2. Chapter 12
Support Reactions

3. Learning Objectives Introduction Types of Loading
Concentrated or Point Load
Uniformly Distributed Load
Uniformly Varying Load
Methods for the Reactions of a Beam
Analytical Method for the Reactions of a Beam
Graphical Method for the Reactions of a Beam
Construction of Space Diagram
Construction of Vector Diagram
Types of End Supports of Beams
Simply Supported Beams
Overhanging Beams
Roller Supported Beams
Hinged Beams
Beams Subjected to a Moment

4. Learning Objectives Reactions of a Frame or a Truss
Types of End Supports of Frames
Frames with Simply Supported Ends
Frames with One End Hinged (or Pin-jointed) and the Other Supported Freely on Roller
Frames with One End Hinged (or Pin-jointed) and the Other Supported on Rollers and Carrying Horizontal Loads
Frames with One End Hinged (or Pin-jointed) and the Other Supported on Rollers and carrying Inclined Loads
Frames with Both Ends Fixed

5. Introduction
In our day-to-day work, we see that whenever we apply a force on a body, it exerts a reaction, e.g., when a ceiling fan is hung from a girder, it is subjected to the following two forces:
Weight of the fan, acting downwards
Reaction on the girder, acting upwards
A little consideration will show, that as the fan is in equilibrium therefore, the above two forces must be equal and opposite. Similarly, if we consider the equilibrium of a girder supported on the walls, we see that the total weight of the fan and girder is acting through the supports of the girder on the walls. It is thus obvious, that walls must exert equal and upward reactions at the supports to maintain the equilibrium. The upward reactions, offered by the walls, are known as support reactions. As a mater of fact, the support reaction depends upon the type of loading and the support.

6. Concentrated or Point Load
Types of Loading
Though there are many types of loading, yet the following are important
from the subject point of view :
Concentrated or point load
Uniformly distributed load
Uniformly varying load
Concentrated or Point Load
A load, acting at a point on a beam is known as a concentrated or a point load as
shown in Fig
In actual practice, it is not possible to apply a load at a point (i.e., at a
mathematical point), as it must have some contact area. But this area being so
small, in comparison with the length of the beam, is negligible.
Fig Concentrated load.

7. Uniformly Distributed Load
A load, which is spread over a beam, in such a manner that each unit length is loaded to the same extent, is known as uniformly distributed load (briefly written as U.D.L.) as shown in Fig The total uniformly distributed load is assumed to act at the centre of gravity of the load for all sorts of calculations.
Fig Uniformly distributed load
Uniformly Varying Load
A load, which is spread over a beam, in such a manner that its extent varies uniformly on each unit length (say from w1 per unit length at one support to w2 per unit length at the other support) is known as uniformly varying load as shown in Fig

8. Methods for the Reactions of a Beam
Sometimes, the load varies from zero at one support to w at the other. Such a load is also called triangular load.
Fig Uniformly varying load.
Methods for the Reactions of a Beam
The reactions at the two supports of a beam may be found out by any
one of the following two methods:
Analytical Method
Graphical Method

9. Analytical Method for the Reactions of a Beam
Fig Reactions of a beam.
Consider a *simply supported beam AB of span l, subjected to point loads W1, W2 and W3 at distances of a, b and c, respectively from the support A, as shown in Fig. 12.4
  Let RA = Reaction at A, and
RB = Reaction at B.
We know that sum of the clockwise moments due to loads about A
 = W1a + W2b + W3c (i)
and anticlockwise moment due to reaction RB about A
= RB l (ii)
Now equating clockwise moments and anticlockwise moments about A,
RB l = W1 a + W2 b + W3 c ( SM = 0)

10. or RB = W1 a + W 2 b + W3 c ..(iii) L Since the beam is in equilibrium, therefore RA + RB = W1 + W2 + W3 ...( SV = 0) and RA = (W1 + W2 + W3) – RB
Graphical Method for the Reactions of a Beam
It is a systematic, but long method, for finding out the reactions of a beam which is done by the following steps :
Construction of vector diagram.
Construction of space diagram.

11. Construction of Space Diagram
It means to construct the diagram of the beam to a suitable scale. It also includes the loads, carried by the beam along with the lines of action of the reactions. Now name the different loads (or forces) including the two reactions according to Bow’s notations.
Construction of Space Diagram
After drawing the space diagram of the beam, and naming all the loads or forces according to Bow’s notations as shown in the figure. The next step is to construct the vector diagram. A vector diagram is drawn in the following steps :
Fig

12. Select some suitable point p, near the space diagram and draw pq parallel and equal to the load PQ (i.e.,W1) to some scale.
Similarly, through q and r, draw qr and rs parallel and equal to the loads QR and RS (i.e., W2 and W3) to the scale.
Select any suitable point o and join op, oq, or and os as shown in Fig (b).
Now extend the lines of action of the loads and the two reactions in the space diagram.
Select some suitable point p1 on the lines of action of the reaction RA. Through p1 draw p1 p2 parallel to op intersecting the line of action of the load W1 at p2.
Similarly, draw p2 p3, p3 p4 and p4 p5 parallel to oq, or and os respectively.
Join p1 with p5 and through o draw a line ot parallel to this line.
Now the lengths tp and st, in the vector diagram, give the magnitude of the reactions RA and RB respectively to the scale as shown in Figs (a) and (b).

13. Types of End Supports of Beams
Though there are many types of supports, for beams and frames, yet the following three types of supports are important from the subject point of view:
Simply Supported Beams
Roller Supported Beams
Hinged Beams
Simply Supported Beams
It is a theoretical case, in which the end of a beam is simply supported over one of its support.
 In such a case the reaction is always vertical as shown in Fig
Fig Simply supported beam

14. Example
A simply supported beam AB of 6 m span is subjected to loading as shown in Fig Find graphically or otherwise, the support reactions at A and B.

15. Overhanging Beams
A beam having its end portion (or portions) extended in the form of a cantilever, beyond its support, as shown in Fig is known as an overhanging beam. It may be noted that a beam may be overhanging on one of its sides or both the sides. In such cases, the reactions at both the supports will be vertical as shown in the figure.
Fig Overhanging beam.

16. Example
A beam AB of span 3m, overhanging on both sides is loaded as shown in Fig Determine the reactions at the supports A and B.
Fig