Effective Length and Slenderness Ratio of Compression Members
The effective length of the column is the vertical height between points of inflection (points of zero bending moment) on the buckled shape. The effective length of the column may be calculated using the formula K x L, which involves multiplying the effective length factor by the column length.
The slenderness ratio is the ratio of its effective length to the radius of gyration (r) of a compression member. It is denoted by.
λ=Klr
Where,
- r= radius of gyration =√(I/A)
- I= moment of inertia
- A = Area of cross-section
As per Clause No 3.8, Table no 3 of IS 800:2007, The Slenderness ratio is limited due to the effect of accidental and construction load is automatic may come into the structure to take care of the unexpected vibration and the bracing member used for temporary support for equipment. The following limits are given by the code for the compression member.
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S. No | Member | Maximum effective slenderness ratio |
1 | A member carrying compressive loads resulting from dead loads and imposed loads. | 180 |
2 | A tension member in which a reversal of direct stress occurs due to loads other than wind or seismic forces. | 250 |
3 | A member subjected to compression forces resulting only from combination with wind/ earthquake actions provided the deformation. | 300 |
4 | A member normally acting as a tie in a roof truss or a bracing system is not considered effective. | 350 |
Modes of Failure in Compression Member
In a compression member, there are several factors due to which compression members fail. The failure depends on the column type (depending upon slenderness ratio).
The compression member fails due to buckling and crushing in the case of the column.
- Crushing or Yielding failure is shown in a short column.
- Buckling failure is further classified into elastic buckling and inelastic buckling. When the compression member translates perpendicular to the applied load, It shows in the long column.
- The intermediate column fails due to inelastic failure. This failure is the combination of yielding and lateral buckling in a column.
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Types of Sections for Compression Members
A hot rolled section is one whole piece of metal made into that shape when it was 'red hot', whereas a welded section is a built-up section made on site by connecting plates.
- Built-up columns are used when the height of the column is such that a rolled section cannot provide a sufficiently large radius of gyration.
- Built-up columns generally consist of two or four shapes connected by cover plates perforated at intervals with access holes. This joining is done with the help of direct welding, Battening System, and a lacing system.
- The most common Examples are the Double channel section(Back to back and Toe to toe channel section), I - section, and the four angles section.
Lacing and Batten column:
The Lacing and Battening are provided to maintain the vertical alignment of the individual member connected. Using Lacing and Battening, the stresses are equally distributed throughout the column section. A laced Column has a lacing element which can be Flat, angle channel Tubular, etc.
Encased column:
A steel column is encased inside the Concrete column to increase the fire resistance of the column and prevent corrosion. The main reason to provide a concrete encased column is to increase the compressive resistance of the column. The expansion of the concrete can be restricted by the steel case, and the buckling of steel will be restricted by the concrete provision.
Design of Compression Members
As per IS 800:1984, The Merchant Rankine Formula has some limitations in the Working Stress Method (WSM). As per IS 800: 2007, due to various imperfection that exists in the real column. Based on statistical test results, Perry Robertson's approach proposes curves a, b, c, and d which consider the imperfection factor. The buckling curve (a,b, c, d) gives the reduction factor value in the resistance of the column. Design axial compressive strength of a member calculated by,
P_{d}= f_{cd}.A_{e}
f_{cd}= (f_{y}/γ_{mo})/Φ+(Φ^{2} -λ^{2 })^{0.5} ≤f_{y}/γ_{mo}
f_{cd} =χ.f_{y}/γ_{mo}
(∵χ = 1//Φ+(Φ^{2} -λ^{2 })^{0.5},Φ= 0.5[1+α(λ-0.2)+ λ^{2})
Where
- f_{cd}= Design stress in compression member
- A_{e}= effective sectional area
- χ = reduction factor
- λ = non dimensional slenderness ratio =fyfcc
- f_{cc}= Eulers buckling stress= π^{2}E/(KL/r)^{2}
- α=Imperfection factor
The imperfection factor depends on the shape of the column cross-section, fabrication process, and direction at which buckling occurs. Buckling class defines the strength and residual stress concerning the category. The hollow section comes in Buckling Class a. The Solid, Angle, T, and Built-up sections come in Buckling Class c.
Buckling class | Imperfection factor |
a | 0.21 |
b | 0.34 |
c | 0.49 |
d | 0.76 |
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