Review of Elastic Bending Theory Fully Plastic Moment

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Charalambides Beams References: AISC Specification and Commentary: Chapter F: Design of

1 Review of Elastic Bending Theory Fully Plastic Moment MORGAN STATE UNIVERSITY SCHOOL OF ARCHITECTURE AND PLANNING LECTURE VII Dr. Jason E. Charalambides 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 2

along their length Loads may be Uniformly Distributed along

is (flexural member) because that is what a beam is mainly

lateral loads in rigid frames (so they also take axial

2 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) 3 Examples of Beams Resisting Gravity Loads Effect of Uniformly Distributed Load applied on beams 4

Examples of Beams Resisting Gravity Loads Effect of point load applied on beams 5 Examples of Beams Resisting Lateral Loads Note the deflected form of the

3 Examples of Beams Resisting Gravity Loads Effect of point load applied on beams 5 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. 6

4 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 V u ΦV n M u ΦM n Check serviceability: Deflection Vibrations 7 Review of Bending Theory - Beams Objective Given a beam subjected to bending moment, determine the following: Bending stress: σ Bending Strains: ε Curvature: φ 8

5 Analysis Stress Strain Law Assuming no residual stresses 9 Analysis Plane Sections remain Plane Strain varies linearly over height of x-section Holds for elastic or inelastic behavior 10

6 Analysis For stresses on x-section: P= σ d A M = yσ d A Holds for elastic or inelastic behavior 11 Elastic Behavior For stresses on x-section: P= σ d A M = yσ d A Φ = curvature (1/in) ε = yφ (plane sections remain plane) For elastic behavior N/A at centroid Φ= Μ ΕΙ (basis of moment-area method) σ max = Μc = M Ι S S= Ι c [Tabulated in AISC manual] 12

7 Elastic Behavior For stresses on x-section: P= σ d A M = yσ d A For elastic behavior Lim. of elastic behavior σ max =F y (In absence of residual stresses) Defining My as the moment at first yield (when σ_max=fy) F y = M y S M y =SF y 13 Example 14

8 Example 15 Example 16

9 Example 17 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 18

10 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 19 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 20

11 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. 21 Example 22

12 Example 23 Example 24

Basic Concepts. Introduction to Composite Design

Basic Concepts. Introduction to Composite Design Introduction to Composite Design MORGAN STATE UNIVERSITY SCHOOL OF ARCHITECTURE AND PLANNING LECTURE IΧ Dr. Jason E. Charalambides Basic Concepts! What is a composite Beam? " Steel W-shape and portion