Forces on buildings: Forces of compression and tension, concept of equilibrium forces and conditions of equilibrium, concept of elasticity and plasticity, Hooke's law, stress - strain relationship of tension and compression. Shear force and bending moment

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1. Compression and Tension

When loads such as dead loads, live loads, wind, and seismic forces act on a building, internal forces develop within structural members to resist these actions.

The two fundamental axial forces are compression and tension.

Axial Compression

━━━━► ┌──────────────────────────────────┐ ◄━━━━
       │       Structural Member         │
       └──────────────────────────────────┘
          Member Shortens

Compression occurs when forces act inward along the longitudinal axis of a member. The member tends to shorten, while its cross-section may expand laterally.

Typical compression members include:

• Columns
• Piers
• Arches
• Structural walls
• Compression chords of trusses

Slender compression members can be susceptible to buckling, a sudden form of lateral instability that may occur before the material reaches its ultimate compressive strength.

Materials such as concrete, stone, and masonry are generally well suited to carrying compressive loads.

Axial Tension

◄━━━━ ┌──────────────────────────────────┐ ━━━━►
       │       Structural Member         │
       └──────────────────────────────────┘
          Member Elongates

Tension occurs when forces act outward along the longitudinal axis of a member. The member elongates and may contract laterally.

Common tension members include:

• Tie rods
• Suspension cables
• Tension members in trusses
• Steel reinforcement

Tension members do not experience classical flexural buckling under pure axial tension. Structural steel and timber are commonly used where high tensile capacity is required.

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2. Structural Equilibrium

A structure is in static equilibrium when all external forces and support reactions are balanced, resulting in zero linear and rotational acceleration.

For a two-dimensional, or coplanar, structural system, equilibrium is satisfied when:

Horizontal Equilibrium

$$\sum F_x = 0$$

The algebraic sum of all horizontal forces must equal zero.

Vertical Equilibrium

$$\sum F_y = 0$$

The algebraic sum of all vertical forces must equal zero.

Moment Equilibrium

$$\sum M_z = 0$$

The algebraic sum of all moments about a reference point must equal zero.

These equations are fundamental for calculating support reactions and internal structural forces.

If the forces or moments are not balanced, a structure may translate or rotate. Structural supports such as pins, rollers, and fixed supports provide reactions that restrain movement and maintain stability.

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3. Elasticity and Plasticity

When a structural material is subjected to an external load, it develops stress and undergoes strain. The resulting deformation depends on the material's mechanical properties and the magnitude of the applied load.

Elasticity

Elasticity is the ability of a material to return to its original shape and dimensions after the applied load is removed, provided the material remains within its elastic range.

Elastic deformation is generally recoverable, meaning the material returns to its original configuration when the load is removed.

Plasticity

Plasticity is the ability of a material to undergo permanent deformation after its elastic limit has been exceeded.

When the load is removed, a plastically deformed material does not completely return to its original shape.

For ductile materials such as mild steel, controlled plastic deformation can provide valuable ductility, energy absorption, and redistribution of stresses before ultimate failure.

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4. Hooke's Law

Within the elastic range of a material, Hooke's Law states that normal stress is directly proportional to normal strain:

$$\sigma \propto \varepsilon$$

Therefore:

$$\sigma = E\varepsilon$$

Where:

• σ = Normal stress

  $\sigma = \frac{P}{A}$

  Common units: N/mm² or MPa

• ε = Normal strain

  $\varepsilon = \frac{\Delta L}{L}$

  Strain is dimensionless.

• E = Young's Modulus (Modulus of Elasticity)

  Represents the stiffness of the material and is commonly expressed in N/mm² or GPa.

A material with a higher Young's Modulus generally undergoes less elastic deformation under the same level of stress.

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5. Stress–Strain Relationship

A stress–strain curve represents the mechanical response of a material as it is subjected to increasing load.

For a typical ductile material such as structural steel, the curve demonstrates the transition from elastic behavior to yielding, plastic deformation, ultimate strength, and fracture.

Stress (σ)
   ▲
   │                 Ultimate Strength (C)
   │                       ┌───┐
   │      Yield Point      │   │
   │          (B) ─────────┘   └──────► Fracture (D)
   │         ╱       Plastic Region
   │        ╱
   │       ╱
   │  (A) ╱
   │     ╱
   └────┴──────────────────────────────► Strain (ε)
        Elastic Region

Important Points on the Stress–Strain Curve

1. Proportional Limit (A)
The point up to which stress is directly proportional to strain and the material follows Hooke's Law.

2. Yield Point (B)
The stage at which significant plastic deformation begins. Beyond this point, permanent deformation remains after unloading.

3. Ultimate Tensile Strength (C)
The maximum engineering stress reached during a tensile test. In a ductile specimen, localized necking generally develops after this point.

4. Fracture Point (D)
The point at which the specimen ultimately breaks under tensile loading.

Compression and Tension Behavior

Material	Compression	Tension
Structural Steel	High strength with ductile behavior	High strength and significant ductility
Concrete	High compressive strength with nonlinear behavior	Relatively low tensile strength and cracking tendency
Stone / Masonry	Generally strong in compression	Generally weak and brittle in tension

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6. Shear Force and Bending Moment

When transverse loads act on a structural member such as a beam, they produce important internal actions known as Shear Force (V) and Bending Moment (M).

These quantities are essential for analyzing and designing beams, slabs, frames, and other flexural structural elements.

Shear Force (V)

Shear force is the algebraic sum of the transverse forces acting on one side of a selected cross-section.

It represents an internal action that tends to cause adjacent portions of a beam to slide relative to one another.

Bending Moment (M)

Bending moment is the algebraic sum of the moments of forces about a selected cross-section.

It causes the member to bend, producing compression on one side of the neutral axis and tension on the other, depending on the direction of bending.

                 Transverse Load (P)
                         │
                         ▼
        ┌─────────────────────────────────┐
        │     Top Fibers — Compression    │
        ├ ─ ─ ─ ─ ─ Neutral Axis ─ ─ ─ ─ ┤
        │     Bottom Fibers — Tension     │
        └─────────────────────────────────┘
          ▲                             ▲
       Reaction R₁                  Reaction R₂

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7. Relationship Between Load, Shear Force and Bending Moment

For a beam subjected to a distributed transverse load, the load intensity, shear force, and bending moment are related through differential equations.

Using a common sign convention:

$$\frac{dV}{dx}=-w(x)$$

$$\frac{dM}{dx}=V(x)$$

Where:

• w(x) = Distributed load intensity
• V(x) = Shear force
• M(x) = Bending moment
• x = Distance along the beam

These relationships form the mathematical basis for developing Shear Force Diagrams (SFDs) and Bending Moment Diagrams (BMDs).

Key Engineering Principle

Since:

$$\frac{dM}{dx}=V$$

the bending moment reaches a local maximum or minimum at a section where:

$$V=0$$

subject to the applicable loading and boundary conditions.

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