Normalized Uniaxial Compressive Strength Data For Mac

Uniaxial Compressive Strength (UCS) and the modulus of elasticity (E) are key parameters in rock engineering. Experimental methods are difficult task for determination UCS and E. There are different methods available for prediction of UCS and E in the literatures 1-11. These methods are based on index tests. SECURITY CLASSIFICATION Of THIS PAGE(1Wha Data Eneod). ABSTRACT (Continued). Maximum shear stress normalized with respect to the unconfined compressive strength. 0 03 + (mac 03 + 1.0 a2 )1/2. A regression.

Contents. Deformation of rocks — elasticity To first order, most rocks obey the laws of linear elasticity. That is, for small strains, the elements of the tensors are related through where.(1) In other words, the stress required to cause a given strain, or normalized length change (Δl k /l l), is linearly related to the magnitude of the deformation and proportional to the stiffnesses (or moduli), M ijkl. Furthermore, the strain response occurs instantaneously as soon as the stress is applied, and it is reversible—that is, after removal of a load, the material will be in the same state as it was before the load was applied. A plot of stress vs.

Robust Scaling of Strength and Elastic Constants and Universal Cooperativity in Disordered Colloidal Micropillars Abstract We study the uniaxial compressive behavior of disordered colloidal free-standing micropillars composed of a. The proposal includes internal functions defining the dependence of the above mentioned four material parameters on the uniaxial compressive strength and the performance parameter.

Strain for a laboratory experiment conducted on rock that obeys such a law is a straight line with slope equal to the modulus. However, in real rocks, the moduli increase as a function of effective stress, particularly at low stress. Some of this increase is reversible (nonlinear elasticity), but some of it is irreversible (plasticity or end-cap compaction). To make matters even more complicated, rocks also exhibit time-dependent behavior, so that an instantaneous stress change elicits both an instantaneous and a time-dependent response.

These anelastic effects can be seen in laboratory experiments, as shown in Fig. 1—This figure shows the loading path and the confining pressure as a function of strain recorded during compaction experiments conducted using two samples of a poorly consolidated, shaley turbiditic sand of Miocene age. Sample 1 was maintained at its saturated condition; Sample 2 was cleaned and dried before testing (modified after Moos and Chang ). At the top of Fig. 1 is shown the stress as a function of time applied in the laboratory to two samples of an upper Miocene turbidite. As in most experiments of this type, a cylindrical rock sample is jacketed with an impermeable soft sleeve and placed in a fluid-filled pressure cell.

The fluid pressure surrounding the sample is increased slowly, and the fluid pressure (confining stress) and sample axial and circumferential strains are measured. To identify the various deformation processes that occur in this unconsolidated sand, the stress is slowly increased at a constant rate and then held constant until the sample stops deforming. Then the pressure is decreased to approximately half of the previous maximum pressure. After that, the pressure is increased at the same rate until the next pressure step is completed. This process continues until the desired maximum pressure is achieved, and then the sample is slowly unloaded and removed from the pressure cell. Stress-strain curve All aspects of typical rock behavior can be seen in the stress-strain curve plotted on the bottom of Fig. At low pressure, the sample is soft, and there is a rapid increase of stiffness with pressure (nonlinear elasticity) owing to crack closure, as well as an increase in stiffness caused by irreversible compaction.

Once the pressure increase stops, the sample continues to deform, with deformation rate decreasing with time (time-dependent creep). The sample is stiffer during unloading than during loading, and during this phase of the experiment, it essentially behaves as a linear elastic material; the permanent strains during loading and creep that occurred through plastic/viscous deformation mechanisms are not recoverable. Reloading follows the (purely elastic) unloading path until the maximum previous pressure has been reached, after which additional plastic deformation begins to occur again as the material resumes following the compaction curve. All of these effects can be seen in situ, including the difference between the loading and unloading response. Measurements of made on this sample during the experiment by measuring the time of flight of pulses transmitted axially along the sample were used to calculate the dynamic shear (G) and bulk (K) moduli with Eq. The implications of the results for pore pressure prediction are discussed later in this chapter.(2) The dynamic bulk modulus calculated from the velocity measurements is higher than the moduli computed from the slopes of the unloading/reloading curves, which in turn are larger than the modulus calculated from the slope of the loading curve. This dispersion (frequency-dependence of the moduli) also is typical of reservoir rocks, and it is the justification for empirical corrections applied to sonic log data to convert from the dynamic moduli measured by the sonic log to static moduli that are used to model reservoir response.

Normalized Uniaxial Compressive Strength Data For Machine

2—This figure, modified from Schutjens et al., In unconsolidated materials, shear-enhanced compaction begins at zero confining stress as soon as the material begins to be loaded (see Fig. In situ, this compaction is the primary cause of the increase in stiffness and decrease in porosity of sediments with burial. The assumption inherent in all standard pore-pressure-prediction algorithms that rock properties are uniquely related to the effective stress is equivalent to assuming that the rock in situ lies along a compaction trend defined by an end cap. If the mean effective stress decreases (for example, because of erosion) or the pore pressure increases, the rock becomes overcompacted. When this occurs, its porosity is no longer in equilibrium with the end cap, and it will behave elastically, as occurred during the unloading stages of the experiment shown in Fig. When a differential load is applied (e.g., σ 1 σ 2 ≅ σ 3), eventually the maximum stress (σ 1) will get so large that the sample either will begin to yield through a process of distributed deformation or will fail because of shear localization and the creation of a brittle failure surface (a fault). 2, the data at the left edge of the plot lie along a limit in the ratio of shear stress to mean stress that is defined by the onset of brittle failure or plastic yielding by shear localization, as discussed next.

Failure models and rock strength that define stress states at which brittle failure occurs follow stress trajectories that lie along the left edge of the data shown in Fig. It is clear that the ability of a rock to carry differential stress increases with confining stress. To establish the exact relationship, rock strength tests are conducted at a number of confining pressures. In these tests, a jacketed, cylindrical sample is loaded into a pressure vessel, a constant confining pressure is established, and an axial load is applied by means of a hydraulic ram. The load is increased slowly by driving the ram at a constant rate, monitoring axial and circumferential strains and maintaining a constant confining pressure, until the sample fails or yields.

An example of an axial stress vs. Axial strain plot from a typical triaxial stress experiment is shown in Fig. 3—Typical plot of axial stress vs. Axial deformation during a triaxial strength test.

Initially, the sample is soft, but as the axial load increases, microcracks begin to close, causing an increase in stiffness. When the axial stress is sufficiently high, inelastic behavior begins to occur.

If the axial load continues to increase, the stress-strain curve will reach a maximum, followed either by catastrophic failure (as shown here) or a roll-over and continued residual strength, for which an increase in deformation can be achieved with no change in axial load (courtesy GeoMechanics Intl. One criterion to define the stress state at failure is the 2D linear Mohr-Coulomb criterion. The defines a linear relationship between the stress difference at failure and the confining stress using two parameters:. S o, the cohesion (or C o, the unconfined compressive strength). Φ, the angle or μi, the coefficient of internal friction, where μ i = tanΦ The equation that defines the 2D linear criterion is τ = S o + μ iσ n. These parameters can be derived from triaxial strength tests on cylindrical cores by measuring the stress at failure as a function of confining pressure. 4 shows graphically how the Mohr-Coulomb parameters are derived.

The upper plot shows a series of Mohr circles, with x-intercepts σ 1 and σ 3 at failure and diameter σ 1 – σ 3, in a plot of shear stress to effective normal stress. The failure line with slope μi and intercept So that just touches each of the circles defines the parameters of the 2D linear Mohr-Coulomb strength criterion for this material. C o is the value of σ1 for σ 3 = 0 of the circle that just touches the failure line. The lower plot graphs σ 1 vs.

Σ 3 and can be used to derive C o directly. 4—Top is a plot of a set of Mohr circles showing the stress state at failure for a series of triaxial strength tests. The results have been fitted to a linear Mohr-Coulomb failure criterion. The lower plot shows axial load at failure vs. Confining stress. S o (or C o) and the coefficient of internal friction, μ i, can be derived easily from these data (courtesy GeoMechanics Intl.

Some of the other strength criteria include the following: Strength Criteria Description Hoek and Brown (HB) criterion like the Mohr Coulomb criterion, is 2D and depends only on knowledge of σ 1 and σ 3, but which uses three parameters to describe a curved failure surface and, thus, can better fit Mohr envelopes than can linear criteria. Tresca criterion A simplified form of the linearized Mohr-Coulomb criterion in which μ i = 0.

It is rarely used in rocks and is more commonly applied to metals, which have a but do not strengthen with confining pressure. Modified Lade criterion Has considerable advantages, in that it, too, is a 3D strength criterion but requires only two empirical constants, equivalent to C o and μ i. Thus, it can be calibrated in the same way as the simpler 2D Mohr-Coulomb failure criterion, but because it is fully 3D, it is the preferred criterion for analysis of wellbore stability. 5—Example plot of a comparison between log-derived C o (gray dots), scratch-test results (solid line), and laboratory measurements of unconfined compressive strength (triangles).

The advantage of scratch testing is that no special core preparation is required. This is in contrast to the extensive preparations required prior to triaxial testing.

The test can be conducted either in the lab or, in principle, on the rig, almost immediately after recovery of core material. No significant damage occurs to the core, which makes this a very attractive substitute for triaxial testing when little material is available. In fact, research is now under way to evaluate the feasibility of designing a downhole tool to carry out this analysis. Penetrometer testing In a penetrometer test, a blunt probe is pressed against the surface of a rock sample using continuously increasing pressure. The unconfined compressive strength is then computed from the pressure required to fracture the sample. As in the case of scratch testing, no special sample preparation is required. In fact, any sample shape can be used for a penetrometer test, and even irregular rock fragments such as those recovered from intervals of wellbore enlargement because of compressive shear failure can be tested.

Recently, methods have been developed to apply penetrometer tests to drill cuttings. Although these have not been widely used, they show considerable promise, and in the future they may become an important component of the measurement suite required to carry out wellbore stability analysis in real time. Estimating strength parameters from other data It is relatively straightforward to estimate C o using measurements that can be obtained at the rig site. Log or measurements of porosity, elastic modulus, velocity, and even activity (GR) have all been used to estimate strength. For example, Fig.

6 shows a lot of C o computed from P-wave modulus (ρ bV p 2) for Hemlock sands (Cook Inlet, Alaska). 7—Crossplots of unconfined compressive strength (top) and angle of internal friction (bottom) vs. GR from log inversion. Dots are laboratory test results. While there is considerable scatter, a clear relationship exists between GR and both parameters.

Because velocity, porosity, and GR can be acquired either using LWD or by measurements on cuttings carried out at the drilling rig from which CEC can also be derived, it is now possible to produce a strength log almost in real time. It is important, however, to recognize that different rock types will have very different log-strength relationships, based on their:. Sand/Shale. Limestone. Dolomite. Age. History.

Consolidation state Therefore, it is important to be careful to avoid applying to one rock type a relationship calibrated for another.

Compressive strength or compression strength is the capacity of a material or structure to withstand loads tending to reduce size, as opposed to tensile strength, which withstands loads tending to elongate. Xport for mac. In other words, compressive strength resists compression (being pushed together), whereas tensile strength resists tension (being pulled apart). In the study of strength of materials, tensile strength, compressive strength, and shear strength can be analyzed independently.

Some materials fracture at their compressive strength limit; others deform irreversibly, so a given amount of deformation may be considered as the limit for compressive load. Compressive strength is a key value for design of structures.

Compressive strength is often measured on a universal testing machine. Measurements of compressive strength are affected by the specific test method and conditions of measurement. Compressive strengths are usually reported in relationship to a specific technical standard.

Introduction[edit]

When a specimen of material is loaded in such a way that it extends it is said to be in tension. On the other hand, if the material compresses and shortens it is said to be in compression.

On an atomic level, the molecules or atoms are forced apart when in tension whereas in compression they are forced together. Since atoms in solids always try to find an equilibrium position, and distance between other atoms, forces arise throughout the entire material which oppose both tension or compression. The phenomena prevailing on an atomic level are therefore similar.

The 'strain' is the relative change in length under applied stress; positive strain characterises an object under tension load which tends to lengthen it, and a compressive stress that shortens an object gives negative strain. Tension tends to pull small sideways deflections back into alignment, while compression tends to amplify such deflection into buckling.

Compressive strength is measured on materials, components,^[1] and structures.^[2]

By definition, the ultimate compressive strength of a material is that value of uniaxial compressive stress reached when the material fails completely. The compressive strength is usually obtained experimentally by means of a compressive test. The apparatus used for this experiment is the same as that used in a tensile test. However, rather than applying a uniaxial tensile load, a uniaxial compressive load is applied. As can be imagined, the specimen (usually cylindrical) is shortened as well as spread laterally. A stress–strain curve is plotted by the instrument and would look similar to the following:

The compressive strength of the material would correspond to the stress at the red point shown on the curve. In a compression test, there is a linear region where the material follows Hooke's law. Hence, for this region, ${\displaystyle \sigma =Eϵ}$, where, this time, E refers to the Young's Modulus for compression. In this region, the material deforms elastically and returns to its original length when the stress is removed.

This linear region terminates at what is known as the yield point. Above this point the material behaves plastically and will not return to its original length once the load is removed.

There is a difference between the engineering stress and the true stress. By its basic definition the uniaxial stress is given by:

${\displaystyle \sigma =\frac{F}{A}}$

where,F = Load applied [N],A = Area [m²]

As stated, the area of the specimen varies on compression. In reality therefore the area is some function of the applied load i.e. A = f(F). Indeed, stress is defined as the force divided by the area at the start of the experiment. This is known as the engineering stress and is defined by,

${\displaystyle {\sigma }_e=\frac{F}{A_0}}$

A₀=Original specimen area [m²]

Correspondingly, the engineering strain would be defined by:

${\displaystyle {ϵ}_e=\frac{l-l_0}{l_0}}$

wherel = current specimen length [m] and l₀ = original specimen length [m]

The compressive strength would therefore correspond to the point on the engineering stress strain curve ${\displaystyle ({ϵ}_e^{\ast },{\sigma }_e^{\ast })}$ defined by

${\displaystyle {\sigma }_e^{\ast }=\frac{F^{\ast }}{A_0}}$

${\displaystyle {ϵ}_e^{\ast }=\frac{l^{\ast }-l_0}{l_0}}$

whereF^* = load applied just before crushing and l^* = specimen length just before crushing.

Deviation of engineering stress from true stress[edit]

In engineering design practice, professionals mostly rely on the engineering stress. In reality, the true stress is different from the engineering stress. Hence calculating the compressive strength of a material from the given equations will not yield an accurate result.^{[clarification needed]} This is because the cross sectional area A₀ changes and is some function of load A = φ(F).

The difference in values may therefore be summarized as follows:

• On compression, the specimen will shorten. The material will tend to spread in the lateral direction and hence increase the cross sectional area.
• In a compression test the specimen is clamped at the edges.^[dubious] For this reason, a frictional force arises which will oppose the lateral spread. This means that work has to be done to oppose this frictional force hence increasing the energy consumed during the process. This results in a slightly inaccurate value of stress obtained from the experiment.^{[citation needed]} The frictional force is not constant for the entire cross section of the specimen. It varies from a minimum at the center, away from the clamps, to a maximum at the edges where it is clamped. Due to this, a phenomenon known as barreling occurs where the specimen attains a barrel shape.c

Comparison of compressive and tensile strengths[edit]

Concrete and ceramics typically have much higher compressive strengths than tensile strengths. Composite materials, such as glass fiber epoxy matrix composite, tend to have higher tensile strengths than compressive strengths. Metals are difficult to test to failure in tension vs compression. In compression metals fail from buckling/crumbling/45deg shear which is much different (though higher stresses) than tension which fails from defects or necking down.

Typical values[edit]

Compressive strength of concrete[edit]

For designers, compressive strength is one of the most important engineering properties of concrete. It is a standard industrial practice that the concrete is classified based on grades. This grade is nothing but the Compressive Strength of the concrete cube or cylinder. Cube or Cylinder samples are usually tested under a compression testing machine to obtain the compressive strength of concrete. The test requirements differ country to country based on the design code. As per Indian codes, compressive strength of concrete is defined as

The compressive strength of concrete is given in terms of the characteristic compressive strength of 150 mm size cubes tested after 28 days (fck). In field, compressive strength tests are also conducted at interim duration i.e. after 7 days to verify the anticipated compressive strength expected after 28 days. The same is done to be forewarned of an event of failure and take necessary precautions. The characteristic strength is defined as the strength of the concrete below which not more than 5% of the test results are expected to fall.^[4]

For design purposes, this compressive strength value is restricted by dividing with a factor of safety, whose value depends on the design philosophy used.

See also[edit]

• Schmidt hammer, for measuring compressive strength of materials

References[edit]

1. ^Urbanek, T; Lee, Johnson. 'Column Compression Strength of Tubular Packaging Forms Made of Paper'(PDF). 34, 6. Journal of Testing and Evaluation: 31–40. Retrieved 13 May 2014.Cite journal requires journal= (help)
2. ^Ritter, m A; Oliva (1990), '9, Design of Longitudinal Stress-Laminated Deck Superstructures', Timber Bridges: Design, Construction, Inspection, and Maintenance(PDF), US Dept of Agriculture, Forest Products Laboratory (published 2010), retrieved 13 May 2014
3. ^Kermani, Majid; Farzaneh, Masoud; Gagnon, Robert (2007-09-01). 'Compressive strength of atmospheric ice'. Cold Regions Science and Technology. 49 (3): 195–205. doi:10.1016/j.coldregions.2007.05.003. ISSN0165-232X.
4. ^'Compressive Strength of Concrete & Concrete Cubes What How CivilDigital '. 2016-07-07. Retrieved 2016-09-20.

• Mikell P.Groover, Fundamentals of Modern Manufacturing, John Wiley & Sons, 2002 U.S.A, ISBN0-471-40051-3
• Callister W.D. Jr., Materials Science & Engineering an Introduction, John Wiley & Sons, 2003 U.S.A, ISBN0-471-22471-5