plane truss element is an axial deformation element oriented arbitrarily in two-dimensional space. 

Definitions: 

s and t = local coordinates along the member's axis and perpendicular to it, respectively.

x and y = global coordinates

d1 and d2 = nodal displacements along s axis (local coordinates)

u1 and u2 = nodal displacements along x axis (global coordinate)

v1 and v2 = nodal displacements along y axis (global coordinate)

L = length of each member

 

Using the following interpolation we can find the deformation at any point on the member based on the known values of nodal displacements:

u(s)=(frac{L-s}{L}-frac{s}{L})egin{pmatrix} d_1 d_2 end{pmatrix};; ; 0leq sleq L

and the element equations in the local coordinate system are as follows:

frac{EA}{L}egin{bmatrix} 1 & -1 1 & -1 end{bmatrix}egin{bmatrix} d_1 d_2 end{bmatrix}=egin{bmatrix} P_1 P_2 end{bmatrix}Rightarrow k_1d_1=r_1

where k1= the local stiffness matrix of element; d1 = local degrees of freedom; r1 = local applied forces; E = elastic modulus of the material; and A = cross-sectional area.

For the sake of brevity, transformation matrices are not described in here.

Final form of element equations are as follows:

frac{EA}{L}egin{bmatrix} l_s^2 & l_sm_s & -l_s^2 & -l_sm_s l_sm_s & m_s^2 & -l_sm_s & -m_s^2 -l_s^2 & -l_sm_s & l_s^2 & l_sm_s -l_sm_s & -m_s^2 & l_sm_s & m_s^2 end{bmatrix}egin{bmatrix} u_1 v_1 u_2 v_2 end{bmatrix} = egin{bmatrix} F_{1x} F_{1y} F_{2x} F_{2y} end{bmatrix}Rightarrow KU=F

where

l_s=cosalpha=frac{x_2-x_1}{L}

m_s=sinalpha=frac{y_2-y_1}{L}

L=sqrt{(x_2-x_1)^2+(y_2-y_1)^2}

 

for solving the above-mentioned equation 

U = K^{-1}F

after solving it, the axial displacements can be obtained from:

egin{bmatrix} d_1 d_2 end{bmatrix} =egin{bmatrix} l_s & m_s & 0 &0  0 & 0 & l_s & m_s end{bmatrix}egin{bmatrix} u_1 v_1 u_2 v_2 end{bmatrix}

and from that, the axial strain is simply the first derivative of the axial displacement, giving constant strain over the element as

epsilon=frac{du}{ds}=frac{1}{L}(-d_1+d_2)

and from the axial strain, one can get to the axial stress of each member using the Hook's law, where

sigma=Eepsilon

and consequently, the force of each member can be obtained from

F = sigma A

 

Note! Equations given above are only for one member, for a plane truss structure containing more than member, stiffness, displacement, and force matrices need to be assembled before solving. On the other hand, sufficient number of boundary conditions are required to get a unique solution. Using the files under download section may help the readers understand the concepts better.



Watch! (Videos):


No videos available for this topic. Suggest one!



Solved sample problems of Plane Trusses:

 



Download Files:


Group 1: 

Nodes file for plane truss analysis
Element files for plane truss analysis

Note! put these three files in the same folder.

 

Group2: 

Nodes file for plane truss analysis
Element files for plane truss analysis

Note! put these three files in the same folder.

 

Group3: 

Nodes for plane truss analysis including initial strains and temperature loads
Elements for plane truss analysis including initial strains and temperature loads
Note! put these three files in the same folder.


Read also:



Share:


Follow our official Facebook page (@civilengineeringbible) and Twitter page (@CivilEngBible) and do not miss the best civil engineering tools and articles!