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1.1.2
Related Math – Introduction to ASD and LRFD
Introduction
The steel detailing practice is bound by current strength design approaches, such as the
approaches found within the 2005 AISC Specification ( www.aisc.org/2005spec). The latest
Specification retains Allowable Stress Design (ASD) by introducing it in form of Allowable
Strength Design. Allowable Strength Design is easily interchangeable with Load Resistance
and Factored Design (LRFD), as will be discussed later in this section.
ASD
Traditional ASD was based on a Stress approach and existed in older AISC Specifications
and Manuals, such as the 1989 AISC ASD Specification and the 9th Edition AISC ASD
Manual. With the release of the 2005 AISC Specification and 14th Edition AISC Manual, an
ASD based on Strength was introduced. Note the differences below:
Allowable Stress Design*
Axial Allowable Stress in ksi
Bending Allowable Stress in ksi
*
stress units are based on kips per square inch (ksi) for both axial and
bending expressions
Allowable Strength Design*
Axial Allowable Strength in kips
Bending Allowable Strength in kip-in
*
strength units are based on kips for axial and kip times inch (k-in) for
bending expressions
LRFD
The Load Resistance and Factored Design (LRFD) approach was originally released in the
1986 AISC LRFD Specification and 1st Edition AISC LRFD Manual and is based on the
same units as Allowable Strength. It is important to note that the terminology Design
Strength pertains strictly to LRFD rather than ASD strength. Refer to the 2005 AISC
Specification for additional information.
Load Resistance and Factored Design*
Axial Design Strength in kips
Bending Design Strength in kip-in
*
strength units are based on kips for axial and kip times inch (k-in)
for bending expressions
Available Strength
As previously discussed, the 2005 AISC Specification establishes ASD and LRFD strength
capacities, such as Allowable Strength and Design Strength, respectively. The terminology
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Available Strength was introduced to simply imply either Allowable Strength or Design
Strength.
Available Strength = Design Strength or Allowable Strength
Nominal Strength
The 2005 AISC Specification uses Nominal Strength expressions in conjunction with
defined factors to establish Allowable Strength and Design Strength. As defined by the
Specification, Nominal Strength is the strength of a structure or component to resist load
effects. To determine Allowable Strength, one must divide the Nominal Strength
expression by a factor of safety, Ω. To determine Design Strength, one must multiple by a
resistance factor, φ. All factors are clearly indicated within the 2005 AISC Specifications.
Allowable Strength = Nominal Strength / Ω
Design Strength = φ x Nominal Strength
ASD / LRFD Relationships
Design Strength values will always be 1.5 times greater than Allowable Strength values. It
follows that LRFD Required Strength will also be greater than ASD Required Strength, as is
evident in the ASCE 7 Loading Specification. This is due to the use of load factors in LRFD
load combinations which may exceed unity.
The interchangeability between ASD and LRFD Available Strengths (namely Allowable
Strength and Design Strength, respectively) in the 2005 AISC Specification is one of many
benefits inherent in using the latest Specification. It should be noted that this
interchangeability only exists between Allowable Strength and Design Strength. One may
not find a constant relationship between Allowable Stress and Design Strength when
comparing older ASD Specifications to the 2005 AISC Specification. It follows that one
should not use older ASD Specifications in the attempt to convert Allowable Stress to
Design Strength, or vice versa.
Design Strength = 1.5 x Allowable Strength
Substituting the above Design Strength and Allowable Strength with the expressions
mentioned in the previous section, we find that:
(φ x Nominal Strength) = 1.5 x (Nominal Strength / Ω)
Hence φ = 1.5 / Ω, which can be rewritten as:
φ x Ω = 1.5
As can be seen, the relationship between the LRFD φ resistance factor and the ASD factor
of safety Ω is 1.5.
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ASD / LRFD Conversions
It is a relatively simple matter to convert Allowable Strength to Design Strength, or vice
versa. One can use a factor of 1.5 to account for the difference between Allowable Strength
and Design Strength, or decide to work directly with the Nominal Strength expression as
shown below:
Allowable Strength to Design Strength
Allowable Strength = Nominal Strength / Ω
Design Strength = 1.5 x Allowable Strength = 1.5 x (Nominal Strength / Ω)
Design Strength to Allowable Strength
Design Strength = φ x Nominal Strength
Allowable Strength = Design Strength / 1.5 = (φ x Nominal Strength) / 1.5
Limit States
As defined in the 2005 AISC Specification, a Limit State is a condition in which a structure or
component becomes unfit for service and is judged either to be no longer useful for its
intended function (serviceability limit state) or to have reached its ultimate load-carrying
capacity (strength limit state).
The basis of strength design is to determine the load effects when a particular strength Limit
State (i.e. yielding, rupture, plastic moment, etc.) will manifest itself. That is, the value of the
Nominal Strength expression indicates when a strength limit state will occur.
It follows that the Available Strength will be at or below the Nominal Strength value to
ensure that the Limit State does not take place. To accomplish this, resistance factors φ
must be unity or less, and factors of safety Ω must be 1.5 or greater, which is the exactly the
case in the 2005 AISC Specification. This is why one must multiply the Nominal Strength by
a resistance factor φ to determine Design Strength, or divide Nominal Strength by a factor
of safety Ω to determine Allowable Strength.
Required Strength
The 2005 AISC Specification defines Required Strength as forces, stresses and
deformations acting on the structural component, determined by either structural analysis, for
the LRFD or ASD load combinations, as appropriate, or as specified by the Specification.
In strength design, one must make sure that the Available Strength is equal to or greater
than the Required Strength. The Required Strength is not determined by the Steel Detailer,
rather, it is usually given in structural steel design drawings and determined by the Owner’s
Designated Representative for Design (i.e. Engineer of Record.)
Required Strength ≤ Available Strength