4.5 Determinate Structures-2

4.5 Determinate Structures-2

Introduction to Moving Loads and Arches

• The previous section focused on structures under fixed loads. In reality, many structures like bridge decks and crane girders are subjected to moving loads.

• Influence Lines are essential tools for analyzing the variation of internal forces at a specific section as a load moves across the structure.

• Arches are efficient structural forms that primarily carry loads through axial compression, minimizing bending moments.

• This section covers the construction of influence lines for determinate structures and the analysis of the fundamental two-hinged arch.

---

1. Influence Lines for Determinate Structures

1.1 Concept of Influence Lines

• Definition: An Influence Line (IL) is a graph that shows the variation of a specific response function (reaction, shear, moment, etc.) at a specific point in a structure due to a unit moving load.

• Unit Load: A dimensionless concentrated load of magnitude 1 (e.g., 1 kN, 1 lb) that traverses the structure.

• Purpose: To determine the most critical position of live loads (traffic, people) to maximize a particular force effect for design.

1.2 Construction by Static Equilibrium (Direct Method)

• This is the primary method for statically determinate structures.

• Procedure:

  1. Identify the response function $\boldsymbol{R}$ (e.g., reaction at A, shear at C, moment at C).

  2. Place a unit load (1) at a variable distance $\boldsymbol{x}$ from a reference point.

  3. Using static equilibrium equations, derive an expression for $\boldsymbol{R}$ as a function of $\boldsymbol{x}$, i.e., $\boldsymbol{R(x)}$.

  4. Plot this function over the entire length of the structure. This plot is the Influence Line for $\boldsymbol{R}$.

1.3 Müller-Breslau Principle (Qualitative ILs)

• Statement: For determinate structures, the influence line for a force response (reaction, shear, moment) is given by the deflected shape of the structure when the restraint corresponding to that force is released and a unit displacement/rotation is introduced in its direction.

• Application:

  • Reaction IL: Remove the support and impose a unit vertical displacement. The resulting rigid-body displacement shape is the IL.

  • Shear IL: Introduce a unit shear displacement (cut and slide) at the section.

  • Moment IL: Introduce a unit hinge rotation at the section.

• This principle allows quick sketching of the shape of the influence line without calculations.

1.4 Influence Lines for Common Response Functions

• For a simply supported beam of length $\boldsymbol{L}$:

Response Function at Section C (distance $\boldsymbol{a}$ from A, $\boldsymbol{b}$ from B)	Influence Line Diagram (ILD) Shape	Key Ordinates
Reaction at A ($\boldsymbol{R_A}$)	A straight line from 1 at A to 0 at B.	$\boldsymbol{R_A = 1 - x/L}$
Shear at C ($\boldsymbol{V_C}$)	Two straight lines. Discontinuity of -1 at C.	Left of C: $\boldsymbol{V_C = -x/L}$. Right of C: $\boldsymbol{V_C = 1 - x/L}$.
Bending Moment at C ($\boldsymbol{M_C}$)	A triangle with peak at C, zero at A and B.	Ordinate at C = $\boldsymbol{ab/L}$.

---

2. Analysis with Point Loads and Uniformly Distributed Loads (UDL)

2.1 Concentrated Point Loads

• To find the value of the response function $\boldsymbol{R}$ due to a set of stationary concentrated loads:

  1. Obtain the Influence Line Ordinate (ILO) $\boldsymbol{y_i}$ at the location of each load $\boldsymbol{P_i}$.

  2. Apply the superposition principle.

     $\boldsymbol{R = \sum P_i \cdot y_i}$

• Finding Maximum Value: For a single moving point load, the maximum response occurs when the load is placed at the peak of the IL diagram.

2.2 Uniformly Distributed Load (UDL) of Finite Length

• For a UDL of intensity $\boldsymbol{w}$ (force/length) extending over a specific segment of the beam:

  1. The value of the response function $\boldsymbol{R}$ is given by the product of the load intensity and the area under the IL diagram over the loaded length.

     $\boldsymbol{R = w \times (\text{Area under IL over loaded segment})}$

• Finding Maximum Value: To maximize $\boldsymbol{R}$, place the UDL over all segments where the IL ordinate is of the same sign. The maximum occurs when the UDL covers the entire positive (or negative) region of the IL.

2.3 Absolute Maximum Effect (for Series of Loads)

• For a series of moving loads (like a truck convoy), finding the absolute maximum shear or moment in the beam (not just at a section) requires:

  1. Determining candidate critical sections.

  2. Placing the load series on the IL for each section to maximize the response.

  3. Using criteria like placing an average load to the left and right of the section equal, or checking when a specific load is at the section's peak.

---

3. Analysis of Two-Hinged Arches

3.1 Arch Fundamentals

• Definition: A curved structural member supporting vertical loads primarily through axial compression, thereby reducing bending moments compared to a straight beam of the same span.

• Ideal Shape: For a given load pattern, an arch can be shaped (e.g., parabolic for UDL) so that under that load, it experiences pure compression with zero bending moment. This is called the funicular shape.

3.2 Two-Hinged Arch

• Supports: Two hinged (pin) supports, which allow rotation but prevent translation.

• Static Determinacy: A two-hinged arch is statically indeterminate to the first degree. (Four total reaction components (2 per hinge) minus 3 equilibrium equations = 1 redundant). However, for symmetrical loading, it can often be treated as determinate.

• Key Feature: The supports develop horizontal thrust ($\boldsymbol{H}$) in addition to vertical reactions. This thrust is the key mechanism that reduces bending.

3.3 Comparison with Simply Supported Beam

• Consider an arch and a simply supported beam of the same span under the same vertical loads.

• Let $\boldsymbol{M^0}$, $\boldsymbol{V^0}$ be the bending moment and shear force in the simply supported beam (called the beam moments).

• For a two-hinged arch with rise $\boldsymbol{h}$, the internal forces at a section with coordinates $\boldsymbol{(x, y)}$ are:

  Horizontal Thrust (Constant):

  $\boldsymbol{H = \frac{\int M^0 y \, ds / EI}{\int y^2 \, ds / EI + \int \cos^2 \theta \, ds / EA}}$

  For an arch with negligible axial deformation, this simplifies. Often, for preliminary analysis, $\boldsymbol{H}$ is taken as the beam moment at crown divided by the rise.

  Internal Forces in the Arch:

  • Axial Force (Compression, usually):

    $\boldsymbol{N = - (V^0 \sin \theta + H \cos \theta)}$

  • Shear Force:

    $\boldsymbol{V = V^0 \cos \theta - H \sin \theta}$

  • Bending Moment:

    $\boldsymbol{M = M^0 - H \cdot y}$

• Interpretation: The arch bending moment $\boldsymbol{M}$ equals the beam moment $\boldsymbol{M^0}$ minus the moment caused by the horizontal thrust $\boldsymbol{H}$. The thrust moment ($\boldsymbol{H \cdot y}$) counteracts the beam moment, often leading to much smaller net bending in the arch.

3.4 Advantages and Behavior

• Advantages: Material efficiency (low bending stress), aesthetic appeal, large clear spans.

• Behavior: Under symmetrical loading, bending moments are small. Under unsymmetrical or moving loads, bending moments develop. Supports must be designed to resist the outward thrust.

---

Summary

• Influence Lines (IL) are graphs showing how a specific force (reaction, shear, moment) at a point varies with the position of a unit moving load. They are constructed using statics for determinate structures.

• The value of the force due to actual loads is found by:

  • Point Loads: $\boldsymbol{R = \sum P_i y_i}$

  • UDL: $\boldsymbol{R = w \times \text{(Area under IL)}}$

• The Müller-Breslau Principle provides a quick way to sketch the shape of ILs for determinate structures.

• A Two-Hinged Arch is an efficient structure that develops horizontal thrust. Its bending moment is significantly less than that of a simply supported beam of the same span under the same load, given by $\boldsymbol{M = M^0 - H \cdot y}$.

• The horizontal thrust (H) is the key redundant force that makes the arch efficient by converting bending action into axial compression.