# Deflection and Bending Moment Diagram in SAP2000

Directional polarity has been the most fundamental thing one should learn before digging deep into the numerical values of any problem. Whether be it in deflection, bending moment, shear forces, torsion, strains or stresses, the directional polarity should be consistent.

For finite element structure, if we use global coordinates the uniformity in the direction is observed but it will be too computationally complicated to analyse in global coordinate. That’s why, local coordinate system was referred to used to analyse any parameter efficiently.

A 2 member continuous beam is taken in consideration for our example. First they are kept with default local axis direction given by SAP2000. Then for our study purpose, we rotate the local axes of the second member by 180 degrees with respect to local axes 1 (red axis), you can see the change in the local axes in the above diagram.

As from theoretical concept, if we apply loading in this beam, no matter what be the local axes, the structure should behave same for the loading condition. The behavior in terms of deflection, shear and moments.

Upon loading 1 kip-ft of uniformly distributed load in this beam, below is the deflection diagram of the beam.

From Figure 3 and 4, we can see the deflection values at the max deflection point of member 1 and member 2. SAP2000 uses global axis for the deflection diagram. So X = U1, Y=U2 and Z=U3 direction is the naming for the deflection axis. The beam should deflection downward with a parabolic curve with vertex at the midspan of each section, thus the value should be a negative value in Z axis. SAP2000 output is U3 at some negative value.

This concludes that SAP2000’s deflection is independent of local axis modification. This is a crucial information for us to determine whether a beam or a column or diagonal member is in negative or positive force or moment state even if local axes are flipped.

Figure 5 is the bending moment diagram for the above beam. As you can see the bending moment is completely flipped in the opposite direction for member 2 compared to member 1.

This flipping in the moment values is to account our change in local axis. If we output this bending moment then it completely creates a haywire in our design. Imagine we are trying to design a RCC beam, and instead of positive moment we record negative moment for a beam in the midspan. If we keep skip think the bottom side does not need tension bar then you will going to have a huge failure in the real world. The whole block fails as it doesn’t have any tension reinforcement in the correct position and the culprit is that flip in the local axis and our wrong interpretation of the moment values.

Thus to be consistent with our positive and negative moment direction deflection diagram helps us to fix our bending moments whether they are positive or negative. SAP2000 writes the cases for these in their knowledge base, how SAP determines what is positive moment and what is negative moment for a deflection case with reference to local and global axis.

SAP2000 defines the frame element internal forces that occurs in the local axes and planes as below:

• P, the axial force
• V2, the shear force in the 1-2 plane
• V3, the shear force in the 1-3 plane
• T, the axial torque (about the 1-axis)
• M2, the bending moment in the 1-3 plane (about the 2-axis)
• M3, the bending moment in the 1-2 plane (about the 3-axis)

Positive M3 bending moments cause compression on the positive 2 face and tension on the negative 2 face. [SAP2000]

From figure 6, one can see clearly that the member in the right side has compression in the top part of the beam and tension in the bottom part of the beam in the first member. And in the middle support the tension is at the top and compression is at the bottom and again the same trend of first member is seen in the second member mid span.

And upon looking at the positive faces, to be consistent over the same horizontal member. Positive 2 face of local axis is XZ plane of the global axis. And according to the SAP2000 if one has compression in the positive 2 face and tension in negative 2 face it is dictated as POSITIVE MOMENT.

To hold this definition and to be consistent over the element, as we have rotated the second member midspan by 180 degrees, the positive 2 face is flipped exactly opposite of first member. Instead of same nature in bending moment, if we plot the bending moment for both of the member we will get two different moment values and like before if we do not interpret it correctly we are going to have lot of problems in design.

CONCLUSION:

Consistency is the most important thing in our civil engineering design. To be consistent in the internal forces, if we get a model, we need to first see the deflection diagram to see the nature of the structure upon the loading condition, and then only fix the local axes direction. If we don’t fix the local axes we might design the member for negative moment, while in real should be designed for positive moment. Using the consistent local axis will help us a lot staying correct over the large design and modelling.