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Figure 2-07a shows a beam supported by reactions R_1 and R_2 and loaded by the concentrated forces F_1, F_2, and F_3. The direction chosen for the y axis is the clue to the sign convention for the forces. F_1, F_2, and F_3 are negative because they act in the negative y direction; R_1 and R_2 are positive.
If the beam is cut at some section location x = x_1, and the left-hand portion removed as a free body, an internal shear force V and bending moment M must act on the cut surface to assure equilibrium. The shear force is obtained by summing the forces to the left of the cut section. The bending moment is the sum of the moments of the forces to the left of the section taken about an axis through the section. Shear force and bending moment are related by the equation
V = dM / dx [eq2-24]
Sometimes the bending is caused by a distributed load. Then, the relation between shear force and bending moment may be written
dV / dx = d^2M / dx^2 = -w [eq2-25]
where w is a downward acting load of w units of force
per unit length.
The sign conventions used for bending moment and shear force in this book are shown in Figure 2-08.
The loading w of equation [2-25] is uniformly distributed. A more general distribution can be defined by the equation
q = LIMIT(del_x->0, del_F / del_x)
where q is called the load intensity; thus
q = -w.
Equations [2-24] and [2-25] reveal additional relations if they are integrated. Thus, if we integrate between, say, x_a and x_b, we obtain
INTEGRAL(V_a,V_b, dV) =
INTEGRAL(x_a,x_b, q * dx) =
V_b - V_a [2-26]
which states that the change in shear force from
a to b is equal to the area of the loading
diagram between x_a and x_b.
In a similar maner
INTEGRAL(M_a,M_b, dM) =
INTEGRAL(x_a,x_b, V * dx) =
M_b - M_a [2-27]
which states that the change in moment from
a to b is equal to the area of the shear-force
diagram between x_a and x_b.
This section of the book is UNDER CONSTRUCTION...
