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SHEAR FORCE & BENDING MOMENT

A beam may be defined as a structural element which has one dimension considerably larger than the other two dimensions, namely breadth and depth, and is supported at few points. The distance between two adjacent supports is

called span. It is usually loaded normal to its axis. The reactions depend upon the type of supports and type of loading. There are various type of supports, types of beams and types of loading are first explained and then the methods of

finding reactions, bending moment and shear forces are illustrated for the following beams:(a ) Simply supported beams (b ) Cantilever beams and (c ) Overhanging beams

TYPES OF SUPPORTSSimple Support(fig.1): If the beam rests simply on a support it is called a simple support. In such case the reaction at the support is at right angles to the support and the beam is free to move in the direction of its axis and also it is free to rotate about the support.

Roller Support(fig.2): In this case, beam end is supported on rollers. In such cases, reaction is normal to the support since rollers can be treated as frictionless.

Hinged Support(fig.3): At a hinged end, a beam cannot move in any direction. However, it can rotate about the support. where VA = F and HA = 0

Fixed Support (fig.4): At such supports, the beam end is not free to translate or rotate. Translation is prevented by developing support reaction in any required direction. where VA = F , HA = 0 & MA= F×d

TYPES OF BEAMS

Simply Supported Beam(fig.5): When both end of a beam are simply supported it is called simply supported beam

Over-hanging Beam(fig.6): If a beam is projecting beyondthe support. It is called an over-hanging beam

Cantilever Beam (Fig.7) : If a beam is fixed at one end and is free at the other end, it is called cantilever beam

Propped Cantilever (Fig.8) : It is a beam with one end fixed and the other end simply supported.

Both Ends Hinged (Fig.9) : In these beams both ends will be having hinged supports.

Continuous Beam (Fig.10) : A beam is said to be continuous, if it is supported at more than two points.

Fixed Beam (Fig.11) : A beam which is from both ends in the wall.

TYPES OF LOADINGConcentrated Loads (Fig.12) : If a load is acting on a beam over a very small length, it is approximated as acting at the mid point of that length and is represented by an arrow as shown in

Uniformly Distributed Load (UDL) (Fig.13) : Over considerably long distance such load has got uniform intensity. It is represented as shown in Fig. 9.13 ( a ) or as in ( b ). For finding reaction, this load may be assumed as total loadacting at the centre of gravity of the loading (middle of the loaded length). For example, in the beam shown in Fig. 9.13, the given load may be replaced by a 20 × 4 = 80 kN concentrated load acting at a distance 2 m fromthe left support.

Uniformly Varying Load (Fig.14) : The load shown in Fig. varies uniformly from C to D. Its intensity is zero at C and is 20 kN/m at D. In the load diagram, the ordinate represents the load intensity and the abscissa represents the position of load on the beam. Hence the area of the triangle represents the total load and the centroid of the triangle represents the centre of gravity of the load. Thus, total load in this case is

And the centre of gravity of this loading is at from D , i.e., 1 + 3 – 1 = 3 m from A.

Q. 1 The beam AB of span 12 m shown in Fig. is hinged at A and is on rollers at B. Determine the reactions at A and B for the loading shown in the Figure.

Solution:

Q. 2 Find the reactions at supports A and B of the loaded beam shown

Solution:

Q. 3 The cantilever shown in Fig. 9.20 is fixed at A and is free at B. Determine the reactions when it is loaded as shown in the Figure.

Soluton:

“Shear Force at a section in a beam (or any structural member) is the force that is trying to shear off the section and is obtained as algebraic sum of all the forces acting normal to the axis of beam either to the left or to the right of the section”.

SIGN CONVENTIONThe shear force that tends to move left portion upward relative to the right portion shall be called as positive shear force.

“Bending Moment at a section in a beam is the moment that is trying to bend the beam and is obtained as algebraic sum of moment of all the forces about the section, acting either to the left or to the right of the section”.

SIGN CONVENTIONThe bending moment that is trying to sag the beam shall be taken as positive bendingmoment.

The graphical representation of shear force in which ordinate represents shear force and the abscissa represents the position of the section is called Shear Force Diagram ( SFD ).

The diagram in which the ordinate represent bending moment the abscissa represent the position of the section is called Bending Moment Diagram (BMD).

Q. 4 Draw shear force and bending moment diagram for the cantilever beam shown

Solution:

Q. 5 Draw shear force and bending moment diagram for the cantilever beam shown

Solution:

Q. 6 The overhanging beam ABC is supported at A and B, the span AB being 6 m. The overhang BC is 2 m (see Fig.). It carries a uniformly distributed load of 30 kN/m over a length of 3 m from A and concentrated load of 20 kN at free end. Draw SF and BM diagrams.

Solution:

Q. 7 Determine the reactions and construct the shear force and bending moment diagrams for the beam shown in Fig. . Mark the salient points and the values at those points.

Solution: