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Buckling of Compressed Bars

If a deformed structure continues to deform in the direction of the initial deflection due to destabilization, it is considered to be in a state of unstable equilibrium. The load, the excess of which causes a loss of stability is critical for the structure and its determination is the purpose of stability calculation.

When the critical load is exceeded, both the general loss of stability leading to the deformation of the entire structure, and local loss of stability causing the deformation of individual elements may occur. With further exceedance of the critical load, the structure can acquire other forms of the loss of stability, however, when making engineering calculations, they are usually limited to the first form corresponding to the minimum critical load.

In the general case, the loss of stability occurs when the energy of the critical load exceeds the potential energy of the deformed structure:

Al > Ud

In this calculation, longitudinally compressed bars of length L with a constant moment of inertia of the cross-section Ix for various boundary conditions are considered. As a result of calculations, the minimum critical load F of the first form of the loss of stability is determined. According to the calculation conditions, the bar stresses should not exceed the yield strength of the material.

For the calculation, the elastic modulus E of the bar should be specified.

Buckling of bars
Buckling of bars under loads

INITIAL DATA

L - Bar length;


Ix - Moment of inertia of the bar cross-section;


E - Young's modulus.

RESULTS DATA

F - Critical buckling force.

Bar length (L)

Moment of inertia (Ix)

Young's modulus (E)

Scheme # 1

Scheme # 2

Scheme # 3

Scheme # 4

Scheme # 5

Scheme # 6

Critical force (F)

BASIC FORMULAS

Critical load:

F = π2*E*Ix / (μ*L) 2,

μ = 1 for scheme # 1;

μ = 2 for scheme # 2;

μ = 0.7 for scheme # 3;

μ = 0.7 for scheme # 4;

μ = 0.5 for scheme # 5;

μ = 1 for scheme # 6.

INITIAL DATA

L - Bar length;


Ix - Moment of inertia of the bar cross-section;


E - Young's modulus.

RESULTS DATA

F - Critical buckling force.

MATERIALS PROPERTIES

Material

Young’s modulus

Pa (psi)

Poisson’s ratio

Steel

1.86÷2.1×1011 (2.7÷3.05×107)

0.25÷0.33

Cast iron

0.78÷1.47×1011 (1.1÷2.1×107)

0.23÷0.27

Copper

1.0÷1.3×1011 (1.45÷1.9×107)

0.34

Tin bronze

0.74÷1.22×1011 (1.1÷1.8×107)

0.32÷0.35

Brass

0.98÷1.08×1011 (1.4÷1.6×107)

0.32÷0.34

Aluminum alloy

0.7×1011 (1.0×107)

0.33

Magnesium alloy

0.4÷0.44×1011 (5.8÷6.4×106)

0.34

Nickel

2.5×1011 (3.6×107)

0.33

Titanium

1.16×1011 (1.7×107)

0.32

Lead

0.15÷0.2×1011 (2.2÷2.9×106)

0.42

Zinc

0.78×1011 (1.1×107)

0.27

Glass

4.9÷5.9×1010 (7.1÷8.5×106)

0.24÷0.27

Concrete

1.48÷2.25×1010 (2.1÷3.3×106)

0.16÷0.18

Wood (along the grain)

8.8÷15.7×1010 (12.8÷22.8×106)

-

Wood (across the grain)

3.9÷9.8×1010 (5.7÷14.2×106)

-

Nylon

1.03×1010 (1.5×106)

-

OTHER CALCULATORS

AREA MOMENTS OF INERTIA
BEAM CALCULATORS
TORSION OF BARS
CIRCULAR FLAT PLATES
BUCKLING
ELASTIC CONTACT
IMPACT LOADS
NATURAL FREQUENCIES
PRESSURED SHELLS
FLUID DYNAMIC
COMPOSITES
SPRINGS
THREAD CONNECTIONS
SHAFT CONNECTIONS
BEARINGS
DRIVES
FATIGUE
HEAT TRANSFER