Euler-Bernoulli Beam Theory in a Simply Supported Beam without Reinforcement

EULER–BERNOULLI BEAM THEORY IN A SIMPLY SUPPORTED
BEAM WITHOUT REINFORCEMENT
Student University
Student Name
Date

Introduction.
For calculations pertaining the three dimensional stress-straining relations such as maximum
load and maximum deflection of a beam, the cross sectional structure and loading design of the
beam will be reflected. (Succar, B. 2009). The beam concept is applied in the planning and
construction of a varied assortment of engineering projects. The beam supports a maximum force
that exerts a transverse and perpendicular force to the horizontal cross sectional alignment of the
beam. In this practical reflection, a simply supported non-reinforced beam supports the load by
bending slightly, depending on the tensile strength and deflection capabilities of the constructing
material such as steel, concrete or plastic. The force in the beam acting perpendicular to its
longitudinal axis is called the shear force, and the beams ability to withstand shear force is used
to decide whether or not to use the beam in construction. The shear force is equal to the load.
Axial force is the force acting parallel to the horizontal alignment of the beam fibers that form
the axis of the beam.
Such a structural member in an element that spans an opening and is supported in different ways.
Roller support and the pins support are the most common support systems. It is assumed that the
loading, supports and longitudinal cross section are symmetrical and equal respectively on this
beam(Halfawy, M.R. 2008).
According to (Engineering ToolBox (2009).), When the beam is exerted by a force that acts
transversely on it, it bends. The beam forms a deflection curve in the region where its fibers
extend. The axis of the beam is formed at the intersection of the neutral surface and the
longitudinal plane where fibers do not extend or contract due to the applied force.

Methodology.
In this assignment, a simply supported concrete beam without reinforcement was considered.
The beam was longitudinally symmetrical with a width of 300mm, a height of 450mm and a
length of 10meters. The concrete material property is 32 megapascals equal 32N/mm2.
From the loading, there is a reaction force at both ends of the support that can be evaluated using
the equations of equilibrium. The moments and the forces of the beam are determined by
illustrating the free-body diagrams of the segments. There are several segments in the beam, the
right and left of the force capacity. A moment of unknown quantity and shear force V act at the
supported points. A moment with a positive vector and force are present.
From the stability equivalences, the shear force is constant but the amount and quantity of the
moment fluctuates along the beam.
V=p/2, m=p/2(x), (0<x<L/2)
Since the beam is divided into two by load, another opposite force to that initially assumed that
can be calculated using the equations of equilibrium.
V=-p/2, m=p/2(l-x), (l/2<x<l)
The result of the analysis can be illustrated by a shear force illustration and a bending momentum
graph. At the point x, there is a positive shear force equal to the force applied by the maximum
load that can be exerted on the beam.
The beam’s bending Stress = σ = Mc / I

where:
M = Bending Moment
c = Largest Vertical (y) Distance from the Neutral Axis to the top or bottom of the section.
I = Moment of Inertia about the Neutral Axis.
So re-arranging this formula and we get: M = σI / c
σ = stress (Pa (N/m2), N/mm 2 with the stress compliance of concrete without reinforcement
being 32MPa.
The Area Moment of Inertia for a rectangular section is calculated as
Moments of inertia at a specific area = w h3 / 12
where
w = width of the beam
h = height
Maximum deflection:
deflection= 5 q L4 / (384 E I)
where
maximum deflection is represented in millimeters, meters or inches.