To determine the load capacity or the size of beam section, it must satisfy the allowable stresses in both flexure (bending) and shear. Shearing stress usually governs in the design of short beams that are heavily loaded, while flexure is usually the governing stress for long beams.
Section force-deformation response & Plastic Moment (Mp)
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.
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.
Steel material follows a typical stress-strain behavior as shown in Figure 3 below.
If the steel stress-strain curve is approximated as a bilinear elasto-plastic curve with yield stress equal to σy, then the section Moment – Curvature (M-φ) response for monotonically increasing moment is given by Figure 4.
Figure 4. Section Moment – Curvature (M-φ) behavior
In Figure 4, My is the moment corresponding to first yield and Mp is the plastic moment capacity of the cross-section.
The ratio of Mp to My 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 Mp: Cross-section subjected to either +σy or -σy at the plastic limit.
See Figure 5 below.
The plastic centroid for a general cross-section corresponds to the axis about which the total area is equally divided, i.e., A1 = A2 = A/2
The plastic centroid is not the same as the elastic centroid or center of gravity (c.g.) of the cross-section.
As shown below, the c.g. is defined as the axis about which A1y1 = A2y2.
For a cross-section with at-least one axis of symmetry, the neutral axis corresponds to the centroidal axis in the elastic range. However, at Mp, the neutral axis will correspond to the plastic centroidal axis.
For a doubly symmetric cross-section, the elastic and the plastic centroid lie at the same point.
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Design of Beams – Flexure and Shear