Fundamental Structural Analysis Equations

Why This Matters

Structural analysis equations form the mathematical backbone of everything you'll do in civil engineering—from sizing a simple beam to analyzing complex bridge systems. These aren't just formulas to memorize; they represent fundamental physical principles about how structures resist loads, distribute forces, and deform under stress. You're being tested on your ability to recognize which equation applies to which situation, understand the assumptions behind each method, and know when those assumptions break down.

The equations in this guide fall into distinct categories: some describe material behavior, others govern equilibrium and compatibility, and still others provide analysis techniques for complex structures. Don't just memorize $\sigma = E\varepsilon$—understand that it only works in the elastic range and why that matters for design. When you see an indeterminate structure on an exam, you should immediately think "energy methods or moment distribution?" Connect each equation to its purpose, and you'll handle any problem thrown at you.

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Equilibrium and Force Balance

These equations express the most fundamental requirement of structural mechanics: a structure at rest must have all forces and moments in balance. Everything else builds on this foundation.

Equilibrium Equations

• $\Sigma F = 0$ and $\Sigma M = 0$—the foundation of all static analysis, stating that forces and moments must sum to zero in every direction
• Three independent equations in 2D ($\Sigma F_x$, $\Sigma F_y$, $\Sigma M$) determine whether a structure is statically determinate or requires additional methods
• Applies universally to free body diagrams at any scale—from entire structures down to infinitesimal elements

Shear and Moment Diagram Relationships

• $\frac{dV}{dx} = -w(x)$ and $\frac{dM}{dx} = V(x)$—these differential relationships connect distributed load, shear, and moment along a beam
• Critical points for maximum moment occur where shear equals zero, essential for identifying design-governing locations
• Area under curves provides a graphical integration method—the area under the shear diagram equals the change in moment

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Material Behavior and Stress-Strain

These equations describe how materials respond to loading—the constitutive relationships that connect force to deformation at the material level.

Hooke's Law

• $\sigma = E\varepsilon$—relates stress to strain through the modulus of elasticity $E$, the fundamental linear elastic relationship
• Only valid below the yield point—once you exceed the elastic limit, permanent deformation occurs and this equation no longer applies
• Material stiffness is captured by $E$; steel ($E \approx 200$ GPa) is roughly 7× stiffer than aluminum ($E \approx 70$ GPa)

Stress Transformation Equations

• $\sigma_{x'} = \frac{\sigma_x + \sigma_y}{2} + \frac{\sigma_x - \sigma_y}{2}\cos(2\theta) + \tau_{xy}\sin(2\theta)$—transforms normal stress to any rotated coordinate system
• Shear stress transformation follows a similar form, essential for finding planes of maximum shear
• Principal stresses occur on planes where shear stress equals zero—critical for applying failure criteria

Mohr's Circle Equations

• Graphical representation of stress transformation where the circle's center is $\frac{\sigma_x + \sigma_y}{2}$ and radius gives maximum shear stress
• Principal stresses appear as the circle's intersection with the horizontal axis—no calculation needed once the circle is drawn
• Maximum shear stress equals the circle's radius: $\tau_{max} = \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2}$

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Beam Bending and Deflection

These equations govern how beams curve and deflect under load—connecting applied moments to the geometry of deformation.

Moment-Curvature Relationship

• $M = EI\kappa$—links bending moment to curvature, where $EI$ is the flexural rigidity of the cross-section
• $\kappa = \frac{1}{\rho}$ where $\rho$ is the radius of curvature—larger moments create tighter curves (smaller radius)
• Increasing $I$ (moment of inertia) reduces curvature for the same moment—this is why I-beams are so efficient

Euler-Bernoulli Beam Equation

• $EI\frac{d^4v}{dx^4} = w(x)$—the governing differential equation relating deflection $v$ to distributed load $w$
• Key assumption: plane sections remain plane and perpendicular to the neutral axis, valid for slender beams where shear deformation is negligible
• Double integration of the moment-curvature relationship yields slope and deflection—boundary conditions determine integration constants

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Energy Methods

Energy principles provide powerful alternative approaches to structural analysis—particularly useful for deflection calculations and indeterminate structures.

Principle of Superposition

• Total response equals the sum of individual responses—analyze each load separately, then add results algebraically
• Only valid for linear elastic systems—material must follow Hooke's Law and displacements must be small
• Dramatically simplifies complex loading by breaking problems into standard cases with known solutions

Virtual Work Equation

• $W_{external} = W_{internal}$—external work by applied forces equals internal work by stresses and strains
• Virtual displacements are imaginary, infinitesimal, kinematically admissible movements used to extract unknown forces or displacements
• Particularly powerful for calculating deflections at specific points without solving the entire deflection curve

Castigliano's Theorems

• First theorem: $\delta_i = \frac{\partial U}{\partial P_i}$—displacement in the direction of load $P_i$ equals the partial derivative of strain energy with respect to that load
• Second theorem applies to complementary energy and is used for force calculations in redundant structures
• Dummy load method extends Castigliano's theorem to find deflections at points where no load exists

Maxwell-Betti Reciprocal Theorem

• Flexibility coefficients are symmetric: $f_{ij} = f_{ji}$—deflection at point $i$ due to unit load at $j$ equals deflection at $j$ due to unit load at $i$
• Reduces computational effort by exploiting symmetry in flexibility matrices
• Validates results—if your calculated flexibility matrix isn't symmetric, you've made an error

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Indeterminate Structure Analysis

These methods solve structures with more unknowns than equilibrium equations—essential for analyzing real-world continuous beams and frames.

Slope-Deflection Equations

• $M_{AB} = \frac{2EI}{L}(2\theta_A + \theta_B - 3\psi) + FEM_{AB}$—relates end moments to joint rotations $\theta$, chord rotation $\psi$, and fixed-end moments
• Displacement method approach: unknowns are joint displacements and rotations, equations enforce equilibrium at joints
• Systematic for frames with sidesway—chord rotation terms capture the effect of lateral displacement

Moment Distribution Method

• Iterative relaxation technique that distributes unbalanced moments at joints until equilibrium is achieved
• Distribution factors $DF = \frac{K}{\Sigma K}$ determine how moment is split among members meeting at a joint, where $K = \frac{I}{L}$
• Carry-over factors (typically 0.5 for prismatic members) propagate moment to the far end of each member

Three-Moment Equation

• $M_A L_1 + 2M_B(L_1 + L_2) + M_C L_2 = -6\left(\frac{A_1 \bar{a}_1}{L_1} + \frac{A_2 \bar{b}_2}{L_2}\right)$—relates moments at three consecutive supports
• Specifically designed for continuous beams—each equation connects three adjacent supports
• Support settlement can be incorporated by adding terms to the right side of the equation

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Moving Loads and Variable Loading

These tools address loads that change position—critical for bridge design and any structure subjected to traffic or crane loads.

Influence Line Equations

• Graphical function showing how a response (reaction, shear, moment) varies as a unit load moves across the structure
• Müller-Breslau principle: the influence line for any response equals the deflected shape caused by removing the constraint associated with that response
• Maximum effects from distributed loads equal the load intensity times the area under the influence line—essential for bridge design

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Quick Reference Table

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Self-Check Questions

1. Both Castigliano's theorem and the virtual work method can find deflections—what distinguishes when you'd choose one over the other, and what assumption must be true for both?

2. You're analyzing a three-span continuous beam. Which two methods could you use, and how does the three-moment equation differ from moment distribution in its approach?

3. A stress element shows $\sigma_x = 100$ MPa, $\sigma_y = -50$ MPa, and $\tau_{xy} = 40$ MPa. Using Mohr's circle concepts, explain how you'd find the principal stresses and maximum shear stress.

4. Compare the Euler-Bernoulli beam equation with the moment-curvature relationship: which is more fundamental, and under what beam geometry would Euler-Bernoulli assumptions break down?

5. An FRQ asks you to find the maximum moment in a bridge girder due to a moving truck load. Explain why you'd use an influence line rather than constructing multiple shear-moment diagrams, and describe how you'd apply the Müller-Breslau principle.