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 .
The column may satisfy
the conditions for which
stress and deformation do
not result in failure, but
failure can still result.
Buckle – suddenly becomes
Taking moments about Q
Use the relation between beam deflection and moment.
This is a linear, homogeneous differential equation with constant coefficients.
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.
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.
Using the dimensions shown, we have
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.