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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXX V, Part B7. Istanbul 2004
2. BEAM DEFLECTION BY INTEGRATION
There are numerous 3D data modelling techniques available,
such as creating a TIN or gridding. Selection of an appropriate
surface model is critical to permit the accurate computation of
an objects deformation. The method chosen to model vertical
deflections in these experiments is based on forming analytical
models representing the physical bending of the beam. The
models are derived from first principles of beam deflection by
integration, which essentially yields low order polynomials (no
higher than a cubic in the experiments presented later). Once
these models are developed, the coefficients of the polynomials
are solved as unknown parameters in a least-squares estimation
process. The observations consist of the several hundred 3D
point samples from each TLS. A single functional model is used
to represent the beam deflection but the parameters of the model
are egtimated for each deflection epoch.
A beam which is subjected to loading will bend into an arc
which can be defined by a curvature function (Beer and
Johnston, 1992). The equation, known as the governing
differential equation for the elastic curve, is shown in Eq. 1. It is
a second-order linear differential equation and is composed of
the beam’s bending moment, M, which is a function of x. the
distance along the beam, divided by the modulus of elasticity,
E. and moment of inertia, I. The bending moment is a reaction
to an applied force which causes a structure to rotate or bend.
This equation holds true for small deflections. Integrating Eq. 1
twice, with respect to x, will yield the function of deflection.
This function will permit the vertical deflections to be
computed. A more detailed explanation of beam deflection by
integration may be sought in Beer and Johnston (1992).
NO (1)
ox ~ El
The modelling process can be demonstrated using an example.
Consider a simply supported beam (i.c. a support point at each
of its ends) consisting of a load point, P, at the centre of the
beam, located at xp. A sketch is shown in Figure 1.
z (m)
M |
X (m)
support beam support
Figure 1, Schematic diagram for the timber beam.
The bending moment, represented by two functions (one each
side of xp), is linear, maximum at Xp and zero at each support
point. Two successive integrations yield a cubic equation. The
&neralised form of the compound cubic polynomial is given in
Eq. 2 and may be adopted for curve fitting the beam shown in
Figure 1. An additional term in the y-axis direction is added to
model any linear tilts about the X-axis ( rotation) that may be
evident in the 3D scan cloud of the beam. Justification for this
term is given later. A detailed derivation of the model (and
curve fitting constraints) can be found in Gordon et al. (2003a).
955
zzi zx Taj tàpotàaàgjy USXEXp o
L = x 2 2)
22 = b39X box ^ bigx boy +agıyiXp SXSL
3. STATISTICAL TESTING OF ESTIMATED
PARAMETERS
The modelling process yields low-order polynomials that are
naturally prone to high parameter correlation, which indicate
that the functional model is not efficient in modelling the
curvature. A test is adopted to assess the significance of the
estimated parameters for each of the solutions. Parameters that
are found to be statistically insignificant should be removed
using an appropriate elimination strategy, such as backward
stepwise selection, to alleviate high coupling. Statistical testing
of parameters is routinely performed to ensure that estimated
models are optimised for the task at hand. Examples in the
geomatics industry may be found in photogrammetry, such as
identification of insignificant additional parameters in aerial
photogrammetry (Jacobsen, 1982). Zhong (1997) shows that
polynomials, used for the interpolation of GPS geoid heights,
can be statistically tested. The author's recommendations
include a methodology for optimal parameter selection for
polynomial models.
3.1 Global Significance Test
The first step in the statistical testing process involves analysing
the overall model for the significance of its parameters.
Following the methodology recommended by Zhong (1997), the
F-test is adopted and a global test statistic. F. is computed
using:
where C;is the covariance matrix of estimated parameters,
^ 2i. : . - . -
00 is the estimated variance factor, r is the degrees of freedom. t
is the number of parameters in x , the parameter vector and F is
Fisher's distribution. Since the least-squares estimation process
involves hundreds of point samples, the degree of freedom is
always very high compared to small number of unknown
parameters, assuming that the observations are uncorrelated.
3.2 Individual Parameter Significance Test
It is possible to extend the analysis to the testing of each
individual parameter using (Zhong, 1997):
2 x . ; D ar.
where x; is the square of the i" estimated parameter, 0; isthe
ES
. ; sa .
variance for that estimated parameter, Gj is the estimated
*.
variance factor and F; is the individual parameter test statistic.
