07-07-2014, 03:31 PM
Computational analysis of multi-stepped beams and beams with
linearly-varying heights implementing closed-form finite element formulation
for multi-cracked beam elements
Computational analysis of multi-stepped.pdf (Size: 587.87 KB / Downloads: 13)
a b s t r a c t
The model where the cracks are represented by means of internal hinges endowed with rotational springs
has been shown to enable simple and effective representation of transversely-cracked slender Euler–Bernoulli
beams subjected to small deflections. It, namely, provides reliable results when compared to
detailed 2D and 3D models even if the basic linear moment–rotation constitutive law is adopted.
This paper extends the utilisation of this model as it presents the derivation of a closed-form stiffness
matrix and a load vector for slender multi-stepped beams and beams with linearly-varying heights. The
principle of virtual work allows for the simple inclusion of an arbitrary number of transverse cracks. The
derived at matrix and vector define an ‘exact’ finite element for the utilised simplified computational
model. The presented element can be implemented for analysing multi-cracked beams by using just
one finite element per structural beam member. The presented expressions for a stepped-beam are
not exclusively limited to this kind of height variation, as by proper discretisation an arbitrary variation
of a cross-section’s height can be adequately modelled.
The accurate displacement functions presented for both types of considered beams complete the derivations.
All the presented expressions can be easily utilised for achieving computationally-efficient and
truthful analyses.
Introduction
Numerous engineering structures are subjected to degenerative
effects during their utilisation. The progressions of cracks can severely
decrease the stiffness of an element and further lead to
the failure of the complete structure. In view of this, it is an important
task of engineers to detect these cracks as soon as possible.
However, the efficiency of structural health monitoring depends
not only on the data measured but also on the qualities and versatilities
of computational models regarding mechanical behaviour
modelling. Undoubtedly, suitable 2D or 3D meshes of finite elements
yield a thorough discretisation of the structure, as well as
of the crack and its surroundings. Although this approach is excellent
when evaluating a structure’s response to a cracked situation
(with all the crack’s details known in advance), it becomes quite
awkward for inverse problems where the potential crack’s details
(presence, location, intensity) are unknown. Consequently, simplified
models are more efficient in such situations.
The model that has been the subject of numerous research in
the past, is the model provided by Okamura et al. (1969). In this
model each crack is replaced by a massless rotational linear spring
of suitable stiffness and the linear moment-rotation constitutive
law is adopted. Each spring connects those neighbouring noncracked
parts of the beam that are modelled as elastic elements.
Okamura et al. introduced the earliest definition for rotational
linear spring stiffness for a rectangular cross-section. In addition,
some other researchers (Dimarogonas and Papadopulus, 1983; Rajab
and Al-Sabeeh, 1991; Ostachowicz and Krawczuk, 1990; Krawczuk
and Ostachowicz, 1993; Sundermayer and Weaver, 1993;
Hasan, 1995; Skrinar and Pliberšek, 2004) have presented their
definitions.
This model was successfully implemented for dynamic analyses.
For a singly-cracked beam Fernández-Sáez and Navarro
(2002) presented closed-form expressions for the approximated
values of fundamental frequencies, whilst for a beam with multiple
cracks several solutions exist for natural frequency calculations: a
technique that reduces the order of the determinantal equation
(Shifrin and Ruotolo, 1999); a transfer matrix-based method leading
to the determinant calculation of a 4 4 matrix (Khiem and
Lien, 2001); a fundamental solutions and recurrences formulaebased
approach for determining the mode-shapes of non-uniform
beams and concentrated masses (Li, 2002). The dynamic response
of a cracked cantilever beam subjected to a concentrated moving
A multi-cracked multi-stepped beam finite element’s mathematical model formulation
The discussed multi-stepped-beam finite element considers
slender elastic homogeneous Euler–Bernoulli beams subjected to
small deflections. The element of total length L is assumed to have
a uniform modulus of elasticity E and width b. It consists of a sequence
of Ns consecutive elastic geometric sections, as shown in
Fig. 1. These sections are numbered from the left-end and each section
of the element is characterised by a different uniform thickness
hj (j = 1,2,. . . ,Ns). The location of the interface between the
section j and adjacent section j + 1 to the right is denoted as Lj
(with Lo ¼ 0 and LNs ¼ L).
Each section can be either non-cracked or cracked with an arbitrary
number of transverse cracks. The cracks are described by a
massless rotational springs. Due to the localised effects of the
cracks, the adjacent non-cracked parts bordering each crack are
modelled as simple elastic sections connected by the rotational
spring.
Each crack is assumed to be open with a uniform depth di. The
spring constant Ki is a function of the corresponding non-cracked
cross-section’s height hj, the relative depth of the crack di = di/hj,
Poisson’s ratio , and the flexural rigidity EIj of the neigbouring
cracked cross-section j. The finite element has altogether Nc cracks
at locations (distances) Li from the left node (i = 1,2,. . . ,Nc).
The finite element has four degrees of freedom altogether:
transverse displacement Y1 and rotation U1 at the left-end (node
1), as well as transverse displacement Y2 and rotation U2 at the
right-end (node 2). Upward displacement and anticlockwise rotations
are taken as positive.
Derivation of stiffness matrix
The four columns of stiffness matrix are obtained from two separate
derivations. Since the finite element has four degrees of freedom
altogether, this consequently means that in order to obtain a
statically determinate structure, two degrees of freedom must be
simultaneously removed. In order to complete this, all the required
stiffness matrix coefficients are obtained from two cantilever substructures:
clamped at both the right and left-ends, respectively.
Computation of exact transverse displacements along the
finite element
In current derivations the transverse displacements interpolation
(or shape) functions were not derived at for two reasons.
Firstly, they were unrequired, as the presented derivations were
accomplished completely without their utilisation. Secondly, standard
interpolation functions are complete polynomials of the third
degree and their implementation, in those situations where either
the transverse-distributed load is applied or the beams have nonuniform
heights, does not produce accurate results. Therefore,
the governing differential equations solutions’ have to be considered
in order to obtain the correct transverse displacements for
both the exposed situations.
Conclusions
This paper studied multi-cracked slender-beams with the transverse
cracks represented by means of internal hinges endowed by
rotational springs. This simplified model was utilised for the derivation
of a corresponding beam finite-element of a stepped
cracked beam. The stiffness matrix, as well as the load vector due
to a uniform continuous load were derived at by implementing
the principle of virtual work. Despite the derivation’s straightforwardness,
all the obtained terms are written entirely in closedsymbolic
forms that make the geometric as well as the damage
parameters clearly observable. The provided expressions for geometric
coefficients can even be numerically evaluated for any geometric
variation of the beam’s height, thus allowing the
computational model to be expanded further than just stepped
beams.
In the cases of multi-stepped beams with multiple cracks, the
presented stiffness matrix, derived at by implementing the principle
of virtual work, produces identical results to the solutions already
proposed in the literature by other authors who have
implemented different approaches. Therefore, in order to expand
the topics, those beams with linearly-varying heights that appear
more frequently, are also covered separately.