Compression Members

ENGR. HEBER JOHN DE VERA

CHAPTER OBJECTIVES:

1. Describe and distinguish compression

members in a structure;
2. Enumerate and explain the limit states that
govern in analysis and design of structural
steel compression members; and
3. Analyze and design structural steel
compression members considering strength
and serviceability requirements in
accordance to the relevant code provisions.
Compression Members
Compression members are structural elements that are subjected to axial compressive
forces; that is, the loads that are applied along a longitudinal axis through the centroid of
the member cross section, and the stress can be taken as , where is considered to be
uniform over the entire cross section. Examples of compression members are columns, truss
members, and frame braces.

The stress in an axially loaded compression

member is given by:

where:
P = magnitude of the load.
A = cross-sectional area of the member normal
to the load.
Limit States in the Design of Compression Members
In analysis and design of compression members,
especially columns, the following member limit
states are to be considered:
• General Buckling – It is the instability of a
compression member due to applied
compression load. General buckling is signified
by either lateral deflection (flexural) or twisting
(torsional) of the member itself.
• Local Buckling – It is the instability of section
components (e.g. flanges, webs) of a
compression member. Local buckling is signified
by the deflection of its section components.
Euler Buckling Theory
General Buckling – is the instability of a
member due to applied axial
compressive load. There are three
considered types of general buckling:
• Flexural Buckling – It is characterized
by lateral bending of the member.
• Torsional Buckling – It is characterized
by twisting of the member.
• Flexural – Torsional Buckling – It is
characterized by both lateral bending
and twisting of the member.
Euler Buckling Theory
Euler buckling theory forms the basis for considering the flexural buckling behavior of
compression members. Assumptions for the formulation of the Euler buckling theory are as
follows:
• The material in which the member is made of is homogeneous (i.e. the stress-strain
relation of the material is the same at all points in its cross-section).
• There are no initial or residual stresses introduced to the member.
• The member is perfectly straight and prismatic.
• The axial compressive load acts through the centroid of the cross-section prior to
bending.
• End conditions are ideal. (In this case, pinned connections at both ends of the member
will be considered first.)
• Small deflection theory of ordinary bending is applicable and shear effects are
neglected (i.e. elastic flexure theory holds true).
• Twisting or distortion of cross-section does not occur.
Euler Buckling Theory
Assuming pinned connection on both ends, the critical load to be applied on the
compression member to prevent buckling is equal to:

where:
• Pcr = critical buckling load
• E = modulus of elasticity of the material
• A = cross-sectional area of the member
• L = full length of the member
• r = radius of gyration of the member
• I = moment of inertia
Example 1
A wide flange column has an unsupported Length Lu = 7.2 m and a modulus of elasticity of
200,000 MPa. Assume that the column has pinned ends.
Properties of wide flange column:
A = 6,437.5 mm2 Ix = 49.6 x 106 mm4
E = 200,000 MPa Iy = 16.64 x 106 mm4

a.) Compute the critical load if the column buckles by bending about its strong axis.
b.) Compute the critical load if the column buckles by bending about its weak axis.
Example 2
A W12 x 50 is used as a column to support an axial compressive load of 145 kips. The length
Is 20 feet, and the ends are pinned. Without regard to load or resistance factors, investigate
this member for stability.
Euler Buckling Theory
For different end conditions, the equation for the critical buckling load is modified
as:

where:
• k = effective length factor
• Note: kL is the effective length of the member, i.e. the expected length of the member
that is affected by buckling.
The term kL/r is referred to as the slenderness ratio. This parameter is used to
determine the susceptibility of a compression member to flexural buckling. Lower
slenderness ratio indicates higher critical buckling load, thus making the member less
susceptible to buckling.
The effective length factor k depends on the end conditions of the member,
which dictates the buckling shape of the member.
Effective Length Factors, K
Euler Buckling Theory
It should also be noted that flexural buckling may occur in any of the two directions,
whether about the strong/major axis of the member (the axis of the cross-section that
has a greater moment of inertia or radius of gyration) or about the weak/minor axis of
the member (the axis of the cross-section that has a lesser moment of inertia or radius
of gyration). The slenderness ratio of a member have two values: one for the strong
axis and the other for the weak axis.
Example 3
The properties of the column are the following:
A = 8,129 mm2
Ix = 178.3 x 106 mm4
Iy = 18.8 x 106 mm4
E = 200,000 MPa

The x-axis has an unbraced length of 8 m which is pinned at the top and fixed at the
bottom to prevent sidesway. The y-axis has an unbraced length of 4 m due to the bracing
at mid-height.

Determine the critical slenderness ratio.

Inelastic Buckling
From the previous section, assumptions for the formulation of Euler buckling theory are
enumerated. However, based on test results on structural steel columns, as well as
general observations and studies on their general behavior, it is observed that the
assumptions laid out for Euler buckling theory are not completely satisfied on all cases. In
other words, Euler buckling theory does not always hold true for all structural steel
columns.

Plot of Comparison of Critical Buckling Load from Euler Buckling Theory and Test Results
Inelastic Buckling
Some of the deviations are as follows:
• In structural members, residual stresses are almost always present, which decreases the
capacity of the member to carry the required loads, which results to lower buckling
load. Residual stresses are caused by:
• Uneven cooling after hot-rolling process;
• Cold bending during fabrication (i.e. bending of members without heat as part of
the fabrication process);
• Punching of holes and cutting operations during fabrication; and/or
• Welding
• The axial compressive load is not always expected to be applied through the centroid
of the cross-section, which introduces additional bending moments on the member.
• Distortion of cross-section may occur due to local buckling, characterized by instability
of the section components such as flanges and webs (i.e. these components may also
bend or warp).
• Other factors such as end restraints/supports and straightness of the member do not
satisfy the assumptions for Euler buckling theory.
Inelastic Buckling
These deviations contribute to the occurrence of the phenomenon on structural steel
columns known as inelastic buckling. Due to this, some theories on inelastic bucking were
developed for structural steel columns, such as the basic tangent modulus theory by
Engesser (1889) and the inelastic column theory by Shanley (1947). These theories form
the basis of the specifications and code provisions developed by AISC for the analysis
and design of structural steel compression members, which the 2015 NSCP Vol. 1 adopts.
Local Buckling of Column Sections
Compression members may fail in two ways (i.e. limit states): general buckling and local
buckling. Local buckling occurs when the section components of a member, such as
flanges and webs, becomes unstable due to applied loads. This occurs due to
slenderness of these components. The susceptibility of these components to be unstable
is measured by its own slenderness measure, known as the width-to-thickness ( ). The
width-thickness ratio of a flange is denoted as , while for a web, it is .
Structural steel sections are composed of either unstiffened elements, stiffened elements,
or both.
• Unstiffened elements – section components that are supported along one edge
parallel to the direction of loading and unsupported on the other edge.
• Stiffened elements – section components that are supported on both edges parallel to
the direction of loading.
Local Buckling of Column Sections
Structural section components subjected to axial compression are classified into two in
terms of their local stability:
• Compact – section components that are not susceptible to local buckling.
• Slender – section components that are susceptible to local buckling.
Structural steel sections which have all of its components classified as compact are
called compact sections. Most hot-rolled I-shaped sections are considered to be
compact.
Local stability checks for structural steel sections may be done by checking the
requirements of Section 502.4 of 2015 NSCP Vol. 1. In case of hot-rolled I-shapes subjected
to axial compression, Tables 502.4.1 & 502.4.2 of 2015 NSCP Vol. 1 may be used. For
flanges, Case 3 should be used. For webs, Case 10 should be used. Components are
classified as compact if . Otherwise, it is slender. For an I-section to be classified as
compact, both its flanges and webs should be compact. Otherwise, it is classified as a
slender section.
Example 4
Check the compactness of the following sections under axial compression:
a) W 14x730
b) W 18x311
Example 4
Check the compactness of the following sections under axial compression:
a) W 14x730
b) W 18x311
Code Provisions for
Analysis and Design of Compression Members
For this module, only compact hot-rolled I-shapes will be considered and used for analysis
and design of compression members. (Provisions for this particular condition is in Section
505.3 of 2015 NSCP Vol. 1.)
In analysis and design of compression members, the following must be satisfied:

where:
LRFD:
ASD:

As a note, members to be designed on the basis of compression must have a slenderness

ratio of at most 200.
Code Provisions for
Analysis and Design of Compression Members
The nominal axial compressive strength of a compression member with compact hot-
rolled I-shaped sections is as follows:

where:
Pn = nominal axial compressive strength fo the member.
Fcr = critical buckling stress of the member
Ag = gross – sectional area of the member.

The critical buckling stress Fcr is calculated as follows:

• When ;

• When ;
• Fy = yield strength of the material
• Fe = elastic critical buckling stress; calculated as
( ⁄)
• kL/r = slenderness ratio of the member.
Procedures for Analysis and Design of Columns
Analysis of Structural Steel Compression Members
These are the steps in analysis of structural steel compression members (doubly symmetric
hot-rolled I-sections with compact flanges & webs):
• Take note of given data and design philosophy to be used. Write also some
assumptions that need to be made.
• Determine the critical slenderness ratio of the member. Check if it is less than 200, as
denoted in Section 505.2 of 2015 NSCP Vol. 1.
• Check if the section is either compact or slender. If the section is compact, provisions
from Section 505.3 of 2015 NSCP Vol. 1 may be used. Otherwise, use other provisions of
Section 505 of 2015 NSCP Vol. 1.
• Assuming that the section is compact, determine the design capacity of the member.
Engr. Heber John de Vera
hjdevera.ce@tip.edu.ph

Contact hours: 1:00PM –

5:00PM (MWF)