a) Let the reactions at supports A and B caused by the load P be R_a and R_b, respectively.  Derive  the formulas 

R_a = Pb/L
R_b = Pa/L

using in each case a single moment equilibrium equation.  

Can you verbalize the formulas so as to memorize their content without relying on symbols?

b) The formulas above are applicable  whether P is applied within or outside segment AB, provided a sign convention is adopted for R_a, R_b, and a or b.  With a and b related by a + b = L, find and state that sign convention.

c) Show that the reactions to an applied moment M are R_a = M/L and R_b = -M/L. What is the sign convention in these formulas? Do the formulas depend on where the moment is applied?

a) On a new sketch of the beam, replace the uniform load and the triangular load, respectively, by equivalent resultant forces, and draw a free-body diagram of the beam.  Apply the formulas of Sec. 1 to determine the reactions.

b) On a new sketch of the beam, replace the total applied loading by its resultants at B, and draw a free-body diagram of the beam. Obtain the reaction at A from that diagram, and check that it is the same as found in part a).

 

c)  Determine the shear force to the immediate right of B by the force transmission approach.  Do the same to the immediate left of B using the reaction at A found earlier.  What is the difference between the two answers, and what is its physical significance? 

3. Resultants of a Spatial Force System

A road sign is formed of a thin plate CDEF, parallel to the (x, z) plane, and attached to a cylindrical post of radius 1 ft centered on the z axis. The plate is subjected to wind pressure p, and its weight is W.  The post weighs w lb/ft.

a) Show on the figure the weight of the plate, and the resultant P of the wind load acting on it. Show a force perpendicular to the plane of the figure as a dot if it points out of the page (tip of the vector) or a cross if it points into the page (tail of the vector).
b) Determine the resultants of the forces acting on the plate, at point A on the z axis. (force resultant components F_x, F_y, F_z, and moment resultant components M_Ax, M_Ay, M_Az) .

c) Determine by the force transmission approach the internal forces of the post at A, and show them on a sketch as acting on the bottom part.

d) Use the results of part c) to determine the support reactions at B, and show them on a sketch.

4. Internal Static Conditions
a) A bar such as AB, which is pinned at both ends, and is not subjected to any external load, is a two-force body subjected to a force at A and a force at B. If the bar is in equilibrium, and F is the force at A, the force at B is -F, and both forces act along line AB.  Why is that so?  What would happen if F and -F did not have a common line of action?

b) Determine the force in bar AB by a single equilibrium equation.

c) If bar AB is subjected to an external load, it ceases to be a two-force member.  Does the problem become statically indeterminate? Explain.

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Force Transmission:

A system being cut into two parts, the internal forces acting on one part are equal, as vectors, to the resultants of the external forces acting on the other part.
Equivalently,
A system being cut into two parts, each part exerts on the other part the resultants of its own external forces.