ABSTRACT
To understand how a beam resists bending moment and how the mechanism of resistance is related to angle change,* we use the metaphor of the crowbar being used to pull a nail from a wall (Figure 11.1). An upward force F applied near the end of the crowbar causes a moment of magnitude Fx at the face of the wall. That moment is balanced by a force couple: T and C. Force C causes compression in the bar. Force T causes tension in the nail and balances force C in the horizontal direction:
C = T (11.1)
The distance between forces C and T we call jd, where j is a coefcient and d is the distance between the nail and the edge of the crowbar. From the requirement of moment equilibrium:
M = F × x = T × j · d (11.2)
To interpret the relationship between the applied and the resisting moments in the beam shown in Figure 11.2a, consider the equilibrium of the segment to the left of section X. Internal forces must balance the moment caused by the reaction with respect to X (Figure 11.2b). Internal stresses distributed over the section can be represented by concentrated forces C (compression) and T (tension), analogous to the forces acting on the crowbar. Again, equilibrium of forces and moments requires
C = T
and
M = F × x = T × j · d
The magnitude of the resultants C and T that balance the moment M and the distance between them (j · d) depends on the distribution of unit stresses. The distance j · d is called the internal lever arm.
