1.10: Problem Solving in Statics
Problem-solving in statics is a crucial aspect of engineering and physics that involves resolving issues associated with bodies in a state of equilibrium. In most cases, problem-solving requires several steps to achieve an accurate result. These steps are crucial to ensuring that the solution is accurate and practical.
The physical situation and mathematical modeling must be considered; however, it is challenging to represent all physical situations using mathematical modeling. With the help of approximations and assumptions, problems can be formulated.
While performing approximations, very small distances compared to larger distances are neglected. For example, the width of a rectangle can be neglected if it is a few orders smaller than the length. Small angle approximations can be used when the angular displacement is smaller than the other dimensions. One example of an approximation is when the force is distributed all over the body or object; this can be considered a point load. The assumptions purely depend upon the accuracy of the result required.
The first step in problem-solving in statics is to formulate the problem. Formulating the problem involves understanding the physical scenario and determining the variables within it. For instance, consider an example of a point load acting on a simply supported beam at a certain distance from one of the supports. In this scenario, the reaction forces and the bending moments at key points need to be estimated.
After formulating the problem, several assumptions are made to solve the problem. Assumptions such as neglecting small quantities compared to large ones, like neglecting the width of the beam compared to the length of the beam, are made. The bending moments at both the supported ends of the beam are assumed to be zero. Another assumption is that the beam is not deformed due to the loading. In addition, appropriate sign conventions that represent the direction of physical quantities are used. These assumptions help simplify the problem and create a more straightforward solution.
The next step in problem-solving in statics is to prepare free-body diagrams. Free-body diagrams are used to analyze the forces acting on the beam at a particular section and to determine mathematical equations for the net force and bending moment. These diagrams help identify the sum of the forces acting on the body and lay out the direction and the equation for the bending moment. They provide a clear picture of the situation under consideration and help determine the next steps to take.
Once the free-body diagrams are made, calculations can then be carried out to determine the reaction forces. Bending moment calculation allows for determination of the bending moments which can then be used to calculate reaction forces. This calculation accurately estimates the reaction forces involved in the scenario.
The bending and shear force diagrams show the variation in the bending and shear forces along the length of the beam. These diagrams help engineers better to understand the state of the body under consideration and identify potential areas of concern.
To represent the results, algebraic symbols can be used. While performing numerical calculations, consistent units need to be used throughout the equations. Another critical step is to verify the answer. Errors in the computing stages can be rechecked by substituting the solutions into the algebraic equations.
