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Strength of Materials - Part 1

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Chapter 1 : Simple Stress and

Strain

Chapter 2 : Principal Stresses

and Strains

Chapter 3 : Bending Moment

and Shear Force Diagrams

Chapter 3 : Part 2

Chapter 4 : Simple Bending of

Beams

Chapter 5 : Torsion

Chapter 6 : Thin Cylinders and

Spheres

Chapter 7 : Columns and Struts

Chapter 8 : Slope and

Deflection

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Chapter 3 : Bending Moment and Shear Force Diagrams (Part 1 )

3.1 CONCEPT OF A BEAMAny member of a machine or structure whose one dimention(length) is very large as cothjared to the other two dimensions(width and thickness) and which can carry lateral or transverseloads in the axial plane is called a beam. A beam may be ofrectangular, square, triangular, hexagonal and circular, etc.cross-sections. A beam is a very important member in structuralmechanics to withstand the transverse loads. A beam may bemade ‘-of timber, flitched beam, i.e. timber reinforced with mildsteel strips, steel, and reinforced conc•rete. The reinforcedconcrete beams are mostly used in building construction, bridgesand flyovers.

3.2 CONCEPTS OF BENDING MOMENT AND SHEAR FORCE 3.2.1 Bending MomentThe bending moment (B.M.) at any point along a loaded beam isthe algebraic sum of the moments of all the vertical forces actingto one side of the point abcut the point. Consider a simplysupported beam AB carrying concentrated loads as shown in Fig.3.1 Lé R and RB be the vertical reactions at supports A and Brespectively. Consider a section x—x at a distance x from end A.The clockwise moment at this sectioh due to all the loads actingon the beam to the left of the section is:

If we consider the forces to the right of the section x—x, thenanticloc4wise moment is:

For equilibrium of the beam, the B.M. given by equations (1) and(2) are equal. The SI units of B.M. are N.m or kN.n. 3.2.2 ShearForceThe shear force (S F) at any point along a loaded beam is thealgebraic sum of all the vertical forces acting to one side of thepoint. Thus, for the beam AB shown in Fig. 3.1, the shear force atcross-section x—x as measured from the left hand side is:

The shear force as measured from the right hand side is:

Since the beam is in equilibrium, the S.F. given by equations (1)

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and (2) are equal. 3.2.3 Sign conventions

3.3 B.M. AND S.F. DIAGRAMS 3.3.1 B.M.DlagramTo draw the B.M.D, the following procedure may be followed:1. Take a sheet of graph paper. Draw the beam along withloading to an appropriate scale.2. Calculate the reactions at the supports by applying theequations of equilibrium, i.e., =0, EF, 0 and EM0 =0.3. Choose a section x-x at a distance x from the left handsupport. The section may be chosen either after everyconcentrated load or before the right hand support. For udi, thesection may be taken within the load.4. Calculate the B.M. beneath every concentrated load. For udi,the B.M. may becalculated along the length of the load. Ignore that term whichbecomes negativeon substituting the value of x.. 5. Draw the B.M.D. for the beam on a convenient scale. Ofcourse, sign convention has to be followed for B.M.Point of Inflexion. It is the point on the beam in the B.M.P. wherethe bending moment becomes zero.Point of Contraflexure. It is the point in the B.M.D where the B.M.changes slopefrom an increasing one to a decreasing one. Contraflexure meansopposite and flexuremeans bending. Some authors consider point of inflexion andpoint of contraflexure to besynonymous. 3.3.2 S.F. DiagramThe first three steps for the S.F.D. are the same as for theB.M.D., and need not be repeated.4. Calculate the S.F. beneath every concentrated load just to theleft and to the right.For u.d.I., the S.F. has to be calculated along the length of theload.5. Draw the S.F.D. to a convenient scale using the signconvention.

3.4 B.M AND S.F DIAGRAMS FOR A SIMPLY SUPPORTEDBEAM

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3.4.2 Uniformly Distributed LoadConsider a beam AB of span 1 simply supported at the ends andcarrying a uniformly distributed load of intensity w per unit lengthas shown in Fig. 3.5 (a)

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3.4.3 Uniformly Varying LoadsConsider a simply supported beam AB of span 1 carrying udiwhich varies from w1 per unit length at end A to w2 per unitlength at end B as shown in Fig. 3.6 (a). The varying load can beconsidered as the sum of two loads, one of uniform intensity w1and the other triangle variation from zero to (w2 — w1).

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3. 5 SIMPLY SUPPORTED BEAM SUBJECTED TO A COUPLE

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3.6 CANTILEVER BEAM 3.6.1 Concentrated Load at Free EndConsider a cantilever beam AB of span 1 carrying a concentratedload P at the free end

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It represents a straight line giving linear variation. The S.F.D isshown in Fig. 3.10 (c). 3.6.3 Uniformly Varying LoadConsider a cantilever beam AB of spand 1 carrying a uniformlyvarying load of intensity zero at the free end to O at the fixedend as shown in Fig. 3.11(a). Consider a section x-x at.

3.7 OVERHANGING BEAMS

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3.7.1 Concentrated Loads

This is a negative B.M. as it produces tension on the top fibres.It is a linearly varyingB.M.Span AB : At a distance x from C near to B,

3.7.2 Uniformly Distributed Load

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3.7.3 UnIformly Varying LoadConsider a beam A13CD with equal overhangs on both sides ofthe supports and carrying uniformly varying load from zero atend A to o per unit length at end D as shown

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It represents a cubic curve. The B.M. is negative as it producestension on the top fibres of the beam.

Span CD:At a distance x from D, we have

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The S.F. is parabolic in nature. The S.F.D. is shown in Fig. 3.14(c).

3.8 RELATIONSHIP BETWEEN LOAD, SHEAR FORCE ANDBENDING MOMENTConsider a simply supported beam carrying audi of intensity wper unit length as shown in Fig. 3.15. Consider an elementarylength of the beam of length ox between cross-Sections

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Therefore, the first derivative of shear force with respect to xat a point gives theintensity of loading at the point.

Therefore, the rte of change of bending moment with respectto x is equai to the shear force. Whenever, bending moment ismaximum or minimum, the shear force is zero.Taking the derivative again, we get

Example 3.1 Draw the bending moment and shear forcediagrams for the simplysupported beam loaded as shown in Fig. 3.16(a).

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Example 3.2 Draw the bending moment and shear forcediagrams for the simply supported beam shown in Fig.3.17(a).

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.The S.F.D. is shown in Fig. 3.18 (c) to a scale of 1 mm = 0.5kN.