4.6 Indeterminate Structures

4.6 Indeterminate Structures

Introduction to Indeterminate Analysis

• Statically indeterminate structures have more unknown forces than available equilibrium equations. Their analysis requires additional conditions of compatibility of displacements.

• These structures offer advantages like increased redundancy, better distribution of loads, and often greater stiffness, but require more sophisticated analysis methods.

• This section covers the fundamental force and displacement methods for analyzing indeterminate structures, their application to arches and continuous beams, and an introduction to plastic analysis.

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1. Force Methods

1.1 Flexibility Method (Force Method)

• Core Concept: Also known as the Compatibility Method. The structure's redundant restraints are removed to create a statically determinate primary structure. The redundants are then treated as unknown forces, and their magnitudes are found by enforcing compatibility conditions (zero displacement) at the locations of the removed restraints.

• General Procedure:

  1. Determine Degree (n): Find the degree of static indeterminacy $\boldsymbol{n}$.

  2. Choose Redundants: Select and remove $\boldsymbol{n}$ redundant forces/moments (reactions or internal forces) to form the primary determinate structure.

  3. Analyze Primary Structure:

     • Under actual loads: Calculate displacements ($\boldsymbol{\Delta_{i0}}$) at redundant locations.

     • Under unit value of each redundant: Calculate flexibility coefficients ($\boldsymbol{\delta_{ij}}$). $\boldsymbol{\delta_{ij}}$ is the displacement at location $i$ due to a unit redundant at location $j$.

  4. Write Compatibility Equations: For each removed restraint, the total displacement must be zero (or a known value, e.g., support settlement).

     $\boldsymbol{\delta_{11}X_1 + \delta_{12}X_2 + ... + \delta_{1n}X_n + \Delta_{1P} = \Delta_1}$ $\boldsymbol{\delta_{21}X_1 + \delta_{22}X_2 + ... + \delta_{2n}X_n + \Delta_{2P} = \Delta_2}$ $\boldsymbol{...}$ $\boldsymbol{\delta_{n1}X_1 + \delta_{n2}X_2 + ... + \delta_{nn}X_n + \Delta_{nP} = \Delta_n}$

     where $\boldsymbol{X_i}$ are the unknown redundants, $\boldsymbol{\Delta_{iP}}$ are displacements due to real loads, and $\boldsymbol{\Delta_i}$ are actual displacements at supports (usually zero).

  5. Solve for Redundants: Solve the system of $\boldsymbol{n}$ equations for $\boldsymbol{X_1, X_2, ..., X_n}$.

  6. Find Final Forces: Use superposition: $\boldsymbol{F = F_P + \sum (X_i \cdot f_i)}$, where $\boldsymbol{F_P}$ is force in primary structure due to loads, and $\boldsymbol{f_i}$ is force due to unit redundant $\boldsymbol{X_i=1}$.

1.2 Application: Two-Hinged Parabolic Arches

• Indeterminacy: A two-hinged arch is indeterminate to the first degree.

• Common Approach: The horizontal thrust (H) at one support is chosen as the redundant.

• Primary Structure: A simply supported curved beam (the arch with one horizontal support reaction released).

• Compatibility Condition: The horizontal displacement at the released support must be zero.

• Formula for Thrust (Neglecting Axial Deformation): Using the Flexibility Method and virtual work, the thrust is derived as:

  $\boldsymbol{H = \frac{\int \frac{M^0 y}{EI} \, ds}{\int \frac{y^2}{EI} \, ds}}$

  where $\boldsymbol{M^0}$ is the bending moment in a simply supported beam of the same span, $\boldsymbol{y}$ is the vertical coordinate of the arch axis, and $\boldsymbol{ds}$ is a length element along the arch.

• For a Parabolic Arch under a uniformly distributed load (UDL) $\boldsymbol{w}$ over the entire span $\boldsymbol{L}$ with rise $\boldsymbol{h}$, the formula simplifies significantly. The funicular shape for UDL is parabolic, leading to:

  $\boldsymbol{H = \frac{w L^2}{8h}}$

  and the bending moment throughout the arch becomes zero (pure compression). For other load cases, bending moments will develop.

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2. Displacement Methods

2.1 Slope Deflection Method

• Core Concept: A classical displacement method where the unknowns are the rotations (slopes) and relative displacements (settlements) of the joints. Equations are written by expressing the end moments of each member in terms of these displacements and then applying joint equilibrium.

• Slope-Deflection Equation: For a prismatic member $\boldsymbol{AB}$ of length $\boldsymbol{L}$, flexural rigidity $\boldsymbol{EI}$, with end rotations $\boldsymbol{\theta_A}$, $\boldsymbol{\theta_B}$, and a relative chord rotation (settlement) $\boldsymbol{\psi = \Delta / L}$:

  $\boldsymbol{M_{AB} = \frac{2EI}{L} \left( 2\theta_A + \theta_B - 3\psi \right) + M_{AB}^F}$ $\boldsymbol{M_{BA} = \frac{2EI}{L} \left( \theta_A + 2\theta_B - 3\psi \right) + M_{BA}^F}$

  where $\boldsymbol{M_{AB}^F}$ and $\boldsymbol{M_{BA}^F}$ are the fixed-end moments due to transverse loads on the member considered fully fixed at both ends.

• Procedure:

  1. Identify unknown displacements (joint rotations, independent joint translations).

  2. Write slope-deflection equations for each member.

  3. Apply joint equilibrium equations ($\boldsymbol{\sum M_{joint} = 0}$) at each joint with unknown rotation.

  4. Apply shear equilibrium equations for storeys with sidesway to get equations for translations.

  5. Solve the system of equations for the unknown displacements.

  6. Back-substitute displacements into slope-deflection equations to find final end moments.

2.2 Moment Distribution Method (Hardy Cross Method)

• Core Concept: An iterative relaxation technique to solve the slope-deflection equations without solving large systems simultaneously. It is particularly efficient for structures without sidesway.

• Key Definitions:

  • Stiffness Factor ($\boldsymbol{K}$): Moment required to produce a unit rotation at one end of a member while the other end is fixed. For a prismatic member, $\boldsymbol{K = 4EI/L}$.

  • Distribution Factor (DF): The fraction of an unbalanced moment at a joint that is distributed to each connecting member.

    $\boldsymbol{DF_{member} = \frac{K_{member}}{\sum K_{joint}}}$

    The sum of DFs at a joint is 1.

  • Carry-Over Factor (COF): The fraction of a moment applied at one end of a member that is "carried over" to the far fixed end. For a prismatic member, $\boldsymbol{COF = 1/2}$.

• Procedure (No Sidesway):

  1. Lock Joints: Assume all joints are fixed against rotation. Calculate fixed-end moments (FEMs) for all loaded members.

  2. Unlock & Distribute: Unlock one joint at a time. The unbalanced moment (sum of FEMs at the joint) is distributed to members according to their DFs.

  3. Carry Over: Half of each distributed moment is carried over to the far end of the member.

  4. Iterate: Repeat steps 2-3 for all joints until the unbalanced moments become negligible.

  5. Summation: Sum all moments (FEMs + distributed moments + carried-over moments) at each end to get the final end moments.

2.3 Stiffness Method (Matrix Analysis)

• Core Concept: The most general and computer-friendly displacement method. The structure's behavior is represented in matrix form: $\boldsymbol{[K]\{D\} = \{F\}}$.

• Key Equation: The global equilibrium equation.

  $\boldsymbol{\{F\} = [K] \{D\}}$

  where $\boldsymbol{\{F\}}$ is the vector of nodal forces, $\boldsymbol{[K]}$ is the global stiffness matrix, and $\boldsymbol{\{D\}}$ is the vector of nodal displacements.

• Procedure:

  1. Discretize the structure into members (elements).

  2. For each element, formulate the element stiffness matrix $\boldsymbol{[k]}$ in local coordinates.

  3. Transform $\boldsymbol{[k]}$ to global coordinates and assemble into the global stiffness matrix $\boldsymbol{[K]}$.

  4. Apply boundary conditions (support restraints) to modify $\boldsymbol{[K]}$ and $\boldsymbol{\{F\}}$.

  5. Solve the system of linear equations $\boldsymbol{[K]\{D\} = \{F\}}$ for the unknown displacements $\boldsymbol{\{D\}}$.

  6. For each element, use $\boldsymbol{\{d\} = [T]\{D\}}$ and $\boldsymbol{\{f\} = [k]\{d\}}$ to find member end forces.

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3. Influence Lines for Continuous Beams

• Influence lines for indeterminate structures like continuous beams cannot be found by simple statics.

• Construction Methods:

  1. Müller-Breslau Principle (Directly Applicable): For an indeterminate structure, the influence line for a force/reaction is the deflected shape of the structure when the corresponding restraint is removed and a unit displacement is applied. The shape must be determined using an indeterminate analysis method (e.g., Moment Distribution).

  2. Flexibility Method: Place a moving unit load, and for each position, solve the indeterminate structure to find the value of the desired response function. Plotting these values gives the IL.

• Characteristics: ILs for continuous beams are curved (not straight-line segments) over each span. They help determine load patterns for maximum positive and negative moments in different spans.

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4. Elementary Plastic Analysis

4.1 Concept of Plastic Hinge and Collapse

• Plastic Hinge: A section in a structure that has yielded sufficiently to develop its full plastic moment capacity ($\boldsymbol{M_p}$) and can rotate indefinitely at constant moment, like a real hinge but with moment resistance.

• Plastic Moment ($\boldsymbol{M_p}$): The maximum moment a cross-section can resist, based on the yield stress ($\boldsymbol{\sigma_y}$) of the material and the plastic section modulus ($\boldsymbol{Z}$).

  $\boldsymbol{M_p = \sigma_y \cdot Z}$

• Collapse Mechanism: A configuration of sufficient plastic hinges that turns the structure (or part of it) into a kinematic mechanism, leading to collapse.

4.2 Theorems of Plastic Analysis

• Lower Bound (Static) Theorem: A load computed from a statically admissible bending moment distribution that nowhere exceeds $\boldsymbol{M_p}$ is less than or equal to the true collapse load.

• Upper Bound (Kinematic) Theorem: A load computed from an assumed collapse mechanism by equating external work to internal work at plastic hinges is greater than or equal to the true collapse load.

• Uniqueness Theorem: A load that satisfies both the static and kinematic theorems is the exact collapse load.

4.3 Method of Analysis

• The most common approach is the Kinematic Method:

  1. Identify possible collapse mechanisms (beam, panel, joint mechanisms).

  2. For each mechanism, apply the virtual work principle:

     $\boldsymbol{\sum (P \cdot \delta) = \sum (M_p \cdot \theta)}$

     where the left side is external work done by loads and the right side is internal energy dissipation at plastic hinges.

  3. Solve for the load ($\boldsymbol{P}$) corresponding to each mechanism.

  4. The lowest load among all possible mechanisms is the true collapse load.

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Summary

• Flexibility Method is a force method where redundants are chosen and found by enforcing compatibility. It is well-suited for structures with low indeterminacy like two-hinged arches, where the thrust formula simplifies for parabolic shapes under UDL.

• Slope Deflection and Moment Distribution are displacement methods. Slope Deflection sets up equilibrium equations directly, while Moment Distribution solves them iteratively. Both are suited for frame analysis.

• The Stiffness Method is the systematic matrix approach underlying modern structural analysis software.

• Influence Lines for indeterminate structures are curved and are constructed using the Müller-Breslau principle in conjunction with an indeterminate analysis.

• Plastic Analysis considers the ultimate strength of structures. It identifies the collapse load through the formation of plastic hinges and the application of the virtual work principle on a plausible collapse mechanism.