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# 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.

***

### 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.

***

### 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.

***

### 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.

***

### 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.

***

#### 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.