1.Mentally cut the body with the point of interest and the plane passing through the section.The body is divided into two parts.
2. One of the separated pieces, for example, the left side I is discarded away II- part will be taken. (Fig. 1.1). In this case, the balance of the remaining left part II is disturbed.
3. Replace the impact of the taken right part II on the removed left part I with internal tension forces . Their magnitude, direction and distribution law are unknown. However, it is known that any system of forces can be reduced to one principal vector and one principal moment. 1.1 – form, v shows their builders (components). They are sometimes referred to as internal force factors.
4. The equilibrium condition of the right part is written.
Each separate type of internal force factors is associated with a separate type of deformation. For example, if only the normal force N is generated in the cross-sections of the bar, and the others are zero, then the bar is under the influence of tension or compression deformation. If only the shear (transverse) force (Q x or Q y ) acts on the cross-sections of the bar, then the shear deformation occurs,if only the torque M b acts on it, then torsion deformation occur. If only M x or M u acts onthe main moment generators in the cross section of the bar ,it is under pure bending deformation. Bar will bein complicated deformation position if several stress forces are acting on the cross section at the same time.The distribution of internal forces by section remains unknown. To solve this problem, it is necessary to check the deformation caused by the external force in the rod. Depending on the deformation, the distribution of internal forces on the cross-sectional surface of the rod is determined.
12-35. Shift. Stress, deformation, potential energy (testing stresses, net displacement, Hooke's law, absolute and relative displacement, potential energy). Shift Stress: In structural analysis, shift stress refers to the internal stress that develops within a material due to external loads or forces applied to the structure. Shift stress is a measure of the intensity of internal forces within the material.
Testing Stresses: Testing stresses refer to the forces or loads applied to a material or structure during experimental testing. These stresses are typically applied gradually or incrementally to measure the response of the material or structure under different loading conditions. Testing stresses can be tensile, compressive, shear, or a combination of these, depending on the type of test being performed.
Net Displacement: Net displacement, also known as total displacement, refers to the overall change in position or deformation of a material or structure resulting from applied forces. It is the vector sum of all displacements experienced by various points within the system. Net displacement accounts for both magnitude and direction and represents the final position relative to the initial position.
Hooke's Law: Hooke's Law is a fundamental principle in solid mechanics that describes the linear relationship between the stress applied to an elastic material and the resulting strain or deformation. According to Hooke's Law, the stress within the elastic limit is directly proportional to the strain. Mathematically, it can be expressed as:
σ = Eε
where σ represents stress, E is the modulus of elasticity (Young's modulus), and ε represents strain. Hooke's Law is valid for many materials as long as they remain within their elastic deformation range.
Absolute and Relative Displacement: Absolute displacement refers to the total change in position of a point or object relative to a fixed reference point or frame of reference. It considers the actual position of the point or object without any reference to other points or objects.
On the other hand, relative displacement refers to the change in position between two points or objects relative to each other. It considers the displacement of one point or object with respect to another. Relative displacement can be expressed in terms of distance, magnitude, and direction.
Potential Energy: In the context of mechanics, potential energy refers to the energy stored within a system due to its configuration or position. It is a form of stored energy that can be converted into other forms, such as kinetic energy or work.
In the case of deformation, potential energy is stored within an elastic material when it is subjected to external forces causing deformation. This potential energy is a result of the work done in deforming the material and is proportional to the amount of deformation. When the external forces are removed, the potential energy is released, and the material returns to its original state.
The potential energy stored in an elastic material can be calculated using the equation:
PE = (1/2) k Δx^2
where PE represents the potential energy, k is the spring constant (related to the material's stiffness), and Δx is the displacement or deformation from the equilibrium position.
13 Determining internal force factors in normal deformations and constructing their diagrams (longitudinal force, torque, transverse force, bending moment, equations, signs, diagram).
When analyzing normal deformations in structures, several internal force factors play a significant role. Here are the commonly considered factors along with their diagrams, equations, signs, and descriptions:
Longitudinal force (Axial force):
Diagram: A straight line along the length of the structure indicating tension or compression.
Equation: N = A * E * ε
Sign convention: Positive for tension, negative for compression.
Description: Longitudinal force refers to the force acting along the axis of the structure, causing it to elongate (tension) or contract (compression).
Torque (Torsional moment):
Diagram: A circular arrow indicating twisting or torque.
Equation: T = J * G * φ / L
Sign convention: Clockwise torque is considered positive (as viewed from the end of the structure), and counterclockwise torque is negative.
Description: Torque is the force that causes a structural member to rotate or twist about its longitudinal axis.
Transverse force (Shear force):
Diagram: A stepped line that changes abruptly at load application points.
Equation: V = -dM/dx
Sign convention: Positive when the upward force is on the left side of the structure.
Description: Transverse force represents the force acting perpendicular to the longitudinal axis of the structure, causing it to bend or shear.
Bending moment:
Diagram: A curved line that changes gradually.
Equation: M = -EI * d²y/dx²
Sign convention: Positive when the concave side of the bending moment diagram is upward.
Description: Bending moment refers to the internal force that causes the structural member to bend about an axis perpendicular to its longitudinal axis.
Note: In the equations above, E represents the modulus of elasticity, A is the cross-sectional area, J is the polar moment of inertia, G is the shear modulus, φ is the angle of twist, L is the length of the member, ε is the axial strain, dM/dx is the rate of change of bending moment with respect to the longitudinal axis, and d²y/dx² is the rate of change of the deflection curve.
