20.2. Nominal Strength

The strength requirement for beams in load and resistance factor design is stated as

$\phi M_n \geq M_u$ (20.10)

where:

• $\phi$ = strength reduction factor; for flexure $= 0.90$
• $M_n$ = nominal moment strength
• $M_u$ = factored service load moment.

This chunk introduces the equation used to determine the strength requirements for beams in load and resistance factor design (LRFD). The equation states that the product of the strength reduction factor ($\phi$) and the nominal moment strength ($M_n$) must be greater than or equal to the factored service load moment ($M_u$). The strength reduction factor accounts for uncertainties in material strength and loading conditions, with a specific value of 0.90 used for flexure. This means that when analyzing beams, engineers can only rely on 90% of the nominal strength to ensure safety and account for possible variabilities.

Consider a bridge carrying vehicles. Engineers calculate how much weight the bridge can safely hold based on materials used—like steel beams—which can support specific loads. To ensure safety, they operate with a margin, only counting on 90% of a beam's theoretical strength in calculations. This is similar to if you were carrying a heavy box; although you know you can lift a certain weight, you only lift 90% of that to avoid injury.

The equations given in this chapter are valid for flexural members with the following kinds of cross-section and loading:

1. Doubly symmetric (such as W sections) and loaded in Plane of symmetry
2. Singly symmetric (channels and angles) loaded in plane of symmetry or through the shear center parallel to the web.

This chunk specifies the types of structural cross-sections for which the previously mentioned strength equations are applicable. Doubly symmetric sections, like wide flange (W) beams, have equal dimensions on both sides of the center line and are loaded symmetrically, ensuring uniform stress distribution. Singly symmetric sections, like channels and angles, may have different dimensions on each side but are still analyzed under similar loading criteria, allowing for efficient design using the same formulas.

Imagine you have two types of shelves: one that is perfectly symmetrical (like a well-balanced scale) and another that is a bit lopsided, like a leaning bookshelf. You can load both types, but if you load the symmetrical shelf evenly, it will bear weight more reliably. The equations in the text only apply accurately to these structured shelves, assuring engineers can effectively distribute weight without risking failure.

The nominal strength M for laterally stable compact sections according to LRFD is

$M_n = M_p$ (20.12)

where:

• $M_p$ = plastic moment strength = $Z F_y$
• $Z$ = plastic section modulus
• $F_y$ = specified minimum yield strength.

Here, the nominal strength ($M_n$) for laterally stable and compact sections is equated to the plastic moment strength ($M_p$). The plastic moment strength is calculated based on the plastic section modulus ($Z$), which is a geometric property indicating the distribution of the beam's cross-section, multiplied by the specified minimum yield strength of the material ($F_y$). This indicates that for certain structural configurations, the maximum load the beam can carry before yielding corresponds directly to these properties.

Think of a strong, sturdy bookshelf made from quality wood. The weight it can hold before bending depends not only on the wood's strength but also on how well the shelves are designed to distribute that weight (like the plastic section modulus). If designed properly, the shelf flexes but doesn’t break, illustrating how beams in structures operate when calculations yield these specific outcomes.

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