A Beginner's Guide to ACI 318-14 Chapter 3 - Strength Analysis © 2018 T. Bartlett Quimby |
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Section 3.3 Equilibrium Last Revised: 04/08/2018 Now that we have the internal forces and their locations for the individual components, we can use equilibrium equations to compute the nominal strengths (i.e. the internal forces) which will cause the neutral axis at failure to be a the location and orientation which we started with. (That last sentence was very precise... you need to make sure that you understand it!) Hopefully you studied the effects of combined bending and axial force in a Mechanics course. For combined bending about two orthogonal axes combined with axial force the neutral axis can be virtually anywhere and at any orientation. This will become clear as we work through the concept and some example problems. The three forces considered when doing the ultimate strength are the three simultaneously acting internal member forces resulting from loading which cause the failure depicted by a cross section with a particularly located and orientated neutral axis.
The three forces/strengths are:
Figure 3.3.1 shows the component forces found by means already discussed and the nominal strengths. This 2D diagram only allows us to compute moment about the centroidal axis perpendicular to the diagram. A similar diagram can be made to depict bending about the orthogonal centroidal axis. Note that the resulting applied forces can always be expressed as a concentric axial force accompanied by moments about the principle axis or as the axial force acting at an eccentricity e that is equal to the moment divided by the concentric force. The force vectors in Figure 3.3.1 are draw as tension positive. It is important to keep the signs correct on the member forces. The moment vector is drawn as compression at "top" of section positive. Once the forces in the concrete and each of the bars are determined, finding the value of each force/strength is easily found using equilibrium equations. Determining Pn The vectors are drawn in tension positive notation and the moment is shown in positive notation for compressing in the "top" of the section. To find Pn, an equilibrium equation for sum of forces in the axial direction is written: 0 = -Pn + TS1 + ... + TSn + Cc Note that it is very important to maintain the proper signs on the actual values of each component force. Tension values are positive and Compressive values are negative. Determining Mnx and Mny To determine the magnitudes of Mnx and Mny it is necessary to know, for each component force, the magnitude (with appropriate sign) and distance between the axis in question and each component force. Using the diagram in Figure 3.3.1, we can write an equilibrium equation for moment about the centroidal axis perpendicular to the drawing using the right hand rule. 0 =-Mn - TS1(yS1) - TS2(yS2)
+ TS3(yS3) - Cc(yc) Where ysi and yc are the distance of the associated forces from the axis of rotation (i.e. the centroidal axis). It is often useful to do these computation is a table similar to:
At this point, it will be useful to follow through the example problems in section 3.6. Take the time to understand the process and how each value was determined. Equilibrium for bending about both Principle Axes
There are many occasions when it is necessary to analysis for strengths that include bending about both principle axes. If biaxial bending is present, then the result is a neutral axis that is not oriented with either principle axis. See Figure 3.3.2 for a typical example. The area of concrete included in Whitney's stress block is the shaded area, less the area of the included bars. It is necessary to compute the area, and centroid, of this region for any given value of location, c, and angle, q. The resulting force, Cc = 0.85f'cAc, will be centered on the center of this area. This resultant force is unlikely to fall on either principle axis, so it will cause moments about each principle axis. Similarly each bar will have its own unique distance, d, measured perpendicular to the compression face reference which will result in a unique strain, stress, and force value for each bar. The same equations presented earlier for computing strain, stress, and force are used. Each bar force will cause moments about both principle axis, unless it falls on one of the axes. Summary All of this can be quite daunting at first. There is no doubt that the computational effort can be quite large, however the basic principles of strain compatibility and equilibrium apply. With the exception of singly reinforced rectangular cross sections, finding strengths involves iteratively varying the location of the neutral axis, c, then computing the resulting concrete and steel strains, stresses, and forces. The resulting forces are then used to compute net axial force and moment about each principle axis. This process relies heavily on the ability to compute concrete areas and centroidal locations as well as determine distances to steel elements, both from the compression face and from the principle axes. Given that shapes and rebar patterns can be complex, the geometrical aspects of the computations can become quite challenging. The sheer number of elements in a cross section can add to the burdens associated with these computations.
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