# Determine the deflection of en-castre beams.

Following on from our previous article on deflection of beams with a combined load, we’re going to look at deflection in a different type of beam.

## What is an en-castre beam?

En-castre beams, or fixed beams, have fixed supports at both ends. So, they don’t allow horizontal, vertical or rotational movement of that support.

The most important characteristic of this beam is that the slope and deflection are zero at both ends.

The below image shows an en-castre beam:

Encastre beams can also be built in walls, as you can see in the image below:

En-castre beams have some benefits, including:

• For the same span and load, fixed beams have lower bending moments than simply supported beams.
• Due to their support fixity, smaller deflections are encountered in fixed beams than in simply supported beams.

However, encastre beams also have some disadvantages.  In encastre beams, if the condition of fixity at the ends is disturbed, for example, by settlement and rotational yield of supports, the consequence is that large bending moments will be induced in the beam to counter this effect. Moreover, in the case where there are changes in temperature, this will create large stresses in the section.

## Structural Action of a Fixed Beam

When a fixed beam is subjected to loads, it bends, developing resisting moments and shears. The deflected shape of the beam is shown in this image:

Note that the slope and deflection at the ends A and B are zero, as they are restrained. The beam ‘sags’ for a part of its length in the middle and ‘hogs’ at the ends. This necessarily produces two points of inflection (contraflexure) corresponding to the change in the bending moment (BM).

## Calculating Deflection of En-castre Beams

A standard data table can be used for the determination of deflection, moments and shear force in a fixed beam:

Let’s check out an example and see how we would go about calculating the deflection.  A fixed beam of 6m span is loaded with point loads of 150 kN at 2m from each support.  We’re going to find the maximum deflection.

Take E = 2 x 108 kN/m2 and I = 8 x 108 mm4

Load = 150 kN = 150000 N

Length of beam = 6m

I = 8 × 108 mm4 = 0.0008 m4

E = 2 × 108 kN/m2

For this loading arrangement case, the table gives the maximum deflection as

max = Pl36EI3a24l2-(al)3

Hence, substituting the values gives

= 1500006362 × 10110.0008322462-(26)3

= 0.03375(0.0833 – 0.037037)

max= 0.00156 m

Keep an eye out for future articles on more deflection calculations.

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