STRUCTURAL BEAM DEFLECTION
The problems bending of beams are the most probably occurs rather than other loading in many physical application and mechanical design. In term of Physics and Engineering, bending or deflection is the amount of structural being displaced under loads which referring to some correlated parameters. For example, application such as shaft, bracket, and bridge beam, axles, lever and many structural applications to be examined for assessment and evaluation. The deflection need to be measured for their application purpose.
At this point of view, the deflection aforementioned is directly related to the slope of a member under load and it can be calculated by integrating the slope’s function of a member under the load. The theoretical formula and equation for moment, displacement and slopes has been identified. Equation derivation and application problem solving was clearly justified the methodological approach to the actual problem was really possible and practicable.
To determine and validate the equation for the moment, slope and deflection of the simply-supported beam application, also the slopes at the ends and maximum deflection.
This project aims to:
- Derive formula and apply mathematical integration to determine structural beam deflection under load.
- Validate the theoretical equation with application analyses based on different variable.
- Identify the maximum and minimum deflection.
Application: Supporting structural, frame, ladder, shaft, bracket, and bridge beam, axles and lever.
Figure 1.0: Simply-supported beam with uniform load
A supported beam at both ends has a uniform distributed load W along the beam, as shown in Figure 1.0 above. A beam of length is loaded by uniform load per meter of beam length. Converting distributed load to concentrated load P = WL.
The reaction at both end support is equal to the load applied as the body was in equilibrium condition. The shear force and bending moment also are symmetrically distributed where submission of shear force at both ends is equal to zero and the amount of bending moment at both ends also zero at each end because the beam is simply supported. For this simply supported beam, the deflections are also zero at each end.
The bending moment equation, for 0≤x ≤L, is,
M = WL/2 x – W/2 x^2
To determine the equation for the slope and deflection of the beam, the slopes at the ends, and the maximum deflection. Integrating equation above as indefinite integral gives,
EI dy/(dx ) = ∫▒〖M dx〗 = WL/4 x^2 – W/6 x^3 + C1
E: Modulus of Elasticity
I: Second moment of area
Where is a consonant of integration evaluated from geometric boundary condition. As we know that the slope is zero at the middle of the beam since the beam and loading are symmetric. As the boundary condition of the problem and verify that the slope is zero at the midspan.
EI y = ∬▒〖M dx〗 = WL/12 x^3 – W/24 x^4 + C1 x+C2
Applying boundary condition: y=0 at x=0 and L.
EI y (L) = WL/12 L^3 – W/24 L^4 + C1 L
Solving for C1 yields, C1= – W/24 L^3
Thus for deflection and slope will respectively be,
y = Wx/24EI (2Lx^2 – x^3 – L^3)
∅= dy/dx= W/24EI (6Lx^2-4x^3-L^3 )
Complete agreement comparing the slope at the right and both end,
at (x=0),∅ = – W/24EI L^3
at (x=l),∅ = W/24EI L^3
For the Ymax, substitute x=L/2 gives dy/dx=0, the maximum deflection occur when dy/dx=0. Substituting x=L/2 gives,
y_max = – 5wL/384EI
The approach has been taken to measure and analyze the two different variable which is modulus of elasticity E, Aluminium Alloy and Carbon Steel and second moment of area I, under the same condition of structural and loading parameter.
Figure 1.1: Simply-supported beam application model.
A simply-supported beam at both ends has a uniform distributed load W along the beam, as shown in Figure above. Modulus of elasticity E= GPa Second moment of area, I = (bh^3)/12 = m^4.
i. Aluminium Alloy, E = 71.7 GN/m2 , cross section b=18cm & h= 15cm.
ii. Carbon Steel, E = 207.0 GN/m2 , cross section b=15cm & h= 12cm.
For both materials identify the moment, deflection, slopes and maximum deflection of the beam.
Derivation of mathematical integration for moment, deflection, slopes and maximum deflection has reached the whole agreement for structural under the load. Modulus of elasticity and second moment of area are the variables that roles to the structural performance and strength. Identification of maximum and minimum deflection describes the range of application level under the operations.
CONTRIBUTION OF THE PROJECT
- Beneficiaries for calculation deflection characteristic on structural.
- Help to determine the materials for application purpose.
- Evaluate and analyses on structural performance and strength.
Richard G. Budynas, 2008, Shigley’s Mechanical Engineering Design, Mc Graw Hill, 1221 avenue of the Amaricas, New York.
Serope Kalpakjian, 2006, Manufacturing Engineering and Technology 5th Edition, Pearson prantice hall, Jurong Singapore.
Gere, James M.; Goodno, Barry J. Mechanics of Materials (Eighth ed.). p. 1083-1087. ISBN 978-1-111-57773-5