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The deflection of beam elements is usually calculated on the basis of the Euler–Bernoulli beam equation while that of a plate or shell element is calculated using plate or shell theory. An example of the use of deflection in this context is in building construction. Architects and engineers select materials for various applications.
Euler–Bernoulli beam theory (also known as engineer's beam theory or classical beam theory) [1] is a simplification of the linear theory of elasticity which provides a means of calculating the load-carrying and deflection characteristics of beams. It covers the case corresponding to small deflections of a beam that is subjected to lateral ...
Macaulay's method (the double integration method) is a technique used in structural analysis to determine the deflection of Euler-Bernoulli beams.Use of Macaulay's technique is very convenient for cases of discontinuous and/or discrete loading.
It is a function of the Young's modulus, the second moment of area of the beam cross-section about the axis of interest, length of the beam and beam boundary condition. Bending stiffness of a beam can analytically be derived from the equation of beam deflection when it is applied by a force.
Let the total deflection of the beam due to these loads be . The adjacent figure shows that, for small displacements, the total deflection of the mid-surface of the beam can be expressed as the sum of two deflections, a pure bending deflection w b {\displaystyle w_{b}} and a pure shear deflection w s {\displaystyle w_{s}} , i.e.,
In building construction, the sagging of beams is called "deflection". The amount of deflection varies in accordance with the beam's stiffness, the span between supports, and the load it carries. [6] Sagging is a common problem in older houses. [7]
The beam has an axis of symmetry in the plane of bending. The proportions of the beam are such that it would fail by bending rather than by crushing, wrinkling or sideways buckling. Cross-sections of the beam remain plane during bending. Deflection of a beam deflected symmetrically and principle of superposition
Direct integration is a structural analysis method for measuring internal shear, internal moment, rotation, and deflection of a beam. Positive directions for forces acting on an element. For a beam with an applied weight w ( x ) {\displaystyle w(x)} , taking downward to be positive, the internal shear force is given by taking the negative ...
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