chap2 - CE 405 Design of Steel Structures – Prof Dr A...

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Unformatted text preview: CE 405: Design of Steel Structures – Prof. Dr. A. Varma Chapter 2. Design of Beams – Flexure and Shear 2.1 Section force-deformation response & Plastic Moment (M p ) • A beam is a structural member that is subjected primarily to transverse loads and negligible axial loads. • The transverse loads cause internal shear forces and bending moments in the beams as shown in Figure 1 below. w P V(x) M(x) x w P V(x) M(x) x Figure 1. Internal shear force and bending moment diagrams for transversely loaded beams. • These internal shear forces and bending moments cause longitudinal axial stresses and shear stresses in the cross-section as shown in the Figure 2 below. V(x) M(x) y d b ε ε σ σ dF = σ b dy V(x) M(x) y d b ε ε σ σ dF = σ b dy Curvature = φ = 2 ε /d ∫ σ = +- 2 / d 2 / d dy b F y dy b M 2 / d 2 / d ∫ σ = +- (Planes remain plane) Figure 2. Longitudinal axial stresses caused by internal bending moment. 1 CE 405: Design of Steel Structures – Prof. Dr. A. Varma • Steel material follows a typical stress-strain behavior as shown in Figure 3 below. σ y ε y ε u σ u σ ε σ y ε y ε u σ u σ ε Figure 3. Typical steel stress-strain behavior. • If the steel stress-strain curve is approximated as a bilinear elasto-plastic curve with yield stress equal to σ y , then the section Moment - Curvature (M- φ ) response for monotonically increasing moment is given by Figure 4. M y M p A: Extreme fiber reaches ε y B: Extreme fiber reaches 2 ε y C: Extreme fiber reaches 5 ε y D: Extreme fiber reaches 10 ε y E: Extreme fiber reaches infinite strain A B C E D Curvature, φ Section Moment, M σ y σ y σ y σ y ε y ε y σ y σ y σ y σ y σ y σ y 2ε y 2ε y 5ε y 5ε y 10ε y 10ε y A B C D E M y M p A: Extreme fiber reaches ε y B: Extreme fiber reaches 2 ε y C: Extreme fiber reaches 5 ε y D: Extreme fiber reaches 10 ε y E: Extreme fiber reaches infinite strain A B C E D Curvature, φ Section Moment, M σ y σ y σ y σ y ε y ε y σ y σ y σ y σ y σ y σ y 2ε y 2ε y 5ε y 5ε y 10ε y 10ε y A B C D E σ y σ y σ y σ y σ y σ y σ y σ y ε y ε y ε y ε y σ y σ y σ y σ y σ y σ y σ y σ y σ y σ y σ y σ y 2ε y 2ε y 2ε y 2ε y 5ε y 5ε y 5ε y 5ε y 10ε y 10ε y 10ε y 10ε y A B C D E Figure 4. Section Moment - Curvature (M- φ ) behavior. 2 CE 405: Design of Steel Structures – Prof. Dr. A. Varma • In Figure 4, M y is the moment corresponding to first yield and M p is the plastic moment capacity of the cross-section. The ratio of M p to M y is called as the shape factor f for the section. For a rectangular section, f is equal to 1.5. For a wide-flange section, f is equal to 1.1....
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This note was uploaded on 04/03/2008 for the course CIVE 410 taught by Professor Chang during the Spring '08 term at Drexel.

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chap2 - CE 405 Design of Steel Structures – Prof Dr A...

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