Structural Analysis and Mechanics

Study lessons

1. 1

   Statics and Beam Analysis: Reactions, Trusses, and Shear-Moment Diagrams

   Master equilibrium, support reactions, truss member forces, and shear and bending-moment diagrams, the computational backbone of nearly every Structural Engineering and Construction item on the CELE.

   15 min
2. 2

   Mechanics of Materials: Stress, Strain, Torsion, and Beam Deflection

   Compute axial and thermal stress, bending stress, torsional shear stress, and beam deflection with the standard formulas the CELE tests most, each worked numerically in SI units.

   15 min
3. 3

   Indeterminate Structures

   Count static and kinematic indeterminacy for beams and frames, recall the standard fixed-end moments, and read off the settled results for propped cantilevers, fixed-fixed beams, and continuous beams.

   15 min
4. 4

   Analysis of Trusses and Frames

   Test a truss for static determinacy and stability with the m + r versus 2j count, solve member forces by the method of joints and the method of sections, spot zero-force members, tell compound from complex trusses, and find reactions and forces in simple determinate frames.

   15 min
5. 5

   Deflection of Beams

   Compute beam deflection by double integration, the moment-area and conjugate-beam methods, the standard maximum-deflection formulas, and superposition, each worked numerically the way the CELE asks it.

   15 min
6. 6

   Columns and Buckling

   Compute the Euler buckling load and critical stress, choose the effective length factor K for the standard end conditions, work the slenderness ratio and radius of gyration, and tell short, intermediate, and long columns apart, each worked numerically.

   15 min
7. 7

   Influence Lines and Moving Loads

   Build influence lines for reactions, shear, and bending moment in determinate beams, apply the Muller-Breslau shortcut, and locate the moving load or wheel train that produces the largest reaction, shear, and moment.

   15 min
8. 8

   Cables and Arches

   Track the constant horizontal pull in a cable, use the general cable theorem for concentrated loads, work the parabolic cable and its tension under a uniform load, and solve the three-hinged arch for its reactions and horizontal thrust, each carried to a number.

   15 min
9. 9

   Slope-Deflection and Moment Distribution

   Build fixed-end moments into the slope-deflection equations and the moment-distribution method, then solve a continuous beam with distribution and carry-over factors, worked all the way to numbers.

   15 min
10. 10

    Plastic Analysis of Structures

    Compute the plastic moment and shape factor, locate plastic hinges, find beam and simple-frame collapse loads by the mechanism (virtual work) method, and apply the lower- and upper-bound theorems, each worked numerically the way the CELE asks it.

    15 min
11. 11

    Approximate Methods for Building Frames

    Solve indeterminate building frames fast with assumed inflection points, using the portal method and the cantilever method for lateral loads and a 0.1L inflection-point model for girders under gravity load.

    15 min
12. 12

    Energy Methods in Structural Analysis

    Store and recover load as strain energy in axial, bending, and torsional members, then find beam and truss deflections with the unit-load (virtual work) method and Castigliano's second theorem, each worked to a number.

    15 min
13. 13

    Introduction to the Matrix Stiffness Method

    Build the member stiffness matrix for a spring and an axial bar, number the degrees of freedom, assemble the structure stiffness matrix by the direct stiffness method, then solve a two-element assembly for its displacements, member forces, and reactions.

    15 min
14. 14

    Structural Dynamics Basics

    Build the single-degree-of-freedom oscillator, compute its natural frequency and period, describe undamped free vibration, sort out damping and damped vibration, and pin down resonance and the dynamic magnification factor, each worked numerically the way the CELE asks it.

    15 min
15. 15

    Combined Loading of Members

    Superpose axial force, bending, and torsion into the normal and shear stresses at a point, then read the principal stresses and the maximum shear stress off Mohr circle, each step worked to a number.

    15 min
16. 16

    Force Method and Consistent Deformations

    Build the force method from redundant selection and the primary structure to compatibility equations and flexibility coefficients, then solve the propped cantilever and two-span continuous beam by consistent deformations, each worked to a number with units.

    15 min
17. 17

    Torsion and Shear Center

    Master torsion of circular shafts with T/J = tau/r = G theta/L, size shafts by allowable stress and transmitted power, and locate the shear center of thin-walled open sections, every step carried to a number with units.

    15 min
18. 18

    Space Trusses and Three-Dimensional Frames

    Classify a space truss with the m + r = 3j count, resolve member forces with direction cosines, march through the method of joints in three dimensions, solve a simple tripod leg by leg, and count the six reaction components a space structure needs to stand up.

    15 min
19. 19

    Shear Walls and Lateral Load Distribution

    Find the rigidity of a shear wall from its height to length ratio, split a story shear between walls by relative rigidity, tell rigid from flexible diaphragms, locate the center of rigidity, and add the torsional shear when the mass and stiffness centers do not line up, each carried to a number with units.

    15 min
20. 20

    Conjugate Beam Method

    Turn deflection into a statics problem by loading an imaginary beam with the M/EI diagram, so conjugate shear returns the real slope and conjugate moment returns the real deflection, each worked to a number for the cantilever and the simple beam.

    15 min
21. 21

    Unit Load Method for Deflections

    Find one deflection or rotation at one point with the principle of virtual work, pairing a real force system against a virtual unit-load system through the beam integral of mM over EI and the truss sum of nNL over AE, each carried to a number.

    15 min
22. 22

    Arches and Funicular Structures

    Read the funicular idea that shapes cables and arches, track the horizontal thrust that defines an arch, solve the three-hinged parabolic arch for its reactions and thrust, and get the bending moment at any section from M = M_beam - H y, each carried to a number with units.

    15 min
23. 23

    Beams on Elastic Foundation

    Model soil as a Winkler spring bed, compute the modulus of subgrade reaction, the characteristic parameter beta and characteristic length, and the deflection, moment, and soil pressure under a point load on a long beam, each worked to a number with units.

    15 min
24. 24

    Three-Moment Equation

    Clapeyron's three-moment equation for continuous beams, with the load terms for uniform and point loads, support settlement terms, and worked support moments for equal and unequal two-span beams solved to numbers with units.

    15 min
25. 25

    Castigliano Theorems

    Store load as recoverable strain energy, then read a deflection or a force straight out of it with Castigliano two theorems, differentiating inside the beam integral and the truss sum and adding a dummy load where no real load sits, each carried to a number.

    15 min
26. 26

    Buckling of Frames and Effective Length

    Review Euler buckling, tell braced non-sway frames from unbraced sway frames, build the joint stiffness ratio G, read the alignment charts for the effective length factor K, compute the critical load of a column in a frame, and note inelastic buckling, each worked to a number with units.

    15 min
27. 27

    Shear Flow in Thin-Walled Sections

    Build shear flow q = VQ/I from equilibrium, turn it into shear stress tau = VQ/(I t) in flanges and webs, size the fasteners of a built-up beam from the connector spacing s = n F / q, and locate the shear center of a channel, every step carried to a number with units.

    15 min

Section practice test

35 test sets covering the whole Structural Analysis and Mechanics section, 1050 original questions with explanations. After you finish the lessons above, prove it here. Randomized every run.