How to Calculate Moments for Beams


Moment, or torque, is a measure of the tendency of a force to cause or change the rotational motion of a body. An example is the turning of a wrench about a bolt. Your hand applies a force that is perpendicular to the wrench handle in order to rotate it. The distance from hand to the point about which the wrench rotates is called the moment arm or lever arm.

You can calculate the magnitude of the moment T about the point where the bolt is by using the formula T= F*l, where F is the perpendicular force exerted by your hand and l is the moment arm. If the force exerted by your hand is at an angle then you will need to use the component that is perpendicular to the moment arm, only. This same principle can also be applied to beams. It is conventional to take counterclockwise torques as positive and vise versa for clockwise moments.

  • Choose the center of the moment of the beam. This is the point that you pick as the location where the moment arm begins. For example take a beam of length s resting on two blocks located at each end. There is the downward force of gravity acting at the beam’s center called W. Two possible locations for the center of the moment are at the location of the blocks.

  • Get the distance and direction of the moment arm. Both the force and the moment arm are vector quantities. A vector quantity has direction, and these directions are indicted by unit vectors. If the center of the moment is located on the left end of the beam then the distance l is s/2 i to the center of the beam, where the force W is. The symbol i a Cartesian unit vector indicating the horizontal direction. If the center of the moment is on the right end then l is –s/2 i.

  • Get the perpendicular component of the force. In the example presented, the perpendicular force is the weight of the beam at its center and is directed downward so it is given by the expression F = -W j, where j is the Cartesian unit vector vector indicating the vertical direction.

  • Calculate the moment using the expressions for the moment arm and force. Since these are vector quantities the magnitude of the moment comes from the cross product of the two, which for the special case of a force that is directed perpendicular to the the moment arm, is the expression T=Fl k, where k is a Cartesian unit vector in a direction perpendicular to both the horizontal and vertical directions. For example, if you choose to take the moment about the left end of the beam then the moment T is (s/2)(-W)= -Ws/2, and if the moment is taken about the right end of the beam then T=(-s/2)(-W) = W*s/2. Note that the moment taken about the left end is negative since this would result in a clockwise rotation.

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