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**Shear Force and Bending Moment**

**SHEAR FORCE AND BENDING MOMENT**

1. SHEAR FORCE at the cross- section of a beam may be defined as the unbalanced vertical force to the right or left of the section.

2. BENDING MOMENT at the cross- section of a beam may be defined as the algebraic sum of the moment of the forces, to the right or left of the section.

3. BEAM is a structural number subjected to transverse loads only.

4. BEAMS can be classified as :

(i) Cantilever

(ii) Simply supported

(iii) Overhanging

(iv) Rigidly fixed OR Built- in

(v) Continuous

5. Shear force and bending moment diagrams: Sign Convention:

(i) Shear force

(ii) Bending moment

SFD and BMD for cantilever beams: (i) Cantilever of length l carrying a concentrated load W at the free end

S_{x }= + W

M_{x }= – Wx

M_{max} = – WL

(ii) Cantilever of length l carrying a uniformly distributed load of 'w' per unit run over the whole length

S_{x} = + wx

S_{max }= + wl

(iii) Cantilever of length l carrying a uniformly distributed load of 'w' per unit run over the whole length and a concentrated load W at the free end S_{x} = wx + W

S_{max} = wl + W

(iv) Cantilever of length l carrying a uniformly distributed load of 'w' per unit run for a distance 'a' form free end form D to B,

S_{x} = + wx

form A S_{x} = + wa

(v) Cantilever of length 'l' carrying a load whose intensity varies uniformly from zero at free end to 'w' per unit run at the fixed end

= area of load diagram between X and B,

M_{x} = Moment of load acting on XB about X = area of the load diagram between X and B × distance of centroid of this diagram form X

(vi) Cantilever carrying a load whose intensity varies uniformly form zero at the fixed end to w per unit run at the free end

SFD and BMD for simply supported beams: (i) Simply supported beam of span l carrying a concentrated load at mid span

S_{x} = + (between AC)

S_{x} = – (between CB)

M_{x} = + x (between CB)

M_{x} = + x (form A to C) (at a distance 'X' form A)

M_{max} = Mc =

(ii) Simply supported beam carrying a concentrated load placed eccentrically on the span

S_{x} = + (form A to D)

= – (form D to B)

M_{x} = + x (form A to D) at a distance 'x' form A

M_{max} = MD =

**NOTE: Maximum B.M. occurs where S.F. changes its sign. **

(iii) Simply supported beam carrying a uniformly distributed load of w per unit run over the whole span

(iv) Simply supported beam carrying a load whose intensity varies uniformly from zero at each end to 'w' per unit run at the mid span

(v) Simply supported beam carrying a load whose intensity varies uniformly from zero at one end to 'w' per unit run at the other end

M_{max} B.M. occurs at x = form end A

M_{max} =

SFD and BMD for simply supported beams with overhang: Simply supported beam with equal overhangs and carrying a uniformly distributed load of 'w' per unit run over the whole length S.F. at any section in EA at a distance x form E,

S_{x} = –wx at any section in A.B,

S_{x} = (w/2)(l + 2a )– wx

B.M. at any section in EA,

at any section in AB.

at x = 'a' and 'a + l' i.e., at A & B,

M_{c} = (w/2)( l^{2} - 4a^{2} )**Case (a) :**

∴ B.M.D. will be as shown in figure above of contraflexure O_{1} & O_{2} are at a distance

form centre.

Thus distance between point of contraflexure O_{1 }O_{2 }=

Thus distance between point of contraflexure

**Case (b) :**

B.M at C = M_{c} = 0 The beam will be subject to only hogging moments.

Points of contraflexure O_{1} & O_{2} will coincide with C.

B.M.D will be as shown in figure (a)

M_{c} is negative ,since l^{2} < 4a^{2}

B. M. will be zero only at ends A and D and at all other sections B.M. will be of hogging type B.M. and S.F due to a couple

**Case (a): **Cantilever There will be no shear force

**Case (b): Simple supported **

Shear force is constant

B.M., M_{x} = – (left of C)

= + (right of C)

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