Building Structures: Column Design and Stability

1. Lecture 5. Building Structures

P R E PA R E D B Y: S .
N I Y E T B AY

2. Compression Members

• Definition
A structural element that primarily resists
compressive forces is referred to in structural
mechanics as a column (strut).
• Function of Columns
Columns are vertical bar-like members
designed to transfer loads:
• from upper structural components
• to the foundation or lower-level elements
• Typical Applications
Columns are used to support:
• floor systems and roofs
• working platforms
• crane-supporting structures
• overpasses and pipelines
• Materials for Columns
The most common materials include:
• steel
reinforced concrete
masonry (brick/stone)
timber
Terminology by Material
Depending on the material:
• steel and reinforced concrete → columns
• timber → posts / struts
• masonry → piers / pillars
• Bar-Like Members
Members with relatively small cross-sectional
dimensions are generally called bars
(members) in analysis.
• Examples of Compression Members
• columns
• top chords of trusses
• selected web members
• arch ribs / arch members

3. Column Design: General Principles

Classification by Structural Behavior
Columns are classified according to the nature of loading:
Axially loaded (centrally compressed) columns
Eccentrically loaded columns
Axially Loaded Columns
An axially loaded column is a member in which the applied force acts through the centroid of the cross-section.
For symmetric cross-sections, the centroid coincides with the geometric center.
Key Feature
Only compressive normal stresses arise
No bending moment is generated from load eccentricity

4. Eccentrically Loaded Columns

• Definition
In eccentrically loaded columns, the applied force does not
pass through the centroid of the section.
• Load Characteristics
• The force acts with an eccentricity e0e_0e0​
• Equivalent interpretation: simultaneous action of:
• Axial force N
• Bending moment M
• Fundamental Relationship
Eccentricity is defined as:
• e0=M/N
• Engineering Implication
• Combined stress state (compression + bending)
• Non-uniform stress distribution across the section

5. Behavior of Axially Loaded Columns

Stress–Strain State
The structural response and failure mode of axially loaded columns depend on:
Behavior of
Axially
Loaded
Columns
material properties
cross-sectional dimensions and shape
member length
end restraints and boundary conditions
General Observation
Despite material differences, compression members exhibit common behavioral features
under load.
Typical Failure Mechanism
In most cases, failure occurs due to loss of overall stability, manifested as buckling
(longitudinal bending).

6. Buckling and Structural Implications

Types of Bending
Longitudinal bending (buckling) – caused by axial compression
Transverse bending – caused by lateral forces applied perpendicular to the member axis
Failure Condition
Failure occurs when stresses in extreme fibers reach limit values, leading to material failure.
Role of Slenderness
Slender members → more prone to buckling
Stocky members → higher stability
Material Influence
Steel and timber columns → typically more slender and flexible
Reinforced concrete and masonry columns → larger sections, lower slenderness
Design Principle
Structural design codes account for safe limits of buckling, forming the basis of column design.

7. Basic Strength Condition for Axially Loaded Columns

Assumption of Stress Distribution
For axially compressed columns, normal stresses σ are assumed to be uniformly distributed across
the cross-section.
Basic
Strength
Condition for
Axially
Loaded
Columns
Ultimate Limit State Criterion
Load-bearing capacity is considered adequate if:
N≤RA
where:
N – design axial force (most unfavorable load)
R – design resistance of the material
A – cross-sectional area
Interpretation
This is the fundamental strength requirement for centrally compressed members.

8. Stability Effect and Buckling Reduction Factor

Practical Behavior of Columns
Real columns are influenced by longitudinal bending (buckling), which
reduces load-bearing capacity.
Modified Design Condition
N≤φRA
where:
φ<1.0 – buckling (stability) reduction factor
Engineering Meaning
Accounts for loss of stability
Applicable to columns of any material
Used as the basic stability design equation

9. Slenderness Ratio and Effective Length

Slenderness Ratio
λ=l0/i
​ where:
l0 – effective (design) length
i – radius of gyration
Effective Length Expression
l0=μl
where:
l – geometric length of the member
μ – effective length factor (depends on end
restraints)
• Key Concept
Boundary conditions significantly influence
buckling behavior and column stability.

10. Radius of Gyration and Section Properties

Radius of Gyration
i= IA
​ where:
I – second moment of area (moment of inertia)
A – cross-sectional area
Engineering Role
Characterizes stiffness of the section relative to buckling
Used in slenderness evaluation
Section Parameters
For standard shapes (rectangular, circular, tubular), area
and inertia are determined by established formulas.

11. Slenderness in Principal Directions

• Directional Slenderness Ratios
• λx=l0x/ix​​,
λy​=​l0y /iy
• Key Concept
• Cross-section dimensions may differ by axis
• Radius of gyration differs: ix≠iy
• Slenderness therefore differs by plane
• Buckling Rule
Buckling tends to occur about the axis with the larger slenderness ratio.

12. Buckling Reduction Factor and Limits

Behavior of Columns
Short columns → stability effects negligible → φ≈1.0
Slender columns → stability critical → φ<1.0
Typical Practical Values
φ≈0.5–0.8
Limiting Slenderness
Design codes introduce a maximum allowable slenderness λlim​, which must not be
exceeded regardless of load level.
Design Implication
Stability considerations may govern design even at relatively low loads.

13. Typical Design Problems for Columns

• Type 1 – Required Section Area
• A≥NφR
• ​ Used for preliminary sizing and selection of
cross-section dimensions.
• Type 2 – Capacity Verification
• N≤φRA
• Used to check safety under given loading
conditions.
• Type 3 – Load-Bearing Capacity
• Φ=φRA
• Determines the maximum admissible axial
force.
• Practical Note
In design practice, sizing and verification
problems are most common and typically
solved together.

14. Eccentrically Loaded Columns – Stress Distribution

Key Difference from Axial Compression
Under eccentric compression, normal stresses are non-uniformly distributed across the cross-section.
Possible Stress Cases
σmin>0, σmax>0 → Entire section in compression
σmin=0, σmax>0 → Neutral boundary condition
σmin<0, σmax>0 → Combined compression and tension
Mechanical Cause
Stress non-uniformity arises due to the simultaneous action of:
Axial force N
Bending moment M
Design Implication
Part of the section may experience tensile stresses, depending on eccentricity.

15. Design Principles for Eccentric Compression

• Strength Conditions
• Compressive stress limitation:
• σmax≤Rc
• ​ If tension occurs:
• σmin≤Rt
• ​ where:
• Rc – design compressive resistance
• Rt​ – design tensile resistance
• Section Optimization Strategy
To improve stability and strength:
• Increase section dimensions in the
direction of bending
• Reduce slenderness and increase
stiffness
• Engineering Note
Analysis of eccentrically loaded
columns is more complex and
strongly dependent on material
behavior (steel, RC, masonry, timber).

16. Steel Columns – Application and Common Forms

Field of Use
Steel columns are widely used in industrial and public buildings, particularly when:
Building heights are significant (e.g., > 10 m)
Heavy crane loads are present
Rapid erection and reduced member size are required
Economic Considerations
Although steel columns may be more expensive than reinforced concrete or masonry, their use is often justified by:
High strength-to-weight ratio
Steel Columns
– Application
and Common
Forms
Smaller cross-sections
Fabrication and installation efficiency
Simplest Structural Form
The most basic solution is a solid column of constant cross-section, typically made from:
Rolled I-sections (wide-flange preferred)
Circular hollow sections (pipes)

17. Types of Solid Steel Column Sections

Typical Cross-Section Configurations
Rolled I-section – most common and efficient
Welded I-section – used for customized geometry
Circular hollow section (pipe) – good torsional properties
Built-up sections from rolled profiles:
• Two channels
• Two angles
• Other combinations
Design Features
Section selection depends on stability and slenderness
Welded sections allow optimization of inertia and stiffness
Hollow sections provide uniform behavior in all directions
Engineering Practice
In structural design courses, analysis often focuses on solid steel columns made from rolled wide-flange
sections

18. Failure Modes of Steel Columns Under Load

Loss of Load-Bearing Capacity May Occur Due To:
Loss of Overall Stability
Global buckling of the member
Most typical for slender columns
Loss of Strength
Caused by section weakening (holes, cutouts,
connections)
Critical when net cross-sectional
area is reduced
May govern before global buckling
in some cases
Loss of Local Stability
Local buckling of plates or flanges
Typical for thin-walled welded sections
Preventive Design Measures
Use of stiffeners (transverse ribs)
Increasing plate thickness
Proper selection of rolled profiles (I-sections, pipes)
Practical Engineering Note
For rolled I-sections and tubular columns, local stability problems are usually
minimized by standardized proportions; therefore, design is often governed by
Failure Modes of
Steel Columns Under
Load

19. Strength and Stability Checks for Steel Columns

Strength Condition
σ=N/An≤Ryγc
​ Where:
σ – normal stress in the column section
N – design axial force
An​ – net cross-sectional area (accounting for holes / reductions)
Ry​ – design yield strength of steel
γc​ – service condition factor

20. Stability Condition

σ=NφA≤Ryγc
​ Where:
φ – buckling reduction factor
A – gross cross-sectional area
In practical design, column dimensions are usually governed by stability, not by material
strength.

21. Slenderness Requirement

• Slenderness Ratio
• λ=lefi≤λlim
• ​ Where:
• lef – effective (design) length of column
• i – radius of gyration of section
• λlim – code-specified limit
• Engineering Interpretation
• Large λ → higher buckling risk
• Small i → more flexible member
Slenderness
Requirement
• Increasing stiffness reduces instability probability
• Even if strength is satisfied, excessive
slenderness is not permitted.

22. Section Selection Procedure (Simplified)

• Main Design Steps
1. Determine axial load
2. Select structural system & boundary conditions
3. Compute effective length:
lef=μl
4. Choose preliminary slenderness λ
5. Determine required radius of gyration:
Section
Selection
Procedure
(Simplified)
i=lefλ
6. ​ Compute required area:
A≥NφRyγc
7. Select section from steel tables
8. Verify stability and strength
• Exact required area rarely matches catalog values →
final check is always mandatory.

23. Structural Components of Steel Columns

Steel Column Design Includes Three Primary Parts:
1. Column Head (Cap / Top Connection)
1. Transfers loads from beams, girders, trusses
2. Requires verification of bearing plate thickness
3. Often strengthened with stiffeners
2. Column Shaft (Member Body)
1. Main load-bearing element
2. Designed from strength & stability requirements
3. Preferably equal stiffness about principal axes
3. Column Base
1. Transfers axial force to foundation
Structural
Components of
Steel Columns
2. Distributes stresses into concrete
3. Ensures anchorage and stability
Engineering Principle
Column detailing is based on results of structural analysis, not
arbitrary proportions.

24. Column Shaft Design Considerations

Key Requirements for Centrally Compressed Columns:
Equal stability in both principal planes
Rational cross-sectional shape
Corrosion protection considerations
Typical Practical Solutions
✔ Rolled I-sections (most common)
✔ Welded I-sections (for large loads)
✔ Tubular sections (good stability, protection issues)
Important Observation
Ordinary rolled I-beams may have different radii of gyration about x–x and y–y
axes → wide-flange sections are preferred.

25. Column Base Design Fundamentals

• Purpose of Column Base
• Transfer load NNN to concrete foundation
• Prevent local crushing of concrete
• Provide fixity and anchorage
• Concrete Bearing Resistance
• Rb,loc=α φb Rb
• ​ Where:
• Rb​ – concrete prism strength
• φb​ – bearing enhancement factor
• α – coefficient (≈ 1.0 for normal concrete)
• Required Bearing Area
• Aloc≥N/Rbγb2
• Practical Design Notes
• Bearing plate thickness (typical): 20–60 mm
• Anchor bolt diameter (typical): 20–30 mm
• Use of stiffened bases reduces plate thickness
• Key Concept
• Base design is governed by concrete behavior,
not steel strength.