4. Structural Mechanics (ACiE04)

4.1 Shear Forces and Bending Moments

Understanding internal forces is crucial for structural analysis and design. Structures under external loads develop internal resisting forces and moments to maintain equilibrium. These internal actions are categorized as axial forces, shear forces, and bending moments.

Axial Forces, Shear Forces, and Bending Moments

Axial Force (N): The internal force acting along the longitudinal axis of a member. It can be tensile (pulling apart) or compressive (pushing together).
Shear Force (V): The internal force acting perpendicular to the longitudinal axis of a member, tending to cause a part of the member to slide past an adjacent part.
Bending Moment (M): The internal moment acting about an axis perpendicular to the longitudinal axis of a member, tending to cause the member to bend.

Sign Conventions

Consistent sign conventions are essential for accurate analysis:

Shear Force: Positive if it tends to rotate the element clockwise. (Alternatively, positive if the force on the left face is upwards and on the right face is downwards).
Bending Moment: Positive if it causes compression in the top fibers and tension in the bottom fibers (sagging). Negative if it causes tension in the top fibers and compression in the bottom fibers (hogging).

Types of Loads

Point Load: A concentrated load acting at a single point on the beam.
Uniformly Distributed Load (UDL): A load spread uniformly over a length of the beam, typically expressed in force per unit length (e.g., kN/m).
Uniformly Varying Load (UVL): A load whose intensity varies linearly over a length of the beam, often triangular or trapezoidal.
Moment: A concentrated external couple applied at a point on the beam.

Load Superposition Principle

The superposition principle states that for a linearly elastic structure, the total response (deflection, stress, internal forces) caused by multiple loads is the sum of the responses caused by each load acting independently. This simplifies analysis by breaking down complex loading scenarios into simpler ones.

Relationships between Load, Shear Force, and Bending Moment

These differential relationships are fundamental:

dV/dx = w: The rate of change of shear force with respect to position x along the beam is equal to the intensity of the distributed load w (positive upwards).
dM/dx = V: The rate of change of bending moment with respect to position x along the beam is equal to the shear force V.
d²M/dx² = w: Combining the above, the second derivative of bending moment with respect to x is equal to the distributed load intensity.

These relationships imply that the shear force is the integral of the load intensity, and the bending moment is the integral of the shear force (or double integral of the load intensity).

Shear Force Diagram (SFD) and Bending Moment Diagram (BMD)

SFD and BMD are graphical representations of the variation of shear force and bending moment along the length of a beam. They are essential for identifying critical sections where internal forces are maximum.

Simply Supported Beams: Typically show linear SFD segments and parabolic BMD segments for UDLs.
Cantilever Beams: Often have constant or linearly varying SFD and linearly or quadratically varying BMD, with maximums at the fixed end.
Overhanging Beams: Combine characteristics of simply supported and cantilever beams, often exhibiting points of contraflexure.

Example: For a simply supported beam with a central point load, the SFD is constant between the load and supports, changing sign at the load. The BMD is triangular, peaking at the load.

Point of Contraflexure

A point of contraflexure (or inflection point) is a location along a beam where the bending moment is zero, and the bending moment diagram changes sign. This indicates a change in the curvature of the beam (from sagging to hogging or vice-versa). It is often found in overhanging or continuous beams.

Maximum Bending Moment Location and Value

The maximum bending moment typically occurs where the shear force is zero or changes sign. Identifying this location is critical for design, as it dictates the required strength of the beam section. The value of the maximum bending moment is calculated by integrating the shear force diagram up to that point or by direct section analysis.

4.2 Stress and Strain Analysis

Stress and strain are fundamental concepts in mechanics of materials, describing the internal forces and deformations within a body subjected to external loads.

Normal Stress and Normal Strain

Normal Stress (σ): The internal force per unit area acting perpendicular to the cross-section. σ = P/A where P is the axial force and A is the cross-sectional area. Tensile stress is positive, compressive stress is negative.
Normal Strain (ε): The deformation per unit length in the direction of the applied load. ε = δ/L where δ is the total deformation (change in length) and L is the original length.

Shear Stress and Shear Strain

Shear Stress (τ): The internal force per unit area acting parallel to the cross-section. In beams, shear stress distribution is given by the flexure formula: τ = VQ/(Ib) where V is the shear force, Q is the first moment of area of the section above (or below) the point where shear stress is calculated, I is the moment of inertia of the entire cross-section, and b is the width of the section at the point of interest.
Shear Strain (γ): The angular deformation, representing the change in angle between two initially perpendicular lines within the material.

Principal Stresses and Principal Planes

When a material is subjected to a complex state of stress (e.g., combined normal and shear stresses), there exist specific planes on which the shear stress is zero. These are called principal planes, and the normal stresses acting on them are called principal stresses (σ1, σ2, σ3). Mohr's circle is a graphical method used to determine principal stresses, principal planes, and the maximum shear stress for a given state of plane stress.

Maximum Shear Stress

The maximum shear stress (τmax) in a plane stress state occurs on planes oriented at 45 degrees to the principal planes. For a 2D stress state (where σ2 is often zero or ignored if out-of-plane), the maximum in-plane shear stress is: τmax = (σ1 - σ3)/2 (where σ1 and σ3 are the maximum and minimum principal stresses, respectively).

Stress-Strain Curves (Mild Steel)

The stress-strain curve for a material like mild steel provides critical information about its mechanical properties:

Proportional Limit: The point up to which stress is directly proportional to strain (Hooke's Law applies).
Elastic Limit: The maximum stress a material can withstand without permanent deformation upon unloading.
Yield Point: The stress at which a material begins to deform plastically (permanently). Mild steel exhibits distinct upper and lower yield points.
Ultimate Stress (Tensile Strength): The maximum stress the material can withstand before necking begins.
Fracture Stress: The stress at which the material breaks.

Hooke's Law and Elastic Moduli

Hooke's Law defines the linear elastic relationship between stress and strain:

Normal Stress and Strain: σ = Eε, where E is Young's Modulus (Modulus of Elasticity), representing the material's stiffness in tension or compression.
Shear Stress and Strain: τ = Gγ, where G is the Shear Modulus (Modulus of Rigidity), representing the material's stiffness in shear.
Relationship between E, G, and Poisson's Ratio (ν): E = 2G(1+ν) where ν is Poisson's ratio, the ratio of transverse strain to axial strain.

Torsion

Torsion refers to the twisting of a shaft caused by an applied torque. For a circular shaft, the torsional stress and deformation are governed by:

Torsion Formula: T/J = τ/r = Gθ/L where T is the applied torque, J is the polar moment of inertia, τ is the shear stress at radius r, G is the shear modulus, θ is the angle of twist, and L is the length of the shaft.
Polar Moment of Inertia (J): For a solid circular shaft of diameter d: J = πd⁴/32 For a hollow circular shaft, J = π(do⁴ - di⁴)/32, where do and di are outer and inner diameters.

Example: Calculating the maximum shear stress and angle of twist in a steel shaft transmitting power.

4.3 Theory of Flexure and Columns

The theory of flexure (or bending theory) describes the behavior of beams under transverse loads, while column theory addresses the stability of slender compression members.

Co-planar and Pure Bending Assumptions

The Euler-Bernoulli beam theory relies on several key assumptions for pure bending (bending without shear):

Plane sections remain plane and perpendicular to the neutral axis after bending.
The material is homogeneous, isotropic, and obeys Hooke's Law.
The beam is initially straight and has a constant cross-section.
The applied loads are within the elastic limit.
The deflections are small compared to the beam's dimensions.
Bending occurs about an axis perpendicular to the plane of the applied moment.

Elastic Curve and Radius of Curvature

The elastic curve is the deformed shape of the beam's longitudinal axis under loading. Its differential equation is central to deflection analysis:

Elastic Curve Equation: d²y/dx² = M/EI where y is the deflection, x is the position along the beam, M is the bending moment, E is Young's Modulus, and I is the moment of inertia of the cross-section.
Radius of Curvature (ρ) and Angle of Rotation (θ): The curvature 1/ρ = d²y/dx². Thus, 1/ρ = M/EI. The angle of rotation (slope) of the elastic curve is dy/dx.

Flexural Stiffness (EI)

The product EI is known as the flexural stiffness of the beam. It represents the resistance of the beam to bending deformation. A higher EI indicates a stiffer beam.

Deflection Methods

Several methods are available to determine beam deflections and slopes:

Double Integration Method: Directly integrates the elastic curve equation d²y/dx² = M/EI twice to obtain slope and deflection equations, using boundary conditions to find integration constants.
Macaulay's Method (or Singularity Functions Method): A modification of the double integration method that uses singularity functions to handle discontinuous loads (point loads, UDLs starting/ending at different points) with a single expression for the bending moment across the entire beam.
Moment-Area Method: Based on two theorems relating the change in slope and the tangential deviation between two points on the elastic curve to the area under the M/EI diagram.
Conjugate Beam Method: An indirect method where an imaginary "conjugate beam" is constructed with specific support conditions, and its shear force and bending moment diagrams correspond to the slope and deflection diagrams of the real beam, respectively.

Bending Stress (Flexure Formula)

The normal stress due to bending (flexural stress) varies linearly across the beam's cross-section, being zero at the neutral axis and maximum at the extreme fibers:

Flexure Formula: σ = My/I where M is the bending moment at the section, y is the distance from the neutral axis to the point where stress is calculated, and I is the moment of inertia of the cross-section about the neutral axis. The maximum bending stress occurs at the extreme fiber, where y = y_max.

Section Modulus (Z)

The section modulus is a geometric property that combines the moment of inertia and the distance to the extreme fiber:

Z = I/y_max The maximum bending stress can then be expressed as σ_max = M/Z. A larger section modulus indicates a greater resistance to bending stress.

Euler's Formula for Long Columns

Euler's formula predicts the critical buckling load (Pcr) for slender columns, assuming ideal conditions (perfectly straight, axially loaded, elastic material):

Pcr = π²EI/(Le)² where E is Young's Modulus, I is the minimum moment of inertia of the cross-section, and Le is the effective length of the column.
Effective Length (Le): Depends on the end conditions of the column:
- Pinned-Pinned: Le = L
- Fixed-Fixed: Le = 0.5L
- Fixed-Pinned: Le = 0.7L
- Fixed-Free: Le = 2L
L is the actual length of the column.

Euler's formula is valid for long columns where buckling occurs elastically. For intermediate or short columns, other formulas like the Rankine-Gordon formula are used to account for both buckling and crushing.

4.4 Determinate Structures-1

Structural analysis begins with classifying structures as determinate or indeterminate. Determinate structures can be fully analyzed using only the equations of static equilibrium.

Degree of Determinacy

The degree of determinacy indicates whether a structure can be analyzed using static equilibrium equations alone. It is determined by comparing the number of unknown reactions and internal forces to the number of available equilibrium equations.

External Determinacy: Relates to the reactions. For a 2D structure, there are 3 equilibrium equations (ΣFx=0, ΣFy=0, ΣM=0). If r is the number of unknown reactions, then:
- r = 3: Statically determinate externally.
- r > 3: Statically indeterminate externally (degree of indeterminacy = r - 3).
- r < 3: Unstable.
Internal Determinacy: Relates to the internal forces in members (e.g., in trusses).
Degree of Indeterminacy for Beams/Frames (2D): Degree of Indeterminacy = (Number of unknown reactions + Number of internal releases) - (Number of equilibrium equations + Number of condition equations) A simplified form for rigid frames is often I = (3m + r) - 3j, where m is members, r is reactions, j is joints. For beams, it's typically r - 3 for external indeterminacy.

Energy Methods

Energy methods provide powerful tools for analyzing deflections and forces in structures, particularly useful for complex loading or geometry. They are based on the principle of conservation of energy.

Strain Energy (U): The energy stored in a deformable body due to elastic deformation.
- Due to Bending: U = ∫(M² dx) / (2EI) where M is the bending moment, E is Young's Modulus, and I is the moment of inertia.
- Due to Axial Force: U = P²L / (2AE) where P is the axial force, L is length, A is area, and E is Young's Modulus.
- Due to Torsion: U = T²L / (2GJ) where T is torque, L is length, G is shear modulus, and J is polar moment of inertia.
Castigliano's Theorem: States that the partial derivative of the total strain energy with respect to a force (or moment) is equal to the displacement (or rotation) at the point of application of that force (or moment) in its direction.

Virtual Work Method (Unit Load Method)

The Virtual Work Method (or Unit Load Method) is a versatile technique for calculating deflections and rotations in structures. It is based on the principle of virtual work, which states that the external virtual work done on a deformable body is equal to the internal virtual work (strain energy) stored in the body.

Deflection (δ): δ = ∫(M m dx) / (EI) (for bending) where M is the bending moment due to the actual (real) loads, and m is the bending moment due to a virtual unit load applied at the point and in the direction of the desired deflection.
Similar formulas exist for axial forces (∫(N n dx) / (AE)) and torsional moments (∫(T t dx) / (GJ)).

Example: To find the deflection at the free end of a cantilever beam under a point load, one would first find the real bending moment function M(x). Then, apply a virtual unit load at the free end and find the virtual bending moment function m(x). Finally, integrate their product over the beam length.

Deflection of Beams and Portal Frames

Using the methods above, deflections can be calculated for various determinate structures:

Deflection of Beams: Standard formulas exist for common cases (e.g., simply supported beam with UDL, cantilever with point load). For more complex loading, the integral methods or virtual work method are applied.
Deflection of Portal Frames: The virtual work method is particularly effective for frames, as it can account for axial, shear, and bending deformations, though bending often dominates.

4.5 Determinate Structures-2

This section extends the analysis of determinate structures, focusing on influence lines and moving loads, as well as the behavior of arches.

Influence Lines for Simple Structures

An influence line (IL) is a diagram that shows the variation of a specific response function (e.g., reaction, shear force at a section, bending moment at a section) as a unit load moves across the structure. Influence lines are crucial for analyzing structures subjected to moving loads, such as bridges.

They are typically drawn for a unit point load (1 unit) moving from one end of the structure to the other.
The ordinate of the IL at any point represents the value of the response function when the unit load is at that point.

Muller-Breslau Principle

The Muller-Breslau principle is a qualitative method for sketching influence lines. It states that the influence line for a reaction component or internal force (shear or moment) at a section is proportional to the deflected shape of the structure when a unit displacement (or rotation) corresponding to that reaction or internal force is applied at that point, while removing the capacity of the structure to resist that force (or moment).

Example: To draw the IL for reaction at support A, remove support A and apply a unit vertical displacement at A. The resulting deflected shape is the IL for reaction at A.

ILD for Simply Supported Beam

Reactions: ILD for a reaction is a triangle, peaking at the support where the reaction is.
Shear at a Section: ILD for shear at a section shows a jump at the section itself, with linear variation on either side.
Moment at a Section: ILD for moment at a section is typically a triangle, peaking at the section.

ILD for Cantilever Beam

Influence lines for cantilever beams are generally simpler. For instance, the IL for the fixed-end reaction is a horizontal line (value 1), and for the fixed-end moment, it's a triangle (value 1 at the free end).

Moving Loads: Maximum Shear and Moment Positions

Influence lines are used to determine the maximum possible shear and moment at a specific section due to a train of moving loads (e.g., a series of point loads or a UDL). To maximize a response:

Maximum Shear: Position the loads such that the heaviest load is just to the right (for positive shear) or just to the left (for negative shear) of the section, and the average load on the segment is maximized.
Maximum Moment: Position the loads such that the section lies between the resultant of the loads and the nearest heavy load, with the resultant and the heavy load equidistant from the section. For UDL, the entire span or specific segments are loaded.

Analysis of Two-Hinged Arches

A two-hinged arch is a determinate structure if the supports are at the same level, but becomes indeterminate if the horizontal thrust is unknown (as it usually is). However, for a two-hinged arch, the horizontal thrust can be determined using compatibility equations (e.g., zero horizontal displacement at one support relative to the other). Once the horizontal thrust (H) is found, the reactions can be calculated using static equilibrium equations.

Horizontal Thrust (H): For a parabolic arch with uniform load w over the full span L and rise h: H = wL² / (8h) (This is a simplified case, generally calculated using virtual work or strain energy methods).
Reactions: Vertical reactions (VA, VB) are found using vertical equilibrium and moment equilibrium, incorporating the calculated H. The internal forces (normal force, shear force, bending moment) at any section can then be determined by taking a section and applying equilibrium equations.

4.6 Indeterminate Structures

Indeterminate structures cannot be fully analyzed using static equilibrium equations alone. Additional compatibility equations, based on deformation, are required. This section covers various methods for analyzing such structures.

Flexibility Method (Force Method)

The flexibility method (or force method) is an approach to analyze indeterminate structures by treating redundant forces (reactions or internal forces that cause indeterminacy) as unknowns. The steps involve:

Determine the Degree of Indeterminacy: Identify the number of redundant forces.
Choose a Primary Structure: Remove the redundant forces to make the structure statically determinate.
Apply External Loads: Calculate displacements at the locations of the redundant forces due to the actual external loads on the primary structure.
Apply Redundant Forces: Apply each redundant force as a unit load on the primary structure and calculate the corresponding displacements (flexibility coefficients).
Formulate Compatibility Equations: Set up equations stating that the total displacement at each redundant location in the primary structure must match the actual displacement (often zero) in the original indeterminate structure. δ_i0 + Σ (f_ij * X_j) = 0 (where δ_i0 is displacement at point i due to actual loads, f_ij is flexibility coefficient, X_j is redundant force).
Solve for Redundant Forces: Solve the system of compatibility equations.
Calculate Final Forces/Moments: Use the calculated redundant forces and equilibrium equations to find all other reactions and internal forces.

Two-Hinged Parabolic Arches (Indeterminate Case)

While often treated as determinate for specific load cases, a two-hinged arch with supports at different levels or complex loading is indeterminate. The horizontal thrust H is often the redundant force. It can be found using the flexibility method by treating H as a redundant force and setting the horizontal displacement at one hinge to zero (relative to the other) via virtual work. H = - (∫(M_0 * m_h dx) / (EI)) / (∫(m_h² dx) / (EI)) where M_0 is moment in primary structure (arch as a beam) and m_h is moment due to unit horizontal thrust.

Slope Deflection Method

The slope deflection method is a displacement method that relates the end moments of a member to the rotations and displacements of its joints. It is particularly suitable for continuous beams and frames.

Slope-Deflection Equation: For a member AB, the end moment M_AB is given by: M_AB = (2EI/L)(2θA + θB - 3ψ) + FEM_AB where θA, θB are rotations at ends A and B, ψ = Δ/L is the chord rotation (due to relative joint displacement Δ), and FEM_AB is the fixed-end moment at A due to applied loads.
Steps:
1. Identify unknown joint rotations (θ) and joint displacements (Δ).
2. Calculate Fixed-End Moments (FEMs) for all members.
3. Write slope-deflection equations for each member end moment.
4. Formulate equilibrium equations at each joint (sum of moments = 0) and for each sway degree of freedom (sum of shear forces = 0 for columns).
5. Solve the system of equations for unknown θ and Δ.
6. Substitute back to find all end moments.

Moment Distribution Method (Hardy Cross)

The moment distribution method is an iterative displacement method for analyzing continuous beams and rigid frames. It simplifies the solution of simultaneous equations by distributing unbalanced moments at joints until equilibrium is achieved.

Distribution Factors (DF): Proportion of an unbalanced moment that each member at a joint resists. DF = (EI/L)_member / Σ(EI/L)_all members at joint.
Carry-Over Factor (COF): The fraction of moment carried over from a distributed end to the far end of the member (typically 0.5 for prismatic members).
Fixed-End Moments (FEM): Moments induced at the ends of a member if both ends were fixed, due to applied loads.
Steps:
1. Calculate FEMs for all members.
2. Calculate DFs for all joints.
3. Lock all joints (assume fixed).
4. Release each joint sequentially: calculate unbalanced moment, distribute it to members using DFs, and carry over half the distributed moment to the far ends.
5. Repeat the distribution and carry-over process until moments converge to a negligible value.
6. Sum all moments (FEMs, distributed, carry-over) at each end to get final end moments.

Stiffness Method (Displacement Method)

The stiffness method (or displacement method) is a powerful matrix-based approach, widely used in computer-aided structural analysis. It formulates equilibrium equations in terms of unknown joint displacements.

Stiffness Matrix: Represents the forces required at the degrees of freedom to produce unit displacements at those degrees of freedom. The global stiffness matrix [K] relates global forces [F] to global displacements [D]: [F] = [K][D].
Steps:
1. Define degrees of freedom (displacements and rotations) for the structure.
2. Determine element stiffness matrices (e.g., for beams, columns).
3. Assemble the global stiffness matrix [K] from element stiffness matrices.
4. Formulate the global force vector [F] (including equivalent nodal forces from distributed loads).
5. Apply boundary conditions to modify [K] and [F].
6. Solve the system of linear equations [F] = [K][D] for the unknown joint displacements [D].
7. Calculate member end forces by substituting [D] back into element force-displacement relations.

Influence Lines for Continuous Beams

Influence lines for indeterminate structures (like continuous beams) are typically curvilinear. They can be determined using the Muller-Breslau principle (qualitatively) or by applying the force method (flexibility method) or stiffness method (quantitatively). For example, to find the IL for a reaction, make that reaction a redundant force, apply a unit displacement, and the resulting deflected shape is the IL.

Elementary Plastic Analysis

Plastic analysis considers the inelastic behavior of structural members, particularly steel, allowing for a more economical design by utilizing the material's full strength beyond the elastic limit.

Plastic Moment (Mp): The maximum bending moment a section can resist when the entire cross-section has yielded. Mp = fy * Zp where fy is the yield stress and Zp is the plastic section modulus.
Plastic Section Modulus (Zp): The first moment of area of the cross-section about the plastic neutral axis. For a rectangular section of width b and depth d, Zp = bd²/4.
Elastic Section Modulus (Ze): Ze = I/y_max.
Shape Factor: The ratio of plastic section modulus to elastic section modulus. Shape Factor = Zp / Ze For a rectangular section, shape factor = 1.5. For an I-beam, it's typically around 1.12.
Load Factor (λ): The ratio of the plastic collapse load to the working (design) load. λ = Collapse Load / Working Load
Mechanism Method: A method in plastic analysis to determine the collapse load by postulating various possible plastic hinge mechanisms and applying the principle of virtual work. The collapse load is the minimum load corresponding to any valid mechanism.

Example: For a fixed-fixed beam with a central point load, the plastic collapse occurs when plastic hinges form at the fixed ends and under the point load, creating a mechanism.

4. Structural Mechanics (ACiE04)