6.1 Behavior and properties

Achievable FE Civil

6. Materials

Behavior and properties

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This chapter covers the following:

Material properties of steel reinforcement
Tensile testing of reinforcement
Plain concrete compressive stress vs. strain behavior
General properties of concrete
Types of portland cement

Material properties of steel reinforcement

Stress-strain behavior of steel reinforcement, showing elastic region, yield point, strain hardening, ductility, and rupture, along with a tensile test setup for measuring elongation. — Material properties of steel reinforcement

The figure shows a typical stress-strain curve for reinforcing steel in axial tension. You’ll usually describe the curve in these regions:

Elastic region: The initial linear portion where Hooke’s law applies.
Yield plateau: Stress stays roughly constant while strain increases.
Strain hardening: Stress increases again after yielding.
Necking and rupture: The final stage where the bar fractures.

Here $E_{s} = 29, 000 k s i$ is the elastic modulus of steel. It’s the slope of the initial linear (elastic) portion of the stress-strain curve. For Grade 60 reinforcement, $f_{y} = 60 k s i$ . The yield strain is:

$ε_{y} = \frac{f _{y}}{E _{s}} = \frac{60 k s i}{29 , 000 k s i} \approx 0.00207$

Rupture strain typically falls in this range:

$0.1 < ε_{r u pt u re} < 0.3$

Tensile testing of reinforcement

The right-hand diagram shows a typical tensile test setup used to measure elongation in a reinforcing bar.

Total bar length: 20 inches
Gage length: 2 inches (between displacement gauges)
Load: Axial tensile load $P$ applied at both ends
Dial gauge: Measures elongation $Δ$ over the 2-inch gage length

The axial stress is:

$σ_{a} = f_{a} = \frac{P}{A}$

where:

$P$ = applied force
$A$ = cross-sectional area of the steel bar

The corresponding strain is:

$ε = \frac{Δ}{L} = \frac{Δ}{2 ^{''}}$

where:

$Δ$ = elongation in the $2 in c h es$ gage length
$L = 2 in c h es$

Some interesting facts on steel

Steel’s resistance to deformation increases through alloying and hardening; hot working generally improves ductility, and tempering reduces brittleness after hardening. Ranked by hardness, the common steel microstructures are:

Martensite - hardest, formed by rapid quenching
Bainite - harder than pearlite, and a good balance of strength and toughness
Pearlite - moderate hardness
Ferrite - softest and most ductile

Electropositive metals such as zinc or magnesium sacrificially protect iron from corrosion, while a more noble metal like gold can accelerate corrosion of exposed iron through galvanic action if its coating is damaged.

Plain concrete compressive stress vs. strain behavior

The figure shows the stress-strain relationship of concrete in compression, including the secant and tangent modulus of elasticity, peak strength, and the standard compression test setup used to measure strain. — Concrete compresive stress and strain relationship

The figure shows the compressive stress-strain curve for concrete, typically measured using a standard concrete cylinder (6" diameter × 12" height). In the test, a compressive force $P$ is applied until the cylinder fails.

Diameter: $D = 6^{''}$
Height: $h = 1 2^{''}$
Gauge length for strain measurement: $8^{''}$

The axial stress is:

$f_{c} = \frac{P}{A}$

where:

$P$ = axial compressive load
$A = \frac{π}{4} D^{2}$ = cross-sectional area

The axial strain is:

$ε = \frac{Δ _{a vg}}{L} = \frac{Δ _{a vg}}{8 ^{''}}$

with:

$Δ_{a vg} = \frac{Δ _{t} + Δ _{b}}{2}$

Where:

$Δ_{t}$ and $Δ_{b}$ = measured deformations at the top and bottom

The curve shape is typically described as:

An initial nearly linear region up to around $0.5 f_{c}^{'}$
An increasingly nonlinear (curving) portion after that, as stress approaches the peak
A maximum stress at $f_{c}^{'}$ (ultimate compressive strength)
A descending branch representing softening or crushing

Common terms on the curve include:

$f_{c}^{'}$ : Peak compressive strength of concrete (psi)
$ε_{c}$ : Strain at $f_{c} = f_{c}^{'}$
$ε_{u}$ : Ultimate strain (crushing failure)

In design, $ε_{u} = 0.003$ . This is the maximum usable concrete strain assumed before crushing.

The modulus of elasticity for concrete is commonly described in two ways.

Tangent modulus of elasticity

The slope of the stress-strain curve at a specific point
Often taken at the origin to represent initial stiffness:

$E_{t} = \frac{d f _{c}}{d ε}_{ε \to 0}$

Secant modulus of elasticity

The average slope from the origin to a defined point (often at $0.5 f_{c}^{'}$ ):

$E_{c, sec} = \frac{0.5 f _{c}^{'}}{ε _{0.5}}$

Concrete crushes when:

$ε = ε_{u} \approx 0.003 (in design)$

This failure is brittle and sudden, which is why concrete is typically used with steel reinforcement.

General properties of concrete

Compressive strength ( $f_{c}^{'}$ )

Obtained from 28-day cylinder tests
Normal strength concrete:

$f_{c}^{'} = 2000 to 8000 p s i$

High-strength concrete:

$f_{c}^{'} = 8000 to 25000 p s i$

Tensile strength ( $f_{t}^{'}$ )

Measured by the splitting tensile strength test:

Typical value:

$f_{t}^{'} \approx 0.10 f_{c}^{'}$

Splitting tensile strength formula:

$f_{t}^{'} = \frac{2 P}{π D L}$

Where:

$P$ = applied load
$D$ = diameter
$L$ = length of cylinder

Modulus of rupture ( $f_{r}$ )

Used to estimate tensile strength in flexure (for example, in beams):

Formula (with $f_{c}^{'}$ in psi):

$f_{r} = 7.5 f_{c}^{'}$

Flexural tensile stress from beam loading:

$f_{b} = \frac{M}{S}$

Where, for a standard $6 \times 6$ in beam ( $b = h = 6$ in) on an 18 in span, loaded by two equal loads $P$ at the third points:

$M = 6 P$ (each load acts 6 in from its support, so the moment between the loads is $P \times 6$ )
$S = \frac{b h ^{2}}{6} = \frac{6 \times ( 6 ) ^{2}}{6} = 36 in^{3}$

Substituting:

$f_{b} = \frac{6 P}{36} = \frac{P}{6} (psi)$

The modulus of rupture is the value of $f_{b}$ at the load that cracks the beam, so for this test $f_{r} = P_{cr a c k} /6$ (psi, with each load $P_{cr a c k}$ in lb). The formula $f_{r} = 7.5 f_{c}^{'}$ estimates that value without testing a beam.

Shear strength ( $V_{c}$ )

For normal weight concrete:

$V_{c} = 2 f_{c}^{'} \cdot b \cdot d$

For lightweight concrete, ACI 318 multiplies $f_{c}^{'}$ by the lightweight modification factor $λ$ :

$V_{c} = 2 λ f_{c}^{'} \cdot b \cdot d$

where $λ = 0.75$ for all-lightweight and $0.85$ for sand-lightweight concrete ( $λ = 1.0$ for normal weight), so a lightweight section carries less shear than an otherwise identical normal-weight one.

Unit weight ( $w_{c}$ )

Normal reinforced concrete:

$w_{c} \approx 150 pcf$

Lightweight concrete:

$w_{c} \approx 120 pcf$

Modulus of elasticity ( $E_{c}$ )

Used in serviceability analysis (for example, deflection).

General expression:

$E_{c} = w_{c}^{1.5} (33) f_{c}^{'}$

For normal weight concrete, taking $w_{c} = 145$ pcf (plain concrete; the 150 pcf above includes the reinforcing steel) gives $14 5^{1.5} \times 33 \approx 57, 600$ , which ACI rounds to the simplified formula:

$E_{c} = 57000 f_{c}^{'}$

Example: modulus of elasticity of concrete

A normal weight concrete mix has $f_{c}^{'} = 4000 p s i$ . What is $E_{c}$ ?

$E_{c} = 57000 f_{c}^{'} = 570004000 \approx 3, 600, 000 p s i$

Answer: $E_{c} \approx 3, 600 k s i$

Watch the units: The empirical formulas for $f_{r}$ , $E_{c}$ , and $V_{c}$ all require $f_{c}^{'}$ in psi. $f_{r}$ and $E_{c}$ come out in psi, and $V_{c}$ comes out in lb when $b$ and $d$ are in inches. Substituting an $f_{c}^{'}$ value given in MPa, ksi, or any other unit produces a meaningless answer - always convert $f_{c}^{'}$ to psi first.

Slump test: workability of concrete

Slump indicates consistency (plasticity)
Typical slump values: $1^{''} to 4^{''}$

Water-cement (W/C) ratio

The water-cement ratio is the ratio of water weight to cement weight in a mix.

Typical range: $0.40 to 0.60$

For full hydration:

$W/C ratio = 0.25 \Rightarrow 25% water by weight of cement$

Higher $W / C$ ratio $\to$ more workable, less strength
Lower $W / C$ ratio $\to$ less workability, higher strength

Hydration

Chemical reaction between cement and water
Produces heat of hydration
Critical for early strength and setting

Types of portland cement

Type I: normal portland cement - general purpose
Type II: modified - moderate heat of hydration and moderate sulfate resistance
Type III: high early strength
Type IV: low heat of hydration - massive structures such as dams
Type V: high sulfate resistance - concrete exposed to soil or groundwater with a high sulfate content

Some interesting facts on cement

Fly ash, a pozzolanic material governed by ASTM C618, reacts with calcium hydroxide (not calcium silicate) produced during cement hydration to form additional cementitious compounds. It also acts as a microfiller, reducing permeability and improving resistance to deicer scaling when used in appropriate proportions.

Material properties of steel reinforcement

Stress-strain curve regions: elastic, yield plateau, strain hardening, necking/rupture
Elastic modulus $E_{s} = 29, 000$ ksi; yield strength $f_{y} = 60$ ksi (Grade 60)
Yield strain $ε_{y} = 0.002$ ; rupture strain $0.1 < ε_{r u pt u re} < 0.3$

Tensile testing of reinforcement

Standard bar: 20" length, 2" gage length for elongation measurement
Axial stress: $σ_{a} = \frac{P}{A}$ ; strain: $ε = \frac{Δ}{2 ^{''}}$
Load applied axially; dial gauge measures elongation

Plain concrete compressive stress vs. strain behavior

Stress-strain curve: initial nonlinear, steep rise, peak at $f_{c}^{'}$ , descending branch (softening/crushing)
Ultimate compressive strength $f_{c}^{'}$ ; design ultimate strain $ε_{u} = 0.003$
Modulus of elasticity: tangent (initial slope), secant (origin to $0.5 f_{c}^{'}$ )

General properties of concrete

Compressive strength $f_{c}^{'}$ : 2000–8000 psi (normal), up to 25,000 psi (high-strength)
Tensile strength $f_{t}^{'} \approx 0.10 f_{c}^{'}$ ; modulus of rupture $f_{r} = 7.5 f_{c}^{'}$
Shear strength $V_{c} = 2 f_{c}^{'} b d$ ; unit weight: normal 150 pcf, lightweight 120 pcf
Modulus of elasticity $E_{c} = 57000 f_{c}^{'}$ (normal weight)
Slump (workability): 1"–4"; water-cement ratio: 0.40–0.60
Hydration: cement + water reaction, produces heat, critical for strength

Types of portland cement

Type I: Normal; Type II: Moderate sulfate resistance; Type III: High early strength
Type IV: Low heat (large structures); Type V: Sulfate resistant

Concrete strength testing requirements

Minimum sampling: once/day, each 150 yd³, or each 5000 ft² (whichever governs)
No test required if <50 yd³ and approved evidence of strength
Strength test: average of 2 (6"x12") or 3 (4"x8") cylinders, tested at 28 days
Acceptance: 3-test average $\geq f_{c}^{'}$ ; no single test $< f_{c}^{'} - 500$ psi (if $f_{c}^{'} \leq 5000$ psi) or $< 0.9 f_{c}^{'}$ (if $f_{c}^{'} > 5000$ psi)
Field-cured cylinders: must be $\geq 85%$ of lab-cured, unless field-cured $> f_{c}^{'} + 500$ psi
Core tests: average of 3 cores $\geq 0.85 f_{c}^{'}$ ; no single core $< 0.75 f_{c}^{'}$

Concrete mix design parameters

Yield: total mass of materials / density of fresh concrete
Absolute volume: mass / (specific gravity × density of water)
Aggregate moisture: SSD condition used; free moisture = total – absorbed moisture
Water-cement ratio $w / c = \frac{mass of water}{mass of cementitious materials}$

Aggregate properties in concrete mix design

SSD: pores filled, no surface water
Oven-dry density: includes all pores, excludes inter-particle voids
SSD density: includes water-filled pores, excludes inter-particle voids
Apparent density: only impermeable portion
Absorption: $(M_{SS D} - M_{O D}) / M_{O D} \times 100%$
Bulk specific gravity (SSD): $G_{b u l k, SS D} = 2.4$ – $2.9$ (normal-weight)
Free moisture: water on aggregate surface above SSD

Concrete maturity

Time-Temperature Factor: $M = \sum (T - T_{0}) Δ t$
- $M$ = maturity index; $T$ = avg. concrete temp; $T_{0}$ = datum temp (usually $3 2^{\circ}$ F)

Concrete exposure categories and classes

Freezing/thawing (F0–F3): increasing exposure to cycles, water, deicing chemicals
Sulfate (S0–S3): based on sulfate content in soil/water, from negligible to severe
Water contact (W0–W2): dry, in contact (low permeability not/required)
Corrosion protection (C0–C2): dry/protected, exposed to moisture, or chlorides

Next | 6.2 Testing, mix design, and durability

Behavior and properties

This chapter covers the following:

Material properties of steel reinforcement
Tensile testing of reinforcement
Plain concrete compressive stress vs. strain behavior
General properties of concrete
Types of portland cement

Material properties of steel reinforcement

The figure shows a typical stress-strain curve for reinforcing steel in axial tension. You’ll usually describe the curve in these regions:

Elastic region: The initial linear portion where Hooke’s law applies.
Yield plateau: Stress stays roughly constant while strain increases.
Strain hardening: Stress increases again after yielding.
Necking and rupture: The final stage where the bar fractures.

$ε_{y} = \frac{f _{y}}{E _{s}} = \frac{60 k s i}{29 , 000 k s i} \approx 0.00207$

Rupture strain typically falls in this range:

$0.1 < ε_{r u pt u re} < 0.3$

Tensile testing of reinforcement

The right-hand diagram shows a typical tensile test setup used to measure elongation in a reinforcing bar.

Total bar length: 20 inches
Gage length: 2 inches (between displacement gauges)
Load: Axial tensile load $P$ applied at both ends
Dial gauge: Measures elongation $Δ$ over the 2-inch gage length

The axial stress is:

$σ_{a} = f_{a} = \frac{P}{A}$

where:

$P$ = applied force
$A$ = cross-sectional area of the steel bar

The corresponding strain is:

$ε = \frac{Δ}{L} = \frac{Δ}{2 ^{''}}$

where:

$Δ$ = elongation in the $2 in c h es$ gage length
$L = 2 in c h es$

Some interesting facts on steel

Martensite - hardest, formed by rapid quenching
Bainite - harder than pearlite, and a good balance of strength and toughness
Pearlite - moderate hardness
Ferrite - softest and most ductile

Plain concrete compressive stress vs. strain behavior

Diameter: $D = 6^{''}$
Height: $h = 1 2^{''}$
Gauge length for strain measurement: $8^{''}$

The axial stress is:

$f_{c} = \frac{P}{A}$

where:

$P$ = axial compressive load
$A = \frac{π}{4} D^{2}$ = cross-sectional area

The axial strain is:

$ε = \frac{Δ _{a vg}}{L} = \frac{Δ _{a vg}}{8 ^{''}}$

with:

$Δ_{a vg} = \frac{Δ _{t} + Δ _{b}}{2}$

Where:

$Δ_{t}$ and $Δ_{b}$ = measured deformations at the top and bottom

The curve shape is typically described as:

An initial nearly linear region up to around $0.5 f_{c}^{'}$
An increasingly nonlinear (curving) portion after that, as stress approaches the peak
A maximum stress at $f_{c}^{'}$ (ultimate compressive strength)
A descending branch representing softening or crushing

Common terms on the curve include:

$f_{c}^{'}$ : Peak compressive strength of concrete (psi)
$ε_{c}$ : Strain at $f_{c} = f_{c}^{'}$
$ε_{u}$ : Ultimate strain (crushing failure)

In design, $ε_{u} = 0.003$ . This is the maximum usable concrete strain assumed before crushing.

The modulus of elasticity for concrete is commonly described in two ways.

Tangent modulus of elasticity

The slope of the stress-strain curve at a specific point
Often taken at the origin to represent initial stiffness:

$E_{t} = \frac{d f _{c}}{d ε}_{ε \to 0}$

Secant modulus of elasticity

The average slope from the origin to a defined point (often at $0.5 f_{c}^{'}$ ):

$E_{c, sec} = \frac{0.5 f _{c}^{'}}{ε _{0.5}}$

Concrete crushes when:

$ε = ε_{u} \approx 0.003 (in design)$

This failure is brittle and sudden, which is why concrete is typically used with steel reinforcement.

General properties of concrete

Compressive strength ( $f_{c}^{'}$ )

Obtained from 28-day cylinder tests
Normal strength concrete:

$f_{c}^{'} = 2000 to 8000 p s i$

High-strength concrete:

$f_{c}^{'} = 8000 to 25000 p s i$

Tensile strength ( $f_{t}^{'}$ )

Measured by the splitting tensile strength test:

Typical value:

$f_{t}^{'} \approx 0.10 f_{c}^{'}$

Splitting tensile strength formula:

$f_{t}^{'} = \frac{2 P}{π D L}$

Where:

$P$ = applied load
$D$ = diameter
$L$ = length of cylinder

Modulus of rupture ( $f_{r}$ )

Used to estimate tensile strength in flexure (for example, in beams):

Formula (with $f_{c}^{'}$ in psi):

$f_{r} = 7.5 f_{c}^{'}$

Flexural tensile stress from beam loading:

$f_{b} = \frac{M}{S}$

Where, for a standard $6 \times 6$ in beam ( $b = h = 6$ in) on an 18 in span, loaded by two equal loads $P$ at the third points:

$M = 6 P$ (each load acts 6 in from its support, so the moment between the loads is $P \times 6$ )
$S = \frac{b h ^{2}}{6} = \frac{6 \times ( 6 ) ^{2}}{6} = 36 in^{3}$

Substituting:

$f_{b} = \frac{6 P}{36} = \frac{P}{6} (psi)$

Shear strength ( $V_{c}$ )

For normal weight concrete:

$V_{c} = 2 f_{c}^{'} \cdot b \cdot d$

For lightweight concrete, ACI 318 multiplies $f_{c}^{'}$ by the lightweight modification factor $λ$ :

$V_{c} = 2 λ f_{c}^{'} \cdot b \cdot d$

Unit weight ( $w_{c}$ )

Normal reinforced concrete:

$w_{c} \approx 150 pcf$

Lightweight concrete:

$w_{c} \approx 120 pcf$

Modulus of elasticity ( $E_{c}$ )

Used in serviceability analysis (for example, deflection).

General expression:

$E_{c} = w_{c}^{1.5} (33) f_{c}^{'}$

$E_{c} = 57000 f_{c}^{'}$

Example: modulus of elasticity of concrete

A normal weight concrete mix has $f_{c}^{'} = 4000 p s i$ . What is $E_{c}$ ?

$E_{c} = 57000 f_{c}^{'} = 570004000 \approx 3, 600, 000 p s i$

Answer: $E_{c} \approx 3, 600 k s i$

Slump test: workability of concrete

Slump indicates consistency (plasticity)
Typical slump values: $1^{''} to 4^{''}$

Water-cement (W/C) ratio

The water-cement ratio is the ratio of water weight to cement weight in a mix.

Typical range: $0.40 to 0.60$

For full hydration:

$W/C ratio = 0.25 \Rightarrow 25% water by weight of cement$

Higher $W / C$ ratio $\to$ more workable, less strength
Lower $W / C$ ratio $\to$ less workability, higher strength

Hydration

Chemical reaction between cement and water
Produces heat of hydration
Critical for early strength and setting

Types of portland cement

Type I: normal portland cement - general purpose
Type II: modified - moderate heat of hydration and moderate sulfate resistance
Type III: high early strength
Type IV: low heat of hydration - massive structures such as dams
Type V: high sulfate resistance - concrete exposed to soil or groundwater with a high sulfate content

Some interesting facts on cement

Achievable FE Civil

6. Materials