Structural Analysis Methods II: Structural Systems - Mechanics of Structures: Variational and Computational Methods - page 246

earlier, this transfer matrix procedure requires two passes of progressive multiplications
of the transfer matrices. In the first pass, Eq. (5.7), is formed, and the initial parameters
z a are evaluated using the boundary conditions. Usually, the values of two of the four
unknown initial parameters

a ,V a ,M a are known by simple observation of the left
end of the beam. In the second pass, using the just determined z a and Eq. (5.6), the state
variables

w

a ,

θ

,V, and M are printed out along the beam. The solution procedure is illus-
trated using a particular beam in the next example problem.
Formulas for the initial parameters for the common boundary conditions are listed in
tables in references such as Pilkey (1994).

w , θ

5.1.4 Summary of the Transfer Matrix Solution Procedure
The notation employed for the transfer matrix method is summarized in Table 5.2. A so-
lution should begin with the modeling of the beam system in terms of elements that con-
nect locations (stations) of point occurrences (for example, applied concentrated forces) or
abrupt changes (for example, a jump in cross-sectional area). For each element, compute the
required section properties such as the moment of inertia I . Then calculate the elements of
the field matrices for each segment, as well as the point matrices for the concentrated occur-
rences. Now compute the global transfer matrix by multiplying, in sequence, all transfer
matrices from the left end to the right end of the beam. That is, calculate U of

U M U M − 1 U 2 U 1 z a

z x = L = z L = ··· U k ··· = Uz x = 0 = Uz 0 (5.12a)

Use this expression to evaluate the unknown initial variables of

a

w
V
M
1

z a = (5.12b)

by applying the boundary conditions to Eq. (5.12a). This can be accomplished by elimin-
ating the unnecessary rows and columns of Eq. (5.12a) and solving the remaining equations.
Finally, the deflection, slope, shear force, and internal moment are computed at all points
of interest using

U j U j − 1 U 2 U 1 z a

z j = ··· U k ··· (5.12c)

TABLE 5.2
Notation for the Transfer Matrix Method
Symbol Definition Remarks

U i Transfer matrix for the i th element (field) z k = U i z j

U M U M − 1 U 2 U 1

U Global or overall transfer matrix that spans several U = ···

elements

U k Point matrix to account for concentrated occurrence,

e.g., a point force or discrete spring, at location k

v k
s k

Displacements
Forces

z k State vector at location k , contains all displacement

and force state variables z k = =

z i Applied loading function vector for the i th element

o r
z i k

Vector of applied loading functions for the i th field,
evaluated at point k