The first crack that appears
in a beam normally is in correspondence of the region where the bending moment
is maximum and the shear force is small; the cracks are aligned with each other
and, more or less, perpendicular to the flexural stress; they are, therefore,
in mode I condition. As visible in a normal load‐deflection plot, at a certain
point, the behavior from linear becomes non‐linear
A longitudinal reinforced
concrete beam in three point bending. First flexural cracks appear in the
region of maximum bending moment (a) accompanied by nonlinearity in load
deflection response (denoted by an asterisk in (b)). More flexural cracks
appear away from the region of maximum moment under increasing load (c), and a
dominant crack propagates towards load point until ultimate failure by crushing
of compressive concrete (d).
Changing the beam dimensions
or the reinforcement, another kind of collapse could be appreciated that
consists into the formation of a secondary crack which crosses the first
flexural cracks. This mode of failure (due to the combination of shear and
normal stresses) is often sudden and unstable and it is called the diagonal
tension mode.
Secondary crack crossing the flexural
cracks leads to sudden brittle failure.
The sliding
displacement of the inclined crack faces bring into play the aggregate
interlock which gives a contribution to the total shear strength. The
contribution of dowel action, aggregate interlock and bond stress due to slip
are very hard to quantify and even fracture mechanics is not able to describe
the crack propagation in a correct way.
Fiber Bridging
The fiber bridging, like the aggregate one,
depends on many parameters and, for simplicity, an isolated fiber is
investigated along a crack. The fiber contributes to dissipate energy to: (1)
matrix fracture and matrix spalling, (2) fiber‐matrix interface debonding, (3) post‐debonding friction between fiber and matrix
(fiber pull‐ out), (4) fiber fracture and (5) fiber abrasion and plastic deformation
(or yielding) of the fiber.
Steel Fiber pull-out
The mechanical behavior of
FRC depends surely on the amount of fiber (which shows benefits from 1 % until
15 %, for engineered cementitious composites ECC), on the orientation of the
fibers and largely on the pull‐out versus load (or load‐slip) behavior of the
individual fibers. In particular, the pull‐out depends on the type and the
mechanical/geometrical properties of the fibers, on the mechanical properties
of the interface between fiber and matrix, on the angle of inclination of the
fiber with respect to the direction of loading and on the mechanical properties
of the matrix. A large amount of literature covers this subject.
(a)
A schematic illustration of some of the toughening effects and crack front
debonding, the Cook Gordon effect, and debonding and sliding in the crack wake.
(b) Matrix spalling and matrix cracking. (c) Plastic bending (deformation) of
inclined fiber during pull‐out
– both at the crack and at the end‐anchor.
The fiber pull‐out behavior is the gradual
deboning of an interface surrounding the fiber, followed by frictional slip and
pull‐out of fiber.
The bond (responsible of the
forces transmission between fiber and matrix) has different components:
‐
the physical and/or chemical adhesion between fiber and matrix;
‐
the frictional resistance;
‐
the mechanical component (arising from the
fiber geometry, e.g. deformed, crimped or hooked‐end);
‐
the fiber‐to‐fiber interlock.
Different deboning models for fiber pullout
The deboning criterion can be described with two different approaches:
- strength‐based criterion (or stress‐based) where it is assumed that the deboning initiates when the interfacial shear stress exceeds the shear strength.
- Fracture‐based criterion that considers the deboning zone as an interfacial crack together with the evaluation of fracture parameters and energy consideration.
Once deboning has taken
place, stress transfer develops owing to frictional resistance that, in its
turn, can be described, as depicted in, with the following different relationships:
- constant friction
- decaying friction (or slip softening)
- Slip hardening friction.
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