University of Khartoum
Faculty of Engineering
Petroleum and Natural Gas Department
Rocks Mechanics
Research about:
Modulus of Rupture
of Rock
Supervisor:
Dr.
BY:
Jood Hashim Mohamed Sultan
INDEX: 178012
July 2021.
Modulus of Rupture of Rock
Definition:
Modulus of rupture, also known as Flexural strength, or bend strength, or transverse
rupture.
Strength is a material property, defined as the stress in a material just before it yields in
a flexure test. The transverse bending test is most frequently employed, in which a
specimen having either a circular or rectangular cross-section is bent until fracture or
yielding using a three point flexural test technique. The flexural strength represents the
highest stress experienced within.
In mechanics, the flexural modulus or bending modulus is a powerful material that is
calculated as a measure of stress pressure on the flexural flexion or inclination of an
object that resists bending.
Flexural Modulus is determined from the slope of the pressure curve formed by the
flexural test (such as ASTM D790) and uses power units in each position.
Ideally, the flexibility or bending of the stiffness mode is similar to the tensile modulus
(Young’s modulus) or the compression modulus of elasticity. In fact, these values may
vary, especially in polymers which are usually viscoelastic (time-dependent) materials.
The alignment of the flexural modulus with Young’s modulus also takes the
corresponding model of pressure and stiffness as the bend models have strong and
compressive pressures. Polymers in particular have different types of compression and
stiffness of the same material.
Flexural modulus of rupture is about 10% to 20% of the compressive strength
depending on the type, size, and volume of the awesome aggregate used in the concrete
block.
Fig.1- The flexural strength is stress at failure in bending. It is equal or slightly larger than the failure
stress in tension
Modulus of Rupture Unit:
The modulus of rupture is a measurement of pressure or force per unit area. Scientists and
engineers use an array of magnitudes for pressure in determining the modulus of rupture.
You can find it expressed in units of Pascal or mega Pascal as well as pounds per square
inch, or psi.
Three-Point Bend Test:
Engineers use a three-point bend test in determining the bond strength, or modulus of
rupture, of a material. In contrast to the center point loading test, this method uses two
different forces along the material of the beam that divide it into three equal parts.
As the applied forces bend the material, be it rock, cement or any other substance, they
keep track of temperature and how particles in the material distribute themselves in
response to the stress. They do this to make sure the material can withstand the pressures
in applications such as foundations for buildings or other projects.
As engineers create graphs of how the material displaces itself in response to different
amounts of force, they study how materials undergo deformation. They can then calculate
Young's modulus and modulus of rupture.
Fig.2- Test fixture on universal testing machine for three-point flex test
Testing method:
The test method for conducting the test usually involves a specified test fixture on a
universal testing machine. Details of the test preparation, conditioning, and conduct
affect the test results.
The sample is placed on two supporting pins a set distance apart .
Modulus of Rupture Formula:
The breakdown Modulus of Rupture Formula varies with a different type of system
loading.
Three Point Bend Test:
#1. First system: a rectangular sample under a load on a three-point bend setup.
3 FL
σ=
2b d2
Here,
𝞼= Modulus of Rupture.
F = load (force) at the cracked point (N).
L= Total length of the support span.
b = Total width of the support span.
d = Total thickness of the support span.
Fig.3- three-point bend test.
Four Point Bend Test:
Fig.4- four-point bend test.
#2. Second system: a rectangle sample under a load on a four-point bend setup, where
the loading span is one third the length of the support span.
FL
σ= 2
bd
Here,
𝞼= Modulus of Rupture.
F = load (force) at the cracked point (N).
L= Total length of the support (outer) span.
b = Total width of the support (outer) span.
d = Total thickness of the support (outer) span.
#3. Third system: a rectangle sample under a load on a four point bend setup, where
the loading span is half the length of the support span.
3 FL
σ=
4 b d2
Here,
𝞼= Modulus of Rupture.
F = load (force) at the cracked point (N)
L = Total length of the support span.
b = Total width of the support span.
d = Total thickness of the support span.
#4. Last system: a rectangle sample under a load on a four point bend setup, where the
loading span is neither one third nor half the support span. Fig. 5.
3 F ( L−Li )
σ=
2 b d2
Here,
𝞼= Modulus of Rupture.
F = load (force) at the cracked point (N).
L= Total length of the support (outer) span.
Li = Total length of the loading (inner) span.
b = Total width of the support (outer) span.
d = Total thickness of the support (outer) span. Fig. 5.
Modulus of Rupture Formula for a circular cross section:
In this case, the load is the external force put on the material of interest. The load force
is applied to the center of a beam of the material elevated slightly above ground. From
this experimental setup, known as the center point loading test, the deformation of the
material in response to stress applied to it. Fig. 6.
16 FL
σ=
3 π d3
𝞼= Modulus of Rupture.
F=¿Load at a given point on the load deflection curve.
L= Length of specimen.
d = The diameter of the specimen.
Fig. 6 - Circular cross section specimen
Standards:
ASTM D790: Standard test methods for flexural properties of unreinforced and
reinforced plastics and electrical insulating materials.
References:
- Bower, A. F. (2009). Applied mechanics of solids. CRC Press.
- Zweben, C., W. S. Smith, and M. W. Wardle (1979), "Test methods for fiber tensile
strength, composite flexural modulus, and properties of fabric-reinforced laminates",
Composite Materials: Testing and Design (Fifth Conference), ASTM International.
- William D. Callister, Jr., Materials Science and Engineering, Hoken: John Wiley &
Sons, Inc., 2003.
Websites:
- https://en.wikipedia.org
- https://slideplayer.com
- https://sciencing.com
- https://sciendirect.com
