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Buckling 05

This document describes a study analyzing the lateral buckling of a simply supported right-angle frame subjected to bending moments. Four cases are analyzed using different element types and densities: beam elements with 8, 16, and 20 elements and plate elements with 64 elements. The results are then compared to analytical and other numerical solutions. For all cases, the buckling loads, deformed shapes, and comparison table are presented. In general, the results converge to the analytical solution as the element density increases.

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Dario Manrique Gamarra
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116 views8 pages

Buckling 05

This document describes a study analyzing the lateral buckling of a simply supported right-angle frame subjected to bending moments. Four cases are analyzed using different element types and densities: beam elements with 8, 16, and 20 elements and plate elements with 64 elements. The results are then compared to analytical and other numerical solutions. For all cases, the buckling loads, deformed shapes, and comparison table are presented. In general, the results converge to the analytical solution as the element density increases.

Uploaded by

Dario Manrique Gamarra
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Buckling-5

Title
Lateral buckling of a simply supported right-angle frame subjected to bending memonts at
both ends

Description
A simply supported right-angle frame is subjected to bending moments M applied at the
centroids of its ends. The buckling loads are determined for the four cases as described
below (Case 1 ~ Case 4). The computed buckling loads are then compared with the
analytical solution [1] and the results from the prominent papers [2, 3].
Case 1: Beam element (total 8 elements: 4 elements for each leg)
Case 2: Beam element (total 16 elements: 8 elements for each leg)
Case 3: Beam element (total 20 elements: 10 elements for each leg)
Case 4: Plate element (total 64 elements)

(a) Beam element model

(b) Plate element model

Structural geometry and boundary conditions

Verification Example

Model

Analysis Type
Lateral torsional buckling
Unit System
N, mm
Dimension
Length

240 mm

Element
Beam element and plate element (thick type without drilling dof)
Material
Youngs modulus of elasticity
Poissions ratio

E = 71,240 N/mm2
= 0.31

Section Property
Beam element : solid rectangle 0.6 30 mm
Plate element : thickness 0.6mm, width 15mm, height 15mm
Boundary Condition
Left end is pinned, and right end is roller.
Load
M = 1.0 Nmm
P = 1/30 N

Buckling-5

Results

Buckling Analysis Results


Case 1: Beam element (total 8 elements)
1st mode

Buckling load

Top view

Isometric view

Verification Example

Case 2: Beam element (total 16 elements)


1st mode

Buckling load

Top view

Isometric view

Buckling-5

Case 3: Beam element (total 20 elements)


1st mode

Buckling load

Top view

Isometric view

Verification Example

Case 4: Plate element (total 64 elements)


1st mode

Buckling load

Top view

Isometric view

Buckling-5

Comparison of Results
Unit: Nmm
Case

Type of element

Timoshenko and Gere [1]


Argyris et al [2]
Saleeb et al [3]
MIDAS Case 1
MIDAS Case 2
MIDAS Case 3
MIDAS Case 4

Beam element
Triangular shell
Beam element
Beam element
Beam element
Beam element
Plate element

No. of total
elements
86
20
8
16
20
64

Critical load for


1st buckling
649.19
624.36
627.37
634.26
622.25
620.82
598.40

Timoshenko and Gere [1] provided the analytical solution using the theory of elastic
stability for the tip cirtical load M cr . Also a number of prominent authors have given
approximate solutions obtained from their geometrically nonlinear ananyses [2, 3].
The analytical solution from Timoshenko and Gere is as follows:

M cr =

1.05
L

EI z GI xx =

I z I xx
1.05
E
2(1 + )
L

where,
L = length of the edge of the square plate
E = Youngs modulus of elasticity
G = shear modulus of elasticity
= poissons ratio

I z = moment of inertia about local z-axis

I xx = torsional moment of inertia


Substituting the material and sectional properties into the above equation gives the
following result:

Verification Example

M cr =
=

I z I xx
1.05
E
L
2(1 + )

1.05
0.54 2.1328
71, 240
240
2(1 + 0.31)

= 649.19 N mm

References
1. Timoshenko, S.P., and Gere, J.M., (1961). Theory of Elastic Stability, McGraw-Hill,
New York.
2. Argyris, J.H., Hilpert, O., Malejannakis, G.A., Sharpf, D.W., (1979). On the
geometrical stiffnesses of a beam in space a consistent V.W. approach, Comp. Meth.
Appl. Mech. Eng., Vol. 20, 10531.
3. Saleeb, A.F, Chang T.Y.P, Gendy A.S., (1992). Effective modeling of spatial buckling
of beam assemblages, accounting for warping constraints and rotation-dependency of
moments, Int. J. Num. Meth. Eng., Vol. 33, 469502.

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