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SHEAR FORCE AND BENDING MOMENT DIAGRAMS CLASS NOTES

 

SHEAR FORCE AND BENDING

MOMENT DIAGRAMS CLASS

NOTES

Shear force and bending moment

diagrams are graphical method to find

their maximum values and their point of

acting. Firstly, design the beam for

 maximum moment alone. Secondly

design beam for maximum shear force.

Among the two designs, larger values of

beam becomes the final design. It

 satisfies for maximum conditions of

shear force and bending moment.

DEFINITION OF A BEAM

Beam is a most important member of a machine, vehicle or a structure.

(i) Normally it is horizontal but it can be in any orientation. It can be at any angle with the horizontal.

(ii) All loads must be perpendicular to the length of the beam. Such perpendicular loads on beam are lateral loads or transverse loads.

(iii) For an inclined load, consider component perpendicular to the length.

(iv)  Width and depth are small as compared to the length of the beam.

TYPES OF BEAMS

Types of beams are as per type of supports, number of supports and types of load.

(i)  Simply supported beam

Support beam on rollers/wheels/rockers.

(ii) Cantilever beam

One end fixed and the other end free.

(iii) Hinged beam

There is a pin connection at one end. Use two pins if hinged at both ends. Pinned ended beam/pivoted ended beam/hinged ended beam are the same thing.

(iv)  Continuous beam

When number of supports is greater than two.

(v) Overhanging beam

A beam not supported on the ends has left overhang and right overhang or both. These overhangs may be equal or unequal.

(v) Fixed beam

 Fix beam at both ends.

(vii) DETERMINATE BEAMS

Find unknown reactions from the equations of equilibrium.

(viii) INDETERMINATE BEAMS

 Unknown reactions are non determinate from the equations of equilibrium.

SPAN OF THE BEAM

The length of the beam is the SPAN.

INTRODUCTION

Find the followings graphically  from Shear force diagram.

(i) Magnitude of absolute maximum shear force

(ii) Location of the absolute maximum shear force on the beam

1. SHEAR FORCE ON A BEAM

It is NET VERTICALLY UPWARDS FORCE OR NET VERTICALLY DOWNWARDS FORCE due to the forces acting on one side of the point. Find the shear force at various points on the beam, then draw the  S.F.D.. From the  loaded beam, find graphically absolutely maximum shear force and its location. Shear Force Diagram shows the variation of the shear force along the span of the beam.

1.1 STEPS TO DRAW THE SHEAR FORCE DIAGRAM

(i) Draw the Free Body Diagram.

(i) Remove the support (or supports)

(ii) Install reactions in place of supports.

Such a sketch of the beam with reactions and actual external loading is FREE BODY DIAGRAM. In the first step, DRAW THE FREE BODY DIAGRAM. Obtain the Free body diagram.

(ii) Install only perpendicular reactions in place of FREE SUPPORT(S).

(ii) Install two unknown reactions (one vertical and one horizontal reaction). These reactions are in place of a hinge or a Pin or a pivot support.

(iii) Install vertical reaction, horizontal reaction and a restoring moment in place of a fixed support. Fixed support is an en-castered support or built-in support.

  1. Determine the value of unknown reactions using the equations of equilibrium i.e.

(i)  ΣFx = 0

(ii) ΣFy = 0

(iii) ΣM = 0

CAUTION: Use equation ΣM = 0 at any ONE END OF THE BEAM and not at any other point.

  1. Now find the shear force at few locations on the beam. Use the definition of shear force on a beam.

  2. Then plot the SFD

  3.  Find the magnitude of absolute maximum shear force

  4. Find the location of the maximum shear force.

2. BENDING MOMENT ON A BEAM

It is the NET CLOCKWISE MOMENT OR NET ANTI-CLOCKWISE MOMENT due to all the forces acting on one side of the point.  (Say on the beam at 1 m from the left end of the beam). Find the bending moment at various points on the beam & draw the BMD. From the BMD, find graphically the absolute maximum bending moment. (neglect the sign) . Also find the location of the absolute maximum moment.

Bending Moment Diagram is one which shows graphically the variation of the bending moment  along the length of the beam.

2.1 STEPS TO DRAW THE BMD

  1. Draw Free Body Diagram.

  2. Remove the support (or supports) and install reactions in place of supports. Such sketch of beam with reactions and actual external loading is FREE BODY DIAGRAM. Therefore, in the first step, DRAW THE FREE BODY DIAGRAM.

  3.  Determine the value of unknown reactions using the equations of equilibrium i.e.

(i)  ΣFx = 0

(ii) ΣFy = 0

(iii) ΣM = 0

CAUTION: Use equation ΣM = 0 at any ONE END OF THE BEAM and not at any other point.

4. Now find the bending moments at few locations on the beam. Use the definition of bending moment on a beam.

5. Then plot the BMD.

6. Find the magnitude of the absolute maximum bending moment.

7. Find the location of the absolute maximum moment.

  SFD and BMD for loaded beams

Fig. Shear force & bending diagrams for a Cantilever

 

Fig.1 SSB WITH UDL

 

Fig. CANTILEVER BEAM WITH UDL  & CONC LOADS

 

Fig. SSB WITH ONLY MOMENT AT 1 m FROM LH SUPPORT

BEAM DESIGN

Utility of SFD and BMD

Design the beam for maximum shear force.

Design means

(i) Select the material of construction

(ii)  Select the method of manufacture

(iii)  To select the shape and contour of the beam

(iv)  To find the dimensions on the basis of

(a)    Strength

(b)   Stiffness

(c)    Stability

(d)   Wear

(v)  To draw a detailed drawing showing the material of construction, method of manufacture, shape and configuration with dimensions.

(vi) All this help fabrication, installation and safe use the beam.

FINAL SELECTION OF THE BEAM

NOTE: Firstly design the beam for maximum shear force. Secondly design the beam for maximum bending moment. Bigger dimensions in the two designs are the final choice for the beam.

 

https://www.mesubjects.net/wp-admin/post.php?post=4231&action=edit               MCQ SFD & BMD

https://mesubjects.net/wp-admin/post.php?post=3142&action=edit                Q. ANS. SFD and BMD

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