Deflection of concrete members follows a bilinear curvature, and it can be affected by many different variables. Concrete members are often internally reinforced and/or tensioned, come in different shapes, and vary with regards to the ingredients and additives in the concrete. Despite these variations, engineers generalized concrete behavior and the ACI Code adopted these general procedures, though difficulties remain. Some research reported discrepancies between actual results and code predicted deflections. The Branson and Bischoff equations are currently in use for calculating Ie, but in reality, they also require calculations of y– and Icr, making the calculations of deflections lengthy and time consuming. In addition, the most recent ACI Code procedure is still dependent on curve fitting rather than a scientific basis.
A new proposed method for calculating concrete deflections is presented herein; this method is based on the general bilinear flexibility of concrete under flexural stress. An earlier paper related to the specific deflections of high-rise concrete shear walls and using the bilinear flexibility method was authored by the two senior authors. In this work, the method is extended in a general way to concrete beams, plates, and to other structural materials.
Ie = effective moment of inertia, in4
Icr = moment of inertia of cracked section transformed to concrete, in4
y– = distance from extreme compression fiber to neutral axis, in
Background
Concrete cracks when subjected to tensile stresses that exceed the modulus of rupture. Since stresses along the beam span are not uniform, the degree of cracking is also non-uniform; more and larger flexural cracks occur at locations of higher flexural stresses. Therefore, a concrete beam even if initially prismatic, becomes non-prismatic when cracked.
The stress magnitude relative to the modulus of rupture is one indicator of flexibility; the larger the ratio, the more cracks cause more flexibility. The amount of longitudinal steel reinforcement is another indicator of flexibility; less rebar causes more flexibility.
Figure 1 shows a typical load deflection curvature for concrete reproduced from Reference 7; the flexibility is bilinear. Below cracking (Point C), gross section properties predict deflections reasonably. However, above cracking, gross properties no longer predict deflections accurately. The larger the spread between the applied load and the load at which cracks initiate, the larger the error. Point A is arbitrary, where the moment Ma is larger than the cracking moment Mcr . Figure 1 also shows the general impact of section geometry on flexural flexibility. Thus, for a given section, the most significant variables affecting flexibility are:
1. The load ratio – Mcr/Ma
2. The area of longitudinal flexural reinforcement.
Concrete deflections follow general accepted formats, but walls are stiffer than beams and stiffer than slabs, as shown in Figure 1.
Flexibilities
Considering the modulus of elasticity E, the product EI is an indication of flexural flexibility. In a cracked beam the value of I varies along the span of the beam: it is Ig where un-cracked and Icr where cracked. For a simply supported beam we focus our analysis on the section at the center of the beam where flexural stresses are the largest. Ie from Branson and Bischoff are shown below; in which they depend on y– and Icr, also shown below.
Branson
Bischoff
As can be seen, to obtain Ie we must first calculate y– then Icr .
E = modulus of elasticity, psi
Ec = modulus of elasticity of concrete, psi
Ig = moment of inertia of gross concrete section, in4
Figure 2 shows a bilinear stress-strain graph but includes a linear equivalent line as if the beam is prismatic, for both before and after cracking (in reality, a cracked beam is non-prismatic, as indicated earlier). The objective is to obtain the equivalent prismatic properties that can be used to predict accurate deflections. Note that concrete cracks at Point C, where stress is equal to vcr. But the goal is to obtain deflections at Point A and we do this directly by using Line 3.
In Figure 2, Line 1 shows the beam flexibility before cracking, Line 2 shows the beam flexibility after cracking, and Line 3 shows the equivalent prismatic flexibility.
Let’s define Ee = λEc, where λ is a coefficient to be calculated. va is the stress at Point A due to flexural loads and εa is the corresponding strain. The objective is to obtain λ as a function of other variables in triangle OAC.
Derivation
Working with triangle OAC in Figure 2, we obtain (see Ref. 3 for derivation):
Where 1/n represents the slope of the upper part of the bilinear graph as Line 2 in Figure 2; vcr is the cracking stress (for concrete); va represents flexural stress. Fy represents yield stress (for steel).
Equations (3), (4) and (5) are the Equations of Bilinear Flexibility and can be evaluated at any point along the beam span and the beam depth.
The ACI Code suggests specific values of n to calculate deflections but for lateral loads only. If Code values are to be used for gravity loads then n values need to be evaluated. ACI Code values at service loads are n = 3 for slabs, n = 2 for beams and cracked walls, but these values are to be used with lateral loads only.
It is instructional to note that λ =1 represents the case of uncracked concrete which occurs when n = 1 or, when va ≤ vcr or when Mcr/Ma > 1.0. When concrete is cracked Mcr /Ma < 1.0, n is greater than 1 and λ is less than 1.
Advantage of Bilinear Flexibility
The Equations of Bilinear Flexibility were developed herein using a method of substitution of bilinear properties with a single equivalent linear property. The variables needed are loads (moments or stresses) and the flexibility coefficient, n. These equations can be used for various concrete structures and materials. It is interesting to note that they are applicable not only for concrete but for all materials with dual flexibilities, including structural steel. They also have implications for seismic analyses, such as when demand exceeds the elastic strength of members and when structural loads get redistributed.
For an engineering office the new equations of bilinear flexibility are simple and save time. They show the correct flexibility when loads exceed cracking without the need to evaluate three equations each time.
Determination of Flexibility Coefficient, n
Deflections of concrete structures are impacted by, among other things geometry, loads, and the quantity of reinforcement, as mentioned earlier. As a result, the value of n, is particular to each design case. This coefficient n may be determined using several possible methods, as follows:
1. Use an approximate n value.
2. Use an n value from tests and experiments.
3. Use n values from calibration with available tools, such as Branson or Bischoff.
Table 1 and Table 2 are used to obtain Figure 3 for an 8-inch-thick one-way flat concrete slab (f’c = 4,000 psi), and calibrated to Bischoff. Mcr /Ma were selected arbitrarily and Ie was calculated for different reinforcement ratios, from Bischoff. The value of n is calculated as below:
Proposed Procedure
1.Design one-way concrete slab and select steel reinforcement ρ = As /bd
2.Enter Figure 3 graph and write down values of n for Mcr /Ma equaling 0.50 thru 0.95. This will establish the slope for Line 2 in the bilinear graph (Figure 2).
3.Obtain values of λ and Ie using Equations 3 thru 5 or Figure 3 graph.
Example
Determine Ie for an 8-inch one-way concrete slab with bottom steel reinforcement of #5 bars at 12˝ o.c. The slab spans 18 feet and support 160 psf of total residential loads.
Solution:
Mcr = 5.0ft − k
Ma = 160 × = 6.48ft − k
From Figure 3:
n = 4.8
Ie = 0.53 × 512 = 271.4 in4
Conclusions
The method of Bilinear Flexibility is presented herein. In this procedure, Bilinear Flexibility is replaced with an equivalent single flexibility. A calibration method was used to obtain flexibility coefficients n. The equivalent single flexibility is obtained with the use of the equation of Bilinear Flexibility which converts n to λ. For accurate results, use appropriate n values, then calculate Ie using the general Equation of Bilinear Flexibility. This simple method is based on the first principles and, therefore, may be appropriate for adoption by building codes.
Future research may focus on the development of n-values considering different, concrete cross- sections, podiums, two-way slabs, etc. Additionally, this method can be used with other materials, and is applicable for post-elastic analyses.■
References
ACI Committee 318, “Building Code Requirements for Structural Concrete (ACI 318-14) and Commentary (ACI 318R-14),” American Concrete Institute, Farmington Hills, MI, 2014, 519 pp.
ACI Committee 318, “Building Code Requirements for Structural Concrete (ACI 318-19) and Commentary (ACI 318R-19),” American Concrete Institute, Farmington Hills, MI, 2019, 623 pp.
Wexler, Neil and Jeoung, Hoonhee, “High-Rise Concrete Shear Walls Subject to Service Loads”, Concrete International, V. 41, Issue. 12, Dec. 1, 2019, pp. 37-41.
Branson, D.E., and Trost, H., “Unified Procedures for Predicting the Deflection and Centroidal Axis Location of Partially Cracked Nonprestressed and Prestressed Concrete Members,” ACI Journal Proceedings, V. 79, No. 2, Mar.-Apr. 1982, pp. 119-130.
Bischoff, Peter H. and Scanlon, Andrew, “Effective Moment of Inertia for Calculating Deflections of Concrete Members Containing Steel Reinforcement and Fiber-Reinforced Polymer Reinforcement,” ACI Journal Proceedings, V. 104, No. 1, Jan.-Feb. 2007, pp. 68-75.
Bischoff, Peter H., “Comparison of Existing Approaches for Computing Deflections of Reinforced Concrete,” ACI Journal Proceedings, V. 117, No. 1, Jan. 2020, pp. 231-240.
Nilson, Darwin, Dolan, “Design of Concrete Structures”, 13th Edition.