Bending Moment
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 an external load applied perpendicularly to a longitudinal axis of the element. By this experiment we 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 the perpendicular 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 the measured bending moment for a beam under various loads.
3. KEYWORDS
Bending moment, hogging, sagging, Datum value, under-slung spring, spring balance and Beam, 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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Bending Moment longitudinal axis are called beams. Beams are important structural and mechanical elements in engineering. 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 pinned at 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 the support. In order to do this the support must exert a "fixing" moment M and a reaction R on the beam. A beam which is fixed at one end in this way is called a Cantilever. If both ends 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 end is called simply supported beam
3. Determinate: A structure is statically determinate when the static equilibrium equations are sufficient for determining the internal forces and reactions on that structure. p
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Bending Moment
4. Indeterminate: A structure is statically indeterminate when the static equilibrium equations are insufficient 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 forces acting within the member which is necessary to balance the force acting externally on it. There are 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 or geometric centre of the cross-sectional area. It acts perpendicular to the area and is developed whenever 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 internal stress contribution acting tangent to the member cross section. The resultant of this stress distribution is called the shear force. C. Bending Moment, M W hen
external moment is applied perpendicular to the axis of a member, the internal distribution of stress is directed perpendicular to the member cross-sectional area and varies linearly from a axis passing the member centroid. The resultant of this stress distribution is called the bending moment. The bending moment is caused by the external loads that tend to bend the body about an axis lying within the plane of the area. D. Torsional moment or Torque, T
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Bending Moment
An external torque tends to twist a circular member about its longitudinal axis. It causes an internal distribution of stress that varies linearly when measured in a radial direction. The resultant 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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Bending Moment 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
Bending Moment: W hen
applied loads act along a beam, an internal bending moment which varies from point to point along the axis of the beam is developed. A bending moment is an internal force that is induced in a restrained structural element when external forces are applied. Failure by bending will occur when loading is sufficient to induce a bending stress greater than the yield stress of the material. Bending stress increases proportionally with bending moment. It is possible that failure by shear will occur before this, although while there is a strong relationship between bending 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 all external forces acting to one side of that section. The forces on either side of the section must be equal in order to counter-act each other and maintain a state of equilibrium. For systems allowed 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 the vector distance at which they are applied. W hen analyzing an entire element, it is sensible to calculate moments at both ends of the element, at the beginning, centre and end of any uniformly distributed loads, and directly underneath any point loads. Of course any pin-joints within a structure allow free rotation, and so zero moment occurs at these points as there is no way of transmitting turning forces from one side to the other. If clockwise bending moments are taken as negative, then a negative bending moment within an element will cause sagging (e.g. a closet rod sagging under the weight of clothes on clothes hangers), and a clockwise moment will cause hogging .It is therefore clear that a point of zero bending moment within a beam is a point of contra flexure . W hen
a beam carries loads, complex stresses build up in the material of the beam. bending that results from the loading causes some beam fibers to: y y y
The
carry tension - these are called tensile forces carry compression - these are called compressive forces Take shear forces.
These
all occur simultaneously.
Internal moment of resistance W hen
a beam bends under load, the horizontal fibers will change in length. The top fibers will become shorter and the bottom fibers will become longer. The most extreme top fibers will be under the greatest amount of compression while the most extreme bottom fibers will be under the greatest amount of tension.
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Bending Moment
5-METHODOLOGY A) Single point loads
Formulae: M1= W x X a
M2= W xX(L-a)
B) Multiple point loads
A
C
W1
W2
W3
R A
B
R B X1
X2
150
mm
X3
L =900mm
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Bending Moment
Formulae: Moment can be calculated experimentally as well as theoretically as: Experimental: M= W c X 150mm Or
M= (net force at C) X (150mm)
Theoretical: About A; M1= W 1 x X1
M2= W 2xX2
M3= W 3xX3
M A = M1+ M2+M3 Similarly About B Percentage Error Percentage error = (theoretical ± experimental) X 100% Experimental A. 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 hanger 300 mm from B. 2. Two
parts of the beam have been aligned using the adjustment on the spring balance and noted the initial ³no 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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Bending Moment 6. These
findings have been shown in the table 1.
PART 2:
Part
2
of this experiment comprises of two parts:
Subpart A)
1.
Without
2.
Similarly, repeated step 1 put 10 N weights on first and third hanger and recorded the reading.
3.
altering the load hangers put a 5 N weight on the second hanger and recorded the balance reading.
Subpart B) 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 1 2 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 1st hanger = 100 mm from A nd 2 hanger = 300 mm from A at C rd 3 hanger = 300 mm from B
ŏ distance = 150 mm
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Bending Moment
Theoretical: Sr.No Unit
Load
Distance from point A of
N
W1 mm
W2 mm
Theoretical Bending Moment W3 mm
W1 N mm
W2 N mm
W3 N mm
1.
10
100
300
600
1000
3000
6000
2.
20
100
300
600
2000
6000
12000
Experimental : Table 1: Spring balance readings for bending moment at C Sr.No
Load W
Unit 1. 2. 3.
N 0 10 20
Balance reading
/ Net force for load
at (Net Force = B.R - Datum Value) W1 W2 W3 N /21 / 5 25 / 9
N /27 / 11 39.5 / 23.5
N /22.5 / 6.5 29.5 / 13.5
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Experimental Bending Moments at W1
W2
W3
N mm
N mm
N mm
750
1650 3525
975 2025
1350
Bending Moment
PART 2:
Subpart A) Datum value
=
16 N
ŏ Distance = 150 mm
st
1 hanger = 100 mm from A nd 2 hanger = 300 mm from A at C rd 3 hanger = 300 mm from B
Theoretical: Sr.No
1. 2.
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
0 10
5 5
0 10
100 100
300
600
300
600
0 1000
1500 1500
0 6000
Experimental : Table 2 a Spring balance readings for bending moment at C Sr.No
Loadings Balance Reading
Net Force
W1
W2
W3
Unit
N
N
N
N
N
1. 2.
0 10
5 5
0 10
22
6
33
17
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Bending Moment N mm 900 2550
Bending Moment
Subpart B) Datum value
=
16.5 N
ŏ Distance = 150 mm
st
1 hanger = 100 mm from A nd 2 hanger = 300 mm from A at C rd 3 hanger = 400 mm from B
Theoretical: Sr.No
1. 2.
Loadings
Distance from A of
Theoretical
W1
W2
W3
W1
W2
W3
W1
N
N
N
mm
mm
mm
N mm
5 5
0 10
12
100 100
300
500 500
500 500
2
300
Bending Moment at W2
W3
N mm
N mm
0 3000
6000
1000
Experimental : Table 2 b Spring balance readings for bending moment at C Sr.No
5.
Loadings(N) Balance Reading
W1
W2
W3
Unit
N
N
N
N
1. 2.
5 5
0 10
12
29
2
33
Net Force N 12.5 16.5
Bending Moment N mm 1875 2475
OBSERVATIONS
6. CONCLUSION
After calculating and observing the values and action of shear force it is concluded that:
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Bending Moment 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 actual values:
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 W hile
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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Bending Moment The
internal load generated within a bending element whenever a pure moment is reacted, or a shear load is transferred by beam action from the point of application to distant points of reaction. 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 beam which curves downwards. Datum Value: The ³no 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. The compression gradually decreases from a maximum at the top of the beam until it is zero at the centre. 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 ³Mechanics of Materials´ fourth edition McGraw Hill. 2. http://www.civilcraftstructures.com/civil-subjects/shear-force-and-bending-moment-asstructural-basics 3. http://www.codecogs.com/reference/engineering/materials/shear_force_and_bending_mom 4. http://www.chest of books.com.
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