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Bernoulli Beam Theory In Beam Reinforcement

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Question:

Discuss about the Bernoulli Beam Theory in Beam Reinforcement.
 
 

Answer:

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.

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 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 ? 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

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.

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