Shear Force & Bending Moment | Civil Engineering SSC JE (Technical) - Civil Engineering (CE) PDF Download

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)( l² - 4a² )

Case (a) :  

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

 
form   centre.
Thus  distance between point of contraflexure O₁ O₂ =

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 O₁ & O₂ will coincide with C.
B.M.D will be as shown in figure (a)

M_c is negative ,since l² < 4a²

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)