The author was inspired by a classic Timoshenko Statics problem with six unknowns, repeated in several university exams. The traditional textbooks provide lengthy solutions, for this and similar problems, forming simultaneous equations requiring several steps. Based on his earlier unpublished work, the author has developed five Simplified Integrated Methods of Solution (SIMS) with a new Polar Vector Notation, to solve any determinate planar Newtonian Engineering Mechanics problem through equations, each with only one unknown. Ten types in basic planar vector systems are identified and solved by the author, applying the five SIMS systematically to yield solutions with least computing steps.
There are four types (T1 to T4) of planar multi-vector systems with two unknowns. T1 has a vector with unknown magnitude and direction related to two or more vectors, each with known magnitude and direction. T2 has Two Vectors, each with unknown direction, related to a vector or resultant, with known magnitude and direction. The first unknown, magnitude of T1 or one of the two directions of T2 is found, by SIM1 through squaring and adding the X, Y components of System's Vector Loop Equation with its single Left Hand Side (LHS) vector containing an unknown direction. The second unknown, the direction of T1 or the other direction of T2 respectively, is found by SIM2 through any inverse trigonometric function with known X and Y components and magnitude of the LHS vector.
T3 has two vectors, each with unknown magnitude, in the system. T4 has one vector with unknown magnitude and another with unknown direction, in the system. SIM3 is applied to T3 and T4 through Perpendicular Component Equation (LCE) to the line parallel to a vector with an unknown magnitude and known angle, eliminating that magnitude. The resulting equation with the second unknown, either magnitude or direction is solved. SIM3 is again applied to the vector of the solved unknown and the first eliminated magnitude is found.
T5 and T6 are three concurrent force Free Body Diagrams (FBDs) with three unknowns. In T5 the pin reaction direction is solved by applying SIM2 to the geometry of concurrency. In T6 the common unknown angle in the two reactions is solved with SIM4, by equating the expressions for the common unknown in the geometry of concurrency.
T7 to T10 are multi-force systems with three unknowns as, one force or distance or direction or moment, and two reactions. By SIM5, Moment Equation about a Canonical Moment Center, usually an intersection of reactions, that eliminates all unknowns except one, is obtained and solved. Examples including one on virtual work, are presented and discussed.
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