MORGAN STATE UNIVERSITY

SCHOOL OF ARCHITECTURE AND PLANNING

LECTURE VII

Dr. Jason E. Charalambides

Review of Elastic Bending Theory

Fully Plastic Moment

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Beams

References: AISC Specification and Commentary:

• Chapter F: Design of Members for Flexure• Chapter G: Design of Members for Shear• Chapter B – Section B4: Local Buckling

AISC Manual Part 3 – Beam Design Aids

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Definitions

What is a Beam? Typically they are horizontal

members Support gravity loads along their

length Loads may be Uniformly Distributed

along the length Loads can be set at point(s) A good term to use is (flexural

member) because that is what a beam is mainly designed for

But beams also participate in resisting lateral loads in rigid frames (so they also take axial loads)

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Examples of Beams Resisting Gravity Loads

Effect of Uniformly Distributed Load applied on beams

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Examples of Beams Resisting Gravity Loads

Effect of point load applied on beams

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Examples of Beams Resisting Lateral Loads

Note the deflected form of the beam due to the transfer of moment from deflected columns

Diagram  indicates  greatly  exaggerated  deflec2ons  on  frame  members  subjected  to  lateral  loading.

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Standard Process for Design of Beams

Steel is designed for minimum cross sectional area set in a way that it will address Moment stress optimally. Shear is somewhat less of an issue: Determine Design loads and

calculate Vu and Mu Compute the nominal strength

Vn and Mn for the beam Address the following

Check serviceability: Deflection Vibrations

The Dom-ino frame design by Le Corbusier during the early 20th century became a paradigm ,a datum in the history of architecture and a reference to the modernist movement.

V u≤ΦV n M u≤ΦM n

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Review of Bending Theory - Beams

Objective Given a beam subjected to bending moment, determine the following:

Bending stress: σ Bending Strains: ε Curvature: φ

The Dom-ino frame design by Le Corbusier during the early 20th century became a paradigm ,a datum in the history of architecture and a reference to the modernist movement.

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Analysis

Stress Strain Law Assuming no residual stresses

The Dom-ino frame design by Le Corbusier during the early 20th century became a paradigm ,a datum in the history of architecture and a reference to the modernist movement.

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Analysis

Plane Sections remain Plane Strain varies linearly over height of x-section

Holds for elastic or inelastic behavior

The Dom-ino frame design by Le Corbusier during the early 20th century became a paradigm ,a datum in the history of architecture and a reference to the modernist movement.

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Analysis

For stresses on x-section:

Holds for elastic or inelastic behavior

P=∫ σ d A

M=∫ yσ d A

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Elastic Behavior

For stresses on x-section:

Φ = curvature (1/in) ε = yφ (plane sections remain plane)

For elastic behavior N/A at centroid (basis of moment-area method)

→ [Tabulated in AISC manual]

P=∫ σ d A

M=∫ yσ d A

Φ=ΜΕΙ

σmax=ΜcΙ

=MS

S= Ιc

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Elastic Behavior

For stresses on x-section:

For elastic behavior Lim. of elastic behavior (In absence of residual stresses)

Defining My as the moment at first yield (when σ_max=Fy)

P=∫ σ d A

M=∫ yσ d A

σmax=F y

F y=M y

SM y=SF y

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Example

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Example

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Example

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Example

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Inelastic Behavior

Neutral Axis Location After the first yield, the N/A is no longer guaranteed to be at the

geometric centroid. However, the N/A can always be located by the requirement:

P=∫ σ d A

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Inelastic Behavior

Relationship between Stress and Moment After the first yield, the formula for stress “σ” no longer applies. However, the moment corresponding to any given stress distribution

can be found from:

M=∫ yσ d A

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Inelastic Behavior of Steel Flexural Members

Consider the bending stresses as curvature increases:

In the absence of strain hardening, the maximum moment which can be sustained occurs when every fiber of the section has yielded.

At that point the section is “fully plastic” and the corresponding moment is Mp

Per AISC Mn≤Mp

The Dom-ino frame design by Le Corbusier during the early 20th century became a paradigm ,a datum in the history of architecture and a reference to the modernist movement.

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Plastic Hinge

How does an element behave when it's extreme top and bottom fibers yield? When a concentrated load is applied this effect is very prominent.

The yielding of the extreme fiber may extend between one third of the span for a member of rectangular cross section, to one eighth of the span for a W shape.

A gradual yielding of the interior fibers proceeds This effect occurs on “compact” sections and only if bracing occurs

that will not allow lateral torsional or local buckling.

The Dom-ino frame design by Le Corbusier during the early 20th century became a paradigm ,a datum in the history of architecture and a reference to the modernist movement.

22

Example

The Dom-ino frame design by Le Corbusier during the early 20th century became a paradigm ,a datum in the history of architecture and a reference to the modernist movement.

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Example

The Dom-ino frame design by Le Corbusier during the early 20th century became a paradigm ,a datum in the history of architecture and a reference to the modernist movement.

24

Example

The Dom-ino frame design by Le Corbusier during the early 20th century became a paradigm ,a datum in the history of architecture and a reference to the modernist movement.