Structural Instability

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STRUCTURAL

INSTABILITY

Columns


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Introduction

A large proportion of an aircrafts structurecomprises thin webs stiffened by slenderlongerons or stringers

Both are susceptible to failure by buckling at abuckling stress or critical stress,

Clearly, for this type of structure, buckling isthe most critical mode of failure

the prediction of buckling loads of columns,thin plates and stiffened panels is extremelyimportant in aircraft design


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Types of structural instability

Two types of structural instability arise:

primary and secondary

Primary:-no change in cross-sectional areawhile the wave length of the buckle is of the

same order as the length of the element solid

and thick-walled columns experience this type

of failure


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S

econdary:-Changes in cross-sectional areaoccur and the wavelength of the buckle is of

the order of the cross-sectional dimensions of

the element

Thin-walled columns and stiffened plates may

fail in this manner


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Euler buckling of columns


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The well-known solution of Equation


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where 2 =PCR/EI and A and B are unknown

constants.

The boundary conditions forthis particular

case are v=0 at z=0 and l. Thus A=0 and B sin

l = 0 For a non-trivial solution (i.e. v=0) then sin l

=0 orl = n where n = 1, 2, 3, . . .


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The smallest value of buckling load, in other words the

smallest value ofP which can maintain the column in a

neutral equilibrium state is obtained by substituting n=1,

hence

Other values ofPCR corresponding to n=2, 3, . . . , are


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The critical stress, CR, corresponding to PCR,

is, from Eq. (8.5)

where le is the effective length of the column. This is the length of apin-ended column that would have the same critical load as that of a column of

length l, but with differentend conditions

The term l/r is known as the

slenderness ratio of the column


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Inelastic Buckling

Above the elastic limit d/d depends upon the value of stress and

whetherthe stress is increasing or decreasing. Thus, in Figure the

elastic modulus at the pointA is the tangent modulus Et if the

stress is increasing but E if the stress is decreasing.


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Determination of reduced elastic modulus


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Continued and get similar

result except E


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The above method for predicting critical loads and stresses

outside the elastic range is known as the reduced modulus

theory


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Effect of initial imperfections

Obviously it is impossible in practice to obtaina perfectly straight homogeneous column andto ensure that it is exactly axially loaded.

An actual column may be bent with someeccentricity of load.

Such imperfections influence to a large degree

the behavior of the column which, unlike theperfect column, begins to bend immediatelythe axial load is applied.


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Let us suppose that a column, initially bent, is

subjected to an increasing axial load P as

shown in Fig.

Initially bent column


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Example 8.2

The pin-jointed column shown in Figure carries a

compressive load P applied eccentrically

at a distance e from the axis of the column.

Determine the maximum bending moment in the

column.


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Stability of beams under transverse

and axial loads

Beams supporting both axial and transverse

loads are sometimes known as beam-columns

or simply as transversely loaded columns.


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Structural Instability

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