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Castigliano's second theorem states that the partial derivative of the total strain energy in an elastic system with respect to a force (or moment) is equal to the displacement (or rotation) at the point of application of that force (or moment) in the direction of the force (or moment).

The torsional shear stress for a circular shaft is $\tau_t = \frac{T r}{J}$, where $T$ is the torque, $r$ is the radius to the point of interest, and $J$ is the polar moment of inertia.

An influence line for a structural response (reaction, shear, moment, etc.) represents how that response varies as a point load moves across the structure and is a tool used for calculating maximum effects due to a series of loads moving across the structure.

Maxwell's reciprocal theorem states that the displacement at point A due to a unit force applied at point B is equal to the displacement at point B due to a unit force applied at point A in the direction of the first unit force.

Mohr's Circle is a graphical method to determine principal stresses, maximum shear stresses, and the orientation of the principal axes by plotting normal and shear stress components at a point in terms of a circle in the stress space.

The governing differential equation for a mass-spring-damper system is $m\ddot{x} + c\dot{x} + kx = F(t)$ where $m$ is mass, $c$ is the damping coefficient, $k$ is the spring constant, and $F(t)$ is the external force as a function of time.

The Portal Method is a simplified analysis technique for estimating the distribution of internal forces and moments in multi-story frame structures by making assumptions on the behavior of the frame under lateral loads.

For a rectangle, the moment of inertia about a centroidal axis parallel to the width is given by $I = \frac{bh^3}{12}$ where $b$ is the width and $h$ is the height of the rectangle.

Compatibility equations ensure that deformations in a structure do not result in gaps or overlaps. An example is $\epsilon_{xy} = \gamma_{xy}/2$, which relates the shear strain $\epsilon_{xy}$ to the engineering shear strain $\gamma_{xy}$.

The critical load for buckling is given by $P_{cr} = \frac{\pi^2 EI}{(KL)^2}$ where $E$ is Young's Modulus, $I$ is the moment of inertia, $K$ is the effective length factor, and $L$ is the actual length of the column.

The slenderness ratio of a column is given by $\lambda = \frac{K L}{r}$, where $K$ is the effective length factor, $L$ is the actual length, and $r$ is the radius of gyration of the cross-section, indicative of the column's tendency to buckle.

The bending stress in a beam is given by $\sigma_b = \frac{M}{S}$ where $M$ is the bending moment and $S$ is the section modulus of the beam's cross-section.

The deflection $y$ at any point along a beam can be calculated by integrating the bending moment equation twice, $y = \int \int \frac{M}{EI} dx^2$, assuming $E$ and $I$ are constant.

Strain is calculated as $\epsilon = \frac{\Delta L}{L_0}$, where $\Delta L$ is the change in length, and $L_0$ is the original length.

The general equations of equilibrium are $\sum F_x = 0$, $\sum F_y = 0$, and $\sum M = 0$, indicating that the sum of forces and moments in any static system must be zero.

The Hardy Cross Method, or moment distribution method, is an iterative process used to analyze statically indeterminate beams and frames. It involves distributing the moment at each joint based on the stiffness of members until convergence is achieved.

Young's Modulus relates stress and strain by $E = \frac{\sigma}{\epsilon}$ provided the material is in the elastic region.

Stress is given by $\sigma = \frac{F}{A}$ where $F$ is the force applied perpendicular to the surface and $A$ is the area over which the force is distributed.

Stresses in curved beams, particularly when the radius of curvature is comparable to the dimensions of the cross-section, are found using the equation $\sigma = \frac{My}{I + Ay^2/R}$ where $M$ is the bending moment, $I$ is the moment of inertia, $A$ is the area of the cross-section, $y$ is the distance from neutral axis, and $R$ is the radius of curvature.

Shear stress is defined as $\tau = \frac{V}{A}$, where $V$ is the shear force applied parallel to the surface area $A$.