Some Welcome Revisions
I would like to commend ACI on their rework of Sections 10.10 through 10.13 in ACI 318-05, the result of which is a single Section 10.10 in ACI 318-08 that, in my opinion, reads much more clearly than previous editions of the 318 document. Section 10.10 covers second-order effects in the analysis and design of members resisting axial/flexural loads. This is not a topic that is easy to convey but I believe ACI has made a great stride in clarifying their intent for the engineer in the latest edition of the 318 code. As a developer of concrete design software I must convert the instructions given in ACI 318 to automated computer processes. The process of implementing a new design code within software is a good litmus test for the clarity of the code content. After all, a machine can't draw its own conclusions if logic is missing.
Section 10.10 is entitled "Slenderness Effects in Compression Members". The way I like to think of this is "what happens when you consider equilibrium on the deformed structure rather than the undeformed structure, and what you need to do if there is a significant difference between the two". Considering equilibrium on the deformed structure (known as second-order analysis) will, in general, produce greater member moments and curvatures than for the equivalent first-order situation. Second-order analysis is a more physically accurate representation of the structure. All structures feel second-order effects, no matter how lightly loaded. Conveniently, in many circumstances the additional response induced from second-order effects is small enough to be neglected. That is, the additional member moments or curvatures are a small percentage of the results given by a first-order analysis.
Types of Second-Order Effects
There are two types of second-order effects that prevail in buildings: those resulting from story drift and those resulting from member curvature. The former is commonly referred to as the P-Δ effect (with a large delta), while the latter is commonly referred to as the P-δ effect (with a small delta). A diagram of each is given below.
Figure 1 - Moment diagrams for a cantilever column for a) First-order effects only and, b) P-Δ effects only.
Figure 2 - Moment diagram for a cantilever column resulting from P-δ effects.
P-Δ effects (large delta) can be captured by one of two analytical means: by an iterative solution or by including a geometric stiffness term in the stiffness formulation. The latter option has the benefit that the solution is mathematically linear, and thus not as computationally rigorous as an iterative solution. The latter option also allows the additive combination of results from multiple load cases. Thus, obtaining results for the combined loads is simply a matter of scaling each of the load case results by the respective load factor and summing. This is not possible with an iterative solution. The iterative solution has the advantage that it can produce the theoretically exact result, even when displacements become large. P-δ effects, in contrast to P-Δ effects, can be captured only by an iterative solution.
Analysis Options
ACI 318-08, Section 10.10 allows the choice of one of three methods to account for second-order effects: nonlinear (inelastic) second-order analysis, elastic second-order analysis, or use of the moment magnification procedure. A discussion of each is given below.
Nonlinear second-order analysis, described in Section 10.10.3, is the most analytically rigorous of the three methods. The most apparent benefit of using this approach is that the member stresses, forces, and deflections that are obtained from the analysis need no manipulation to be used for design purposes. Second-order effects and material nonlinearity are accounted for directly in the analysis. If performing an inelastic second-order analysis, Section 10.10 is essentially reduced to 10.10.3. That is, moment magnification, variable stiffness multipliers to account for cracking, and identifying frames as sway or non-sway are all unnecessary.
Elastic second-order analysis, described in Section 10.10.4, is substantially less computationally rigorous since material response is considered elastic, and is thus less iteration-intensive than inelastic second-order analysis. This requires less computation time than inelastic analysis. The drawback is that material nonlinearity must be accounted for by scaling the elastic stiffness of members in some way. Section 10.10.4.1 provides a simplified means of accounting for cracked regions along members by multiplying the Ig or Ag value of a section by a multiplier less than 1.0. This is an inexact approach. Strictly speaking, the stiffness multiplier used for a member applies only to a single load condition. If you remove load from the structure less cracking will occur. If you load the structure at the same magnitude in a different direction, the member forces will change and thus so will the distribution of cracking in each member. Similarly, service-level conditions dictate different stiffness multipliers than ultimate-level conditions. This effect is captured automatically in one model when performing an inelastic analysis. With an elastic second-order analysis, as with inelastic second order analysis, moment magnification and the identification of stories as sway or non-sway is unnecessary. However, unlike inelastic second-order analysis, stiffness multipliers must be assigned to members to simulate the effects of member cracking.
The moment magnification procedure, described in Section 10.10.5, is the least analytically rigorous of the three procedures. The benefit of this method is that it requires only a first-order elastic analysis of the structure. The stresses, forces, and deflections from this analysis are then scaled according to code-mandated procedures to account for second-order effects. There is a good deal of book-keeping that goes along with this procedure. To conduct this analysis such that an economical design is attained (without making too many simplifying assumptions), it is a good idea to utilize a software package that handles all the ACI slenderness calculations. This procedure requires the identification of stories and frames as sway or non-sway. The sway designation essentially means that the lateral drift of the story is large enough that the P-Δ component of second-order response needs consideration. With a non-sway story or frame, the P-Δ effect is assumed small enough to be neglected.
Putting it into Practice
So which of these procedures should be utilized? That is a question I am hoping practicing engineers reading this article will answer. The decision will likely depend on the type of building under consideration, as well as the loads to which the structure is subjected. As developers of engineering software, we here at Bentley face the challenge of accommodating a wide range of structural configurations within our analysis and design routines. However, what works for a twenty-story cast-in-place office structure will likely not suffice for a single story precast warehouse, and vice-versa. It is through feedback from and interaction with our clients that we hope to identify the best means of implementing the intent of ACI within our software. I encourage anyone involved in the design of concrete building structures to post a follow-up to this article with his or her own opinions and conclusions. Feedback from the engineering community has always been, and will continue to be, a key ingredient to the success of Bentley's software solutions.
REFERENCES
American Concrete Institute. Building Code Requirements for Structural Concrete (ACI 318-05) and Commentary, American Concrete Institute, Farmington Hills, MI
American Concrete Institute. Building Code Requirements for Structural Concrete (ACI 318-08) and Commentary, American Concrete Institute, Farmington Hills, MI
Lindeburg, Michael R. Civil Engineering Reference Manual for the PE Exam (Eighth Edition), Professional Publications, Inc., Belmont, CA