bending moments in beams and plates have a straightforward interpretation as the stress resultants in a cross-section of the structural element. For example, in a beam in the figure, the bending moment vector due to stresses in the cross-section A perpendicular to the x-axis is given by Expanding this expression we have, We define the bending moment components as The internal moments are computed about an origin that is at the neutral axis of the beam or plate and the integration is through the thickness () Example In the beam shown in the adjacent figure, the external forces are the applied force at point A () and the reactions at the two support points O and B ( and ). For this situation, the only non-zero component of the bending moment is where is the height in the direction of the beam. The minus sign is included to satisfy the sign convention. In order to calculate , we begin by balancing the forces, which gives one equation with the two unknown reactions, To obtain each reaction a second equation is required. Balancing the moments about any arbitrary point X would give us a second equation we can use to solve for and in terms of . Balancing about the point O is simplest but let's balance about point A just to illustrate the point, i.e. If is the length of the beam, we have Evaluating the cross-products: If we solve for the reactions we have Now to obtain the internal bending moment at X we sum all the moments about the point X due to all the external forces to the right of X (on the positive side), and there is only one contribution in this case, We can check this answer by looking at the free body diagram and the part of the beam to the left of point X, and the total moment due to these external forces is If we compute the cross products, we have Thanks to the equilibrium, the internal bending moment due to external forces to the left of X must be exactly balanced by the internal turning force obtained by considering the part of the beam to the right of X which is clearly the case. Sign convention In the above discussion, it is implicitly assumed that the bending moment is positive when the top of the beam is compressed. That can be