Shear force diagram and bending moment diagram

Shear force diagram (SFD): Graph of shear force V vs x .
Bending moment diagram (BMD): Graph of bending moment M vs x .
Example 1.5.1: Draw the shear force and bending moment diagrams for the beam shown in Fig.
1.5.1.
               
            Fig. 1.5.1. SFD and BMD for a simple beam with a concentrated load.

(a) Determine reactions by considering the equilibrium of the entire beam.
+􀀴ΣMB = 0 : 0 A −R L + Pb = , A
R Pb
L
⇒ = .
0: A +􀀴ΣM = 0 B −Pa + R L = , B
R Pa
L
⇒ = . (1.5.1.1)
(b) Cut the beam to the left of the load P at distance x from A (i.e., 0 < x < a ). From the freebody
diagram of the left-hand part of the beam,
A
V R Pb
L
= = , A
M R x Pbx
L
= = . (1.5.1.2)
Shear force is constant from A to point of application of load P . Bending moment varies
linearly with x .
(c) Cut the beam to the right of the load P at distance x from A (i.e., a < x < L ). From the freebody
diagram of the left-hand part of the beam,
V Pb P Pa
L L
−
= − = , M Pbx P(x a) Pa 1 x
L L
= − − = ⎛ − ⎞ ⎜ ⎟
⎝ ⎠
. (1.5.1.3)
Shear force is constant. Bending moment is a linear function of x .
Note: It is easier to obtain eqn (1.5.1.3) by considering the right-hand part of the beam as a free
body.

Example 1.5.2: Draw the shear force and bending moment diagrams for the beam shown in Fig.
1.5.2.
                     
                    Fig. 1.5.2. SFD and BMD for a simple beam with a uniform load.

(a) Determine reactions by considering the equilibrium of the entire beam.
+􀀴ΣMB = 0 : ( ) 0
A 2
−R L + qL ⎛ L ⎞ = ⎜ ⎟
⎝ ⎠
,
A 2
⇒ R = qL .
0: A +􀀴ΣM = ( ) 0
2 B
− qL ⎛ L ⎞ + R L = ⎜ ⎟
⎝ ⎠
,
B 2
⇒ R = qL . (1.5.2.1)
(b) Cut the beam at distance x from A (i.e., 0 < x < L ). From the free-body diagram of the lefthand
part of the beam,
0 : Y + ↑ ΣF = 0 A R − qx −V = ,
2
⇒V = q⎛ L − x ⎞ ⎜ ⎟
⎝ ⎠
.
0: A +􀀴ΣM = ( ) 0
2
− qx ⎛ x ⎞ −Vx +M = ⎜ ⎟
⎝ ⎠
, ( )
2
⇒ M = qx L − x . (1.5.2.2)
The SFD is a straight line. Slope of line is −q . BMD is a parabolic curve. Slope of the BMD
is equal to the shear force V . The maximum BM occurs when dM 0
dx
= (i.e., at the crosssection
when V = 0 ).

Example 1.5.3: Draw the shear force and bending moment diagrams for the beam shown in Fig.
1.5.3. Determine the maximum normal stress due to bending.
                        
                                     Fig. 1.5.3. SFD and BMD for a beam.



Example 1.5.4: Draw the shear force and bending moment diagrams for the beam shown in Fig.
1.5.4. Determine the maximum normal stress in sections just to the left and just to the right of
point D. S = 2.08×106 mm3 about the X − X axis.
                              
                                           Fig. 1.5.4. SFD and BMD for a beam.