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Research Papers

# Prediction of Formation of Wavy Surfaces in Rolled Plates by Post-Buckling Analysis

[+] Author and Article Information
Jinseok Kim

Department of Mechanical Engineering, Pohang Institute of Science and Technology, Pohang 790-784, Korea

Shrikant Pattnaik1

Department of Mechanical Engineering, University of Cincinnati, Cincinnati, OH 45221-0072mailspp@gmail.com

Jay Kim

Department of Mechanical Engineering, University of Cincinnati, Cincinnati, OH 45221-0072

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1

Corresponding author.

J. Appl. Mech 77(4), 041005 (Apr 09, 2010) (8 pages) doi:10.1115/1.4000908 History: Received November 21, 2008; Revised October 19, 2009; Published April 09, 2010; Online April 09, 2010

## Abstract

Excessive wavy surfaces formed by a cold- or hot-rolling process in a thin plate degrades the value of the plate significantly, which is called the flatness problem in the industry. It is a result of post-buckling due to the residual stress caused by the rolling process. Because the buckling occurs in a very long, continuous plate, a unique difficulty of the problem as a buckling problem is that the buckling length is not given but has to be found. In many previous works, the length that gives the lowest critical load of the plate for the given load profile was taken as the buckling length. In this work, it is shown that this approach is flawed, and a new approach is developed to solve the flatness problem by extending a classic post-buckling analysis method based on the energy principle. The approach determines the buckling length and amplitude without using any unfounded assumptions or hypothesis. Using simple displacement functions, approximate solutions are obtained in closed forms for the plate subjected to a linearly distributed residual stress. The new solution approach can be used to determine the condition for the maximum rolling production that does not cause the flatness problem.

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## Figures

Figure 1

Buckling wave formed in a rolled plate; (a) a very long plate where the wave is formed. Definitions of the buckling length (L), buckling amplitude (δ) and plate thickness (h) are shown. (b) The free body diagram of the plate section of one buckling length. Coordinate system and the edge load Nx(y) are shown.

Figure 2

The concept to determine the buckling length. (a) N0,cr is the critical load calculated as a function of the plate of length. The previous approach based on the minimum critical load determines L1 as the buckling length assuming that the axial load in the plate increases from zero while the plate length is kept constant (illustrated as the path “a-b-c” in Fig. 2). (b) The actual path of the axial load-plate length will describe a path similar to the path illustrated as “1–2” in Fig. 2.

Figure 3

Illustration to explain the process of the buckling wave formation. In the plate that keeps moving through the rollers, the length of the unbuckled part of the plate is initially very small while the axial load due to the residual stress is quite high (point 1 in Fig. 2). As the rolling continues, this unbuckled length will increase and the axial load will also change until the load becomes large enough to cause finite buckling deflection, reaching to point 2 in Fig. 2.

Figure 4

Buckling amplitude as a function of the buckling load N0

Figure 5

Symmetric (solid line) and antisymmetric (dotted) two-term solution

Figure 6

Buckling amplitude obtained for a symmetric mode

Figure 7

Comparison of C1 and C2 for the edge compression case; symmetric mode (left) and antisymmetric mode (right)

Figure 8

Comparison of buckling amplitude calculated for the edge compression case using the symmetric mode function: one-term solution (dotted) and two-term solution (solid)

Figure 9

Comparison of C1 and C2 for the center compression case

Figure 10

Comparison of buckling amplitude calculated for the center compression case: one-term solution (dotted) and two-term solution (solid)

Figure 11

Effect of magnitude of the load N0 on flatness. Solid line: symmetric edge buckling; dotted line: antisymmetric edge buckling; dashed line: center buckling.

Figure 12

Effect of the plate thickness on flatness. Solid line is symmetric edge buckling; dotted line: antisymmetric edge buckling; dashed line: center buckling.

Figure 13

Effect of the plate width on flatness. Solid line: symmetric edge buckling; dotted line: antisymmetric edge buckling; dashed line: center buckling.

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