A Beginner's Guide to ACI 318-14

Chapter 3 - Strength Analysis

© 2018 T. Bartlett Quimby

Basic Principles

Strain Compatibility

Equilibrium

Interaction
Diagrams

Special Cases

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Section 3.2

Strain Compatibility

Last Revised: 04/08/2018

The essence of strain compatibility is that plane sections remain plane and the steel strain is the same as the concrete strain at all locations. These assumptions require that the reinforcing steel be bonded to the concrete so that the strains in the steel and adjacent concrete are always equal. This concept allows us to establish relationships which result in easily computable internal stresses and forces.

Plane Sections Remain Plane

Figure 3.2.1
Linear Strain Diagram

The assumption that stress is proportional to the distance from the neutral axis can be illustrated using the general section shown in Figure 3.2.1.

Please note, that for this discussion, negative values indicate compression and positive values indicate tension.

Note that the neutral axis location is measured from the most extreme fiber in compression. The value of "c", the shortest distance of the neutral axis from the compression face, can be any value from zero (pure tension on the section) to infinity (pure compression on the section). The neutral axis need not be located on the section. Any value of "c" between zero an infinity represents an internal force condition that includes both axial force and bending, combined.

The following discussion keys on determining a strength condition for an assumed location of c. In order to find a particular nominal strength, a particular value of c is needed. Unfortunately, however, it is not reasonable to compute this location directly except in a few special cases (such as singly reinforced rectangular beams).  In most cases, the desired condition is determined by iterative analysis where the value of c is modified until the desired condition is reached. This means that a spreadsheet or computer program of some type is extremely useful in performing strength calculations.

The location of the neutral axis is a function of the relative values of the moments about the principle centroidal axes (not shown in the figure) and the axial force which cause this strain condition to exist.  If you recall from Mechanics, the location of the neutral axis depends of the effects of the combined forces.

In this case we do not yet know what the combined forces are that cause this strain condition, but we will find them by working backward from a known neutral axis location and orientation. The resulting forces are determined from equilibrium equations and, for each location and orientation of neutral axis, represent one of a infinite number of failure conditions for the section.

Steel Stress and Bar Force

For the purposes of reinforced concrete strength determination, steel is considered to be a bi-linear material. That means that Hooke's Law applies until the stress reaches the specified yield stress, fy.  Beyond that, the stress is taken to be equal to the specified yield stress, fy. This applies to both tension and compression. Figure 3.1.2 illustrates the idealized stress-strain relationship.

Figure 3.1.2
Reinforcing Steel Idealized
Stress-Strain Diagram

Mathematically the stress in a steel bar can be expressed as:

fsi = max[min[fy, Esesi], -fy]            (Equation 3.2.1)

Where:

  • fsi = actual steel stress in the ith bar
  • esi = actual steel strain in the ith bar
  • i = bar number
  • fy = specified yield strength
  • Es = steel modulus of elasticity (29,000 ksi)

Having a defined strain diagram, we can determine the strain in each piece of steel in the cross section, if the concrete is bonded to the steel such that they deform equally. Bar location "d", like "c", is measured from the extreme compressive fiber. This permits us to use a similar triangles approach to finding the strain in the ith bar. Refer to Figure 3.1.3 for a graphical representation of this process.

Figure 3.1.3
Determination of Bar Forces

ei = (-0.003/c) * di            (Equation 3.2.2)

To find the stress in each bar the stress-strain relationship shown above in Equation 3.2.1 is used.

To determine the force in each bar, we simply multiply the bar's stress by its area:

Ti = fsi * Asi             Equation (3.2.3)

Where:

  • Ti = force in in the ith bar
  • fsi = actual steel stress in the ith bar
  • Asi = the area of the ith bar

Concrete Stress and Compressive Force

It needs to be noted that ACI 318-14 22.2.2.2 states that concrete in tension is to be neglected in these strength calculations. This is because the concrete in the tensile zone is assumed to be cracked as the member approaches its ultimate strength. What this means is that we only consider the compression region when computing the internal forces at the ultimate load condition.

Figure 3.1.4
Idealized Stress-Strain relationship for
concrete in compression
 

Figure 3.1.5
Idealized Determination of Concrete Force
 

Figure 3.1.6
Approximate Determination of Concrete Force
 

Unlike steel, concrete has a non-linear stress-strain relationship, which complicates the computation of internal compressive force on the concrete. Figure 3.1.4 shows an idealized stress-strain relationship for concrete in compression. This relationship can be used to determine the stress, fci, using the strain, ei, at any location, i.

Applying this non-linear stress-strain relationship to the linear strain results in non-linear stress similar to the idealized stress distribution shown in Figure 3.1.5. Integrating over the compressive face of the section results in a net compressive force, Cc, located some distance dc from the extreme compressive fiber.

dc is determined by a center of force integration.

Note that the area of concrete in compression, Ac, does not include the area occupied by the reinforcing bars located in the compression area.

In trying to apply this approach, it will become quickly obvious that the mathematics is somewhat intensive and dependent on obtaining a well defined stress-strain relationship for concrete.  Given all the assumptions that have been made along the way, this level of effort is not justified if an reasonable approximation can be made.

ACI 318-14 22.2.2.3 allows such an approximation to be made as long as it is in substantial agreement with test results. ACI 318-14 22.2.2.4 provides the approximation most frequently used.

The approach presented in ACI 318-14 22.2.2.4 recognizes that the result needs to be a force approximately equal to that which the integration would yield. The resultant force must also be located at approximately the same location that the center of force integration would yield.

This approximation is based on work done by Whitney in the 1930s and the resulting stress distribution is frequently called "Whitney's Stress Block".

To deal with the need for an equivalent force, it has been found that a stress of 0.85 of the specified 28-day compressive stress, f'c, "uniformly distributed over a portion of the compression zone bounded by edges of the cross section and a line parallel to the neutral axis located a distance a from the fiber of maximum strain..." yields a compressive force in substantial agreement with test results.

The distance "a" is determined by ACI 318-14 equation 22.2.2.4.1. The term, b1, is depended on the concrete property f'c. ACI 318-14 Table 22.2.2.4.3 gives values for b1.  This table can also be state mathematically as:

b1 = max[0.65, min[0.85, 0.85-(0.05)*(f'c-4000)/1000]]            (Equation 3.2.4)

By using a partial area located as shown and using a uniform stress the center of force is located at the center of area bounded as specified. This center of force is in substantial agreement with test results, thus this approximation meets the requirements.

Again, note that the area of concrete in compression, Ac, does not include the area occupied by the reinforcing bars.
 

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