Excerpt

## Abstract

In this paper, two types of steel frames, steel frame without side sway permission and another with side sway permission are created in Abaqus with 10 multiple slenderness ratio of the columns by changing the length every time starting from 1 M and ending with 10 M length of the columns,Twenty models of steel frames with single story and single bay were created, the models are with the same 2D dimensions and material properties, the cross section of the steel is (0.5*0.5) M ,and the supports are fixed, two equal forces P= 1000 N are exerted on the frames in the position mentioned in fig 6, a beam section was defined for the frame integrated before analysis with Young modulus of elasticity E=1*107 N/M2 , and shear modulus G = 3.8*106 N/M2 and poisons ratio ν = 0.3. .

A linear perturbation step is created for buckling and 10 eigenvalues are requested for analysis, a standard quadratic beam element type is generated with global seeding of 0.6, and 20 Jobs are created for every situation and conclusions have been obtained, the critical buckling loads of the frames fall in the ranges between the Euler loads forms which has been proved for each type of frames and this scientific approach was verified in this research, in addition to that the relation between the length of the column and the eigenvalues that represent the critical loads of buckling verified, and the simulations of the mode shapes of buckling of the steel frames were identified adopting finite element analysis which shows the amount of loads necessary to reach each mode shape of buckling for each type of steel frames mentioned before .

## Keywords

Euler column **,** stiffness matrix, critical buckling load, eigenvalues and eigenvectors

## Introduction to buckling

If a beam element is under a compressive load and its length if the orders of magnitude are larger than either of its other dimensions such a beam is called a column. Due to its size its axial displacement is going to be very small compared to its lateral deflection called buckling.

Slender or thin‐walled components under compressive stress are susceptible to buckling and is called “Euler buckling” where a long slender member subject to a compressive force moves lateral to the direction of that force, as illustrated in Figure 1. The force, F, necessary to cause such a buckling motion will vary by a factor of four depending only on how the two ends are restrained. Therefore, buckling studies are much more sensitive to the component restraints that in a normal stress analysis. The theoretical Euler solution will lead to infinite forces in very short columns, and that clearly exceeds the allowed material stress. Thus in practice, Euler column buckling can only be applied in certain regions and empirical transition equations are required for intermediate length columns. For very long columns the loss of stiffness occurs at stresses far below the material failure.

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**Figure.1** Long columns fail due to instability

Quite often the buckling of column can lead to sudden and dramatic failure. And as a result, special attention must be given to design of column so that they can safely support the loads **.**

Buckling can be related to the singularity of the tangent stiffness matrix, which in turn consists of two parts. The first part is the material stiffness matrix which is related to the deformational stiffness of the components, taking into account the connectivity of components in the current geometric configuration of the structure. For linear elastic components, the material stiffness is identical to the linear elastic stiffness, but updating the structural geometry to include the effect of any displacements. The second part is the geometric stiffness matrix, which is related to the component forces, and in some cases to the applied loading, taking into account the effect of a change in geometry from the current configuration. For typical structures, the material stiffness is positive for all deformation modes, mathematically referred to as positive-definite, whereas the geometric stiffness can admit negative values for certain modes, depending on the component forces and applied loading. It is therefore the effect of a negative geometric stiffness that can lead to a singular overall tangent stiffness matrix, and hence buckling.

## Stability concept

The question of the stability of various forms of equilibrium of a compressed bar can be investigated by using the same theory as used in investigating the stability of equilibrium configurations of rigid-body systems (Timoshenko and Gere, 1963). Consider three cases of equilibrium of the ball shown in Figure.2. It can be concluded that the ball on the concave spherical surface (a) is in a state of stable equilibrium, while the ball on the horizontal plane (b) is in indifferent or neutral equilibrium. The ball on the convex spherical surface (c) is said to be in unstable equilibrium.

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**Figure.2** Ball equilibrium cases

The compressed bar shown in Figure.3 can be similarly considered. In the state of stable equilibrium, if the column is given any small placement by some external influence, which is then removed, it will return back to the un-deflected shape. Here, the value of the applied load P is smaller than the value of the critical load Pcr. By definition, the state of neutral equilibrium is the one at which the limit of elastic stability is reached. In this state, if the column is given any small displacement by some external influence, which is then removed, it will maintain that deflected shape. Otherwise, the column is in the state of unstable equilibrium.

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**Figure.3** Column equilibrium cases

## Euler Column

The Euler column is the axially loaded member shown in Figure.4 which is very idealized and is assumed to have a constant cross sectional area and to be made of homogeneous material. In addition, four assumptions are made:

1- The ends of the member are simply supported. The lower end is attached to an immovable hinge, and the upper end is supported so that it can rotate freely and move vertically but not horizontally.

2- The member is perfectly straight, and the load is applied along its Centroid axis.

3- The material obeys Hooke's law.

4- The deformations of the member are small enough so that the curvature term can be approximated to derive the Euler load P which is the buckling load.

Due to imperfections no column is really straight. At some critical compressive load it will buckle. To determine the maximum compressive load (Buckling Load) we assume that buckling has occurred as shown in Figure.4.

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**Figure.4** Buckling load

**[...]**

- Quote paper
- Nazim Nariman (Author), 2013, Finite element analysis of the buckling critical loads in un-braced steel frames with multiple slenderness ratio configurations, Munich, GRIN Verlag, https://www.grin.com/document/262659

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