Finite Element Methods (FEM)
Finite Element Methods for Members Parallel to the Global X-Axis
The finite element method is one of popular modern methods used for the analysis of structural members. The following statements are also true about the Finite Element Method:
~ Most popular method so far in structural mechanics developed since the 1960’s in parallel by:
§ CIVIL ENGINEERS developed the DIRECT STIFFNESS METHOD
§ MATHEMATICIANS developed the variational principle, and the principle of virtual work.
~ The direct stiffness method is used for sparse structures like trusses and beams.
Key concepts in finite element methods include: element stiffness matrix, global stiffness matrix, assembly, support (boundary) conditions.
Where:
F1 = Force pushing into bar at node 1 and causing compression of bar; hence, let us designate it as a compressional force with a negative sign. This however, will be resisted by an equal and opposite reaction N.
F2 = Force pulling out of bar at node 2 and causing tension in the bar; hence, let us designate it as a tensile force with a positive sign. This will be resisted by an equal and opposite reaction.
That is:
F1 = N
F2 = -N
We know from elementary principles that:
Stress σ = F/A
Where: F = force, and A = Area
Also:
Strain Ɛ = ∆L/L
Where:
∆L = change in length, and L= original length.
But young modulus E = stress/strain.
Therefore,
E = (F/A) / (∆L/L).
E = F/A * L/∆L
E = FL/A∆L
EA∆L/L = F = EA(U2-U1)/L
F1 = -EA(U1-U2)/L
F2 = EA(U1-U2)/L
The above solution is only applicable to bars that are parallel to the global x – axis.
Finite Element Methods for Inclined Members
F1 = F1cosθ / F1sinθ = F1x / F1y
F2 = F2cosθ / F2sinθ = F2x / F2y
Hence,
Where c = cosθ,
s = sinθ.
Similarly for the displacements,
Multiply both sides by the transpose of the transformation matrix ([{T^t}])
Considering the right hand side
Recall from trig. Identities that [co{s^2}theta {rm{ }} + {rm{ }}si{n^2}theta {rm{ }} = {rm{ }}1]
From equation 2,
Recall that
This is the elemental force-displacement rotation for a truss member. This holds true for one truss member.
