Structural analysis-II (CE-503 (A)) - Unit 1 Short Notes

I. MOMENT DISTRIBUTION METHOD

A. Fundamental Concepts

• Distribution Factor (DF): Ratio of stiffness of a member to the sum of stiffnesses of all members meeting at a joint.

  For a beam/column with far end fixed: stiffness = $$\displaystyle \frac{4EI}{L} $$; far end pinned: stiffness = $$\displaystyle \frac{3EI}{L} $$.

$$DF_{ij} = \frac{K_{ij}}{\sum K_{ij}}$$

where $$\displaystyle K_{ij} $$ is stiffness of member $i$ at joint $j$.

• Carry-Over Factor (COF): Fraction of moment carried over to the far end when a moment is applied at one end.

  For prismatic member:

  • Far end fixed → $$\displaystyle COF = 0.5 $$

  • Far end pinned → $$\displaystyle COF = 0 $$

• Fixed-End Moments (FEM): Moments at ends of a member when both ends are fixed, due to external loads.

  Key cases (span $L$, load $w$ or $P$):

  • UDL: $$\displaystyle M_{AB} = \frac{wL^2}{12} $$, $$\displaystyle M_{BA} = -\frac{wL^2}{12} $$

  • Point load at midspan: $$\displaystyle M_{AB} = \frac{PL}{8} $$, $$\displaystyle M_{BA} = -\frac{PL}{8} $$

  • Temperature: $$\displaystyle M_{AB} = \frac{EI \alpha \Delta T}{L} $$ (symmetric), sign depends on heating/cooling.

  • Sinking support: $$\displaystyle M_{AB} = -\frac{6EI \Delta}{L^2} $$, $$\displaystyle M_{BA} = \frac{6EI \Delta}{L^2} $$.

• Relative Stiffness: For members with uniform $EI$, stiffness $$\displaystyle \propto \frac{I}{L} $$.

  For beams: $$\displaystyle K = \frac{I}{L} $$; for columns: $$\displaystyle K = \frac{I}{L} $$ (consider joint rotations).

  [!TIP] When joint is rigidly connected, use $$\displaystyle \frac{4EI}{L} $$ for far-end fixed; if pinned, use $$\displaystyle \frac{3EI}{L} $$.

B. Application to Continuous Beams

1. Procedure:

   • Calculate DFs at each joint (except supports).

   • Compute FEMs for all spans under given loads.

   • Lock all joints (apply FEMs).

   • Release joints one by one: apply unbalanced moment = -(sum of fixed-end moments + carried-over moments), distribute using DFs, carry over half to far ends.

   • Repeat cycles until moments are negligible (typically < 1% of largest moment).

2. Stopping Criteria: Iterate until the algebraic sum of moments at any joint is nearly zero.

3. Drawing Diagrams:

   • Bending Moment Diagram (BMD): Sum of FEMs + distributed moments + carry-overs at each section. Sketch on tension side.

   • Shear Force Diagram (SFD): Compute from loads and support reactions (obtained from moment equilibrium).

C. Application to Rigid Frames

• Frames with Sway: Lateral displacement causes additional moments. Two methods:

  1. Sway Correction: Analyze for non-sway loads first, then apply artificial joint rotations to eliminate sway, compute sway moments, and combine.

  2. Direct Analysis: Include column sidesway effects by treating frame as having an additional degree of freedom.

• Joints: Rigid joints distribute moments; pinned/roller supports have zero moment.

• Treatment of Supports: Fixed ends have no rotation; pinned ends have zero moment but rotation allowed.

D. Numerical Problems

• Symmetric Loading: Exploit symmetry to reduce calculations (half structure).

• Asymmetric Loading: Full analysis required.

• Different Stiffnesses: Use actual $EI/L$ values for DFs and COFs.

II. KANI'S METHOD (ROTATION INVERSION METHOD)

A. Basic Principles

• Based on rotation distribution and inversion of moments at joints.

• No separate carry-over step; inversion implicitly accounts for carry-over.

• Distribution factors are same as in moment distribution: $$\displaystyle DF = \frac{K_i}{\sum K} $$.

B. Procedure for Beams and Frames

1. Calculate DFs at each joint.

2. Start at one joint (usually leftmost), apply unbalanced moment (sum of FEMs) and distribute.

3. Move to next joint: the distributed moment from previous joint becomes the "inverted" moment (opposite sign) at this joint, added to existing FEMs, then distribute.

4. Continue until all joints processed; repeat cycles if needed.

5. Final moments are the algebraic sum of all distributed moments at each joint end.

C. Comparison with Moment Distribution Method

III. FLEXIBILITY METHOD (FORCE METHOD)

A. Introduction

• Redundant: Force (reaction or internal) removed to make structure statically determinate.

• Primary Structure: Statically determinate base structure.

• Degree of Indeterminacy: $$\displaystyle r = \text{number of redundants} $$ (static indeterminacy).

B. Flexibility Coefficients

• Definition: $$\displaystyle f_{ij} $$ = displacement at $i$ due to unit load at $j$.

• Calculation: Use unit load method:

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

where $$\displaystyle m_i, m_j $$ are moments from unit loads.

• Muller Breslau Principle: To find $$\displaystyle f_{ij} $$, remove restraint at $i$, apply corresponding displacement (unit), the resulting deflected shape gives influence line for $$\displaystyle f_{ij} $$.

C. Compatibility Equations

• For each redundant $$\displaystyle X_k $$: $$\displaystyle \sum_{j=1}^{r} f_{kj} X_j + \Delta_k^0 = 0 $$

  where $$\displaystyle \Delta_k^0 $$ is displacement at $k$ due to external loads (from primary structure).

• Solve simultaneous equations for $$\displaystyle X_j $$.

D. Application to Beams and Frames

• Selection of Redundants: Choose reactions or internal forces (e.g., moment at fixed end).

• Numerical Examples: Typically 1-2 redundants for continuous beams.

IV. STIFFNESS METHOD (DISPLACEMENT METHOD)

A. Introduction

• Degrees of Freedom (DOF): Independent displacements/rotations (e.g., joint rotations, translations).

• Unknowns: Usually joint displacements.

• Stiffness Coefficient $$\displaystyle k_{ij} $$: Force at $i$ due to unit displacement at $j$ (with others zero).

B. Formation of Equilibrium Equations

• Assemble global stiffness matrix $[K]$ from element stiffness matrices.

• Apply boundary conditions (modify $[K]$ and force vector $\{F\}$).

• Solve $$\displaystyle [K]\{D\} = \{F\} $$ for displacements $\{D\}$.

• Compute reactions from element stiffness equations.

C. Application to Beams and Frames

• Beam Elements: Local stiffness matrix (2 DOF per node: transverse displacement, rotation).

$$\begin{bmatrix} k_{11} & k_{12} \\ k_{21} & k_{22} \end{bmatrix} = \frac{EI}{L^3} \begin{bmatrix} 12 & 6L \\ 6L & 4L^2 \end{bmatrix}$$

(for far ends fixed).

• Frame Elements: Include axial effects; transform to global coordinates using transformation matrix.

• Assembly: Connect nodes, sum stiffness contributions.

D. Comparison with Flexibility Method

V. PLASTIC ANALYSIS (ULTIMATE LOAD THEORY)

A. Fundamental Concepts

• Load Factor $\lambda$: Factor by which working load is multiplied to get ultimate load.

• Factor of Safety (FOS): $$\displaystyle \frac{\text{Ultimate load}}{\text{Working load}} $$ (in load factor form).

• Plastic Hinge: Zone where section yields fully, rotation occurs at constant moment $$\displaystyle M_p $$.

• Mechanism: Structure becomes kinematically unstable when enough plastic hinges form (for beam: $n+1$ hinges for $n$ spans).

• Theorems:

  • Static Theorem: Any moment distribution satisfying equilibrium and $$\displaystyle M \leq M_p $$ gives upper bound collapse load.

  • Kinematic Theorem: Collapse load from mechanism analysis is lower bound; true collapse load when both bounds coincide.

B. Section Properties

• Plastic Modulus $$\displaystyle Z_p $$: $$\displaystyle Z_p = \frac{A}{2} (y_1 + y_2) $$ where $$\displaystyle y_1, y_2 $$ are distances from PNA to extreme fibers.

• Plastic Neutral Axis (PNA): Axis dividing section into equal areas (for symmetric sections, coincides with elastic NA).

• Shape Factor $$\displaystyle S_f = \frac{Z_p}{Z_e} $$; typical values:

  • Rectangle: $$\displaystyle S_f = 1.5 $$

  • I-section: $$\displaystyle S_f \approx 1.1 - 1.2 $$

  • T-section: $$\displaystyle S_f \approx 1.2 - 1.4 $$

  • Circle: $$\displaystyle S_f = \frac{16}{3\pi} \approx 1.697 $$

C. Calculation for Standard Sections

• Rectangular ($b \times d$):

  $$\displaystyle Z_e = \frac{bd^2}{6} $$, $$\displaystyle Z_p = \frac{bd^2}{4} $$, $$\displaystyle S_f = 1.5 $$.

• I-section: Sum of $$\displaystyle Z_p $$ for flanges and web.

• Circular (radius $R$):

  $$\displaystyle Z_e = \frac{\pi R^3}{4} $$, $$\displaystyle Z_p = \frac{4R^3}{3} $$, $$\displaystyle S_f = \frac{16}{3\pi} \approx 1.697 $$.

D. Analysis of Beams for Collapse

• Mechanism Method:

  1. Assume plastic hinge locations (usually at maximum moment points: supports, midspan, point loads).

  2. Write geometry compatibility (relative rotations).

  3. Compute collapse load $$\displaystyle w_u $$ from virtual work: $$\displaystyle \sum M_p \theta = \sum P \delta $$.

• Propped Cantilever: Hinges at fixed end, propped support, and midspan (for UDL) → collapse load $$\displaystyle w_u = \frac{8M_p}{L^2} $$.

• Continuous Beams: Identify span hinges and possible hinge at interior supports; consider different mechanisms.

E. Plastic Design of Steel Beams

• Given ultimate load $$\displaystyle w_u $$, compute required $$\displaystyle M_p = \frac{w_u L^2}{8} $$ (simply supported) etc.

• Select I-section with $$\displaystyle Z_p \geq \frac{M_p}{f_y} $$.

• Check compactness per codal provisions.

VI. INFLUENCE LINES FOR INDETERMINATE STRUCTURES

A. Muller Breslau Principle

• Statement: The influence line for any reaction/force is the deflected shape of the structure when the corresponding restraint is removed and a unit displacement (rotation or translation) is applied.

• Procedure:

  1. Remove restraint corresponding to the function (e.g., for reaction, remove support).

  2. Apply corresponding unit displacement (e.g., vertical translation for reaction).

  3. Determine deflected shape = influence line.

• Note: For indeterminate structures, the deflected shape is not linear; requires analysis (often using moment distribution or flexibility).

B. Influence Lines for Indeterminate Beams

• Reaction: Remove support, apply vertical displacement, compute using slope-deflection or moment distribution.

• Shear Force: Cut section, apply vertical displacement discontinuity.

• Bending Moment: Cut section, apply rotation discontinuity.

• Effect of Supports: Fixed ends cause curved influence lines; pinned/roller supports may have kinks.

C. Influence Lines for Frames

• Consider joint rotations; use slope-deflection or moment distribution to compute ordinates.

• For moment at a joint: remove moment restraint, apply unit rotation.

D. Numerical Examples

• Double Overhanging Beam: Use moment distribution to find reactions, then IL for shear/moment at section.

• Continuous Beams: Apply Muller Breslau with fixed ends; use conjugate beam or moment distribution.

• Propped Cantilever: Fixed at one end, propped at other; IL for reaction at prop: remove prop, apply vertical displacement, solve as indeterminate.

VII. ANALYSIS OF FRAMES UNDER LATERAL LOADS

A. Wind and Earthquake Loads

• Characteristics: Dynamic, directionally variable, magnitude depends on height/terrain.

• Codal Provisions (BIS):

  • Wind: IS 875 (Part 3) – basic wind speed, risk coefficient, terrain, height, structure factor.

  • Earthquake: IS 1893 – seismic zone, importance factor, response spectrum, base shear.

• Equivalent Static Method:

  • Base Shear $$\displaystyle V_b = A_h W $$, where $$\displaystyle A_h = \frac{Z I S_a}{R g} $$ (seismic) or from wind pressure.

  • Distribute $$\displaystyle V_b $$ to floors: $$\displaystyle F_i = \frac{w_i h_i}{\sum w_j h_j} V_b $$.

• Dynamic Analysis: Response spectrum (for multiple DOF) or time history (advanced).

B. Sway Frames

• Moment Distribution / Stiffness Method: Include sidesway by introducing joint translations as unknowns or using sway correction.

• Portal Method (Approximate):

  • Assumptions:

    1. Points of contraflexure at mid-height of columns.

    2. Shear distributed equally among columns on each floor.

    3. Moment at exterior column = 2/3 of total moment at that level; interior column = 1/3.

• Cantilever Method: For tall frames, assume frame acts as cantilever; shear distributed proportional to $I/h$ of columns.

C. Combined Gravity and Lateral Load Analysis

• Sequential Analysis:

  1. Analyze for gravity loads (dead + live) → obtain fixed-end moments.

  2. Analyze for lateral loads with gravity FEMs as fixed-end moments (no joint translations from gravity).

• P-Δ Effects: Secondary moments due to joint displacements under lateral load; significant for tall flexible frames; may require iterative analysis.

VIII. SECTION PROPERTIES AND DESIGN (RELATED TO PLASTIC ANALYSIS)

A. Elastic Section Modulus ($$\displaystyle Z_e $$)

• $$\displaystyle Z_e = \frac{I}{y_{\max}} $$, where $I$ is second moment of area, $$\displaystyle y_{\max} $$ distance to extreme fiber.

• For rectangle: $$\displaystyle Z_e = \frac{bd^2}{6} $$.

B. Plastic Section Modulus ($$\displaystyle Z_p $$)

• $$\displaystyle Z_p = \sum A_i y_i $$ for each half about PNA.

• For rectangle: $$\displaystyle Z_p = \frac{bd^2}{4} $$.

C. Shape Factor

• $$\displaystyle S_f = \frac{Z_p}{Z_e} $$; indicates extra moment capacity beyond elastic theory.

• [!TIP] Shape factor > 1 for most sections; circle has highest (~1.7).

D. Calculation for Circular Section (Radius $R$)

• Elastic Modulus: $$\displaystyle I = \frac{\pi R^4}{4} $$, $$\displaystyle y_{\max} = R $$ → $$\displaystyle Z_e = \frac{\pi R^3}{4} $$.

• Plastic Modulus: PNA through center.

$$Z_p = 2 \int_0^R y \cdot 2\sqrt{R^2 - y^2} dy = 4 \int_0^R y \sqrt{R^2 - y^2} dy$$

Let $$\displaystyle u = R^2 - y^2 $$, $$\displaystyle du = -2y dy $$:

$$\int y \sqrt{R^2 - y^2} dy = -\frac{1}{2} \int \sqrt{u} du = -\frac{1}{3} u^{3/2}$$

Evaluate $0$ to $R$: $$\displaystyle \frac{R^3}{3} $$. So $$\displaystyle Z_p = 4 \cdot \frac{R^3}{3} = \frac{4R^3}{3} $$.

• Shape Factor: $$\displaystyle S_f = \frac{4R^3/3}{\pi R^3/4} = \frac{16}{3\pi} \approx 1.697 $$.

IX. KEY DEFINITIONS (FROM EXAM PATTERN)

1. Load Factor: Factor by which working loads are multiplied to obtain ultimate loads for plastic design. $$\displaystyle \boxed{\lambda = \frac{\text{Ultimate load}}{\text{Working load}}} $$.

2. Plastic Modulus ($$\displaystyle Z_p $$): Geometric property of cross-section; sum of first moments of areas above and below PNA about PNA. $$\displaystyle \boxed{Z_p = \frac{A}{2}(y_1 + y_2)} $$.

3. Plastic Neutral Axis (PNA): Axis that divides the cross-section into two equal areas under plastic moment.

4. Relative Stiffness: Stiffness of a member relative to others at a joint, used in moment distribution. For beam/column: $$\displaystyle \frac{I}{L} $$ (or $$\displaystyle \frac{4EI}{L} $$ if far end fixed).

5. Distribution Factor (DF): Fraction of unbalanced moment distributed to a member at a joint. $$\displaystyle \boxed{DF_i = \frac{K_i}{\sum K}} $$.

6. Factor of Safety (FOS): Ratio of ultimate strength to allowable stress. In plastic design, often incorporated via load factor. $$\displaystyle \boxed{FOS = \frac{f_u}{f_{\text{allow}}}} $$.

[!TIP] In exams, clearly define each term with formula where applicable. For plastic modulus and PNA, include a simple sketch of section showing PNA and areas.