Sources and References

Primary texts — Hibbeler, R. C. Structural Analysis, 10th ed. Pearson, 2017. Kassimali, A. Structural Analysis, 6th ed. Cengage, 2019.

Supplementary texts — McGuire, W., Gallagher, R. H., and Ziemian, R. D. Matrix Structural Analysis, 2nd ed. Wiley, 1999. Ghali, A., Neville, A. M., and Brown, T. G. Structural Analysis: A Unified Classical and Matrix Approach, 7th ed. CRC Press, 2017. West, H. H. and Geschwindner, L. F. Fundamentals of Structural Analysis, 2nd ed. Wiley, 2002.

Online resources — MIT OpenCourseWare 1.571 Structural Analysis and Control; MIT OCW 1.050 Engineering Mechanics I; Stanford CE-221 lecture archives; American Society of Civil Engineers open publications on analysis practice; SAP2000 and Abaqus technical reference manuals (publicly available chapters).

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Chapter 1: Determinacy, Indeterminacy, and Stability

1.1 Static Determinacy

A structure is statically determinate if its reactions and internal forces can be found from equilibrium alone. For a two-dimensional frame with \(r\) reaction components, \(b\) members, and \(j\) joints, determinacy of a frame is typically

\[ 3b + r = 3j + c, \]

where \(c\) counts internal condition equations (pins, rollers within members). If the left side exceeds the right, the structure is indeterminate to the degree of the excess. A truss uses \(b + r = 2j\).

1.2 Stability

Counting equations is necessary but not sufficient. A structure can be geometrically unstable if its constraints cannot resist motion for some loading direction. Three reactions that are concurrent or parallel give an unstable two-dimensional frame even though the count is correct. Stability must be checked by inspection or by the non-singularity of the stiffness matrix in matrix methods.

1.3 Why Indeterminacy Matters

Indeterminate structures are typical in modern buildings because redundancy provides alternative load paths and smaller member forces. The same loading produces roughly half the maximum bending moment in a continuous beam over three supports as in a simply supported beam of the same span. The cost is that analysis requires compatibility, not only equilibrium, and that support settlement, temperature, and fabrication tolerances all induce internal forces.

Chapter 2: Force (Flexibility) Method

2.1 Principle

Select the primary structure by removing redundants equal in number to the degree of indeterminacy. Compute the displacements at the released locations under the applied load (\(\delta_{iL}\)) and under unit redundants (\(\delta_{ij}\)). Impose compatibility:

\[ \delta_{iL} + \sum_j \delta_{ij}X_j = 0, \]

where \(X_j\) are the unknown redundants. The resulting linear system yields the redundants; all other forces follow from equilibrium.

2.2 Unit Load Method for Displacements

The flexibility coefficients \(\delta_{ij}\) are computed by the principle of virtual work. For a frame structure in bending,

\[ \delta_{ij} = \int_0^L\frac{M_i M_j}{EI}\,dx, \]

where \(M_i\) and \(M_j\) are moment diagrams under unit loads at locations \(i\) and \(j\). Axial and shear deformation contributions are added for trusses and deep beams.

2.3 Worked Propped Cantilever

A cantilever of length \(L\) with uniform load \(w\) and a prop at the free end is once indeterminate. Choose the prop reaction \(R\) as redundant. Primary structure: cantilever loaded by \(w\) and upward \(R\) at the tip. Free-end deflection from \(w\) downward is \(wL^4/(8EI)\); from unit \(R\) upward is \(L^3/(3EI)\). Compatibility: \(-wL^4/(8EI) + R\,L^3/(3EI) = 0\), giving \(R = 3wL/8\). The fixed-end moment follows as \(-wL^2/8\) and the prop-end moment is zero.

Chapter 3: Displacement (Stiffness) Method

3.1 Principle

Identify the kinematic unknowns (joint rotations and translations that are not prescribed). Write equilibrium at each unknown degree of freedom in terms of member stiffness. The system \(\mathbf{K}\mathbf{u} = \mathbf{F}\) is solved for displacements; member forces follow.

3.2 Slope-Deflection Equations

For a prismatic beam member between joints A and B of length \(L\) with constant \(EI\), the end moments for rotations \(\theta_A, \theta_B\) and chord rotation \(\psi = \Delta/L\) are

\[ M_{AB} = \frac{2EI}{L}(2\theta_A + \theta_B - 3\psi) + M^F_{AB}, \]\[ M_{BA} = \frac{2EI}{L}(\theta_A + 2\theta_B - 3\psi) + M^F_{BA}, \]

where \(M^F\) are fixed-end moments from intermediate loads. Equilibrium at each joint sums end moments to zero or to an applied couple. The stiffness approach reduces to the slope-deflection equations for continuous beams and rigid frames without sway.

3.3 Moment Distribution

Hardy Cross’s moment distribution method iteratively balances joint moments. At each joint, the unbalanced moment is distributed among connecting members in proportion to their stiffness, and half of each distributed moment is carried over to the far end of each member. The process converges rapidly for most structures and was the standard hand method before computers.

Chapter 4: Influence Lines

4.1 Influence Lines for Determinate Structures

An influence line plots the value of a given response (reaction, shear, moment) as a unit load moves across the structure. For a simply supported beam of length \(L\), the influence line for the moment at midspan is triangular with peak \(L/4\) at midspan.

Maximum response under a moving system of wheel loads is obtained by positioning the loads to maximize the sum of (load × ordinate) over the influence line. For AASHTO truck loads on highway bridges, this positioning is standardized.

4.2 Influence Lines for Indeterminate Structures

Müller-Breslau’s principle states that the influence line for any response in a linear structure is the deflected shape produced by releasing the corresponding constraint and imposing a unit relative displacement. For the midspan moment in a two-span continuous beam, release the moment at midspan (insert a hinge), rotate one side by a unit angle relative to the other, and the resulting deflected shape, measured perpendicular to the beam axis, is the influence line.

This is conceptually elegant but requires the solution of an indeterminate problem, which in practice is done by computer.

Chapter 5: The Matrix Stiffness Method

5.1 Member Stiffness Matrix

For a plane frame member between nodes 1 and 2 of length \(L\), cross section \(A\), and moment of inertia \(I\), with local axes aligned along the member, the local \(6\times 6\) stiffness matrix relates end forces \(\{F_x^1, F_y^1, M^1, F_x^2, F_y^2, M^2\}\) to end displacements. Its non-zero structure is

\[ \mathbf{k}_{local} = \begin{bmatrix} EA/L & 0 & 0 & -EA/L & 0 & 0 \\ 0 & 12EI/L^3 & 6EI/L^2 & 0 & -12EI/L^3 & 6EI/L^2 \\ 0 & 6EI/L^2 & 4EI/L & 0 & -6EI/L^2 & 2EI/L \\ -EA/L & 0 & 0 & EA/L & 0 & 0 \\ 0 & -12EI/L^3 & -6EI/L^2 & 0 & 12EI/L^3 & -6EI/L^2 \\ 0 & 6EI/L^2 & 2EI/L & 0 & -6EI/L^2 & 4EI/L \end{bmatrix}. \]

5.2 Coordinate Transformation

A rotation matrix \(\mathbf{T}\) transforms local to global coordinates. The member stiffness in global coordinates is \(\mathbf{k} = \mathbf{T}^T\mathbf{k}_{local}\mathbf{T}\).

5.3 Assembly and Solution

The global stiffness matrix is assembled by summing member contributions at shared degrees of freedom. Applying boundary conditions (fixing rows and columns at restrained DOFs), the reduced system is solved for unknown displacements. Member forces are recovered by \(\mathbf{f} = \mathbf{k}\mathbf{u}\) member by member.

5.4 Numerical Considerations

The stiffness matrix is symmetric and sparse but generally not diagonally dominant. Banded and skyline solvers exploit its sparsity; modern frame analysis packages use sparse direct solvers such as LU or Cholesky decomposition. Condition number deteriorates for structures with widely disparate member stiffnesses or near-mechanism configurations; warnings of ill conditioning should always be taken seriously.

Chapter 6: Approximate Methods for Frames

Before computers, and still for preliminary design, approximate methods allow hand calculation of indeterminate frames.

6.1 Portal Method for Lateral Loads

For rectangular frames under lateral load, assume inflection points at the midheight of columns and midspan of beams, and distribute the horizontal shear so that interior columns take twice the shear of exterior columns. This gives a statically determinate frame with moments accurate to about 15% for typical low-rise frames.

6.2 Cantilever Method

For taller frames, the cantilever method assumes that axial stresses in columns vary linearly with distance from the centroid of the columns, like the flexure formula for a solid beam. This captures the flexural cantilever action that dominates tall-building behaviour better than the portal method.

Chapter 7: Trusses and Arches

7.1 Determinate Truss Analysis

Method of joints and method of sections find member forces in determinate trusses. Zero-force members are identified by inspection and reduce computation.

7.2 Deflection of Trusses

For a truss, virtual work gives joint deflection

\[ \Delta = \sum \frac{n_i N_i L_i}{A_i E_i}, \]

where \(N_i\) are forces under the applied load, \(n_i\) are forces under a unit load at the deflection location and direction, \(L_i\) are lengths, and \(A_i E_i\) are axial rigidities.

7.3 Arches

Three-hinged arches are determinate and analyzed directly. Two-hinged and fixed arches are indeterminate, analyzed by the force method. The horizontal reaction \(H\) of a parabolic three-hinged arch of span \(L\) and rise \(h\) under uniform load \(w\) is \(H = wL^2/(8h)\), balanced by compression that flows along the arch axis.

Chapter 8: Computer Applications and Interpretation

8.1 Modelling Choices

Commercial software (SAP2000, ETABS, RISA, RFEM, Abaqus) shields the engineer from the algebra but raises the stakes for modelling choices: rigid versus pinned connections, effective flange widths, rigid diaphragms, supports that approximate the real behaviour, and so on. An incorrect model returns wrong answers in plausible-looking plots.

8.2 Verification

Every computer analysis should be verified against hand calculations at the level of gross behaviour: total applied load equals total reaction, maximum moments agree with back-of-envelope approximations, deflections match span-over-depth ratios expected for the system. A beam spanning 6 m that deflects 2 mm under a 1 kN point load at midspan implies unrealistic stiffness and should trigger investigation.

8.3 Nonlinear Effects

Most practical analysis is linear elastic, but second-order (P-delta) effects, material nonlinearity, and construction-stage analysis are increasingly required by codes for tall or slender structures. The stiffness method extends naturally: at each load increment the tangent stiffness matrix is updated, and iterative methods such as Newton-Raphson converge to the nonlinear equilibrium.