The Galerkin method is applied to analyze the elastic large deflection behavior of metal plates subject to a combination of in-plane loads such as biaxial loads, edge shear and biaxial inplane bending moments, and uniformly or nonuniformly distributed lateral pressure loads. The motive of the present study was initiated by the fact that metal plates of ships and ship-shaped offshore structures at sea are often subjected to non-uniformly distributed lateral pressure loads arising from cargo or water pressure, together with inplane axial loads or inplane bending moments, but the current practice of the maritime industry usually applies some simplified design methods assuming that the non-uniform pressure distribution in the plates can be replaced by an equivalence of uniform pressure distribution. Applied examples are presented, demonstrating that the current plate design methods of the maritime industry may be inappropriate when the non-uniformity of lateral pressure loads becomes more significant.
Ships and ship-shaped offshore structures are composed of metal plate elements, and the accurate computation of nonlinear behavior of the plate elements in deck, bottom and side shells up to the ultimate limit state is a basic requirement for the structural safety assessment. The plate elements in ships and ship-shaped offshore structures are generally subjected to combined inplane and lateral pressure loads. Inplane loads include biaxial compression/tension, biaxial inplane bending and edge shear, as shown in Figure
In-plane load components applied in a plate element.
Roll motion of a vessel causing nonuniformity of lateral pressure load distribution in the hull cross-section.
Non-uniformly distributed lateral pressure loads in a plate element.
A large number of studies have been available in the literature, for example [
The elastic large deflection behavior of a plate element with initial imperfections can be analyzed by solving two differential equations, one representing the equilibrium condition and the other representing the compatibility condition [
Equations (
With the homogeneous solution
In the present paper, the procedure described in [
In (
Some applied examples are now presented. A simply supported rectangular plate of
Also, the lateral pressure loads are taken as
Configuration of the initial deflection in the plate.
The added deflection of the plate due to applied loads can be expressed as follows.
Figures
Deformed shapes of the plate when
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0.1 | 0.1 |
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0.1 | 0.15 |
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0.1 | 0.2 |
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0.1 | 0.3 |
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0.1 | 0.4 |
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(a) The axial compressive load versus total deflection curves at
Figure
The axial compressive loads versus total deflection at
It is found that the current practice of the maritime industry with an average magnitude of applied pressure loads underestimates the lateral deflection of the plate compared to the real condition of the pressure load application, that is, with a non-uniform distribution. The difference between the two cases becomes larger and larger as the non-uniformity of lateral pressure loads becomes more significant.
The underestimation of the plate deflection due to external pressure loads gives rise to the overestimation of plate strength performance which may lead to unsafe design at optimistic side.
The aim of the present paper has been to analyze the elastic large deflection behavior of metal plates subject to combined inplane and lateral pressure loads. When lateral pressure loads applied in plate elements are non-uniform, the current practice of the maritime industry applies some simplified design methods in which the non-uniform pressure distribution in the plates is replaced by an equivalence of uniform pressure distribution.
In the present paper, the Galerkin method was used to solve the nonlinear governing differential equations of plate elements under non-uniformly distributed lateral pressure loads in addition to inplane loads. Some applied examples are presented, demonstrating that the current practice of the maritime industry, that is, with an average magnitude of applied pressure loads as an equivalence, results in a great underestimation of lateral deflection calculations when the non-uniformity of lateral pressure loads becomes more significant. The underestimation of the plate deflection may lead to unsafe design of plates at optimistic side.
Thin plates buckle in elastic regime, while stocky plates may buckle in elastic-plastic or plastic regime. The present paper deals with elastic behavior only, and further studies are then recommended to take into account the effect of plasticity which is dominant in thick plates. Also, it is highly desirable for practical design purpose to develop a simpler method or design formulation.
The coefficients
Length of the plate
Breadth of the plate
Plate bending rigidity
Young’s modulus
Airy’s stress function
Maximum half-wave number of the assumed added deflection function in the
In-plane bending moment in the
In-plane bending moment in the
Lateral (out-of-plane) pressure load on the surface area
Axial force in the
Axial force in the
Thickness of the plate
(Added) deflection of the plate due to the action of external loads
Initial deflection of the plate
Total deflection of the plate
Axis direction normal to the
Aspect ratio of the plate (
Poisson’s ratio
Bending stresses in the
Normal stresses in the
Mean stresses in the
Shear stress.
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant no. K20903002030-11E0100-04610).