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Engineering Statics Open and Interactive

Daniel W. Baker, William Haynes

Section 4.2 Scalar Addition of Moments

Since moments are vectors, they add according to the rules of vector addition. In three dimensions this usually involves finding the \(x\text{,}\) \(y\text{,}\) and \(z\) components of two or more moments and adding them to find the components of the resultant moment.
A simpler and more interactive approach using scalar components is available for moments in two dimensions.

Subsection 4.2.1 Scalar Components

In Subsection 3.3.2 we saw that vectors can be expressed as the product of a scalar component and a unit vector. For example, a \(\N{100}\) force acting in the \(-\jhat\) direction, i.e. straight down, can be represented by
\begin{equation*} \vec{F} = F_y\ \jhat = -\N{100}\ \jhat\text{,} \end{equation*}
where \(F_y=\N{-100}\) is the scalar component and \(\jhat\) is the unit vector. The absolute value of \(F_y\) determines the magnitude of the force, and unit vector along with the sign of \(F_y\) determines its direction. The negative sign here indicates that the direction of \(\vec{F}\) is opposite to the direction of \(\jhat\text{.}\)
We have previously established two sign conventions for scalar components; we now add a third:
  • For forces acting in the \(x\) direction, right is positive, corresponding to the direction of \(\ihat\text{.}\)
  • For forces acting in the \(y\) direction, up is positive, corresponding to the direction of \(\jhat\text{.}\)
  • For moments acting in the \(x\)-\(y\) plane, counterclockwise moments are positive, corresponding to the direction of \(\khat\) according to the right-hand rule.

Example 4.2.1. Sign Conventions.

For each scalar component, determine whether the moment is clockwise or counterclockwise, and apply the right-hand rule to determine the direction of the corresponding moment vector.
  1. \(\displaystyle M_1 = \Nm{30}\)
  2. \(\displaystyle M_2 = \kNm{-400}\)
  3. \(\displaystyle M_3 = \Nm{25} \circlearrowright\)
  4. \(\displaystyle M_4 = \ftlb{-100} \circlearrowright\)
Answer.
  1. \(+\khat\) or CCW
  2. \(-\khat\) or CW
  3. \(-\khat\) or CW
  4. \(+\khat\) or CCW
Solution.
By the right-hand rule, counterclockwise moments in the \(x\)-\(y\) plane are vectors that point in the \(+\khat\) direction. The corresponding scalar component has a positive sign. Clockwise moments point in the \(-\khat\) direction and have a negative sign.
  1. \(M_1\) is positive, so it is CCW and the vector points in the \(+\khat\) direction.
  2. \(M_2\) is negative, so it is CW and the vector points \(-\khat\) direction.
  3. The arrow on \(M_3\) establishes the positive direction as CW. A positive scalar component means that the direction is CW and the vector points in the \(-\khat\) direction.
  4. The arrow on \(M_4\) establishes the positive direction as CW, but the negative sign on the scalar component means that the actual moment is CCW. The vector points in the \(+\khat\) direction.
In three dimensions, moments, like forces, can be resolved into components in the \(x\text{,}\) \(y\text{,}\) and \(z\) directions.
\begin{equation*} \vec{M} = M_x\,\ihat + M_y\, \jhat + M_z\, \khat\text{.} \end{equation*}
This means that the three scalar components are required to fully specify a moment in three dimensions, whereas only one is required for moments in a plane.

Warning 4.2.2.

Be careful not to mix up magnitudes with scalar components.
  • Both are scalar values with units.
  • Magnitudes are never negative. Scalar components may be positive or negative.
  • Scalar components always have an associated sign convention. It may be implied or specifically indicated. If the sign convention is unstated, assume that counter-clockwise moments in the \(x\)-\(y\) plane are positive.
  • There is no special symbol or notation to indicate whether a quantity represents a vector magnitude or a scalar moment, so pay attention to context.

Subsection 4.2.2 Addition of Moments

Scalar components are most useful for adding and finding the resultant of several moments. When moments are expressed in terms of scalar components, vector addition reduces to algebraic addition of the scalar components.
To add several moments in a plane using scalar components, simply append a positive sign to the magnitude of counter-clockwise moments and a negative sign to clockwise moments to create scalar components, then add the scalar components normally.
The resulting algebraic sum of the scalar components will be either positive, negative, or zero, and this sign indicates the direction of the resultant moment. If the result is positive the resultant acts in the direction of the positive unit vector, and in the opposite direction if it is negative.

Example 4.2.3. Scalar addition.

Find the resultant of the following three moments:
\(\vec{M_1} = \kNm{25} \circlearrowright\text{,}\) \(\vec{M_2} =\kNm{40} \circlearrowleft\text{,}\) and \(\vec{M_3} = \kNm{30} \circlearrowright\)
Answer.
\(\vec{M} = \kNm{15}\) Clockwise
Solution.
Manually attaching the positive or negative signs according to the right-hand rule turns the magnitudes into scalar moments:
\begin{gather*} M_1 = \kNm{-25}\\ M_2 =\kNm{+40}\\ M_3 = \kNm{-30} \end{gather*}
Adding these moments algebraically gives the resultant scalar moment.
\begin{align*} M \amp= M_1 + M_2 + M_3\\ \amp = (\kNm{-25}) + (\kNm{40}) + (\kNm{ - 30})\\ \amp = \kNm{-15} \end{align*}
The negative sign indicates that the direction of the resultant moment is clockwise.
Interpreting the scalar moment as a vector gives:
\begin{equation*} \vec{M} = \kNm{15}\circlearrowright \end{equation*}
The corresponding magnitude of \(\vec{M}\) is
\begin{equation*} |\vec{M}| = \kNm{15} \end{equation*}
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