Unit
- Equilibrium equations: A structure in static equilibrium must satisfy sum of forces in x, y, and sum of moments equal to zero. This principle is the basis for all analysis methods. In two-dimensional problems, three equations are available for each free body diagram.
- Types of supports: Supports restrict degrees of freedom. A pinned support prevents translation in two directions but allows rotation. A roller support prevents translation perpendicular to its base only. A fixed support prevents all translation and rotation, providing three reaction components.
- Stability and determinacy: A structure is externally stable if its supports prevent rigid body motion. For a plane structure, the number of unknown reactions must equal the number of equilibrium equations for static determinacy. If reactions exceed three, the structure is externally indeterminate.
- Free body diagrams: Isolating a structural element by removing supports and replacing them with reaction forces enables the application of equilibrium equations. Correctly drawing free body diagrams is a critical first step in any analysis.
- Internal forces: Axial force, shear force, and bending moment are the three internal stress resultants. These vary along the length of a member and are determined by cutting sections and applying equilibrium to the subassembly.
- Sign conventions: Standard sign conventions for internal forces must be consistently applied. Positive axial force corresponds to tension, positive shear causes clockwise rotation of the element, and positive bending moment compresses the top fiber.
- Load types: Concentrated loads act at a point, distributed loads spread over an area or length, and moments cause rotation without translation. Each load type produces distinct internal force distributions.
- Idealization of structures: Real structures are idealized using centerline dimensions, rigid joints, and simplified support conditions. This idealization is essential for practical analysis while maintaining accuracy within engineering tolerance.
- Superposition principle: For linear elastic structures, the total response due to multiple loads equals the sum of responses from each load applied separately. This principle enables complex load cases to be broken into simpler components. It holds only when material behavior is linear and deformations are small.
- Method of sections: Used to determine internal forces in specific members of a truss. A cut is made through the truss, isolating a portion, and equilibrium equations are applied to the resulting free body diagram. This method is efficient for finding forces in a few members without solving the entire structure.
- Influence lines: A graphical representation of how a response quantity, such as reaction, shear, or moment, varies as a unit load moves across the structure. Influence lines are crucial for moving load analysis, like on bridges, to determine critical load positions that produce maximum effects for design.
Code Example
Python
import numpy as np
# Example: Solve for reactions of a simply supported beam
# Beam length L, point load P at distance a from left support
L = 10.0 # meters
P = 50.0 # kN
a = 4.0 # meters from left support
# Equilibrium equations: sum vertical forces, sum moments
# R_A + R_B = P
# R_B * L - P * (L - a) = 0 (moment about A)
A = np.array([[1, 1], [0, L]])
b = np.array([P, P * (L - a)])
reactions = np.linalg.solve(A, b)
print('R_A = {:.2f} kN'.format(reactions[0]))
print('R_B = {:.2f} kN'.format(reactions[1]))This script solves for vertical reactions of a simply supported beam with a single point load. The coefficient matrix encodes the equilibrium equations, and NumPy's linear algebra solver returns the reaction forces.
System Architecture
Pattern: Free Body Diagram Decomposition
Components:
- Structure: The entire assembly, idealized as sticks and nodes.
- Supports: Points where the structure contacts the ground, providing reactions.
- Free Body: Subset of the structure isolated for equilibrium analysis.
- Forces: Applied loads and reaction components drawn on the free body.
Quick Revision
- Equilibrium requires sum of forces and moments equal zero.
- Support types (pinned, roller, fixed) provide different reaction components.
- Static determinacy occurs when unknown reactions equal equilibrium equations.
- Free body diagrams are essential for isolating and analyzing structural members.