Carbon steel cantilever beams are widely used in many applications in aerospace, civil and mechanical engineering. Pitting corrosion is a phenomenon which places severe limitations on the design of such applications. As such, understanding this phenomenon and the methods to deal with it, are of a great importance. This paper presents numerical investigation by using F. E. (Finite Element) simulation on the load carrying capacity of corroded cantilever beams with pitting corrosion damage. The pitting corrosion hole shape has been modeled using ASTM G46 Standard Guide. Several different cases of pitting corrosion, represented by hemispherical holes, were modeled and examined by using ANSYS computer program. Clamped edge constraint was used on one end, while the other end was free. In these F. E. models, element of Solid95 was used and comparison to Bernoulli-Euler theory was made. The effect of the radius of the pitting corrosion holes on the stresses in the beam was examined in comparison to yield stress. It has been found that the M. S. (Margin of Safety) has been reduced gradually with increasing radii. Agreement with Bernoulli-Euler theory has been achieved only for small radii. Moreover, three methods of pitting corrosion repairs were examined, together with Bernoulli-Euler theory comparison: 1) Regular surface repair; 2) Extension surface repair; and 3) “Handy Removal”. It was found that extension surface repair has the highest M. S. value.
Pitting corrosion is a critical problem in many fields such as civil engineering, ocean engineering and aircraft integrity design. In some cases, it can cause the formation of fatigue cracks, increase in the internal stresses and strength reduction. Pitting corrosion phenomenon, including other types of corrosion, has been investigated experimentally by Hoeppner [
In addition, F. E. simulations and numerical calculations have been made on the subject for different geometries of mechanical components. For instance, Chatterjee et al. [
Additionally, studies that include both experimental and numerical simulations were conducted by Potisuk et al. [
A thorough investigation that included F. E. analysis together with experimental data was done by Ruwan [
In contrary to many of the recent studies, this article concentrates on F. E. analysis rather than experimental data. In addition, a comparison to Bernoulli-Euler theory with the presence of pitting corrosion is performed. The last part of this article suggests three repair methods of pitting corrosion damage and comparison to Bernoulli-Euler theory is included.
In this study, a simulation of corroded cantilever beam has been done by using F. E. analysis and compared to Bernoulli-Euler theory. The pitting corrosion hole has been modeled by using hemispherical shape. The influence of hemispherical corrosion radius has investigated and comparison to Bernoulli-Euler theory was made. Eventually, three methods for repair of corrosion damage have been proposed and examined by using F. E. method and compared to Bernoulli-Euler theory.
The Beam in
The width and height of the section are represented by the parameters (b, h) and P is the force that is applied on the right end of the cantilever beam area. The left end of the cantilever beam is fully constrained and L represents the cantilever span. These geometric parameters are summarized in
The F. E. M. model has been created by using ANSYS 10.0 program. The model includes geometry, mechanical properties of the carbon steel and appropriate mesh selection and refinement.
The elements that were used to create the basic model are Solid95. According to ANSYS 10.0 information documents [
to model curved boundaries. The element is defined by 20 nodes having three degrees of freedom per node: translations in the nodal x, y, and z directions. The element may have any spatial orientation. SOLID95 has plasticity, creep, stress stiffening, large deflection, and large strain capabilities.
The mesh refinement must satisfy the need for a fine mesh to give an accurate stress distribution in a reasonable analysis time. The optimal solution is to use a finer mesh in areas of high stress: in the hemi spherical hole of the pitting corrosion and in the supports regions, respectively (
Total load of 73575N was applied on 9 nodes connected to area in the right end of the beam. Also, the left end surface area is restrained in every possible direction.
The material model that was used is AISI 1025 carbon steel. The yield and ultimate tensile stresses considered for the beam according to MIL-HDBK-5H [
Additionally, three different repair methods were established and modeled by F. E. M. The first repair method as illustrated in
Shell elements model for the first repair is illustrated in
rections, and rotations about the x, y, and z-axes. SHELL- 181 is well-suited for linear, large rotation, and/or large strain nonlinear applications. Change in shell thickness is accounted for in nonlinear analyses.
The second repair is an extension of the first repair by creating thick layer plate (0.06 × 0.06 × 0.01) that covers not only the hole but also the regions around it. The model was built by SOLID95 elements as illustrates in
The third repair method is called “Handy Removal” and is based on removing the corrosion by mechanical means. The removal geometrical model situation is shown in
Comparison between the repairs above and the influence of pitting corrosion on material strength will be discussed in Section 5.
The Bernoulli Euler equations of deflection and bending stress respectively are given by:
while is the deformation as function of axis and is the second moment of inertia.
while is the bending moment and is the maximum perpendicular distance to the neutral axis that in our case is .
Substitution of and in Relations (1)-(2)
while leads to:
Comparison between Bernoulli-Euler theory (Equations (3) and (4)) and pitting corrosion F. E. model for the stress and deflection will be will be made in the next sections.
In order to make credible comparison to Bernoulli-Euler theory, model calibration should be made. The F. E. model calibration is made of SOLID95 elements and no
corrosion pitting is modeled. The model is bounded in one end and the other end is subjected to a bending force on its area. The displacement in axis Y direction (see
F. E. model together with maximum deflection and principal stress results are presented in
The influence of the corrosion pitting on the beam’s strength has been examined by F. E. analysis. Three ratios of pitting corrosion hemisphere were modeled independently. It was found that the maximum principal stress is obtained on the circular shape of the corrosion according to Figures (8)-(10). Corrosion diameter sizeincreasing leads to M. S. (margin of safety) decreasing and deflection increasing (see
In addition, agreement with Bernoulli-Euler theory and pitted corrosion beam F. E. model as shown in
Comparison between shell and solid elements in case of 30 mm diameter shows that solid elements are more accurate in cases where the thickness is more critical and out of plane stresses and deflection play a main role as shown in
The sensitivity of the corrosion pitting repair has also been examined by F. E. analysis. Three kinds of pitting co-
rrosion repair were modeled independently. The diameter that was chosen to be repaired was 30 mm. Results of the three models are shown in Figures (12)-(14).
Comparisons between these repairs for principal stress, deflection and M. S. parameters are summarized in
The surface extension repair method was found to be with maximal M. S. value while handy removals repair method was found to be with minimal M. S. value. The handy removal repair method is based on cross section
reduction that causes to highly stress concentration value and therefore it’s the least effective method to use.
In addition, agreement with Bernoulli-Euler theory for these repairs as shown in
SOLID 95
SHELL 181
homogeneously behavior of the stress flow that leads to concentration reduction.
F. E. analysis is very effective tool to use in order to understand the pitting corrosion mechanical behavior. ANSYS program is used in this study since it presents a plain and simple way to study the behavior of cantilever beam pitting corrosion.
The influence of hemispherical pitting corrosion shape
on cantilever beam has been studied by F. E. analysis in the context of stress failure (comparing to yield stress) and maximum deflection allowance. Three types of hemisphere radii were examined (30 mm, 60 mm, 80 mm). The M. S. decreasing is caused by corrosion diameter increasing since cross section reduction causes to stress concentration.
Also, compatibility between maximum principal stress
and deflection to Bernoulli-Euler theory was found only for small radius of the hemispherical corrosion shape (30 mm). Possible explanation was given by saying that Bernoulli assumption (“cross-sectional planes during bending deformation remain planes and perpendicular to the neutral axis”) is no longer necessarily valid for increasing diameter of pitted corrosion.
Moreover, examination of pitting corrosion repair was examined by using F. E. analysis. Three methods of repair have been investigated: 1) Regular surface repair; 2) Extension surface repair; and 3) “Handy Removal”.
Due to cross section reduction, the removal repair method is found to be with minimal M. S. value while surface extension repair method is with the maximal M. S. value.
In addition, agreement with Bernoulli-Euler theory for the three repairs was found only for extension surface repair (3.8% error) but for regular surface and handy removal repairs it was found to be inadequate (about 27% and 40% error respectively). Possible explanation for this phenomenon is due to the repair surface effectiveness; by connecting to as many nodes as possible, the repair surface area is large enough to cause homogeneously behavior of the stress flow that leads to concentration reduction.
The author gratefully acknowledges the financial support by Technion—Israel institute of Technology.