4.1 |
Introduction |
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When a structure is subjected to the action of applied loads each member undergoes deformation due to which the axis of structure is deflected from its original position. The deflections also occur due to temperature variations and lack-of-fit of members. The deflections of structures are important for ensuring that the designed structure is not excessively flexible. The large deformations in the structures can cause damage or cracking of non-structural elements. The deflection in beams is dependent on the acting bending moments and its flexural stiffness. The computation of deflections in structures is also required for solving the statically indeterminate structures. |
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In this chapter, several methods for computing deflection of structures are considered. |
4.2 |
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The moment-area method is one of the most effective methods for obtaining the bending displacement in beams and frames. In this method, the area of the bending moment diagrams is utilized for computing the slope and or deflections at particular points along the axis of the beam or frame. Two theorems known as the moment area theorems are utilized for calculation of the deflection. One theorem is used to calculate the change in the slope between two points on the elastic curve. The other theorem is used to compute the vertical distance (called tangential deviation) between a point on the elastic curve and a line tangent to the elastic curve at a second point. |
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Consider Figure 4.1 showing the elastic curve of a loaded simple beam. On the elastic curve tangents are drawn on points A and B . Total angle between the two tangents is denoted as
 . In order to find out
 , consider the incremental change in angle  over an infinitesimal segment
 located at a distance of
 from point B . The radius of curvature and bending moment for any section of the beam is given by the usual bending equation.
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(4.1) |
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where R is the radius of curvature; E is the modulus of elasticity; I is the moment of inertia; and M denotes the bending moment. |
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The elementary length  and the change in angle
 are related as, |
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(4.2) |
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