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

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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 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 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
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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 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 y y y y 10ε y 10ε y A B C D E Figure 4. Section Moment - Curvature (M- φ ) behavior. 2
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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. Calculation of M p : Cross-section subjected to either + σ y or - σ y at the plastic limit. See Figure 5 below. Plastic centroid. A 1 A 2 σ y σ y σ y A 1 σ y A 2 y 1 y 2 (a) General cross-section (b) Stress distribution (c) Force distribution 2 2 1 1 2 1 y 2 1 2 y 1 y A of centroid y A of centroid y , Where ) y y ( 2 A M 2 / A A A 0 A A F = = + × σ = = = = σ - σ = (d) Equations Figure 5. Plastic centroid and M p for general cross-section. The plastic centroid for a general cross-section corresponds to the axis about which the total area is equally divided, i.e., A 1 = A 2 = A/2 The plastic centroid is not the same as the elastic centroid or center of gravity (c.g.) of the cross-section. 3
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CE 405: Design of Steel Structures – Prof. Dr. A. Varma As shown below, the c.g. is defined as the axis about which A 1 y 1 = A 2 y 2.
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