A rectangular column with a 4.4 -m effective length is made...

A rectangular column with a 4.4 -m effective length is made of glued laminated wood. Knowing that for the grade of wood used the adjusted allowable-stress for compression parallel to the grain is $\sigma_{C}=8.3 \mathrm{MPa}$ and the adjusted modulus $E=4.6 \mathrm{GPa}$ determine the maximum allowable centric load for the column.

Key Concepts

Modulus of Elasticity

The modulus of elasticity, also known as Young’s modulus, measures the stiffness of a material in response to elastic deformation under load. It quantifies the relationship between stress and strain in the elastic region of the material’s behavior and is a crucial factor in determining the critical buckling load of a column. A higher modulus indicates a stiffer material that is less prone to buckling under axial loads.

Euler Buckling

This concept describes the critical load at which a slender, perfectly straight column under axial compression becomes unstable and buckles. Euler’s formula uses the modulus of elasticity, the column’s moment of inertia, and the effective column length to determine the load at which buckling will occur. It is fundamental for understanding design limits in columns where buckling, rather than material failure, governs the maximum allowable load.

Allowable Compressive Stress

The allowable compressive stress is the maximum stress that a material can safely withstand under compression without experiencing failure or excessive deformation. In design, the actual stress from a load is compared with this allowable value. For wooden columns, this adjusted value accounts for factors such as material variability and long-term loading effects, ensuring that design remains within safe stress limits.

Effective Length

The effective length of a column represents the length over which buckling is assumed to occur and accounts for the boundary conditions at the column’s ends. It may differ from the actual physical length depending on the degree of fixity or rotational restraint provided by the supports. This concept is critical in buckling analysis, as it directly influences the critical load through Euler’s formula.

Transcript

00:01 Here we have been given the material property of the wood.

00:04 Young's modulus of wood is given as 12 giga pascal and the yield stress for this wood is given as 55 megapascal.

00:19 And the cross sectional area is given as 100 mm square.

00:31 So now let us determine the critical buckling load.

00:35 So it will be pcr is equal to 5.

00:38 Square ei divided by effective length square.

00:45 So here, since the column is fixed at the bare end and free at top end, so the effective length will be 2l.

00:54 So let us put this 5 square multiplied by the value of e is given as 12 gigapascal, that is 2l multiplied by 10 to the power of 9 pascal and the value of i is bh cubed divided by 12, where b is equals to 0 .1 and h is also 0 .1 cube divided by 2l divided by effective length is now the effective length is twice of the original length so 2 multiplied by 2 meter so this square now after calculation we get the critical or the buckling load is equal to 61 .68 kilo newton 61 .68 kilo newton now this is part 1.

01:49 In part 2, we need to check for yielding in y direction.

01:55 Check yield in y -y direction.

02:04 So let us find out the critical stress.

02:08 It will be critical force divided by the area.

02:11 So the critical force is 61 .68 divided by area.

02:16 Area is 0 .1 multiplied by 0 .1 meter square...

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