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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 13th 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