Expanding the force/displacement eqn. (9.39) to include terms associated with the y- direction, requires the insertion of zeros in the stiffness matrix of eqn. (9.41) and hence becomes identical to eqn. (9.14).

Element stress matrix in local coordinates

For the case of a linear rod element, substituting from eqn. (9.35) into eqn. (9.36) gives the element stress as a(e) = |t_! 1]{H, (9 42)

Again, by inserting zeros in the matrix, to accommodate terms associated with the y- direction, eqn. (9.42) becomes identical to eqn. (9.15).

Transformation of element stiffness and stress matrices to global coordinates

The element stiffness and stress matrices obtained above can be transformed from local to global coordinates using the procedures of § 9.7.1 to give the results previously obtained, namely the stiffness matrix of eqn. (9.23) and stress matrix of eqn. (9.26).

Formation of structural governing equation and assembled stiffness matrix

Section 9.7.1 has covered the combination of individual element stiffness contributions, necessary to analyse an assemblage of elements representing a complete framework. Equilibrium and compatibility arguments were used to form the structural governing eqn. (9.28) and hence the assembled stiffness matrix, eqn. (9.29). Now, the alternative principle of virtual work will be used to derive the same equations.

Eqn. (9.37) gives the element internal virtual work in local coordinates as

Summing all such contributions for the entire structure of m elements, gives m

Wi = ("}T / [ß(e)]T£(e)[ßw] dv {h'w}) (9.43)

Summing the contributions over all n nodes, the external virtual work will be given by n

where [~p\ is the column matrix of all nodal virtual displacements for the structure and (fj is the column matrix of all nodal forces. Substituting from eqns. (9.43) and (9.44) into the equation of the principle of virtual work, eqn. (9.10) gives m .

0 = {p}T{P} - / [fl<e>]TE<e>[fl<e>] dv {w'(e)}) (9.45)

Relating the virtual displacements in local and global coordinates via the transformation matrix [r(e)], gives

{mw} = [r<e)]{p(e)] and {u'ie)}T = {p(e)}T[r(e)]T

Summing the contributions and recalling {/?} denotes the nodal displacements for the entire structure, gives m m mm

and the assembled stiffness matrix m

It follows from eqn. (9.46) since {/?) is arbitrary and non-zero, that

which is the structural governing equation and the same as eqn. (9.28), and implies nodal force equilibrium.

9.8. A simple beam element

As with the previous treatment of the rod element, the two approaches using fundamental equations and the principle of virtual work will be employed to formulate the necessary equations for a simple beam element.

9.8.1. Formulation of a simple beam element using fundamental equations

Consider the case, similar to §9.7.1, in which the deformations, member stresses and reactions are required for planar frames, excepting that the member joints are now taken to be rigid and hence capable of transmitting moments. The behaviour of such frames can be represented as an assemblage of beam finite elements. A typical simple beam element is shown in Fig. 9.26, the physical and material properties of which are taken to be constant throughout the element. As with the previous rod element, changes in properties and load application are only admissible at nodal positions. In addition to u and v translatory freedoms, each node has a rotational freedom, 0, about the z axis, giving three dof. per node. Hence, axial, shear and flexural deformations will be represented, whilst torsional deformations which are inappropriate for planar frames will be ignored.

V'jv'j VjVj

V'jv'j VjVj

Element stiffness matrix in local coordinates

The differential equation of flexure appropriate to a beam element can be written as d2v/dxa = N'/EI (9.48)

in which v' denotes the displacement in the local / direction, N' is the element moment, E is the modulus of elasticity and I is the relevant second moment of area of the beam. The first derivative of the moment with respect to distance x' along a beam is known to give the

§9.8 Introduction to the Finite Element Method 335 shear force, V', i.e. dN'/dx' = V' (9.49)

Similarly, the first derivative of the shear force will give the loading intensity, co', i.e. dV'/dx = oJ (9.50)

Differentiating eqn. (9.48) and utilising eqn. (9.49) gives d3v'/dx'3 = V'/EI (9.51)

Differentiating again and utilising eqn. (9.50) gives d4v'/dx,4 = oj /EI (9.52)

Integrating eqn. (9.52), recalling that loads can only be applied at the nodes, and hence co' — 0, gives d3v'/dx'3 = C, = V'/EI, (from eqn. 9.51) (9.53) Further integration gives d2v'/dx'2 = C\x! + C2 = N'/EI, (from eqn. 9.48) (9.54)

and v = C,jc'3/6 + Czxa/2 + C3x' + C4 (9.56) With reference to Fig. 9.26, it can be seen that v'(0 ) = v'i, v'(L) = v'j, dv'/dx\Q) = 6'i, dv'/dx'(L) — 9'j

v'j = C,L3/6 + C2L2/2 + C3L + C4 = C,L3/6 + C2L2 / 2 + 9\L + v\ (9.59)

An expression for C2 can now be obtained by multiplying eqn. (9.60) throughout by L/3 and subtracting the result from eqn. (9.59), (to eliminate Ci), to give r/j - O'jL/3 = C2(L2/2 - L2/3) + 9\(L - L/3) + v] = C2L2/6 + 9\2L/3 + v\

Rearranging, C2 = 6(1/. - v\)/L2 - 6(9'jL/3 + 9\2L/3)/L

Rearranging eqn. (9.60) and substituting from eqn. (9.61) gives

C, = (2/L2)[(9'j - 9]) - 6(-4 + v'j)/L + 2(20; + %)]

Substituting for constant C\ from eqn. (9.62) into eqn. (9.53) gives shear force V = EIC\ = 12E7(t.; - v))/L3 + 6EI(0\ + O'^/L1

Substituting for constants C] and C2 from eqns. (9.62) and (9.63) into eqn. (9.54) gives the moment

= 6EI(2x' - L)(v\ - v'j)/L3 + 6E!x(Q\ + Q'f)/L2 - 2£7(26>' + 6))/L (9.64)

Note that the shear force, eqn. (9.63) is independent of distance x' along the beam, i.e. constant, whilst the moment, eqn. (9.64) is linearly dependent on distance x', consistent with a beam subjected to concentrated forces.

It follows that the nodal shear force and moments are given as

V"(0) = V'(L) = 12£7(u,' - v'j)/Û + 6EI($'i + <9})/L2 N'(0) = 6EI(-v', + v'j)/L2 - 2EI{20\ + d))/L N'(L) = 6EI(v'i - v'j)/L2 + 2El(d\ + 20'^¡L

The shear force and moments given by eqns. (9.65)-(9.67) use a Mechanics of Materials sign convention, namely, a positive shear force produces a clockwise couple and a positive moment produces sagging. To conform with the sign convention shown in Fig. 9.26, the following changes are required:

Writing in matrix form, eqns. (9.65)-(9.67) become

Combining eqn. (9.68) with eqn. (9.12) gives the matrix equation relating element axial and shear forces and moments to the element displacements as


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