Shear force and bending moments in Beams Hareesha

Shear force and bending moments in Beams Hareesha N G Lecturer, Dept of Aeronautical Engg DSCE, Blore-78

Types of supports • Beams are usually described by the manner in which they are supported. – Pinned or Hinged support – Roller support – Fixed support Pinned or Hinged support • The essential feature of a pin support is that it prevents translation at the end of a beam but does not prevent rotation. • A pinned support restrict the movement of beam horizontally and vertically but the axis of the beam can rotate in the plane of the figure. • Consequently, a pin support is capable of developing a force reaction with both horizontal and vertical components (HA and RA), but it cannot develop a moment reaction.

Roller support • The roller support prevents translation in the vertical direction but not in the horizontal direction; • Hence this support can resist a vertical force (RB) but not a horizontal force. • Of course, the axis of the beam is free to rotate. • The vertical reactions at roller supports and pin supports may act either upward or downward, and the horizontal reaction at a pin support may act either to the left or to the right.

Fixed support (or clamped support) • At the fixed support, the beam can neither translate nor rotate, whereas at the free end it may do both. • Consequently, both force and moment reactions may exist at the fixed support.

Types of Loads 1. 2. 3. 4. Concentrated load or point load Uniformly distributed load Uniformly Varying load Couple 1. Concentrated load or point load • When a load is applied over a very small area it may be idealized as a concentrated load, which is a single force (P 1 and P 2 in figure) 2. Uniformly distributed load (UDL) • When a load is spread along the axis of a beam, it is represented as a distributed load. • Distributed loads are measured by their intensity, which is expressed in units of force per unit distance (for example, newtons per meter). • A uniformly distributed load, or uniform load, has constant intensity q per unit distance as shown in figure.

3. Uniformly Varying load(UVL) • A varying load has an intensity that changes with distance along the axis; for instance, the linearly varying load of Fig. has an intensity that varies linearly from q 1 to q 2. 4. Couple • Another kind of load is a couple, illustrated by the couple of moment M 1 acting on the overhanging beam (Fig. c).

Types of Beams – Simply supported beam – Cantilever beam – Over hanging beam Simply supported beam • For instance, a beam with a pin support at one end a roller support at the other is called a simply supported beam or a simple beam. • Thus, end A of the beam of Fig. a. cannot move horizontally or vertically but the axis of the beam can rotate in the plane of the figure. • Consequently, a pin support is capable of • • developing a force reaction with both horizontal and vertical components (HA and RA), but it cannot develop a moment reaction. At end B of the beam (Fig. a) the roller support prevents translation in the vertical direction but not in the horizontal direction; Hence this support can resist a vertical force (RB) but not a horizontal force. Of course, the axis of the beam is free to rotate at B just as it is at A. The vertical reactions at roller supports and pin supports may act either upward or downward, and the horizontal reaction at a pin support may act either to the left or to the right

Cantilever beam The beam shown in Fig. b, which is fixed at one end and free at the other, is called a cantilever beam. • At the fixed support (or clamped support) the beam can neither translate nor rotate, whereas at the free end it may do both. • Consequently, both force and moment reactions may exist at the fixed support. •

Over hanging beam • This beam is simply supported at points A and B (that is, it has a pin support at A and a roller support at B) but it also projects beyond the support at B. • The overhanging segment BC is similar to a cantilever beam except that the beam axis may rotate at point B

Shear force and bending Moment • The force which is trying to shear off the section is called shear force. • Shear Force at a section in a beam is the force that is trying to shear off the section and is obtained as the algebraic sum of all the forces including the reactions acting normal to the axis of the beam either to the left or to the right of the section. • The moment which bends the section is called bending moment. • Bending moment at a Section in a beam is the moment that is trying to bend it and is obtained as the algebraic sum of the moments about the section of all the forces including the reaction acting on the beam either to the left or to the right of the section

Sign conventions • • • Clock wise (CW) moment is taken as Negative Counter clock wise (CCW) moment in taken as positive A positive shear force acts clockwise against the material A negative shear force acts counterclockwise against the material. A positive bending moment compresses the upper part of the beam A negative bending moment compresses the lower part.

• Reactions: • • • What are reactions? How do they occur? Why do they occur? How to find? Types of reactions? H, V, Couple • Illustrate with example. P 2 P 1 RA RB 100 100