Strength of materials i egce201 1
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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 egce201 1

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

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

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.


Strength of materials i egce201 1

Derivation

Taking moments about Q

Use the relation between beam deflection and moment.

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


Strength of materials i egce201 1

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.


Strength of materials i egce201 1

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.


Strength of materials i egce201 1

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

Buckling axis


Strength of materials i egce201 1

Buckling axis (continued)

Using the dimensions shown, we have


Critical stress

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.


Strength of materials i egce201 1

Extension of Euler’s buckling


Strength of materials i egce201 1

Example I


Strength of materials i egce201 1

Example II


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