How do you choose the best shape factor for plastic design of beams?

1 What is shape factor?

The shape factor, also known as the plastic section modulus ratio, is defined as the ratio of the plastic section modulus Zp to the elastic section modulus Ze of a beam cross-section. The plastic section modulus is the moment of area of the section about the plastic neutral axis, which is the axis that divides the section into two equal areas under uniform stress. The elastic section modulus is the moment of area of the section about the elastic neutral axis, which is the axis that passes through the centroid of the section. The shape factor indicates how much the section can deform plastically before reaching its ultimate moment capacity. The higher the shape factor, the more efficient the section is in plastic design.

2 How to calculate shape factor?

The shape factor can be calculated by using the formulas for Zp and Ze for different types of beam sections, such as I-sections, T-sections, circular sections, etc. Alternatively, the shape factor can be obtained from tables or charts that provide the values for common beam sections. For example, the shape factor for a rectangular section is 1.5, which means that it can resist 50% more moment in plastic design than in elastic design. The shape factor for a circular section is 1.7, which means that it can resist 70% more moment in plastic design than in elastic design.

3 How to choose the best shape factor?

The best shape factor for plastic design of beams depends on several factors, such as the loading condition, the span-to-depth ratio, the slenderness ratio, and the design code. Generally, the higher the shape factor, the better the section is in plastic design, because it can utilize more of its material strength and ductility. However, there are some trade-offs and limitations that need to be considered. For example, sections with high shape factors tend to have large depths and widths, which may increase the self-weight and the deflection of the beam. Sections with high shape factors may also require more reinforcement or stiffeners to prevent local buckling or lateral-torsional buckling. Moreover, some design codes may impose upper limits on the shape factor or the plastic moment capacity of the beam, depending on the ductility class or the seismic zone of the structure.

4 How to optimize the shape factor?

One way to optimize the shape factor for plastic design of beams is to use composite sections, which are made of two or more materials with different properties, such as steel and concrete. Composite sections can achieve higher shape factors than single-material sections, because they can take advantage of the different stress-strain characteristics of each material. For example, a composite section with a steel flange and a concrete web can have a higher shape factor than a steel I-section, because the concrete web can resist more compression and the steel flange can resist more tension in plastic bending. Another way to optimize the shape factor is to use variable cross-sections, which are sections that change their dimensions along the length of the beam. Variable cross-sections can reduce the self-weight and the deflection of the beam, and also increase the plastic moment capacity at critical regions, such as mid-span or supports.

5 How to verify the shape factor?

The shape factor is an idealized parameter that assumes uniform stress distribution and perfect plasticity of the beam section. In reality, there may be some deviations and uncertainties that affect the actual performance of the beam in plastic design. Therefore, it is important to verify the shape factor by using experimental tests, numerical simulations, or analytical methods that account for the nonlinear behavior and the failure modes of the beam. For example, the shape factor can be verified by using strain gauges, load cells, or displacement sensors to measure the stress, force, and deformation of the beam under different loading scenarios. The shape factor can also be verified by using finite element analysis, plastic hinge analysis, or moment-curvature analysis to model the nonlinear response and the ultimate capacity of the beam.

6 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?