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

This article discusses Euler-Bernoulli Beam Theory in a Simply Supported Beam without Reinforcement. It covers the stress-strain relations, maximum load, and deflection of a beam. The methodology, results, and discussion are also presented. The subject is relevant to civil engineering and construction courses. The college or university is not mentioned.

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.

E = modulus of elasticity
The deflection is the primary consideration as a factor in beam consruction in which it must be
stronger than the required maximum load.
The calculated values are:
 Load per unit area,repseneted as q is 5 (N/mm)
 Total capacity that can be subjected on beam is 50 (kN)
 Length is 10m converted to 10000mm
 Calculated moment of Inertia denoted I was 337500000 (mm4)
 Young’s modulus of elasticity symbolized as E is 320000 (N/mm2)
 Assuming that the beam does not bend, neutral axis represented as y is 450 (mm)
 The force present on the right and left supports is 25 (kN)
 Maximum Deflection when a load of 50 (kN) is applied at the midpoint of the beam is
6.03 (mm)
Result and Discussion.
A simply supported beam of concrete with a modulus of elasticity of 32MPa has a maximum
loading capacity of 50kN and deflection of 6.03milimeters.
The simple beam theory is also called the Euler–Bernoulli theory which is simple modification
of the linear theory of elasticity. This theory affords the modalities to calculate the deflection and
the maximum force that can be exerted on a beam before it collapses. This theory illustrates the
mechanism in to which the beams deflection relates to the applied load on a beam. This

calculation, involving the deflection of a constant, fixed beam, is used widely in construction and
civil engineering (Eastman et al 2014).
Apart from the ability to determine deflection, the beam equation based on the simple beam
theory defines forces and moments, therefore can be applied to illustrate stresses in the beam.
The bending moment and the shear forces generate stresses in the beam, with maximal effect
along lines of neutral axis of the beam, as long as the cross-sectional area the beam is constant.
The principal maximum stress generated by the stress due to the shear force is in some general
area within the beam. However, since stress concentrations occur at the surfaces, bending
moments stresses are to be considered more than shear stresses in a beam.
With an expression for the strain in deflection of the neutral surface of a beam to share common
features with tensions in a Euler–Bernoulli beam thus to the bend. To acquire the relationship,
assumptions are made that normal to the neutral surface constantly are normal during the
deformation and that the deflections are insignificant. In expression, considering the assumptions
made, a conclusion that the beam bends in an arc of radius Ƿ and that the neutral surface remains
intact during the deformation. Considering the length of segment above the point of neutral
surface to the point of application of the load, bending of the beam creates a curvature that gives
the axial strain in the beam as a function of distance from the neutral surface. However, a
relationship of the radius of the arch to the beam deflection has to be determined (Sanders &
Kelly Sanders, R. and Kelly, D., 20082008).
Taking a point from the non-bending fibers of the beam, the slope made by the this surface to the
perpendicular line of the beam forms an angle encountered in the beam theory. Using this
contemplation, the beam has a strain that may be expressed as a function. Considering the

concrete beam without reinforcement as a consistent isotropic rectilinear supple material, the
stress created relates with the strain. This stress is equated to the Young’s modulus by an
equation that can be a derivative of the relationship between the bending moments and the axial
stress. Since the shear stress varies with the relation with the bending moment, a strain
relationship with the shear force is derived into existence and can be calculated using the Euler–
Bernoulli beam theory.
In a simply supported beam, the boundaries of the beam represent supports are also areas point
loads, moments and distributed loads. These supports are representation of a force on the beam
that is used to position of the beam and reduce mobility and rotaion. These point supports split
the beam into segments that influence the beam equation to form a continuous solution.
External distribution of loads or point loads on a beam are modeled with a Dirac delta function.
In Euler–Bernoulli beams such as this simply supported beam, the shear forces and the bending
moments are determined using the static balance of forces and moments. In extreme cases where
the number of independent equilibrium equations exceed the number of reactions, we have a
statistically indeterminate beam.
Euler–Bernoulli beams theory is established upon the kinematic assumptions to allow it extended
to analyze and explain more advanced three dimensional transverse loading on beams. However,
this theory disregards the effects of the transverse shear strain. It therefore downplays the effects
of deflections and exaggerates natural frequencies. This assumption plays a significant role in
determining shear strain forces in thick beams, hence the simple beam theory is not suitable.

Conclusion
As from the report above, the following are the conclusions:
According to (Volk et al 2014),The response of the strain stress of the young concrete in the
compression varies evidently from the mature concrete. This is to note that the young concrete
are not demonstrating the strain relaxing and they doesn’t have the vibrant strain which would
corresponding to the highest strain. Putting into consideration the associations which were taken
from the mature concrete, the contingent on the strain value which corresponds to the stress
which is the highest , this means that they are not able to predetermine the feedback for the age
of the concerete which is a day in terms of age.
At the point when the solid age is no less than one day, the CSA A23.3 (2014) exact conditions
for the flexible modulus and the modulus of burst precisely measure esteems saw by Khan
(1995).
Regular rearranging approximations embraced for flexural investigation don't matter to solid that
is short of what one day old. Utilizing a versatile split examination to register the yield minute
might be off base in light of the fact that the presumption of a direct flexible reaction in pressure
isn't right. For the related low compressive quality of cements that are short of what one day old,
it is reasonable for the expert to check the greatest c/d constrain in A23.3 (CSA 2014) is fulfilled.
Utilization of the Todeschini and Modified Hognestad connections overestimates the underlying
unbending nature in the rising part existing apart from everything else shape reaction at

exceptionally youthful ages. At the point when the solid age surpasses 14.5 hours, this distinction
winds up irrelevant. The occasion bend reaction is best anticipated by the Modified Hognestad
relationship since it predicts less strain softening than the Todeschini relationship(Succar, B.
2009).
Ascertained redirections are especially delicate to the expected splitting minute when the
connected minute is low or, for gently fortified individuals, where the proportion of the breaking
to-connected minute and related length of the broke locale is touchy to the modulus of crack
accepted. At the point when the connected minute 141 surpasses the breaking minute for the vast
majority of the part length, run of the mill for shafts with higher fortification proportions, the
viable snapshot of dormancy approaches the broke snapshot of inactivity, independent of the
modulus of crack accepted, and the different redirection figuring strategies give a sensible gauge
of the avoidance. When utilizing the suggested decreased moduli of burst, the processed
outcomes are predictable and moderate. Utilizing the full modulus of crack with the Branson
Equation gives the minimum preservationist and slightest precise outcomes.

References.
Engineering ToolBox, (2009). Beams - Supported at Both Ends - Continuous and Point Loads.
[online] Available at: https://www.engineeringtoolbox.com/beam-stress-deflection-d_1312.html
[Accessed Day Mo. Year].
Mancuso, C., and Bartlett, F.M. 2016. Instantaneous Deflections of Concrete Slabs Computed
Using Discretized Analysis, Proceedings of the 5th International Structural Specialty
Conference, STR 855, London, ON, Canada, pp. 10.
Mancuso, C., and Bartlett, F.M. 2014. Built-up Wrought-iron Compression Member with
Missing Stitch Rivets, Electronic Proceedings, 37th IABSE Symposium Report. 37th

International Association for Bridge and Structural Engineering Symposium, Madrid, Spain, pp.
304-305.
Succar, B., 2009. Building information modelling framework: A research and delivery
foundation for industry stakeholders. Automation in construction, 18(3), pp.357-375.
Eastman, C.M., Eastman, C., Teicholz, P. and Sacks, R., 2011. BIM handbook: A guide to
building information modeling for owners, managers, designers, engineers and contractors. John
Wiley & Sons.
Sanders, R. and Kelly, D., 2008. Dealing with risk in scientific software development. IEEE
software, 25(4).
Succar, B., Sher, W. and Williams, A., 2013. An integrated approach to BIM competency
assessment, acquisition and application. Automation in Construction, 35, pp.174-189.
Volk, R., Stengel, J. and Schultmann, F., 2014. Building Information Modeling (BIM) for
existing buildings—Literature review and future needs. Automation in construction, 38, pp.109-
127.
Halfawy, M.R., 2008. Integration of municipal infrastructure asset management processes:
challenges and solutions. Journal of Computing in Civil Engineering, 22(3), pp.216-229.