1.0 SHEAR FORCE AND BENDING MOMENT
1.1 Introduction
The algebraic sum of the vertical forces at any section (point) of a beam to the right or left of the section
is known as shear force. It is abbreviated as SF. Similarly, the algebraic sum of the moments of all the
forces acting to the right or left of the section is known as bending moments. It is abbreviated as BM.
1.2 Shear Force and Bending Moment Diagrams
A shear force diagram (SFD) is one which shows the variation of shear force along the length of the beam.
Similarly, a bending moment diagram (BMD) is one which shows the variation of bending moment along
the length of the beam.
1.3 Sign conventions for SFD and BMD
Recall section 4.4.1
1.3.1 Shear Force
Shear forces are the unaligned forces pushing one part of a body in one direction and another part of the
body in the opposite direction. Figure 5.1 shows a simply supported beam AB carrying a load of 10𝑘𝑁 at
its mid-point. The reactions at supports will be equal to 5𝑘𝑁, hence 𝑅𝐴 = 𝑅𝐵 = 5𝑘𝑁. Imagine the beam
to be divided into two portions by the section 𝑥 − 𝑥. The resultant of the load and reaction to the left of
𝑥 − 𝑥 is 5𝑘𝑁 vertically upwards. Note in this case, there is no load acting to the left of section 𝑥 − 𝑥. The
resultant of the load and reaction to the right of section 𝑥 − 𝑥 is 5𝑘𝑁 downwards. The resultant force
action on any one of the parts normal to the axis of the bam is called the shear force at eh section 𝑥 − 𝑥.
Hence, the shear force at section 𝑥 − 𝑥 is 5𝑘𝑁.
Figure 5.1: Illustration of Shear Force
The shear force at a section will be considered positive when the resultant of the forces to the left of the
section is upwards or the right of the section is downwards. Similarly, the shear force at a section will be
considered negative if the resultant of the forces to the left of the section is downwards, or to the right of
,the section is upwards. For section 𝑥 − 𝑥, the resultant force to the left of the section is upwards, hence
the shear will be positive.
1.3.2 Bending Moment
Bending moment is the reaction induced in a structural element when a moment is applied to it causing
it to bend. The bending moment at any section is considered positive if the bending moment at that
section is such that it tends to bend the beam to a curvature having concavity (sagging) at the top as
shown in Figure 5.2 (a). Similarly, the bending moment at a section is considered negative if the bending
moment of that section is such that it tends to bend the beam to a curvature having convexity (hogging)
at the top as shown in Figure 5.2 (b).
Figure 5.2: Illustration of Bending Moment
Consider the simply supported beam in Figure 5.3 carrying a load of 10𝑘𝑁 at its mid-point. Reactions 𝑅𝐴
and 𝑅𝐵 are equal and have magnitude of 5𝑘𝑁. Imagine the beam to be divided into two portions by
section 𝑥 − 𝑥 and let 𝑥 − 𝑥 be at a distance of 1𝑚 from A.
Figure 5.3
The moments of all the forces (i.e. load and reaction) to the left of 𝑥 − 𝑥 at the section 𝑥 − 𝑥 is
𝑅𝐴 × 1𝑚 = 5𝑘𝑁𝑚 (𝑐𝑙𝑜𝑐𝑘𝑤𝑖𝑠𝑒). Also, the moments of all forces (i.e. load and reaction) to the right of
𝑥 − 𝑥 at section 𝑥 − 𝑥 is 𝑅𝐵 × 3𝑚 (𝑎𝑛𝑡𝑖𝑐𝑙𝑜𝑐𝑘𝑤𝑖𝑠𝑒) − 10𝑘𝑁 × 1𝑚 = 15𝑘𝑁𝑚 − 10𝑘𝑁𝑚 =
5𝑘𝑁𝑚 (𝑎𝑛𝑡𝑖𝑐𝑙𝑜𝑐𝑘𝑤𝑖𝑠𝑒). Hence, the tendency of the bending moment at 𝑥 − 𝑥 is to bend the beam so as
to produce concavity (sagging) at the top as shown in Figure 5.4 (a).
, Figure 5.4
The bending moment at a section is the algebraic sum of the moments of forces and reactions acting on
one side of the section. Hence, BM at 𝑥 − 𝑥 is 5𝑘𝑁𝑚. The bending moment will be considered positive
when the moment of the forces and reaction on the left portion is clockwise, and on the right portion is
anticlockwise. In Figure 5.4 (a), the bending moment at section 𝑥 − 𝑥 is positive. Similarly, the BM will be
considered negative when the moment at the forces and reactions on the left portion is anticlockwise and
on the right portion is clockwise as shown in Figure 5.4 (b). In Figure 5.4 (b), the bending moment at
section 𝑥 − 𝑥 is negative.
1.4 Important Points for Drawing SFD and BMD
In 5.2, we mentioned that SFD is one which shows the variation of the shear force along the length of the
beam and BMD shows variation of the bending moment along the length of the beam. In SFD and BMD,
the shear forces or bending moments are represented by y-axis, whereas the beam is represented by the
x-axis. The following are the important points for drawing SFD and BMD:
(i) Consider the left or right portion of the section.
(ii) Add the forces (including reaction) normal to the beam on one of the portions. If the right portion
of the section is chosen, a force on the right portion acting downward is positive while a force
acting upwards is negative. If the left portion of the section is chosen, a force on the left portion
acting upwards is positive while a force acting downwards is negative.
(iii) The positive values of shear force are plotted above the baseline (length of the beam), and the
negative values below the baseline. Conversely, the positive values of bending moment are
plotted below the baseline while the negative values are plotted above the baseline.
(iv) The SFD will increase or decrease suddenly (kink) by a vertical straight line at a section where
there is a vertical point load.
(v) The shear force between any two vertical loads will be constant hence the SFD diagram between
two vertical loads will be horizontal.
(vi) The bending moment at the two supports of a simply supported beam and at the free end of a
cantilever will be 0.
1.1 Introduction
The algebraic sum of the vertical forces at any section (point) of a beam to the right or left of the section
is known as shear force. It is abbreviated as SF. Similarly, the algebraic sum of the moments of all the
forces acting to the right or left of the section is known as bending moments. It is abbreviated as BM.
1.2 Shear Force and Bending Moment Diagrams
A shear force diagram (SFD) is one which shows the variation of shear force along the length of the beam.
Similarly, a bending moment diagram (BMD) is one which shows the variation of bending moment along
the length of the beam.
1.3 Sign conventions for SFD and BMD
Recall section 4.4.1
1.3.1 Shear Force
Shear forces are the unaligned forces pushing one part of a body in one direction and another part of the
body in the opposite direction. Figure 5.1 shows a simply supported beam AB carrying a load of 10𝑘𝑁 at
its mid-point. The reactions at supports will be equal to 5𝑘𝑁, hence 𝑅𝐴 = 𝑅𝐵 = 5𝑘𝑁. Imagine the beam
to be divided into two portions by the section 𝑥 − 𝑥. The resultant of the load and reaction to the left of
𝑥 − 𝑥 is 5𝑘𝑁 vertically upwards. Note in this case, there is no load acting to the left of section 𝑥 − 𝑥. The
resultant of the load and reaction to the right of section 𝑥 − 𝑥 is 5𝑘𝑁 downwards. The resultant force
action on any one of the parts normal to the axis of the bam is called the shear force at eh section 𝑥 − 𝑥.
Hence, the shear force at section 𝑥 − 𝑥 is 5𝑘𝑁.
Figure 5.1: Illustration of Shear Force
The shear force at a section will be considered positive when the resultant of the forces to the left of the
section is upwards or the right of the section is downwards. Similarly, the shear force at a section will be
considered negative if the resultant of the forces to the left of the section is downwards, or to the right of
,the section is upwards. For section 𝑥 − 𝑥, the resultant force to the left of the section is upwards, hence
the shear will be positive.
1.3.2 Bending Moment
Bending moment is the reaction induced in a structural element when a moment is applied to it causing
it to bend. The bending moment at any section is considered positive if the bending moment at that
section is such that it tends to bend the beam to a curvature having concavity (sagging) at the top as
shown in Figure 5.2 (a). Similarly, the bending moment at a section is considered negative if the bending
moment of that section is such that it tends to bend the beam to a curvature having convexity (hogging)
at the top as shown in Figure 5.2 (b).
Figure 5.2: Illustration of Bending Moment
Consider the simply supported beam in Figure 5.3 carrying a load of 10𝑘𝑁 at its mid-point. Reactions 𝑅𝐴
and 𝑅𝐵 are equal and have magnitude of 5𝑘𝑁. Imagine the beam to be divided into two portions by
section 𝑥 − 𝑥 and let 𝑥 − 𝑥 be at a distance of 1𝑚 from A.
Figure 5.3
The moments of all the forces (i.e. load and reaction) to the left of 𝑥 − 𝑥 at the section 𝑥 − 𝑥 is
𝑅𝐴 × 1𝑚 = 5𝑘𝑁𝑚 (𝑐𝑙𝑜𝑐𝑘𝑤𝑖𝑠𝑒). Also, the moments of all forces (i.e. load and reaction) to the right of
𝑥 − 𝑥 at section 𝑥 − 𝑥 is 𝑅𝐵 × 3𝑚 (𝑎𝑛𝑡𝑖𝑐𝑙𝑜𝑐𝑘𝑤𝑖𝑠𝑒) − 10𝑘𝑁 × 1𝑚 = 15𝑘𝑁𝑚 − 10𝑘𝑁𝑚 =
5𝑘𝑁𝑚 (𝑎𝑛𝑡𝑖𝑐𝑙𝑜𝑐𝑘𝑤𝑖𝑠𝑒). Hence, the tendency of the bending moment at 𝑥 − 𝑥 is to bend the beam so as
to produce concavity (sagging) at the top as shown in Figure 5.4 (a).
, Figure 5.4
The bending moment at a section is the algebraic sum of the moments of forces and reactions acting on
one side of the section. Hence, BM at 𝑥 − 𝑥 is 5𝑘𝑁𝑚. The bending moment will be considered positive
when the moment of the forces and reaction on the left portion is clockwise, and on the right portion is
anticlockwise. In Figure 5.4 (a), the bending moment at section 𝑥 − 𝑥 is positive. Similarly, the BM will be
considered negative when the moment at the forces and reactions on the left portion is anticlockwise and
on the right portion is clockwise as shown in Figure 5.4 (b). In Figure 5.4 (b), the bending moment at
section 𝑥 − 𝑥 is negative.
1.4 Important Points for Drawing SFD and BMD
In 5.2, we mentioned that SFD is one which shows the variation of the shear force along the length of the
beam and BMD shows variation of the bending moment along the length of the beam. In SFD and BMD,
the shear forces or bending moments are represented by y-axis, whereas the beam is represented by the
x-axis. The following are the important points for drawing SFD and BMD:
(i) Consider the left or right portion of the section.
(ii) Add the forces (including reaction) normal to the beam on one of the portions. If the right portion
of the section is chosen, a force on the right portion acting downward is positive while a force
acting upwards is negative. If the left portion of the section is chosen, a force on the left portion
acting upwards is positive while a force acting downwards is negative.
(iii) The positive values of shear force are plotted above the baseline (length of the beam), and the
negative values below the baseline. Conversely, the positive values of bending moment are
plotted below the baseline while the negative values are plotted above the baseline.
(iv) The SFD will increase or decrease suddenly (kink) by a vertical straight line at a section where
there is a vertical point load.
(v) The shear force between any two vertical loads will be constant hence the SFD diagram between
two vertical loads will be horizontal.
(vi) The bending moment at the two supports of a simply supported beam and at the free end of a
cantilever will be 0.
