Structural analysis-II (CE-503 (A)) - Unit 5 Short Notes

UNIT 5: ADVANCED ANALYSIS OF INDETERMINATE STRUCTURES

I. KEY DEFINITIONS AND CONCEPTS

[!TIP] Exam Distinction:

• Load Factor (ultimate loads) is for Plastic/Limit State Design.

• Factor of Safety (material strength) is for Elastic/Working Stress Design.

II. FORCE METHOD (FLEXIBILITY METHOD)

Principle: The structure is statically indeterminate. The compatibility of displacements (e.g., continuity at supports) is enforced by releasing redundant forces and solving for them using flexibility coefficients ($$\displaystyle f_{ij} $$).

General Procedure:

1. Determine Degree of Indeterminacy ($n$).

2. Select $n$ Redundant Reactions and remove them to obtain a statically determinate primary structure.

3. Calculate Flexibility Coefficients $$\displaystyle f_{ij} $$: displacement at $i$ due to unit load at $j$.

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

where $$\displaystyle m_i, m_j $$ are **bending moment diagrams** due to unit redundants.

4. Write Compatibility Equations: $$\displaystyle \sum_{j=1}^{n} f_{ij} X_j + \Delta_{iP} = 0 $$, where $$\displaystyle \Delta_{iP} $$ is displacement at $i$ due to external loads.

5. Solve for redundants $$\displaystyle X_j $$.

6. Draw Final BMD using superposition.

Application to Continuous Beams: Common redundants are support moments or vertical reactions. Use Clapeyron's Theorem (Three-Moment Equation) as a shortcut for spans with uniform $EI$.

III. DISPLACEMENT METHODS

A. MOMENT DISTRIBUTION METHOD (MDM)

Principle: Iterative process of balancing and carrying over moments at joints until convergence. Assumes fixed ends initially.

Step-by-Step Procedure:

1. Calculate Fixed-End Moments (FEM): For each span due to external loads (use standard table).

2. Calculate Distribution Factors (DF): $$\displaystyle DF_{ij} = \frac{K_{ij}}{\sum K} $$ at each joint. For far ends, $$\displaystyle DF = 0 $$.

3. Balance Moments: At each joint (except fixed supports), apply balancing moment = $-$ (Unbalanced Moment). Distribute it to connected members using DF.

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

5. Repeat steps 3 & 4 until moments are negligible.

6. Sum Moments: Final moment at each end = FEM + All Balancing + All Carry-Over moments.

   • Sagging (+ve) moment: tension at bottom.

   • Hogging (-ve) moment: tension at top.

[!TIP] Key Points:

• Joint with fixed support: No distribution (DF = 0 for fixed end).

• Sway frames: Require separate sway analysis (apply a hypothetical joint translation, calculate shear, equate to external horizontal load).

• Sign Convention: Clockwise moments are +ve (consistent with FEM tables).

B. KANI’S METHOD

Fundamentals: Similar to MDM but uses stiffness factor $$\displaystyle K = \frac{I}{L} $$ (not $EI/L$) and distribution factors based on relative stiffness. No carry-over factor—the entire distributed moment is considered to act at the joint.

Procedure:

1. Calculate stiffness $$\displaystyle K = I/L $$ for each member.

2. Compute Distribution Factors at each joint: $$\displaystyle DF_{ij} = \frac{K_{ij}}{\sum K_{joint}} $$.

3. Initial Unbalanced Moments = Fixed-End Moments (FEM).

4. Distribute the unbalanced moment at a joint to all connected members using DF.

5. Rotate the members by the distributed amount. The moment at the far end of a member is not half, but is found from the rotation compatibility of the adjacent joint in the next cycle.

6. Iterate until moments stabilize.

[!TIP] Difference from MDM:

• Kani’s uses $$\displaystyle K = I/L $$, MDM uses $$\displaystyle K = EI/L $$.

• Kani’s has no carry-over; moments are distributed directly based on joint rotation equilibrium.

C. STIFFNESS METHOD (SLOPE-DEFLECTION / MATRIX)

Formulation for Beam Elements:

For a beam element $ij$ (length $L$, $EI$ constant):

$$\begin{bmatrix} M_{ij} \\ M_{ji} \end{bmatrix} = \frac{EI}{L} \begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix} \begin{bmatrix} \theta_i \\ \theta_j \end{bmatrix} + \begin{bmatrix} FEM_{ij} \\ FEM_{ji} \end{bmatrix}$$

where $FEM$ are fixed-end moments due to loads.

Analysis of Beams:

1. Write slope-deflection equations for all members.

2. Apply joint equilibrium ($$\displaystyle \sum M_{joint} = 0 $$) to get equations in $\theta$.

3. Apply boundary conditions (e.g., $$\displaystyle \theta = 0 $$ at fixed support, known rotations).

4. Solve for unknown rotations.

5. Compute member end moments.

6. Draw BMD.

IV. PLASTIC ANALYSIS

A. FUNDAMENTAL CONCEPTS

• Plastic Hinge: A zone in a beam where yielding occurs across the entire cross-section, allowing rotation at constant moment $$\displaystyle M_p $$.

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

• Collapse Load (Lower Bound Theorem): The load at which a mechanism forms with $$\displaystyle M \leq M_p $$ everywhere. The actual collapse load is the lowest such load from all possible mechanisms.

• Shape Factor ($$\displaystyle S_f $$): Ratio of plastic moment capacity to yield moment capacity. Measures the reserve strength beyond elastic limit.

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

- For rectangular section: $$\displaystyle S_f = 1.5 $$

- For I-section (flange dominant): $$\displaystyle S_f \approx 1.1 - 1.2 $$

Section Moduli for Standard Sections:

• Rectangular (b×h): $$\displaystyle Z_e = \frac{bh^2}{6} $$, $$\displaystyle Z_p = \frac{bh^2}{4} $$

• I-Section: $$\displaystyle Z_e $$ and $$\displaystyle Z_p $$ from tables (flange/web contributions).

B. PLASTIC DESIGN OF BEAMS

Design for UDL on Simply Supported Beam:

• Plastic Hinges form at supports and midspan (3 hinges → mechanism).

• Collapse Load: $$\displaystyle M_p = \frac{w_u L^2}{8} $$ → $$\displaystyle w_u = \frac{8 M_p}{L^2} $$

Propped Cantilever with UDL:

• Hinges at fixed end (A), prop (B), and midspan (C) or another point.

• Mechanism: 3 hinges needed. Typically at A, B, and C.

• Collapse Load: Use virtual work or equilibrium.

$$w_u = \frac{16 M_p}{L^2} \quad \text{(if hinge at midspan)}$$

C. PLASTIC ANALYSIS OF FRAMES

• Beam-Column: Hinges form at beam ends (high moments) and possibly at column tops/bottoms.

• Gable Frame: Mechanisms involve panel hinges (beam sides) and corner hinges.

• Procedure: Identify potential hinge locations, assume a mechanism, apply virtual work:

$$\delta_{ext} = \sum \theta_i \cdot M_{pi}$$

where $$\displaystyle \delta_{ext} $$ is external work, $$\displaystyle \theta_i $$ are hinge rotations.

V. INFLUENCE LINES FOR INDETERMINATE STRUCTURES

A. MULLER BRESLAU PRINCIPLE

Concept: The influence line for a reaction (or shear, moment) in a statically indeterminate structure is obtained by:

1. Removing the restraint corresponding to the desired reaction/shear/moment.

2. Applying a unit displacement (rotation for moment, translation for reaction) in the positive direction of the quantity.

3. The deformed shape of the released structure (subject to its static indeterminacy) is the influence line.

Procedure:

• For Reaction $$\displaystyle R_A $$: Release $A$, apply vertical unit displacement upward. The resulting deflected shape is ILD for $$\displaystyle R_A $$.

• For Shear $S$: Cut the section, apply unit shear displacement (relative vertical displacement of left/right parts).

• For Moment $M$: Release the moment restraint (hinge), apply unit rotation.

[!TIP] Key: The ILD is not the elastic curve—it's the kinematic deformation under the unit "action."

B. INFLUENCE LINES FOR SPECIFIC STRUCTURES

1. Double Overhanging Beam (supports at B, C):

• Reaction at B: Release vertical at B, apply unit ↑. ILD: Linear segments with breaks at overhang ends.

• Shear at section n: Cut at n, apply unit relative displacement. ILD: Two lines meeting at n with slope ±1.

• Moment at n: Insert hinge at n, apply unit rotation. ILD: Two triangles meeting at n.

2. Propped Cantilever (fixed at A, prop at B):

• Reaction at Fixed End A: Release vertical at A, apply unit ↑. ILD: Starts at 1 at B, goes linearly to -1 at A (since $$\displaystyle R_A + R_B = P $$).

• Moment at Midspan (C): Insert hinge at C, apply unit rotation. ILD: Two lines from supports meeting at C with slopes determined by static equilibrium of released structure.

3. Continuous Beam:

• Reaction at Support: Release the support, apply unit displacement. ILD is piecewise linear, with slopes determined by span lengths.

• General: For a beam with $n$ spans, ILD for a reaction has $n$ linear segments. Use three-hinge concept or Muller Breslau directly.

VI. ANALYSIS OF SPECIFIC STRUCTURAL SYSTEMS

VII. SPECIAL TOPICS: TALL BUILDINGS & LATERAL LOADS

Wind & Earthquake Effects:

• Wind: Creates lateral pressure → shear and overturning moments. Causes story drift.

• Earthquake: Inertial forces $$\displaystyle F = m \cdot a_g $$ (where $$\displaystyle a_g $$ = ground acceleration). Causes dynamic response.

• P-Δ Effect: Secondary moments due to lateral displacement × gravity loads. Significant in tall, flexible frames.

BIS Codal Provisions (IS Codes):

• IS 1893 (Part 1): Criteria for earthquake resistant design – provides seismic zone map, importance factor, response spectrum, lateral force calculation.

• IS 875 (Part 3): Wind loads on buildings – gives wind pressure $$\displaystyle p_z = K_z K_a K_d p_z $$ based on height, terrain, risk coefficient.

• Analysis Methods: Permitted: Equivalent Static, Response Spectrum, Time History.

• Design Checks: Story drift limits (e.g., $$\displaystyle \frac{\Delta}{h} \leq 0.004 $$ for IS 1893), torsion considerations.

[!TIP] Exam Focus:

• Know IS 1893 for earthquakes, IS 875(3) for wind.

• Understand P-Δ effect and its significance.

• Be able to compute base shear $$\displaystyle V_b = A_h \cdot W $$ (from IS 1893).