22.3. Examples

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 example, we are tasked with determining if an 'AW12 120' beam is adequate for the given loads and moments. The beam is not braced against sideways motion, making it a straightforward case. The problem involves checking both the strength and stability of the beam by calculating if it can handle the loads applied to it without failing. We assume that the beam is supported in a way that allows for some flexibility (perfectly pinned at both ends), and it is made of a particular type of steel (A36). It is also critical to note that the moment being considered (the bending effect from the loads) might occur about the weak axis of the member.

Imagine holding a pencil horizontally at both ends. If you push down in the middle, the pencil may bend. This scenario is similar to our beam, where it must handle a weight without breaking or bending too much, and we want to be sure it's strong enough at the points where it can move.

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: Lp = 13 ft, Lr = 75.5 ft.

In engineering, we often use charts and equations to find critical dimensions. Here, 'L' represents length parameters necessary for our calculations. The length 'Lp' indicates the length for plastic behavior of the beam while 'Lr' indicates the length concerning its elastic behavior. By referencing equations or tables in a manual, we can quickly identify these lengths for standard beam sections like the 'W 12 120'. For this beam, we find that 'Lp' is 13 feet and 'Lr' is 75.5 feet. These lengths help in determining if the beam will bend or buckle under load.

Think of a long rubber band: if you stretch it too far, it can snap. Just like knowing how much you can safely stretch a rubber band, engineers need to find specific lengths for beams to ensure they won't fail under pressure.

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. Using this equation, we must first calculate the plastic and elastic moment capacity values, Mp and Mr.

Understanding how a beam reacts to loads is crucial, especially when deciding if it will buckle (bend excessively) or remain stable. In this case, we determine that our beam's effective length leads us to a scenario known as 'inelastic buckling,' indicating that it will fail differently than a perfectly elastic material. The nominal moment capacity, denoted as 'M', is key to assessing the beam's ability to resist bending. The first step is to calculate the values for plastic and elastic moment capacities, referred to as 'Mp' and 'Mr', by using a specific equation (Eq. 6-5). This process involves substituting known values into the equation to check the beam's adequate strength against the applied moments.

Consider a tall tower. It can sway with the wind, but if the wind gets too strong, it might buckle. This example is similar as engineers need to calculate what loads a beam can withstand without buckling, much like ensuring a tower sways safely without collapsing.

Using Eq. 6-5 (assuming C is equal to 1.0), we find: Mn = Cb[Mp - (Mp - Mr)] Llb / (L - Lp). Therefore, the design moment capacity is as follows: Mb = 0.90(Mn) = 496.3 k.ft.

Once we have 'Mp' and 'Mr', we apply them in the provided equation to find the nominal moment capacity 'Mn'. This calculation also includes the lengths we found earlier, which influence how the beam behaves under load. After computing 'Mn', we adjust it to get 'Mb', the design moment capacity, which ensures we account for safety factors in design. For this beam, we arrive at a design moment capacity of approximately 496.3 k-ft, meaning that's the maximum moment the beam can safely endure.

Think of 'Mb' like the maximum weight a bridge can carry. Engineers do careful calculations to make sure the bridge can hold heavier cars without buckling, just like how we've calculated our beam's ability to support loads without failing.

Now consider the effects of moment magnification on this section. Based on the alternative method since the member is not subjected to sidesway (Mlt = 0), we have: Mu = B * Mu1 = 1 nt B = Cm / (1 - Qu / Pe).

Moment magnification occurs when the actual moment a beam experiences is greater than the expected due to instability or lack of bracing against lateral movements. In our case, we know the beam isn't experiencing sideways motion (Mlt = 0), allowing us to simplify our calculations. We derive a factor 'B' using specific parameters where 'Qu' represents applied loads and 'Pe' is the critical buckling load. These calculations help us understand if our design still stands strong when considering potential real-world variables.

Imagine walking on a tightrope; the moment you lean to one side, the tension increases. Similar principles apply here: a beam's ability to carry moments changes depending on how it’s supported or loaded, which we factor into our calculations for safety.

Therefore, calculating the B magnifier we find: B = 1 / (1 - (Qu) / (Pe)). Calculating the amplified moment as follows: Mu = B * Mu1. Therefore, the adequacy of the section is calculated from Eq. as follows: Pu + 8 * MuX < 1.0.

With the moment magnification factor 'B' calculated, we proceed to calculate our amplified moment 'Mu'. This amplified value gives us a true representation of the forces acting on the beam. Finally, we check if the beam's capacity can handle the applied loads without exceeding safe limits. We perform a strict comparison against the allowed values using Pu and check if the equation holds true. If the calculations prove that our combined loads are less than safe limits, the beam is adequate and safe for use.

Consider a seesaw: if too many kids sit on one side, it could snap. Before letting kids play, you must ensure the seesaw can handle their collective weight within safe limits. Similarly, we check our beam calculations to ensure it can bear the loads without breaking!

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