22.3.1. Verification

Example 22-1: Verification, (?) AW12 120 is used to support the loads and moments as shown below, and is not subjected to sidesway. Determine if the member is adequate and if the factored bending moment occurs about the weak axis. The column is assumed to be perfectly pinned (K = 1.0) in both the strong and weak directions and no bracing is supplied. Steel is A36 and assumed C = 1.0.

In this section, we are introduced to a verification example where a specific beam, designated as AW12 120, is evaluated for its adequacy in supporting loads. The key factors to consider include whether it resists bending about its weak axis and the conditions under which it operates, such as being pinned and unbraced. The properties of the material, which in this case is A36 steel, will also come into play, affecting the beam's capacity to carry loads.

Imagine a bridge beam that supports the weight of cars and pedestrians. Just like engineers check whether a bridge beam can hold the expected weight safely, in this example, we check if the AW12 120 can handle the loads placed on it without failing.

The values for L and L can be calculated using Eq. 6-1 and 6-2 respectively or they can be found in the beam section of the LRFD manual. In either case these values for a W 12 120 are: L = 13 ft, L = 75.5 ft. Since our unbraced length falls between these two values, the beam will be controlled by inelastic buckling, and the nominal moment capacity M can be calculated from Eq. 6-5.

In this part of the verification process, we calculate two critical lengths related to the beam's behavior: Lp (13 ft) and Lr (75.5 ft). This informs us about the unbraced length of the beam, which indicates how it will respond to loads. If the unbraced length falls between the two calculated lengths, we determine that the beam's performance is governed by inelastic buckling, leading us to further calculate the nominal moment capacity (Mn) using specific equations.

Think of this like ensuring a tall, skinny pencil can stand upright on a table; if it's too tall (akin to our maximum length) or too skinny, it may buckle under too much pressure. Knowing these lengths helps us confirm if our 'pencil' can stand straight under load.

The plastic and elastic moment capacity values, Mp and Mr, are calculated as follows: Mp = F * Z = (36 ksi)(186 in³ ft) = 558 k.ft; Mr = (Fy * S) = (36 ksi)(163 in³) = 353.2 k.ft.

Next, we compute two key moment capacities of our beam: Mp (plastic capacity) and Mr (elastic capacity). The equations used involve two material properties—stress (F) and section modulus (Z for plastic, S for elastic). Calculating these helps us understand how much bending moment the beam can resist before it yields, which is essential for ensuring safety and adequacy.

Consider a road that can support a certain weight based on its materials; similarly, we calculate the limits of our beam’s strength to ensure it doesn’t buckle under unexpected loads.

Using Eq. 6-5 (assuming C is equal to 1.0), we find: Mn = Cb[ Mp - (Mp - Mr)] Llb / (Lp - Lr) = 1.0[558 - (558 - 353.2)] k.ft (75.5 - 15) ft = 551.4 k.ft.

Now we calculate the nominal moment capacity (Mn) using a specific equation that takes into account the Cb (bracing coefficient) and includes our previously calculated values. This step is crucial in determining how effective and safe the beam is under expected loads and conditions.

Imagine adjusting the support under a bridge based on traffic patterns; similarly, we adjust our moment calculations based on what we've learned about the beam's capacity to ensure it's able to handle the expected pressures.

Therefore, the design moment capacity is as follows: Mb = 0.90(551.4) k.ft = 496.3 k.ft. Now consider the effects of moment magnification on this section. Based on the alternative method, since the member is not subjected to sidesway (Mu = 0), we find: applying the moment magnification factor B = Cm...

Finally, we calculate the design moment capacity (Mb) by applying a reduction factor to Mn. After this, we take into account the effects of moment magnification that occur when loads change. By calculating B (the moment magnification factor), we can assess how much the moment will increase under various conditions, which is essential for accurate assessment.

It's like anticipating heavier traffic on a once quiet road; as conditions change, we must predict how much more capacity we need for the road to handle heightened demands and apply safety measures accordingly.

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