Shear Force in a Beam (Edited)1

 BFC 2091 2091 Structure Structure Lab – Shear Force In A Beam

TITLE: SHEAR FORCE IN A BEAM  ________  ___________ ________ _________ ________ ________ ________ _______ ________ _________ ________ ________ ________ _______ ________ _________ ________  ____  1.0

OBJECTIVE

1.1

To exa examin minee how how shea shearr forc forcee vari varies es wit with h an incr increa easi sing ng poi point nt load load..

1.2

To exam examin inee how she shear ar for force vari varies es at the the cut cut posit positio ion n of the beam beam for  for  various loading conditions.

2.0

LEARNING OUTCOME

2.1

The app appli licat catio ion n of engi enginee neeri ring ng know knowle ledge dge in in prac practi tical cal app appli lica cati tion. on.

2.2 2.2

To enha enhanc ncee techn echnic ical al comp compet eten ency cy in str structu uctura rall engin ngineeerin ering g throu hroug gh laboratory application.

2.3 2.3

To com commu muni nica catte eff effec ecti tive velly in in gr group. oup.

2.4

To ident identif ify y probl problem em,, solvin solving g and findi finding ng out out appro appropr priat iatee solut solution ion thro throug ugh h laboratory application.

3.0

INTRODUCTION AND THEORY

A beam is a horizontal structural element that is capable of  withstanding load primarily by resisting bending The bending force induced into the material of the beam as a result of the external loads, own weight, span and external reactions to these loads is called a bending moment moment . If the ends of a beam are restrained longitudinally by its support or if a beam is a component of a continuous frame, axial force may also develop. If the axial force is small, the typical situation for most beams can be neglected when the member is designed. In the case, of reinforced concrete beams, small values of axial compression actually produce a modest increase (on the order of 5 to 10 percent) in the flexural strength of the member.

To design a beam, the engineer must construct the shear and moment curves to determine the location and magnitude of the maximum values of these forces.

Wong Siew Hung AF040176 

 BFC 2091 2091 Structure Structure Lab – Shear Force In A Beam

Except for short, heavily loaded beams whose dimensions are controlled by shear  requirements, the proportion of the cross section are determined by the magnitude of the maximum moment in the span. After a section is sized at the point of  maximum moment, the design is completed by verifying that the shear stresses at the point of maximum shear usually adjacent to a support are equal to or less than the allowable shear strength of the material. Finally, the deflection produced by service loads must be checked to ensure that the member has adequate stiffness. Limits on deflection are set by structural codes.

To provide this information graphically, we construct shear and moment curves. curves. These curves, which preferably preferably should be drawn to scale, consist of values of shear and moment plotted as ordinates against distance along the axis of the  beam. Although we can construct construct shear and moment curves by cutting cutting free bodies at inter interval valss along along the axis axis of a beam beam and writ writing ing equat equation ion of equil equilib ibri rium um to establish the values of shear and moment at particular section, it is much simpler to construct these curves from the basic relationships that exist between load, shear  and moment.

Bending moment at any section of a beam is defined to be the algebraic sum of the moment at the sectioning developed by vertical components of external forces applied on the beam by considering the left or the right of assumed section, or unbalanced moment at the sectioning, to the left or the right of the assumed section. Variation of bending moment along beam can be visualized by Bending Moment Diagram (BMD), which is defined as a diagram that shows variations of   bending moment moment along the beam considered. considered. The final step step in the design of a beam is to verify that it does not deflect excessively. Beams that are excessively flexible undergo undergo large large deflecti deflections ons that can damage damage attache attached d nonstru nonstructur ctural al constru constructi ction: on:  plaster,  plaster, ceiling, ceiling, masonry masonry walls, walls, and rigid rigid piping piping for example example may crack. crack. Since most beams are span short distances, say up to 30 or 40 ft, are manufactured with a constant cross sections, to minimize cost, they have excess flexural flexural capacity at all sections except the one at which maximum moment moment occurs.

Wong Siew Hung AF040176 

 BFC 2091 Structure Lab – Shear Force In A Beam

Beams are typically classified by the manner in which they are supported. A beam supported by a pin at the one end and a roller at the other end is called a simply supported beam. If the end of the simply supported beam extends over a support, it is referred to as a beam with an overhang.

A cantilever beam is fixed at the one end against translation and rotation. Beams are supported by several intermediate support are called continuous beam. If   both ends of a beam are fixed by the support, the beam is termed fixed ended. Fixed ended beams are not commonly constructed in practice, but the values of end moments in them produced by various types of load are used extensively as the starting point in several methods of analysis for indeterminate structures.

Fig. 1 : Shear Force and Bending Moment

Wong Siew Hung AF040176 

 BFC 2091 Structure Lab – Shear Force In A Beam

Fig. 2 : Change of Shape due to Shear Force

There are a number of assumptions that were made in order to develop the  Elastic Theory of Bending . These are: 3.1 The beam has a constant, prismatic cross-section and is constructed of a flexible, homogenous material that has the same Modulus of Elasticity in  both tension and compression (shortens or elongates equally for same stress). 3.2 The material is linearly elastic; the relationship between the stress and strain are directly proportional. 3.3 The beam material is not stressed past its proportional limit. 3.4 A plane section within the beam before bending remains a plane after bending (see AB & CD in the image below). 3.5 The neutral plane of a beam is a plane whose length is unchanged by the beam's deformation. This plane passes through the centroid of the cross-section. Theory

 Part 1

Wong Siew Hung AF040176 

 BFC 2091 Structure Lab – Shear Force In A Beam

W

a R A

‘cut’ L

R B

Figure 1

Shear force at left of the section, Sc = W ( L-a ) …………..equation 1 L Shear force at the right of the cut section, Sc = -Wa …………equation 2 L

 Part 2

Use this statement : “The shear force at the ‘cut’ is equal to the algebraic sum of the force acting to the left or right of the cut” 

4.0

APPARATUS

Wong Siew Hung AF040176 

 BFC 2091 Structure Lab – Shear Force In A Beam

Figure 1 : Measuring Force Machine

Figure 2 : Load

Figure 3 : Data Analysis (Group members)

5.0

PROCEDURE 5.1

 Part 1

Wong Siew Hung AF040176 

 BFC 2091 Structure Lab – Shear Force In A Beam

5.1.1

Check the Digital Force Display meter reads zero with no load.

5.1.2

Place a hanger with a 100g mass to the left of the ‘cut’.

5.1.3

Record the Digital Force Display reading in Table 1. Repeat using any masses between 200g and 500g. Convert the mass into a load in  Newton (multiply by 9.81). Shear Force at the cut (N) = Displayed Force.

5.1.4

Calculate the theoretical Shear Force at the cut and complete the Table 1.

5.2

 Part 2

5.2.1

Check the Digital Force Display meter zero with no load.

5.2.2

Carefully load the beam with the hangers in any positions and loads as example in Figure 2, Figure 3 and Figure 4 and complete Table 2.

5.2.3

Record the Digital Force Display reading where : Shear Force at the cut (N) = Displayed Force.

5.2.4

Calculate the support reaction (R A and R B) and calculated the theoretical Shear Force at the cut.

140mm

R A

‘cut’

R B

Wong Siew Hung AF040176 

 BFC 2091 Structure Lab – Shear Force In A Beam

W1 = 200g (1.96N)

Figure 2

R A

220mm

W1

W2

‘cut’

R B

260mm

Where ; W1 & W2 any load between 100g to 500g

Figure 3

R A

220mm

W1

‘cut’

R B

W2 400mm Where ; W1 & W2 any load between 100g to 500g

Figure 4

6.0

Mass *(g)

RESULT

Load (N)

Force (N)

Theoretical Shear Force Experimental Shear Force

(N)

(N)

Wong Siew Hung AF040176 

 BFC 2091 Structure Lab – Shear Force In A Beam

0 100 200 300 400 500

0 0.981 1.962 2.943 3.924 4.905

0 0.6 1.2 1.8 2.3 2.8

0 0.6 1.2 1.8 2.3 2.8

0 0.401 0.803 1.204 1.605 2.01

* Use any mass between 200g to 500g  Table 1

No

2 3 4

Mass 1

Mass2

W1

W2

Force

(g)

(g)

(N)

(N)

(N)

200 200 200

0 300 300

1.962 1.962 1.962

0 2.943 2.943

Theoretical

Experimental

Shear Force

R A (N)

RB (N)  

(N)

- 0.50 2.60 0.70

Shear Force (Nm)

- 0.50 2.60 0.70

2.586 2.185 1.248

- 0.624 2.720 3.657

- 0.624 2.720 0.713

Table 2

7.0

DATA ANALYSIS

7.1

For Table 1 (Part 1)

 From Figure 1; W

a

‘cut’ L Wong Siew Hung AF040176 

 BFC 2091 Structure Lab – Shear Force In A Beam

R A

R B

For :Mass, g =100

Load, N =100 x 9.81 / 1000 = 0.981 N Force , N = 0.6 N Experimental shear force , N = displayed forced (shear force at a cut , N ) = 0.6 Theoretical shear force N, Sc = W (L-a) / L = 0.981 x (0.44 – 0.26) / 0.44

= 0.401 N For ; Mass, g = 200

Load, N = 200 x 9.81 / 1000

= 1.962 N

Force, N = 1.2 N Experimental Shear Force, N

= Displayed Force

(Shear Force at a cut, N)

= 1.2 N

Theoretical Shear Force, N, Sc = W (L – a) / L = 1.962 x (0.44 – 0.26) / 0.44 = 0.803 N

For ; Mass, g = 300

Load, N = 300 x 9.81 / 1000

= 2.943 N

Force, N = 1.8 N Experimental Shear Force, N

= Displayed Force

(Shear Force at a cut, N)

= 1.8 N

Theoretical Shear Force, N, Sc = W (L – a) / L = 2.943 x (0.44 – 0.26) / 0.44

Wong Siew Hung AF040176 

 BFC 2091 Structure Lab – Shear Force In A Beam

= 1.204 N

For ; Mass, g = 400

Load, N = 400 x 9.81 / 1000

= 3.924 N

Force, N = 2.3 N Experimental Shear Force, N

= Displayed Force

(Shear Force at a cut, N)

= 2.3 N

Theoretical Shear Force, N, Sc = W (L – a) / L = 3.924 x (0.44 – 0.26) / 0.44 = 1.605 N

Fore ; Mass, g = 500

Load N = 500 x 9.81 / 1000 = 4.905 N Force , N = 2.8 N Experimental shear force , N = displayed force (shear force at a cut , N ) = 2.8 N Theoretical shear force , N, Sc = W (L-a) /L = 4.905x (0.44-0.26 )/0.44 = 2.01 N

7.2

For Table 2 (Part 2)

 From Figure 2;

140mm

R A

‘cut’

R B

W1 = 200g (1.962N)

Wong Siew Hung AF040176 

 BFC 2091 Structure Lab – Shear Force In A Beam

Force, N = - 0.9N Experimental Shear Force, N

= Displayed Force

(Shear Force at a cut, N)

= - 0.9 N

∑M = 0, ∑Fx = 0, ∑Fy = 0 ∑MB = 0 ;

-2943 (0.58) + R A (0.44) = 0

R A = 3879.41N

∑Fx = 0, ∑Fy = 0

; RB  + 3879.41 –2943= 0

R B = -936.41 N

Theoretical Shear Force, N

= - Wa/ L = - (2943) x (0.14) / 0.44 = - 0.936 N

 From Figure 3 ;

R A

220mm

W1

W2

‘cut’

R B

260mm

Wong Siew Hung AF040176 

 BFC 2091 Structure Lab – Shear Force In A Beam

Force, N = 2.0 N Experimental Shear Force, N

= Displayed Force

(Shear Force at a cut, N)

= 2.0 N

∑M = 0, ∑Fx = 0, ∑Fy = 0 ∑MA = 0 ;

R B (0.44) – 2.943(0.26) – 1.962(0.22) = 0 R B = 1.197 / 0.44 R B = 2.720 N

∑Fx = 0, ∑Fy = 0

; RA  – 1.962 – 2.943 + 2.720 = 0 R A = 1.962 – 2.943 – 2.720 R A = 2.185 N

Theoretical Shear Force,  N

(

= −

−

(

= −

W 1 a  L

)

−

(

−

W 2 a  L

1.962 x0.22 −

0.44

)

−

) (

2.943 x0.26 −

0.44

)

= 0.981 + 1.739 = 2.720 N

 From Figure 4 ;

R A 240mm

W1

‘cut’ W2

RB  

400mm

Wong Siew Hung AF040176 

 BFC 2091 Structure Lab – Shear Force In A Beam

Force, N = 0.70 N Experimental Shear Force, N

= Displayed Force

(Shear Force at a cut, N)

= 0.70 N

∑M = 0, ∑Fx = 0, ∑Fy = 0 ∑MB = 0 ;

-1.962 (0.22) – 2.943(0.04) + R A (0.44) = 0 R A = 0.549 / 0.44 R A = 1.248 N

∑Fx = 0, ∑Fy = 0

; RB  + 1.248 – 1.962 – 2.943 = 0 R B = 1.962 + 2.943 – 1.248 R B = 3.657 N

Theoretical Shear Force,  N

=

(

=

[

−

(

W 2  L  L

−

a

]

)

−

2.943[ 0.44 −

(

−

−

W 1 a  L

0.4]

0.44

)

)

−

(

1.962 x 0.22 −

0.44

)

= - 0.268 – (-0.981) = 0.713 N

8.0

8.1

DISCUSSION

Part 1 8.11

Derive equation 1

 From Figure 1; W

a

‘cut’

Wong Siew Hung AF040176 

 BFC 2091 Structure Lab – Shear Force In A Beam

L

R A

R B

Let ; ∑MB = 0 ( R A x L ) – W ( L –a ) = 0 R A = W ( L –a ) L Since the force at the cut is equal to the algebraic sum of the force acting to the left or right of the cut; Therefore, SC = R A Sc = W ( L –a ) L

Let ; ∑MA = 0 ( -R B x L ) – ( W x a ) = 0 R B = ( - W x a ) L Therefore ;

SC = ( - W x a ) L

Where,

W = Load a = Cut section from R A L = Length from R A to R B

This equation is used to determine the value of Shear Force by theory. W is a load place upon the ‘cut’ section with the length of a. L is total length from R A to R B.

Wong Siew Hung AF040176 

 BFC 2091 Structure Lab – Shear Force In A Beam

8.12

Plot a graph, which compare your experimental result to those you calculated using theory.

 Please see graph 1, as attached.

8.13

Comment on the shape of the graph. What does it tell you about how Shear Force varies due to an increased load?

From the Shear Force versus Load graph we plotted in this experiment, a linear graph was obtained for both Experimental Shear Force and Theoretical Shear Force values. Both graphs are linear and go through the origin (0,0) which tell us that, Shear Force does not exist when no load was applied on the beam. From the graph, we can notice that, when the load applied on the beam was increase, the Shear Force will also increase. This indicate that, Shear Force is linearly proportional (positive) to the load apply on the beam :

Shear Force α Load

8.14

Does the equation you used accurately predict the behavior of the beam?

Yes, the equation, Sc = W(L – a) / L that we used in this experiment for  Theoretical Shear Force calculation accurately predict the behavior of the  beam. This is because, from the Graph 1 plotted, we can notice that, when the load we placed at the beam was increased, the value of Shear Force also increased. This indicate that, Shear Force is linearly proportional (positive) to the load apply on the beam.

Wong Siew Hung AF040176 

 BFC 2091 Structure Lab – Shear Force In A Beam

 Example ; From the experiment, when a 2.453 N load was applied on the beam at the ‘cut’, the Experimental Shear Force obtained was 1.40 N. From the calculation done for Theoretical Shear Force by using the Sc = W(L – a)/L equation, the Shear Force we obtain was 1.45 N. This indicates that, this equation can accurately predict the behaviors of the beam.

8.2

Part 2 8.21

Comment on how the results of the experiments compare with those calculated using the theory?

From the experiments done by our group, we found that, there is only a small difference between the values of Experimental Shear Force and the Theoretical Shear Force. For figure 2 and figure 3, the value of the Experimental Shear Force is almost the same compare to the Theoretical Shear Force. While for the figure 4, the value of the Theoretical Shear Force is higher than the value of the Experimental Bending Moment. Referring to this results, we conclude that the differences between the value of the experiment and theory was probably cause by the mistake done by our  group member when taking the value for the force when it was hang on the  beam.

8.22

Does the experiment proof that the shear force at the ‘cut’ is equal to the algebraic sum of the forces acting to the left or right of the cut. If  not, why?

Yes, the experiment proof that the shear force at the ‘cut’ is equal to the algebraic sum of the forces acting to the left or right of the cut. This is  because, from the value of W1, W2, R A and R B , we can conclude that, W1 + W2 = R A + R B

 For the example, from data in the table 2,

Wong Siew Hung AF040176 

 BFC 2091 Structure Lab – Shear Force In A Beam

 Figure 2 W1 + W2 = R A + R B 1.962 N + 0

= 2.586 N + (-0.624 N) = 1.962 N

 Figure 3 W1 + W2 = R A + R B 1.962 N + 2.943

= 2.185 N + 2.720 N = 4.905 N

 Figure 4 W1 + W2 = R A + R B 1.962 N + 2.943

= 1.248 N + 3.657 N = 4.905 N

8.23

Plot the shear force diagram for load cases in Figure 2,3 and 4.

 Please see graph 2 and 3 as attached.

8.24

Comment on the shape of the graph. What does it tell you about how Shear Force varies due to various loading condition?

From GDR Graph for Figure 2 we obtained in Graph 2, we can noticed that when a loading, -1.962 N is put at the end of the beam (left side of R A), the value of the shear force cause by this load is negative. Reaction Force at A is equal to 2.586 N and therefore the total Shear Force at this point is + 0.624 N. Negative force of -0.624 N at B balances the Shear Force at A and thus, total Shear Force at B is zero.

Wong Siew Hung AF040176 

 BFC 2091 Structure Lab – Shear Force In A Beam

From GDR Graph for Figure 3 we obtained in Graph 2, when a loading, -1.962 N and -2.943 N are both place at the length of 220 mm and 260 mm from the right side of R A, calculation reveal that reaction force at A is + 2.185 N and reaction force at B is + 2.720 N. The graph also indicates that Shear Force on the negative part is equivalent to the positive part, that is equal to zero.

From GDR Graph for Figure 4 we obtained in Graph 3, we can conclude that, when a loading of 1.962 N and 2.943 N are both place 240 mm and 400 mm from the right side of R A, calculation reveal that reaction force at A is + 1.248 N and reaction force at B is + 3.657 N. The graph also tells us that Shear Force on the negative part is equilibrium to the positive part, that is zero.

From both GDR graph obtained from the Graph 2 and Graph 3, the shape of  the graph is close at the both end of the origin. This indicate that Shear  Force will change according to the load apply to the beam. This happens to ensure that Shear Force at left side is equal to the Shear Force at the right side to create equilibrium.

9.0

CONCLUSION

From this experiment, our group managed to examine how shear force varies with an increasing point load. We also managed to examine how shear force varies at the cut position of the beam for various loading conditions.

For part one experiment, we conclude that, when the load we place at beam is increase, the Shear Force will also increase. Thus, we conclude that, Shear Force is linearly proportional (positive) to the load apply on the beam.

Wong Siew Hung AF040176 

 BFC 2091 Structure Lab – Shear Force In A Beam

While for the part two experiment, we conclude that, from the GDR graph draw by our group in this experiment, we noticed that, Shear Force normally will happen at any point on the beam when a load is apply at the ‘cut’. The result from the experiment also indicate that Shear Force at the ‘cut’ section is equal to the forces acting at both right and left side of the ‘cut’ section on the beam.

10.0

REFERENCES

Yusof Ahamad (2001). “Mekanik Bahan Dan Struktur.” Malaysia: Universiti Teknologi Malaysia Skudai Johor Darul Ta’zim.

R. C. Hibbeler (2000). “Mechanic Of Materials.” 4th. ed. England: Prentice Hall International, Inc.

Wong Siew Hung AF040176