1. The essence of the science of strength of material. Concept of bars, plates and shells

The cross-sectional displacement of a bar during tension or compression can be calculated using Hooke's law, which relates the stress and strain of a material under a given load.

The potential energy stored in a material due to deformation under tension or compression can be calculated using the strain energy equation, which relates the applied load, the deformation, and the material properties.

The internal forces in a structure under tension or compression can be determined using the equations of equilibrium and the conditions of static equilibrium. The condition of strength is satisfied when the internal stresses do not exceed the allowable stress.

Longitudinal deformation refers to the change in length of a material under tension or compression, which can be calculated using Hooke's law. Hooke's law states that the strain is proportional to the stress.

Transverse deformation refers to the change in width or thickness of a material under external loads, which is described by Poisson's ratio. Poisson's ratio is a material property that relates the transverse strain to the longitudinal strain.

The universal equation of the bent axis of a beam relates the bending moment, shear force, and distributed load along the length of the beam.

The differential equation of the bent axis of a beam describes the relationship between the bending moment, shear force, and distributed load at each point along the length of the beam. The solution of the differential equation involves the integration of the shear force and distributed load to obtain the bending moment.

The differential relations between bending moment, shear force, and distributed load describe the relationship between these internal forces along the length of the beam.

17. 
    Internal forces in bending refer to the forces that occur within a beam or other structural member as a result of an external bending moment. These forces include normal stress (compression and tension), shear stress, and bending stress. The sign rules and diagrams for internal forces in bending depend on the type of force and the location within the beam. For example, the sign of the bending moment changes as you move from the top to the bottom of the beam, and the sign of the shear force changes as you move from left to right.
    
18. 
    Supports in structural analysis are used to resist external loads and maintain equilibrium in a system. Different types of supports include fixed supports, which resist both vertical and horizontal movement; hinged supports, which resist vertical movement but allow for rotation; and roller supports, which allow for both vertical and horizontal movement. The determination of support reactions depends on the type of support and the external loads applied to the system.
    
19. 
    A statically indeterminate system is a structural system that cannot be analyzed using only equilibrium equations. This is because the number of unknown reactions exceeds the number of equations available. Statically indeterminate systems require additional equations, such as compatibility equations, to be solved.
    
20. 
    Static indeterminate systems in tension and compression are analyzed using methods such as the strain energy method or the stiffness method. These methods involve the use of additional equations to determine the unknown reactions in the system. The calculation of static indeterminate systems involves finding the internal forces and deformations in the system under external loads.
    
21. 
    Geometric properties of flat cross-sectional surfaces include the area of the section, the location of the centroid (center of gravity), and the moments of inertia (both about the x and y axes and the polar moment of inertia). These properties are used to analyze the behavior of beams and other structural members under different loading conditions.
    
22. 
    The static moment of a section is a measure of the section's resistance to bending. It is equal to the product of the area of the section and the distance from a reference axis to the centroid of the section. The static moment changes when the axes move parallel because the distance between the centroid and the reference axis changes.
    
23. 
    The central axis of a section is the line through the centroid of the section that is perpendicular to both the x and y axes. The coordinates of the center of gravity of the section can be determined by dividing the moment of the area about one axis by the total area of the section.
    
24. 
    Moments of inertia of a section are a measure of the section's resistance to bending and torsion. They depend on the shape and size of the section, as well as the location of the axes of rotation. The moments of inertia change when the axes move parallel because the distance between the centroid and the axis changes.
    
25. 
    Moments of inertia of simple forms, such as rectangles and circles, can be determined using formulas. For more complex shapes, the moment of inertia can be found by dividing the section into smaller parts with known moments of inertia and summing them up.
    
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