Steel design. Compression members. Flexural buckling.
Buckling is the main mode of failure in compression members.
In this lecture, we will discuss the notion of buckling and how it impacts the design of steel members using the AISC specifications.
Flexural buckling is a sudden lateral deflection of the member when subjected to a significant compressive force.
Suppose a column is supported by a pin, and a roller at its ends.
When the compressive force in the column reaches a certain value, the member suddenly deflects laterally like this:
We call this phenomenon buckling.
It is important to note that buckling is not a gradual deflection due to a lateral load,
rather it is a sudden occurrence due to a compressive force.
Using the famous Euler buckling equation,
We can determine the magnitude of the force that causes buckling.
In this equation, L is the length of the member, I is the moment of inertia of the beam's cross-section,
and E is the material's modulus of elasticity.
So, for a steel member having a length of 12 feet
and a standard cross-section of w8x21, the buckling load is:
When the magnitude of the compressive force reaches 135 kips, theoretically speaking,
the member buckles like this:
Note that the member buckles about the weaker axis,
the axis that has the smaller moment of inertia.
So we can say that the nominal strength of this specific member against buckling is 134.9 kips.
We get the buckling stress if we divide the buckling load by the member's cross-sectional area.
This is the compressive stress in the member at the onset of buckling.
The moment of inertia (I) divided by the cross-sectional area (A) is related to the radius of gyration (r) of the section.
Therefore, we can rewrite the buckling stress equation as:
We refer to L/r as the slenderness ratio.
Here is the buckling stress graph as a function of the slenderness ratio.
This graph gives us a theoretical model for predicting the compressive stress that causes the member to buckle.
For example, at this slenderness ratio
the member buckles when the compressive stress reaches this value.
However, the available experimental data does not perfectly match the theoretical Euler buckling equation.
That data shows that the buckling stress is less than the value provided by the Euler equation.
There are two main reasons for this mismatch between the theory and the experimental data.
The theory assumes a perfectly straight compression member.
In practice however such perfection cannot be achieved.
Due to the fabrication process, steel members generally have a slight crookedness that adversely impacts their buckling strengths.
Furthermore, steel members with standard sections embody residual stress even when no load is present.
For example, a standard I-shaped member has a residual stress profile like this:
As we can see, the ends of the two flanges and the center of the web experience residual compressive stress.
Consequently, compressive stress in these areas grows and spreads faster than the theory suggests,
causing the member to buckle well below the value given by the Euler equation.
The AISC specification provides two curves for determining the buckling strength of compression members.
In this region, the material is assumed to behave elastically.
However, in this region, we assume inelastic Behavior due to the significance of the residual stress.
The equation for the elastic part of the curve is:
This is equation E3-3 given in the AISC manual.
And the equation of the inelastic region of the curve is:
To determine the slenderness ratio value at the intersection of these two regions,
we can equate the two AISC equations and solve for L/r.
A close examination of the elastic and inelastic curves reveals two intersection points.
The two curves intersect at 4.71 times the square root of E/Fy,
and 5.02 times the square root E/Fy.
AISC uses the smaller value as the bifurcation point between the two curves.
So, if the slenderness ratio of the member is less than this value,
we use this equation to determine the buckling strength of the member.
Otherwise, the buckling strength needs to be calculated using this equation.
To determine the design compressive strength of the member,
we need to multiply the nominal buckling strength
by the cross-sectional area and the strength reduction factor (theta_c).
For buckling, theta_c is 0.90.
Fcr is given by this pair of equations, where Fe is the Euler buckling equation.
Let's calculate the design compressive strength for a structural member having a length of 15 feet
using two standard sections: a w8x21 section and an L8x6x1 section.
In this example, we assume the member is pin supported at both ends.
We will discuss how other boundary conditions impact the solution in future lectures.
The pertinent properties of the two standard sections are given in the AISC manual.
For this example, we need the cross-sectional area and the radii of gyration about the x and y axes for each section.
Using the smaller radius of gyration,
we can determine the slenderness ratio for each section.
For the W-shaped section L/r =142.86
For the angle L/R = 104.65
Assuming a modulus of elasticity of 29 000 ksi and Fy of 36 ksi,
the bifurcation value for L/r becomes 133.68.
So, if the slenderness ratio is less than 133.68, we use this equation to determine the nominal buckling strength.
Otherwise, we need to calculate the nominal strength using this equation.
For w8x21 cross-section, the Euler buckling equation yields:
Hence, the nominal buckling strength becomes:
Therefore, the column has a design strength of 68 kips.
If the member has an L8x6x1 cross-section, the Euler buckling equation gives:
Hence, the nominal buckling strength becomes:
Yielding a design strength of 238 kips.
We will discuss the effect of various boundary conditions
on the buckling strength formulation starting next lecture.