Structural analysis-II (CE-503 (A)) - Unit 2 Short Notes

UNIT 2: Advanced Structural Analysis Methods

1. Fundamental Definitions and Concepts

Key Terminology for Indeterminate Analysis:

$$P_u = \gamma_L \times P_{service}$$

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| Factor of Safety (FoS) | Ratio of material strength (yield/ultimate) to applied stress. Used in elastic/working stress design. |

$$FoS = \frac{\text{Strength}}{\text{Applied Stress}}$$

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| Plastic Modulus (Z_p) | Section property representing the first moment of area about the Plastic Neutral Axis (PNA). Used to calculate plastic moment capacity (M_p). |

$$M_p = f_y \times Z_p$$

<br> For symmetric I-section: $$\displaystyle Z_p = \frac{A_f}{2}(y_f + \frac{d}{2}) $$ |

| Plastic Neutral Axis (PNA) | The axis in a cross-section that divides it into two equal plastic areas (compression = tension) when the section yields fully. May not coincide with the elastic neutral axis. | | Relative Stiffness (K) | Stiffness of a member relative to others in a joint, used in Moment Distribution. For prismatic member: $$\displaystyle K = \frac{EI}{L} $$. Often, $EI$ is taken constant, so $$\displaystyle K \propto \frac{1}{L} $$. |

$$K_{AB} = \frac{EI_{AB}}{L_{AB}}$$

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| Distribution Factor (DF) | Fraction of an unbalanced moment at a joint that is distributed to a connecting member. Sum of DFs at a joint = 1. |

$$DF_{AB} = \frac{K_{AB}}{\sum K_{connected}}$$

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| Shape Factor (S_f) | Ratio of plastic moment capacity (M_p) to yield moment capacity (M_y) for a given cross-section. Represents reserve strength beyond initial yield. |

$$S_f = \frac{M_p}{M_y}$$

<br> For rectangular section: $$\displaystyle S_f = 1.5 $$ |

[!TIP] Exam Alert: Distinguish clearly between Load Factor (for ultimate loads) and Factor of Safety (for working stress). Remember PNA location for unsymmetric sections (e.g., T-section) requires area balance, not centroid.

2. Moment Distribution Method

Principle: An iterative, relaxation technique for analyzing statically indeterminate beams and frames. Based on fixed-end moments (FEM) and the principle of moment equilibrium at joints.

Core Procedure:

1. Calculate Fixed-End Moments (FEM): For each member under given loads, assuming joints are fixed.

2. Compute Distribution Factors (DF): For each joint based on relative stiffness $$\displaystyle K = \frac{EI}{L} $$.

3. Balance Joints: Apply unbalanced moment at each joint, distribute to connected members using DF.

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

5. Repeat: Continue cycles of balancing and carrying over until moments become negligible.

6. Sum Moments: Final moment at each end = FEM + Sum of all distributed/carry-over moments.

7. Draw Diagrams: Use final end moments to construct Bending Moment Diagram (BMD) and Shear Force Diagram (SFD). Deflected shape follows the sign convention (sagging +ve).

Kani’s Method (Modified Moment Distribution):

• Key Difference: No carry-over factor. Distribution factor is based on stiffness but the method accounts for joint rotation effects more directly.

• Procedure:

  1. Calculate rotation factors for each member end.

  2. Start from the top of a frame (or a free end) and work downwards, distributing moments considering the carry-over effect implicitly.

  3. Particularly efficient for sway frames.

• Application: Used for rigid frames with side sway.

  DiagramCANVAS: Show a simple portal frame with a horizontal load, indicating distributed moments at joints using Kani's method arrows.

Analysis of Rigid Frames:

• Without Sway: Treat as beams with joint translations prevented.

• With Sway: Requires additional analysis to determine the sway moment due to joint translation. This can be done by:

  • Method of Joint Translation: Apply a hypothetical force to cause unit sway, calculate moments, and scale.

  • Direct use of Kani’s Method: Which inherently handles sway.

[!TIP] Common Pitfall: Forgetting to include FEM due to joint translation (sway) in frames under lateral loads. Always check for side sway possibility.

3. Flexibility Method (Force Method)

Concept: Also called the Force Method or Method of Consistent Deformations. Redundant forces are treated as unknowns. Compatibility of deformations is enforced.

Step-by-Step Procedure:

1. Determine Degree of Indeterminacy (n).

2. Select Primary Structure: Remove n redundant constraints (e.g., a support reaction, a fixed end moment) to make the structure statically determinate and stable.

3. Calculate Influence Coefficients (δ_ij): Deformation (deflection/rotation) at the location of redundant i due to a unit load applied in the direction of redundant j. Use M/EI diagrams and moment-area or conjugate beam methods.

$$\delta_{ij} = \int \frac{m_i m_j}{EI} dx$$

*   Where $$\displaystyle m_i $$, $$\displaystyle m_j $$ are bending moment diagrams due to unit loads for redundants *i* and *j*.

4. Form Compatibility Equations: Total deformation at redundant location must be zero (or equal to support settlement).

$$\delta_{i0} + \sum_{j=1}^{n} \delta_{ij} X_j = \Delta_i$$

*   $$\displaystyle \delta_{i0} $$: Deformation due to **external loads**.

*   $$\displaystyle X_j $$: Unknown redundant forces.

*   $$\displaystyle \Delta_i $$: Given displacement (often 0).

5. Solve Equations: Find redundants $$\displaystyle X_j $$.

6. Find Final Reactions/Moments: Superpose effects of external loads and redundants on primary structure.

Application to Continuous Beams:

• Common redundants: support reactions or fixed-end moments.

• Influence lines for indeterminate structures (see Unit 5) are used conceptually to find $$\displaystyle \delta_{ij} $$.

[!TIP] Exam Strategy: The flexibility method becomes cumbersome for highly indeterminate structures ($$\displaystyle n > 3 $$) due to solving large systems. Best for structures with few redundants.

4. Stiffness Method (Displacement Method)

Concept: Also called the Displacement Method. Unknown joint displacements (rotations, translations) are the primary unknowns. Equilibrium equations are formulated at joints.

Key Steps:

1. Define Degrees of Freedom (DOF): Identify independent joint rotations and translations.

2. Form Element Stiffness Matrices ([k]): For a beam element in local coordinates:

$$[k] = \frac{EI}{L^3} \begin{bmatrix} 12 & 6L & -12 & 6L \\ 6L & 4L^2 & -6L & 2L^2 \\ -12 & -6L & 12 & -6L \\ 6L & 2L^2 & -6L & 4L^2 \end{bmatrix}$$

*   Rows/Columns correspond to: $$\displaystyle [v_i, \theta_i, v_j, \theta_j]^T $$.

3. Assemble Global Stiffness Matrix [K]: Assemble all element [k] matrices into a global matrix for all DOFs.

4. Form Equilibrium Equations:

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

*   $\{D\}$: Vector of unknown displacements.

*   $\{F\}$: Vector of equivalent joint loads (from external loads, fixed-end moments converted to joint loads).

5. Apply Boundary Conditions: Modify [K] and {F} for supports (settlements, fixed joints).

6. Solve for Displacements: $$\displaystyle \{D\} = [K]^{-1} \{F\} $$.

7. Find Member End Forces: Use $$\displaystyle \{f\} = [k] \{d\} $$ for each element, where $\{d\}$ are the element's end displacements from global $\{D\}$.

Assumption: Constant EI for each member simplifies matrix formation.

[!TIP] Key Insight: Stiffness method is systematic and suitable for computer implementation. The number of equations equals the number of unrestrained DOFs.

5. Influence Lines for Indeterminate Structures

Müller-Breslau Principle (MBP): The influence line for any reaction, shear, or moment in an indeterminate structure is obtained by:

1. Remove the constraint corresponding to the desired quantity (e.g., remove support for reaction, cut section for shear/moment).

2. Impose a displacement (rotation or translation) of unit magnitude in the positive direction of the quantity, consistent with the structure's redundancy/constraints.

3. The resulting deformed shape (within the elastic range) is the influence line for that quantity.

Procedure for Constructing ILD:

1. Identify the degree of indeterminacy (n).

2. For the desired quantity (e.g., reaction at A), the MBP requires introducing n additional displacements to maintain compatibility.

3. The final shape is a combination of the basic determinate ILD and correction shapes due to the structure's continuity.

4. Ordinates are found by similar triangles or moment equilibrium.

Application Examples:

• Continuous Beam: ILD for a support reaction involves a vertical displacement at that support and relative rotations at adjacent spans due to continuity.

• Propped Cantilever: ILD for reaction at prop involves a vertical displacement at prop and a rotation at the fixed end (since it's indeterminate to 1st degree).

• Overhanging Beam: Similar to continuous beam, but with overhanging segments.

[!TIP] Critical Point: For indeterminate structures, MBP gives the correct qualitative shape, but quantitative ordinates often require verification using equilibrium (e.g., $$\displaystyle \sum M = 0 $$ about a point) because the "unit displacement" assumption doesn't directly give scale.

6. Plastic Analysis

Core Concepts:

• Plastic Hinge: A localized zone where full plastic moment (M_p) has developed, allowing rotation while moment remains constant. Forms at points of maximum moment.

• Mechanism Formation: Structure becomes unstable when sufficient plastic hinges form to create a kinematic mechanism. Collapse load is the load causing this mechanism.

• Plastic Collapse Theorem: The collapse load is the lowest load that can cause a mechanism.

Analysis of Beams:

1. Calculate M_p for the section: $$\displaystyle M_p = f_y \times Z_p $$.

2. Locate Potential Plastic Hinges: At points of maximum moment (supports, mid-span under UDL, points of contraflexure).

3. Form Mechanisms: Try different hinge combinations. A simply supported beam needs 1 hinge (at mid-span). A propped cantilever (indeterminate to 1st degree) needs 2 hinges (e.g., at fixed end and prop, or fixed end and mid-span).

4. Apply Virtual Work: For the mechanism,

   • External Work (W_e) = Internal Work (W_i)

$$W_e = P_u \times \delta_{point\ of\ load}$$

*   

$$W_i = \sum M_p \times \theta_{hinge}$$

*   Solve for **collapse load** $$\displaystyle P_u $$.

Section Properties:

• I-Section: Plastic modulus $$\displaystyle Z_p $$ found by dividing section into rectangles, summing $A \times y$ from PNA.

  • Shape Factor $$\displaystyle S_f \approx 1.12 - 1.15 $$ for typical rolled I-sections.

• Circular Section:

  • $$\displaystyle Z_p = \frac{4R^3}{3} $$

  • $$\displaystyle S_f = \frac{16}{3\pi} \approx 1.698 $$

[!TIP] Design Application (Nov 2023): To design an I-section for a given ultimate load, first find required $$\displaystyle M_p $$, then select a section with $$\displaystyle Z_p \geq M_p/f_y $$. Check shape factor if using elastic moment for initial sizing.

7. Analysis of Frames under Lateral Loads

Effects of Wind/Earthquake: Cause side-sway (lateral displacement) and torsional effects in frames. Induce significant bending moments in columns and beams.

BIS Codal Provisions (IS 1893, IS 875):

• Wind Load: Calculated based on basic wind speed, terrain category, height, and structure type (low-rise vs. high-rise). Applied as static equivalent pressure.

• Earthquake Load: Response Spectrum Method for regular structures; Time History for irregular/critical. Seismic weight includes dead load + imposed load (as per code percentage). Base shear $$\displaystyle V_b = A_h \times W $$, where $$\displaystyle A_h $$ is horizontal acceleration coefficient.

Analysis of Frames with Side Sway:

• Moment Distribution: Requires sway analysis. Apply a hypothetical force to prevent sway, analyze, then remove force and apply equal and opposite force to cause sway. Combine results.

• Stiffness Method: Side sway is a translational DOF. Naturally included in the global stiffness matrix formulation.

• Kani’s Method: Efficient for sway frames as it handles joint translations directly.

[!TIP] Important: In moment distribution for sway frames, FEM due to sway (from the hypothetical force) must be calculated and included in the balancing cycles.

8. Comparative Study and Integrated Applications

Selection Criteria:

• Hand Calculation, Few Redundants: Flexibility Method.

• Hand Calculation, Frames/Beams (n moderate): Moment Distribution (or Kani’s for sway).

• Computer Analysis, Complex Structures: Stiffness Method (Finite Element Method).

Integrated Use: In complex structures, different methods may be used for different parts (e.g., stiffness method for overall frame, flexibility for local connections). Plastic analysis is used for ultimate load design after elastic analysis for service loads.

[!TIP] Exam Question Strategy: When asked to "analyze," check the structure's indeterminacy and loading. For a continuous beam (n=1 or 2), both Flexibility and MD are viable. For a sway frame, Kani’s or Stiffness is better. Always state your assumptions (constant EI, prismatic members).