46504823 Bending Moment Experiment

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EXPERIMENT 2B: SHEAR FORCE AND BENDING MOMENT

1. ABSTRACT

Performance-based design approach, demands a thorough understanding of axial forces.Bending characterizes the behavior of a slender structural element subjected to anexternal load applied perpendicularly to a longitudinal axis of the element. By this experimentwe can verify the limit load for the beam of rectangular cross-section under pure bending.Moments at the specific points are calculated by the method of statics or by multiplying theperpendicular force or load and the respective distance of that load from the pivot point.

2. OBJECTIVE

The objective of this experiment is to compare the theoretical internal moment with themeasured bending moment for a beam under various loads.

3. KEYWORDS

Bending moment, hogging, sagging, Datum value, under-slung spring, spring balance andBeam, Neutral axis.

4. THEORY

Bending Moments:

Bending Moment at AA is defined as the algebraic sum of the moments about the section of all

forces acting on either side of the section.

Definition of a Beam:

Members that are slender and support loadings that are applied perpendicular to their


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longitudinal axis are called beams. Beams are important structural and mechanical elements inengineering. Beams are in general, long straight bars having a constant cross-sectional area,often classified as to how they are supported. For example, a simply supported beam is pinnedat one end and roller-supported at the other etc.Types of Beams:

1. Cantilever:

A Built-in support is frequently met. The effect is to fix the direction of the beam at thesupport. In order to do this the support must exert a "fixing" moment M and a reaction Ron the beam. A beam which is fixed at one end in this way is called a Cantilever. If bothends are fixed in this way the reactions are not statically determinate.

2. Simply Supported:

A beam that has hinged connection at one end and roller or pin connection in other endis called simply supported beam

3. Determinate:

A structure is statically determinate when the static equilibrium equations are sufficient fordetermining the internal forces and reactions on that structure.

p


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4. Indeterminate:

A structure is statically indeterminate when the static equilibrium equations areinsufficient for determining the internal forces and reactions on that structure.

Types of Internal Loadings:

The design of a structural member, such as a beam, requires an investigation of the forcesacting within the member which is necessary to balance the force acting externally on it. Thereare generally four types of internal loading that can be resisted by a structural member:

Types of Loadings :A. Normal Force, N

This force acts along the member on longitudinal axis and passes through the centroid orgeometric centre of the cross-sectional area. It acts perpendicular to the area and is developedwhenever the external loads tend to push or pull on the two segments of the body.

B. Shear Force, V

If the external force is applied perpendicular to the axis of a member, it causes an internalstress contribution acting tangent to the member cross section. The resultant of this stressdistribution is called the shear force.

C. Bending Moment, M

When external moment is applied perpendicular to the axis of a member, the internaldistribution of stress is directed perpendicular to the member cross-sectional area and varieslinearly from a axis passing the member centroid. The resultant of this stress distribution iscalled the bending moment. The bending moment is caused by the external loads that tend tobend the body about an axis lying within the plane of the area.

D. Torsional moment or Torque, T


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An external torque tends to twist a circular member about its longitudinal axis. It causes aninternal distribution of stress that varies linearly when measured in a radial direction. Theresultant of this stress distribution is called the torsional moment.

Types of External Loads:

External loads are of two types:

Concentrated Load:

A Concentrated load is one which can be considered to act at a point although of course in

practice it must be distributed over a small area (normally vertical or incline loads). (Unit in kN)


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Distributed Load:

A Distributed load is one which is spread in some manner over the length or a significant

length of the beam. It is usually quoted at a weight per unit length of beam. It may either be

uniform or vary from point to point. (Unit in kN/m)

Convention:

The sign convention depends on the direction of the stress resultant with respect to the

material against which it acts. It is used for both shear force and bending moments in

analyzing the directions. Positive (+ve) bending moments always elongate the lower section of

the beam and negative (-ve) would elongate the mid-section upward of the beam.

Bending moments are considered positive when the moment on the left portion is clockwise

and on the right anticlockwise. This is referred to as a sagging bending moment as it tends to

make the beam concave upwards at AA. A negative bending moment is termed hogging


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Bending Moment:

When applied loads act along a beam, an internal bending moment which varies from point topoint along the axis of the beam is developed. A bending moment is an internal force that isinduced in a restrained structural element when external forces are applied. Failure by bendingwill occur when loading is sufficient to induce a bending stress greater than the yield stress ofthe material. Bending stress increases proportionally with bending moment. It is possible thatfailure by shear will occur before this, although while there is a strong relationship betweenbending moments and shear forces, the mechanics of failure are different.

A bending moment may be defined as; the sum of turning forces about that section of allexternal forces acting to one side of that section. The forces on either side of the section mustbe equal in order to counter-act each other and maintain a state of equilibrium. For systemsallowed to rotate, then the equivalent force would be referred to as torque.

Moments are calculated by multiplying the external vector forces (loads or reactions) by thevector distance at which they are applied. When analyzing an entire element, it is sensible tocalculate moments at both ends of the element, at the beginning, centre and end of anyuniformly distributed loads, and directly underneath any point loads. Of course any pin-jointswithin a structure allow free rotation, and so zero moment occurs at these points as there is noway of transmitting turning forces from one side to the other.If clockwise bending moments are taken as negative, then a negative bending moment withinan element will cause sagging (e.g. a closet rod sagging under the weight of clothes on clotheshangers), and a clockwise moment will cause hogging .It is therefore clear that a point of zerobending moment within a beam is a point of contra flexure.

When a beam carries loads, complex stresses build up in the material of the beam.

Thebending that results from the loading causes some beam fibers to:

y carry tension - these are calledtensileforcesy carry compression - these are calledcompressiveforcesy Takeshearforces.

These all occur simultaneously.

Internal moment of resistance

When a beam bends under load, the horizontal fibers will change in length. The top fibers willbecome shorter and the bottom fibers will become longer. The most extreme top fibers will beunder the greatest amount ofcompressionwhile the most extreme bottom fibers will beunder the greatest amount oftension.


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5-METHODOLOGY

A) Single point loads

Formulae:

M1=Wx X a M2=WxX(L-a)

B) Multiple point loads

A W1 C W2 W3 B

RA RB

X1

X2 150 mm

X3

L =900mm


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Formulae:Moment can be calculated experimentally as well as theoretically as:

Experimental:

M= Wc X 150mm

Or M= (net force at C) X (150mm)

Theoretical:

About A;

M1=W1 x X1 M2=W2xX2 M3=W3xX3

M A= M1+ M2+M3

Similarly About B

Percentage ErrorPercentage error= (theoretical experimental) X 100%

ExperimentalA.PROCEDURE

Bending moment experiment has been divided into two parts:Part 1 Single point load

Part 2 Multiple Point loads

PART 1:

1. First hanger has been positioned 100 mm from point A, second hanger in the groove

just to the right of the section C and the third hanger300 mm from B.

2. Two parts of the beam have been aligned using the adjustment on the spring balance

and noted the initial no load reading.

3. Placed a 10 N weight on the first hanger, realigned the beam and recorded the reading.

4. Similarly, repeated step 3 with second and third hanger.

5. Also, repeated the whole procedure till step 3 using a 20 N weight.


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6. These findings have been shown in the table 1.

PART 2:

Part2

of this experiment comprises of two parts:

Subpart A)

1. Without altering the load hangers put a 5 N weight on the second hanger and recorded

the balance reading.

2. Similarly, repeated step 1 put 10 N weights on first and third hanger and recorded the

reading.

Subpart B)

3. Unloaded the beam, and moved the third hanger to 400 mm from B and after aligning

the beam, recorded the new datum value.

4. For this new load value, placed 10 N on the first and 12 N on the third hanger.

5. Shifted a 10 N load from third hanger to the second hanger and recorded the reading.

6. Findings have been shown in the tables 2a and 2b for subparts A and B respectively.

B.READINGS & CALCULATIONS

PART 1:

Datum Value = 16 N distance = 150 mm1st hanger= 100 mm from A2

nd hanger=300 mm from A at C3

rd hanger=300 mm from B


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Theoretical:

Experimental :

Table 1:Spring balance readings for bending moment at C

Sr.No LoadDistance from point A

ofTheoretical Bending

Moment

Unit NW1 W2 W3 W1 W2 W3

mm mm mm N mm N mm N mm

1. 10 100 300 600 1000 3000 6000

2. 20 100 300 600 2000 6000 12000

Sr.No LoadBalance reading/ Net force for load

at(Net Force = B.R - Datum Value)

Experimental BendingMoments at

W W1 W2 W3 W1 W2 W3

Unit N N N N N mm N mm N mm

1. 0 /- /- /-2. 10 21 / 5 27 / 11 22.5 / 6.5 750 1650 9753. 20 25 / 9 39.5 / 23.5 29.5 / 13.5 1350 3525 2025


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PART 2:

Subpart A)

Datum value = 16 N

Distance = 150 mm1st hanger= 100 mm from A2


rd hanger=300 mm from B

Theoretical:

Experimental :

Table 2 a

Spring balance readings for bending moment at C

Sr.No LoadingsBalance Reading Net Force Bending Moment

W1 W2 W3

Unit N N N N N N mm

1. 0 5 0 22 6 9002. 10 5 10 33 17 2550

Sr.No Loadings Distance from A of Theoretical Bending Moment at

W1 W2 W3 W1 W2 W3 W1 W2 W3

N N N mm mm mm N mm N mm N mm

1. 0 5 0 100 300 600 0 1500 0

2. 10 5 10 100 300 600 1000 1500 6000


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Subpart B)

Datum value = 16.5 N

Distance = 150 mm1st hanger= 100 mm from A2


rd hanger= 400 mm from B

Theoretical:

Experimental :

Table 2 bSpring balance readings for bending moment at C

5. OBSERVATIONS

6. CONCLUSION

After calculating and observing the values and action of shear force it is concluded that:

Sr.No Loadings Distance from A of Theoretical Bending Moment at

W1 W2 W3 W1 W2 W3 W1 W2 W3

N N N mm mm mm N mm N mm N mm

1. 5 0 12 100 300 500 500 0 60002. 5 10 2 100 300 500 500 3000 1000

Sr.No Loadings(N)Balance Reading Net Force Bending Moment

W1 W2 W3

Unit N N N N N N mm

1. 5 0 12 29 12.5 1875

2. 5 10 2 33 16.5 2475


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The bending moment is at maximum when the shear force is zero or changes sign. For every member the internal forces are described by shear force and bending moment.

7. SOURCES OF ERROR

Following were the possible errors which produced a mark difference from the actualvalues:

Making the beam less stable.

Unstable positioning of loads i.e., not placing the loads on the exact middle or on the

marked lines.

Reading error called parallax error.

Possibly the distance between the loads and span was not exactly equal.

Disturbing the load while applying the force.

8. GENERAL PRECAUTIONS

While carrying out this experiment several precautions must be kept in mind so that the

possibility of divergence from the accurate result is minimized.

Avoid parallax error.

Avoid disturbance from the surroundings.

Make sure that the beam is in the balanced position then take the readings.

Make sure that there should not be zero error in the spring balance. If any then subtract

from the final result. Always and every time first measure the datum value.

It is good practice to see the balance level of the beam from a certain distance.

Make sure that in screwing/unscrewing your hand must not disturb the balance level.

Neither put heavy loads first nor over load the beam.

9. DEFINITIONS OF KEYWORDS

Beam:A beam is defined as a structural member designed primarily to support forces acting

perpendicular to the axis of the member.Shear Force:The force parallel or along the cross-section of any member.Span:The length of the beam is called the Span.Bending Moment:


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The internal load generated within a bending element whenever a pure moment is reacted, ora shear load is transferred by beam action from the point of application to distant points ofreaction.Hogging & Sagging:

Hogging and sagging describe the shape of a beam or similar long object when loading is

applied. Hogging describes a beam which curves upwards, and sagging describes a beamwhich curves downwards.Datum Value:The no load value- obtain when only the hangers are suspended having no loads.Spring balance:The vertical spring above the beam used for tensioning/adjustments and load measurement.Neutral axis:From the top fiber of a beam to the central fiber, the fibers are in compression. Thecompression gradually decreases from a maximum at the top of the beam until it is zero at thecentre. The centre is called the neutral axis (N/A). From the neutral axis to the bottom fiber,the fibers are in tension. The tension gradually increases from zero at the centre to a maximum

at the bottom fiber.

10.REFERENCES

1. Beer, Johnston and Dewolf Mechanics of Materials fourth edition McGraw Hill.2. http://www.civilcraftstructures.com/civil-subjects/shear-force-and-bending-moment-as-

structural-basics3. http://www.codecogs.com/reference/engineering/materials/shear_force_and_bending_mom 4. http://www.chest of books.com.

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46504823 Bending Moment Experiment