<< Back to Axial, Shear, & Moment Diagrams
Step 1: Find the support reaction forces/moments.
We should always begin by drawing the free body diagram (FBD). Knowing all the externally applied forces and moments, we can determine the reactions at the support by summing the forces and moments.
From the FBD, we know that there is only one point load. Therefore, there is one vertical, opposite reaction and a moment caused by that point load at the fixed support. To find these reactions, we can sum the forces and moments. For consistency, up forces and counterclockwise moments are positive.
Note that RA is a positive value, meaning that it is pointing upwards, according to our sign convention. It also matches the direction of the arrow in our current FBD. However, if we selected the reaction force RA to point downwards in the FBD, the value would become negative, meaning that the actual reaction force is pointing upwards. Either way gives us the same result. Therefore, the main takeaway is that the positive and negative signs have to be consistent with the FBD and chosen sign convention.
Step 2: Determine axial/shear forces.
To determine the internal axial (A) and shear (V) forces, we are going to look at four segments. Generally, we can find the axial and shear forces along each segment by summing all forces.
By now, we should know that the assigned direction of A or V in the FBD should not matter. If their calculated values are negative, the forces actually go in the opposite direction, and we just made the wrong 50/50 guess. But, what do the directions of the forces mean? Do outward arrows correspond to tensile forces? Well, let us recall our positive sign convention for frames.
According to this sign convention, axial tensile forces are positive. This means that by default, the axial force (A) is drawn in the direction that creates tension on the segment in FBDs. Similarly, the shear force (V) is drawn in the positive direction as shown above.
Mathematically, we can say the force at distance x from the denoted origin in each segment is:
Since there is just one point load on this frame, notice the pattern; there is no shear force (V) where there is axial force (A), and there is no axial force (A) where there is shear force (V).
Step 3: Draw axial/shear force diagrams.
Once we have the force equations for each segment, we can plot the axial and shear force diagrams (AFD and SFD). Note that the axial force at the support is equal to the reaction. Also, as stated in the list of basic rules, the AFD and SFD do not have to be continuous.
Step 4: Determine bending moment.
Similarly, to determine the bending moments (Mz), we are going to look at the same four segments. Generally, we can find the bending moment along each segment by summing all the moments. Mz is drawn in the positive direction in the FBDs.
Mathematically, we can say the bending moment at distance x from the denoted origin in each segment is:
Step 5: Draw bending moment diagram.
Once we have the bending moment equation for each segment, we can plot the bending moment diagram (BMD). As stated in the list of basic rules, the BMD is continuous, even for frames.
Recall that shear is the rate of change or derivative or slope of moment. Therefore, when shear force is zero, moment is constant. Also, we can determine the critical points in the BMD by finding the area under the SFD.