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Strength of Materials I EGCE201 กำลังวัสดุ 1. Instructor: ดร.วรรณสิริ พันธ์อุไร ( อ . ปู ) ห้องทำงาน : 6391 ภาควิชาวิศวกรรมโยธา E-mail: [email protected] โทรศัพท์ : 66(0) 2889-2138 ต่อ 6391. Columns. Members that support axial loads. Columns fail as a result of an instability .

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Strength of Materials I EGCE201 กำลังวัสดุ 1

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Strength of Materials I EGCE201กำลังวัสดุ 1

Instructor: ดร.วรรณสิริ พันธ์อุไร (อ.ปู)

ห้องทำงาน: 6391 ภาควิชาวิศวกรรมโยธา

E-mail: [email protected]

โทรศัพท์: 66(0) 2889-2138 ต่อ 6391


Columns

  • Members that support axial loads.

  • Columns fail as a result of an instability.

The column may satisfy

the conditions for which

stress and deformation do

not result in failure, but

failure can still result.

Buckle – suddenly becomes

sharply curved


Euler’s Formula

  • Buckling is an instability related to deflection.

  • One would like to determine the smallest value of P, the critical buckling load, which is known as Euler’s formula.


Derivation

Taking moments about Q

Use the relation between beam deflection and moment.

This is a linear, homogeneous differential equation with constant coefficients.


Derivation (continued)

By setting p2=P/EI, the relation above becomes

The general solution for this equation is

Using the B.C.’s for ends A and B, we find that for y=0 at x=0, B=0.

Next, one consider the boundary condition y=0 at x=L, which yields

The possible solutions are A=0 and sin pL=0. If A=0,y=0 the column is straight.


Derivation (continued)

Examine the second solution

which is satisfied if

Using p2=P/EI and solving for P, the following relation is established.

The smallest value of P occurs when n=1. Setting n=1, one obtains the

critical buckling load, which is known as Euler’s formula.


The area moment of inertia (I) defines the axis about which buckling will occur.

Buckling axis


Buckling axis (continued)

Using the dimensions shown, we have


Critical stress

  • The stress corresponding to Pcr is called the critical stress and is denoted as scr.

  • The inertia can be represented in terms of the radius of gyration by I=Ar2where A is the cross-sectional area of the column and r is the radius of gyration. Using this definition for inertia, the critical stress is written as

The quantity L / r is called the slenderness ratio of the column. The min r=Imin

and should be used when computing the critical stress.


Extension of Euler’s buckling


Example I


Example II


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