PRESENTED BY: Arpita Patel(130450131028) Patel priya(130450131039) Branch: computer science 3rd Semester 2nd Year (2014- 2015)

Introduction Calculation of deflections is an important part of structural analysis Excessive beam deflection can be seen as a mode of failure. Extensive glass breakage in tall buildings can be attributed to excessive deflections Large deflections in buildings are unsightly (and unnerving) and can cause cracks in ceilings and walls. Deflections are limited to prevent undesirable vibrations

Beam Deflection Bending changes the initially straight longitudinal axis of the beam into a curve that is called the Deflection Curve or Elastic Curve

Beam Deflection Because the y axis is positive upward, the deflections are also positive when upward. Traditional symbols for displacement in the x, y, and z directions are u, v, and w respectively.

Beam Deflection

Basic concept of elastic beam The beam lies on elastic foundation when under the applied external loads , the reaction forces of the foundation are proportional at every point to the deflection of the beam at this point. This assumption was introduced first by Winkler in 1867.

Consider a straight beam supported along its entire length by an elastic medium and in the subjected to vertical forces acting plane of symmetry of the cross section.

Elastic-Beam Theory Consider a differential element of a beam subjected to pure bending. The radius of curvature  is measured from the center of curvature to the neutral axis Since the NA is unstretched, the dx=d

Elastic-Beam Theory The fibers below the NA are lengthened The unit strain in these fibers is:

Elastic-Beam Theory Below the NA the strain is positive and above the NA the strain is negative for positive bending moments. Applying Hooke’s law and the Flexure formula, we obtain: The Moment curvature equation

The product EI is referred to as the flexural rigidity. Since dx = ρdθ, then In most calculus books

The Double Integration Method Once M is expressed as a function of position x, then successive integrations of the previous equations will yield the beams slope and the equation of the elastic curve, respectively. Wherever there is a discontinuity in the loading on a beam or where there is a support, there will be a discontinuity. Consider a beam with several applied loads. The beam has four intervals, AB, BC, CD, DE Four separate functions for Shear and Moment

The Double Integration Method Relate Moments to Deflections

The Euler–Bernoulli equation describes the relationship between the beam's deflection and the applied load

The curve describes the deflection of the beam in the direction at some position (recall that the beam is modeled as a one-dimensional object). is a distributed load, in other words a force per unit length (analogous to pressure being a force per area); it may be a function of , , or other variables.

Note that is the elastic modulus and that is the second moment of area of the beam's cross-section. must be calculated with respect to the axis which passes through the centroid of the cross-section and which is perpendicular to the applied loading.[N 1] Explicitly, for a beam whose axis is oriented along x with a loading along z, the beam's cross-section is in the yz plane, and the relevant second moment of area is where it is assumed that the centroid of the cross-section occurs at y = z = 0.

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