In the past, our teams placed strain gages at critical locations on fiberglass prototypes and concrete canoes. In every case, the highest strain was measured directly under the bow paddler during 2-man races in a direction perpendicular to the longitudinal axis of the canoe (see Fig. 1). To determine the service load, Pcr, we assume that this critical section is in pure bending and load a flat test plate, having a cross section identical to that used in the prototype, until the critical strain is reached. The strain in the test plate is determined by placing a strain gage on the surface, in the x-direction, at point C (see Fig. 2). Knowing the load allows us to compute the maximum stress when we test an unreinforced concrete plate in a similar manner. A detailed step-by-step example calculation follows.
Figure 2 shows a schematic diagram for the third point bending (TPB) test which includes a prismatic beam of length, L, having a rectangular cross section with a base, b, and height, h. The beam is supported by a pin at point B and a roller at point D. Concentrated loads of equal magnitude are applied at points A and E by placing a load, P, at the center of a loading platen FG. The lengths of sections AB, BD, and DE are equal to L/3 so that the test complies with ASTM C78/C78M-10e1 (ASTM 2010); the standard established for a configuration typically referred to as “third point” or “four point” (pure) bending.
Fig. 2. Third point bending (TPB) test
Figure 3 shows the free body diagram for the configuration illustrated in Fig. 2. The reactions at the inner supports, labeled as RBx, RBy, and RD, can be determined by satisfying equilibrium.
Fig. 3. Free body diagram for third point bending (TPB) test
Summing forces along the x axis,
ΣFx=0= RBx and RBx =0 .
Taking moments about point B, where positive is counter clockwise,
ΣMB=0= RDL3-P22L3+ P2L3 and RD=P2 .