22.3 - Examples

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.

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.

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.

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.

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.

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.