experiment : verify the reactions at the support of a semply sported beam

 EXPERIMENT NO 2

Objective : To verify the reactions at the support of a semply sported beam 

Apparatus: 

1. UTM or Beam apparatus with movable knife frame(bending fixture), 

2. Vernier calipers, dial gauge, and a Tape measure. Calipers should be used to measure the width 

and thickness of the beam. Dial gauge will be used to measure the deflection of the beam. The 

tape measure is used to measure

3. Metal beam. The beam should be fairly rectangular, thin and long .Specific dimensions are dependent to the size of the test frame and available weights

 THEORY

• BEAM 

A beam is a structural member used for bearing loads. It is typically used for resisting vertical loads, shear forces and bending moments.

 

 Types of Beams:

Beams can be classified into many types based on three main criteria. They are as follows:

• Based on geometry:

 Straight beam – Beam with straight profile

 Curved beam – Beam with curved profile

 Tapered beam – Beam with tapered cross section

 Based on the shape of cross section:

i. I-beam – Beam with ‘I’ cross section

ii. T-beam – Beam with ‘T’ cross section

iii. C-beam – Beam with ‘C’ cross section

• Based on equilibrium conditions:

 Statically determinate beam – For a statically determinate beam, equilibrium conditions alone can be used to solve reactions.

 Statically indeterminate beam – For a statically indeterminate beam, equilibrium conditions are not enough to solve reactions. Additional deflections are needed to solve reactions.

• Based on the type of support:

 Simply supported beam

 Cantilever beam

 Overhanging beam

 Continuous beam

 Fixed beam

Classification of beams based on the type of support is discussed in detail below:

• Simply supported beam:

A simply supported beam is a type of beam that has pinned support at one end and roller support at the other end. Depending on the load applied, it undergoes shearing and bending. It is the one of the simplest structural elements in existence.

The following image illustrates a simply supported beam.

Simply Supported Beam (SSB)

 

Cantilever beam:

A cantilever beam is fixed at one end and free at other end. It can be seen in the image below.

Cantilever Beam

 

• Overhanging beam:

A overhanging beam is a beam that has one or both end portions extending beyond its supports. It may have any number of supports. If viewed in a different perspective, it appears as if it is has the features of simply supported beam and cantilever beam.

Overhanging Beam

 

• Continuous beam:

A continuous beam has more than two supports distributed throughout its length. It can be understood well from the image below.

Continuous Beam

 

• Fixed beam:

As the name suggests, fixed beam is a type of beam whose both ends are fixed.

Fixed Beam

 

How many support reaction on the simply supported beam?

There are Primarily 4 types of supports. There reactions are as follows

• Loads 

We all know, every machine in this world is built for the purpose of doing some particular work. For doing any type of work, one or many types of forces are transferred from many parts of the machine. Hence, while designing any particular part of machine, one should know all types of forces/ loads coming or acting on that particular part.

A load can be defined as the combination of one or more forces acting on the body. SI unit of the load is ‘Newton’. There are many ways, we can classify the types of loads; but for calculation of various types of stresses and strains in ‘strength of materials’, the types of loads can be-

 Axial load/ Direct load- When the axis of the body and the line of action of the load coincides with each other then it is called as ‘axial or direct load’. This load always passes through the center of gravity of all cross-sections of the body and acts.

 Axial load can be further classified into-

I)Tensile load- When axial load acting on the body is of pull type and the body is subjected to tension, then it is called as tensile load (Fig. 1). The dimension which is parallel to the axis of the body through which the load is acting tends to increase in such a case.

ii) Compressive load- When axial load acting on the body is of push type and the body is subjected to compression, then it is called as compressive load (Fig. 2). The dimension which is parallel to the axis of the body through which the load is acting tends to decrease in such a case.

 Non-Axial load/ Eccentric load- When the axis of the body and the line of action of the load does not coincide with each other then it is called as ‘non-axial or eccentric load’ (Fig. 3). This load always acts at a distance from the center of gravity of all cross-sections of the body.

 Shear load- When two equal and opposite loads are acting tangentially on the cross-section under consideration of the body, it is called as shear load (Fig. 4). Cutting of sheet metal is a good example of a shear load. The body tends to shear off the section due to shear load.

 Bending load- When two or more loads are acting on the body in such a way that the body tends to bend, then the loads are called as bending loads. These loads can be point/ concentrated; uniformly distributed over a length of the body (udl) or uniformly varying over a length (zero at one end and uniformly increasing to the other end) (Fig. 5).

 Torsional/ twisting load- When two equal loads act like a couple on the body due to which it tends to twist, then the couple is called as twisting/ torsional loads (Fig. 6). Various cross-sections along the body rotate through different angles due to torsion.

 Another classification of loads based on the  magnitude variation is-

1)Static Load- Static load can be defined as a force which does not change its magnitude and direction with respect to time. Example- loads acting on the column of the house.

2) Fluctuating/ dynamic Load- Fluctuating/ dynamic load can be defined as a force which varies in its magnitude and direction with respect to time. Most of the machine components fail due to fluctuating/ dynamic loading.

There are three types of load. These are;

 Point load that is also called as concentrated load.

 Distributed load

 Coupled load

Point Load

Point load is that load which acts over a small distance. Because of concentration over small distance this load can may be considered as acting on a point. Point load is denoted by P and symbol of point load is arrow heading downward (↓).

 Distributed Load

Distributed load is that acts over a considerable length or you can say “over a length which is measurable. Distributed load is measured as per unit length.

Example

If a 10k/ft load is acting on a beam having length 10′. Then it can be read as “ten kips of load is acting per foot”. If it is 10′ then total point load acting is 100Kips over the length.

 Types of Distributed Load

 Distributed load is further divided into two types.

 Uniformly Distributed load (UDL)

 Uniformly Varying load (Non-uniformly distributed load).

• Uniformly Distributed Load (UDL)

Uniformly distributed load is that whose magnitude remains uniform throughout the length. For Example: If 10k/ft load is acting on a beam whose length is 15ft. Then 10k/ft is acting throughout the length of 15ft.

Uniformly distributed load is usually represented by W and is pronounced as intensity of udl over the beam, slab etc.

• Uniform Distributed Load To Point Load

Conversion of uniform distributed load to point load is very simple. By simply multiplying the intensity of udl with its loading length.  The answer will be the point load which can also be pronounced as Equivalent concentrated load (E.C.L). Concentric because converted load will acts at the center of span length.

Mathematically, it can be write as;

Equivalent Concentrated load = udl intensity(W) x Loading length

Uniformly Varying Load (Non – Uniformly Distributed Load)

It is that load whose magnitude varies along the loading length with a constant rate.

Uniformly varying load is further divided into two types;

 Triangular Load

 Trapezoidal Load

Triangular Load

Triangular load is that whose magnitude is zero at one end of span and increases constantly till the 2nd end of the span. As shown in the diagram;

 

Trapezoidal Load

Trapezoidal load is that which is acting on the span length in the form of trapezoid. Trapezoid is generally form with the combination of uniformly distributed load (UDL) and triangular load. As shown in the diagram below;

Coupled Load

Coupled load is that in which two equal and opposite forces acts on the same span. The lines of action of both the forces are parallel to each other but opposite in directions. This type of loading creates a couple load.

Coupled load triy to rotate the span in case one load is slightly more than the 2nd load. If force on one end of beam acts upward then same force will acts downwards on the opposite end of beam.

Coupled load is expressed as kip.m, kg.m, N.m, lb.ft etc.

BENDING MOMENT

Bending Moment is mostly used to find bending stress. It is the internal torque holding a beam together (stopping the left and right halves from rotating - if it was to break in half!)

• Definition of Bending Moment

Bending Moment is the torque that keeps a beam together (anywhere along the beam).

It is found by cutting the beam, then calculating the MOMENT needed to hold the left (or right) half of the beam stationary.

If this is done for the other (left) side you should get the same answer - but opposite direction.

Bending Moment in 19 seconds...

 

http://youtu.be/BiaQEsGk9Ws

In the video above, the wooden plank has been cut through at mid span. The only thing holding it together is the spring loaded hinge. This hinge applies a clockwise moment (torque) to the RHS, and a counter-clockwise moment to the LHS.

 

The hinge is applying a moment to BOTH sides of the beam. This is called Bending Moment. You can't normally see it happening unless the beam breaks, but bending moment is being applied everywhere along the length of the beam.

 

 

Yuri van Gelder: Reuters

The gymnast pushes each arm downwards - hard. He is applying a moment to each arm, turning himself into a "beam" between each ring. The longer the arms the greater the bending moment - which is why the wrist is turned inwards, slightly reducing the length of each arm.

• Positive Bending Moment

 

• Simply supported beam loaded from the top. Positive bending..

This type of bending is common - where the load is pushing down and reactions at the end push upwards. This is called positive bending.

In a more strict sense, positive bending is a sagging beam.

Positive bending is whenever the beam tends to sag downwards. Negative bending bows upwards - called hogging.

 

• Positive Bending Moment: http://examcrazy.com

To calculate the Bending Moment at any location along the beam, we "cut" the beam at that point, then do a moment equation for ONE SIDE of the beam (left or right - whichever is easier). You don't do both sides because the moments balance each other and you will get ZERO. (Because it is equilibrium - of course)

• Shear Force

We can cut the beam to find the Bending Moment (by doing moment equation of ONE side).But what if we don't know where the maximum is? Are we going to pick lots of places to "cut" until we eventually find it?

There is another way. We can use shear force to find bending moment, using a diagram method.

But first - a definition of shear force in a beam.

Imagine ONLY the sliding aspect of the beam, not the bending. This is a bit wierd since beams don't slide apart, they bend apart. But anyway, we still need to get this shear force thing happening.

Imagine the beam as a stack of magnets. They can slide OK but not bend.

 

Now if one hand pushes up and the other down, it slides. (shears)

 

Of course, it doesn't matter which magnet slides, they all want to slide. This one just happened to have the least friction, but in fact every slice has the same shear force.

• Positive Shear Force

In a diagram we would show it like this;

 

When the left hand side (LHS) goes up then this is called positive shear force.

A Shear Force Diagram is a graph of the shear force all the way along a beam.

Procedure

1. To calibrate the two spring as they are not accurate. On each of these spring put dead weight of 0.5,1,1.5,2,3,4,kgf and corresponding to each load take the reading of each of the spring balance and plote the calibration curve as: 

X axse : 1unit = 1k

Y axse : 1unit = 1kg 

                 Calibration curve for RA & RB

2. Place the beam and take the initial reading of both the spring balances, as this has to be deducted from all the readings to take care of mass less beam. 

3. Suspend weight at different points of beam.

4. Note down the spring reading and measure the distance of weight from one end of the beam.

5. Take at least six reading by keeping the weight at different points of beam.

6. Make use of equation (i) & (ii) to calculate RA & RB.

7. From the calibration curve find out actual load for the observed load reading at all the points.

8. Finally find out the % error in RA &RB in between the calculated load and calibration load.

Observation table 

Initial reading of left spring  R1    =      kgf

Initial reading of right spring  R2  =      kgf

Total length L of beam                =      Cm

S.no. W1 W2 L1 L2 RA1 RB2 RA' RB' %error RA %error  RB

1. 2 2 16 86 1.4 1.4 2 2 42.8 42.8

2. 3 3 21 81 2.2 2.2 3 3 36.36 36.36

3. 4 4 26 76 3 3 4 4 33.33 33.33

4. 4 4 31 71 3.2 3.2 4 4 25 25

5. 4.5 4.5 36 66 3.4 3.4 4.5 4.5 32.35 32.35

             For % error = (RA'-RA)×100/RA'

                            & = (RB'-RB)×100/RB'

Experimentally and compare these values with exact velue given by above relation.

Note: The above relation have been derived for mass less beam however if we deduct initial reading for all the subsequent readings the difference will be effect of additional load.