Modeling, Analysis and Design
SNAPColumn™
is a family member of SNAPBridge™: Bridge Analysis and Design Suite, which is
used for the Analysis and Design of Reinforced Concrete Columns.
Copyright © SAS Suite, LLC and Lubin Gao,
Ph.D., PE. All Rights Reserved.
This chapter describes the methodology used in SNAPColumn™.
Table
of Contents
service load
design and strength design method
load and
resistance factor design method.. 3
capacity/Strength/resistance
Analysis of column sections
Nominal Strength(Resistance) Interaction Diagram/Surface
design capacity/strength/resistance
capacity/strength/resistance
check
consideration on slenderness effect
Column is a structural component primarily taking axial compression loads. Bridge pier columns, drilled shafts, arch ribs are the examples. These structural components shall be designed to undertake the axial loads and most-commonly the bending moment either in one direction or two direction including any dead and live load, self-weight of the structure, temperature and shrinkage effects, and earthquake loads in accordance with the design specifications.
There are different types of columns. The type of an column is generally determined and selected on the basis of the aesthetics, site condition, cost-efficiency etc. however, they should all meet the general requirement for safety and functionality.
SNAPColumn™ is designed to help design engineers to speed their design, increase productivity, simplify the design procedure, and reduce their tedious trial-error time.
Design specifications used in SNAPColumn™ are AASHTO Standard Specifications for Highway Bridges and LRFD Bridge Design Specifications.
For
Service Load Design Method (Allowable Stress Design), Articles 8.15.1, 8.15.2
and 8.15.4 of AASHTO Standard Specifications for Highway Bridges applies.
For
Strength Design Method (Load Factor Design), Articles 8.16.1, 8.16.2, 8.16.4
and 8.16.5 of AASHTO Standard Specifications for Highway Bridges applies.
For Load
and Resistance Factor Design, Articles 5.7.1, 5.7.2, 5.7.3 and 5.7.4 applies.
For
Service Load Design Method (Allowable Stress Design) and Strength Design Method
(Load Factor Design) of compression members, Articles 8.7 and 8.16.2 of AASHTO
Standard Specifications for Highway Bridges lists the design assumptions for
reinforced concrete members including columns. The assumptions are
Article
8.7.1: “The modulus of elasticity, Ec, for concrete may be taken as
wc1.5 33√f’c in psi for values of wc
between 90 and 155 pounds per cubic foot. For normal weight concrete (wc=145pcf),
Ec may be considered as 57,000√f’c.”
Article
8.7.2: “The modulus of elasticity, Es,
for non-prestressed steel reinforcement may be taken as 29,000,000 psi.”
Article
8.16.2.1: The design of the members
shall be based “on the satisfaction of the applicable conditions of equilibrium
of internal stresses and compatibility of strains.”
Article
8.16.2.2: “The strain in reinforcement
and concrete is directly proportional to the distance from the neutral axis.”
Article
8.16.2.3: “The maximum usable strain at
the extreme concrete compression fiber is equal to 0.003.”
Article
8.16.2.4: “The stress in reinforcement
below its specified yield strength shall be Es (Modulus of Elasticity) times
the steel strain. For the strains greater than that corresponding to fy, the
stress in the reinforcement shall be considered independent of strain and equal
to fy.”
Article
8.16.2.5: “The tensile strength of the
concrete is neglected in flexural calculation”
Article
8.16.2.6: “The concrete compressive
stress/strain distribution may be assumed to be rectangle, trapezoid, parabola …..”
Rectangular distribution of compressive stress is used in this software.
Article
8.16.27: “A compressive stress/strain
distribution, which assumes a concrete stress of 0.85f’c uniformly distributed
over an equivalent compression zone bounded by the edge of the cross section
and a line parallel to the neutral axis at a distance a = b1c from the fiber of maximum
compressive strain, may be considered to satisfy the requirements of Article
8.16.2.6. The distance c from the fiber of maximum strain to the neutral axis
shall be measured in a direction perpendicular to that axis. The factor b1 shall be taken as 0.85 for concrete
strength, f’c, up to 4000psi. For strengths above 4000psi, b1 shall be reduced continuously at a
rate of 0.05 for each 1000psi of strength in excess of 4000psi but b1 shall not be taken less than
0.65.”
Article
8.18.1.1: “The area of longitudinal reinforcement for compression members shall
not exceed 0.08 times the gross area, Ag, of the section.”
Article
8.18.1.2: “The minimum area of
longitudinal reinforcement shall not be less than 0.01 times the gross area,
Ag, of the section. When the cross section is larger than that required by
consideration of loading, a reduced effective area may be used. ….”
For Load
and Resistance Factor Design, Articles 5.4.2, 5.4.3 and 5.7 of AASHTO LRFD
Bridge Design Specifications lists the design assumptions for compression
members. The assumptions are
General
Assumptions:
Article
5.4.2.4: “In the absence of more
precise data, the modulus of elasticity, Ec, for concretes with unit weights
between 0.09 and 0.155 kcf, may be taken as: wc1.5 33,000√f’c.” f’c and
Ec are in ksi. wc is in kcf.
Article
5.4.3.2: “The modulus of elasticity,
Es, of bars and deformed wires shall be assumed as 29,000ksi.”
Assumptions
for Service Limit State:
Article
5.7.1:
“the
strains in the concrete vary linearly, except…,
the
modular ratio, n, is rounded to the nearest integer number,
the
modular ratio is not less than 6.0, and
an
effective modular ratio of 2n is applicable to permanent loads and …”
Assumptions
for Strength and Extreme Event Limit States
Article
5.7.2.1: “Factored resistance of
concrete components shall be based on the conditions of equilibrium and strain compatibility,
the resistance factors as specified in Article 5.5.4.2, and the following
assumptions:
Strain is
directly proportional to the distance from the neutral axis, except …
If the
concrete is unconfined, the maximum usable strain at the extreme concrete
compression fiber is not greater than 0.003,
If the
concrete is confined, a maximum usable strain exceeding 0.003 may be utilized
if verified,
Except ….,
the stress in the reinforcement is based on a stress-strain curve
representative of the steel, …..
The
tensile strength of the concrete is neglected, and
The
concrete compressive stress-strain distribution is assumed to be rectangular,
parabolic, or any other shape which results in a prediction of strength in
substantial agreement with the test results.
“
Article
5.7.2.2:
“The
natural relationship between concrete stress and strain may be considered
satisfied by an equivalent rectangular concrete compressive stress block of
0.85f’c over a zone, bounded by the edges of the
cross-section and a straight line located parallel to the neutral axis at the
distance a=b1c
from the extreme compression fiber. The distance c shall be measured
perpendicular to the neutral axis. The factor b1 shall be taken as 0.85 for concrete strengths not exceeding
4.0ksi. For concrete strengths exceeding 4.0ksi, b1 shall be reduced at a rate of 0.05 for each 1.0ksi of strength in
excess of 4.0ksi, except that shall not
be less than 0.65.”
Article
5.7.4.2: The maximum area of prestress and non-prestressed longitudinal reinforcement
for noncomposite compression components shall meet the requirements of
Equations (5.7.4.2-1) and (5.7.4.2-2). The minimum area of prestressed and
non-prestressed longitudinal reinforcement for noncomposite compression
components shall meet the requirement of Equation (5.7.4.2-3).
SNAPColumn™ offers a user-friendly interface for users to develop structural models visually.
The X-Y coordinate system is used. The global X-axis is in the horizontal direction to the right. Y-axis is upward. Right-hand rule is applied for Z-axis.
Elements are used to define the boundary of the column sections for the irregular types of column sections. Each element has 2 nodes.
Users draw the typical section of the column directly on the screen using mouse. The software also provided commands to let users to modify the typical section conveniently.
Users can define any loads and load combinations using Loads definition command. When users click the command, the Load Definition dialog box will appear.
Users can define as many load cases as necessary. There is no limit on number of load cases or combinations.
SNAPColumn(TM) will calculate the earth pressure, self-weight, contributing weight of soil, shear and moment in the backwall, shear and moment in spread footing/pile cap, bearing pressure for spread footing foundation, pile reactions for pile foundation.
SNAPColumn(TM) uses the following conventions:
Compressive
axial loads are positive and tensile axial loads are negative.
The
right-hand rule is used to determine the compression and tension faces of the
section. A positive moment about the x-axis, Mx, produces tension at the top
face of the section and compression at the bottom face. A positive moment about
the y-axis, My, produces tension at the left face of the section and
compression at the right face.
All
moments are referenced to the geometric centroid of the gross concrete section
(neglecting the reinforcement).
For a
short column having a low slenderness ratio where buckling does not control the
failure, the strength of the column is governed by the material properties, the
column section geometry and reinforcement. The strength of a short column is
achieved when the extreme concrete compression fiber reaches the maximum usable
strain. There are infinite number of combinations of axial load and bending
moment at which the section reaches its failure. These maximum axial loads and
bending moments constitute a bounding curve or surface which defines the
section strength failure diagram or failure surface, so-called strength
interaction curve or interaction surface.
The
software determines the strength interaction curve for uniaxial bending or the
strength interaction surface for biaxial bending of the column section. The
following control points are first computed
Po: nominal axial load strength of a section at
zero eccentricity, It can be calculated by Equation (8-31) of AASHTO Standard
Specifications and Equation (5.7.4.4-2) of AASHTO LRFD Bridge Design
Specifications.
Pure
Flexure: nominal moment strengths
around positive and negative direction of one axis considered for uniaxial
bending and around positive and negative direction of both axis. The nominal
strength(s) can be calculated in accordance with the equations in Article
8.16.3 of AASHTO Standard Specifications, or the equations in Article 5.7.3 of
AASHTO LRFD Bridge Design Specifications for rectangular and flanged sections.
For irregular sections, the strain compatibility, force equilibrium and other
design assumptions can be used to compute the nominal axial load and bending
moment strength at the balanced condition.
Pure
Tension: nominal tension strength of a section at zero eccentricity. It can be
easily calculated assuming only reinforcement contributes to tension strength.
Balanced
Strain Condition: Balanced strain condition exists at a section when the
tension reinforcement reaches the strain corresponding to its specified yield
strength, just as the concrete in compression reaches its assumed usable
strain. The nominal axial load and moment(s) at the balanced condition may be
calculated on the basis of the design assumption.
The
nominal strengths at the points on the interaction curve/surface except the
above-mentioned control points are calculated based on the basic assumptions.
The
maximum nominal axial load strength should meet the requirement of Equation
(8-29) or (8-30). The maximum usable strength reduction factor is 0.85 for
spiral columns and 0.8 for tied columns.
In
accordance with Article 8.15.4 of AASHTO Standard Specifications, the combined
flexural and axial load capacity of compression members shall be taken as 35%
of that nominal strength determined with the provisions of Article 8.16.4, which means that the design
capacity diagram can be obtained by scaling 35%. the nominal strength
interaction diagram discussed above.
The
combined flexural and axial load design strength of compression members can be
obtained by applying the strength reduction factor to the nominal strength
interaction diagram discussed above. In accordance with the specifications, the
strength reduction factor shall be as follows:
Flexural
and Axial Tension: φ = 0.9
Axial
Compression with
Spirals: φ = 0.75
Ties: φ = 0.7
The
strength reduction factor is increased linearly from the value for compression
members (0.75 for spiral column and 0.7 for tied columns) to the value for
flexure (0.9) as the design axial load strength, φ Pn, decreases from
0.1f’cAg or φPb, which ever is smaller, to zero.
The
combined flexural and axial load factored resistance of compression members can
be obtained by applying the resistance factor to the nominal
strength(resistance) interaction diagram discussed above. In accordance with
the specifications, the resistance factor shall be as follows:
Flexural
and Axial Tension: φ = 0.9
Axial
Compression: φ = 0.75
(Except as specified in Article 5.10.11.4.1b for Seismic Zones 3 and 4)
For
compression member with flexure, the vale of resistance factor is increased
linearly to the value for flexure as the factored axial load resistance,
φPn, decreases from 0.1f’cAg to zero.
The design
of a structural component such as a column is always a trial-error process. The
dimension of the column is always determined by some site constraints and by means
of some simple procedure or design guide. Experienced designers normally prefer
to decide the column dimension and reinforcement according to sample projects
and good detailing practice, in most cases the initial trial is good enough.
Generally speaking, “design” software which has the function to determine the
column size, rebar details etc. is not a practical tool. Because of this
reason, the “design” features are not implemented in this software. But the
following design checks are automatically performed
For Service Load Design or Load Factor
Design, Article 8.18.1.1 and 8.18.1.2 of AASHTO Standard Specifications define
the maximum and minimum area of longitudinal reinforcement for compression
members.
For Load
and Resistance Factor Design, Article 5.7.4.2 of AASHTO LRFD Bridge Design
Specifications defines the limits for reinforcement.
This
software automatically checks these limits.
Article
8.16.4.3 of AASHTO Standard Specifications and Article 5.7.4.5 of AASHTO LRFD
Bridge Design Specifications provide approximate equations to check the capacity/strength/resistance of
compression members under biaxial bending. Those are simplified procedures
designers can use when there is no more precise analysis. This software is a
true biaxial bending column analysis tool. The column
capacity/strength/resistance is checked against the strength interaction diagram directly. It is a more precise
method than the method provided in the two AASHTO design specifications. The
software also provides the strength check in accordance with the approximate
method of the specifications.
Serviceability Limit State control for cracking of the column section is checked in accordance with the code requirements.
Article
8.16.5.1 of AASHTO Standard Specifications and Article 5.7.4.1 of AASHTO LRFD
Bridge Design Specifications list the general requirements for slenderness
effect consideration. Both of the specifications provide a approximate
procedures to evaluate the slenderness effect in compression members. More precise slenderness effects can be computed by a second-order (or
P-Delta) structural analysis. Both the
approximate method and the P-Delta analysis are used to compute the forces in a
structural compression member considering its slenderness effects, which are
not related to the design capacity/strength or resistance of a column. Because
this software is primarily developed for engineers to get the design
capacity/strength or resistance of compression members, and further check the
strength vs loads users inputted into the software, the function to evaluate
the slenderness effects on loads is not implemented in this software.
The
general procedure designing a compression member considering slenderness effect
is as follows:
Determine
a preliminary dimension and reinforcement for the compression member
considered,
Determine
the loads in the compression member with slenderness effect to be considered
through the combination of a first-order linear analysis and an approximate
moment magnification factor analysis, or a second-order (P-Delta) analysis,
Input the
column geometry, material properties, rebars and loads into this software and run
the software,
Check the
results graphically and/or in text format and make any adjustment if necessary,
and repeat to achieve a satisfied solution.