Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev

Civil Engineering SSC JE (Technical)

Civil Engineering (CE) : Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev

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Chapter 5 

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

 Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev
Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev
Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev
Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev

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

(i) Shear force
Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev

(ii) Bending moment
Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev

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

M= – Wx

Mmax = – WL

 Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev
Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev

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

 Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev
Smax = + wl
Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev

(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 Sx = wx + W

 Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev
Smax = wl + W
Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev
Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev
Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev

(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,

Sx = + wx

 Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev
form A Sx = + wa
Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev
Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev

(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

 Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev

= area of load diagram between X and B,  

 Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev

Mx = 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

 Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev
Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev
Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev

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

 Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev
Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev
Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev
Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev

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

 Sx = + Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev (between AC)
Sx = – Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev  (between CB)
Mx = + Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev x (between CB)
Mx = +Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev x (form A to C) (at a distance 'X' form A)
Mmax = Mc = Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev
Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev

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

 Sx  =  +Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev  (form A to D)
= –Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev (form D to B)
Mx = + Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev x (form A to D) at a distance 'x' form  A
Mmax = MD = Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev
Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev

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 

Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev
Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev
Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev
Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev
Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev

(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

 Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev
Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev
Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev
Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev
Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev
Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev

(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

 Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev
Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev
Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev

Mmax B.M. occurs at x = Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev  form end A
MmaxShear Force and Bending Moment Civil Engineering (CE) Notes | EduRev

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,
Sx = –wx at any section in A.B,

 Sx = (w/2)(l + 2a )– wx

B.M. at any section in EA,

 Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRevShear Force and Bending Moment Civil Engineering (CE) Notes | EduRevShear Force and Bending Moment Civil Engineering (CE) Notes | EduRev

at any section in AB.
Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev

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

 Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev
Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev
Mc = (w/2)( l2 - 4a2 )
Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev
Case (a) :  Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev
Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev

∴  B.M.D. will be as shown in figure above of contraflexure O1 & O2 are at a distance

 Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev
form   centre.
Thus  distance between point of contraflexure OO=
Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev

Thus  distance between point of contraflexure

Case (b) :
Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev
B.M at C = Mc = 0 The beam will be subject to only  hogging moments.
Points of contraflexure O1 & O2 will coincide with C.
B.M.D will be as shown in figure (a)

 Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev
Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev
Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev
Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev
Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev

Mc is negative ,since l2 < 4a2
Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev

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

 Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev

Case (b): Simple supported 
Shear force is constant

Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev

B.M., Mx = – Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev (left of C)
= +Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev (right of C)

Shear Force and Bending Moment Civil Engineering (CE) Notes | EduRev

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