Deflection of Beams and Shafts
(Chapter 12)
[Sections to be covered: 12.1, 12.2, 12.3, 12.5, 12.7, 12.9, 12.4 and 12.8]
Elastic curve: The deflection diagram (that assists in visualizing the deformed shape) of the longitudinal axis (that passes through the centroid of each cross-sectional area) of the beams is called the elastic curve.
Assumptions
(i) Cross-sections remain plane and perpendicular to the longitudinal axis of the beam, during elastic deformation. Sections rotate compared to the vertical undeformed plane, but do not distort out-of-plane
(ii) Longitudinal axis, which lies within the neutral surface, does not experience and change in length.
(iii) The beam bends along the longitudinal axis of the beam, in a + ve or - ve manner. Perpendicular to this longitudinal axis, the beam obeys the position effect and deforms in a contrary manner. At locations where the tensile stresses occur, the transverse section contracts, and at locations where the compressive stresses occur, the transverse section expands.
This shows that + ve moment causes a - ve stress on the + ve side of the axis.
When s tends to zero, the curve AC and straight line AC tend to be the same
Since ds = d,
Eqn. (8) becomes,
Summing up vertical forces, i.e.,
When w(x) is a continuous function, equation (16) can be used to solve the problem of deflection of beams. When w(x) is not continuous additional procedures have to be developed to handle the problem.
Macaulay or Singularity Functions
Take the moment at a section, where all the loads acting on the beam are included in the b.m. equation.
are called singularity functions
….., 2, 1, 0, -1, -2, ….
We can write the shear and load equation by,
When the index of the function is less than zero, it does not exist.
i.e., do not exist
When the index of the zero or greater than zero, it exists at and beyond the parameter within the function.
i.e.,
Method of superposition
As long as the deformations in a structure are elastic, and the deformations are small, the deflection due to a series of loads, acting on the structure, can be obtained by summing up the response of the structure to each of those loadings.
(a)
(b)
With the condition that
(c)
Vertical deflection at c,
= vc cantilever+B a
Horizontal deformation at c
= - vhB
Deformation of Given Structures
12.87 :
At B,
Compatibility condition
At B,
(Compatibility condition)
12.117
At B,
Moment-Area Method
It is a semigraphical procedure for finding the slope and deflection at specific points on the elastic curve of the beam.
B/A = Angular (or tangential deviation of
angles) deviation between A & B
This can be stated as: “The angular deviation between B and A is equal to the area under the M/EI diagram between the points A and B”.
Enlarge section between S and T on elastic curve
Tangential deviation of point B on elastic curve from the tangent drawn from A to meet the vertical line through B (at B)
= Vertical deviation of the tangent at a point A on the elastic curve with respect to the tangent drawn from B
tA/B = Moment of the area under the M/EI diagram between A and B, computed about point A.
When tA/B is +ve, point A on the elastic curve is above the point O, the intersection of the vertical line from A to the tangent drawn from B.
When tD/C is – ve, point D on the elastic curve is below point O, the intersection point of the vertical line from D to the tangent drawn from C.
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