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Structural Materials & Testing

8% of exam

Engineering properties of concrete, steel, timber, and masonry; stress-strain behavior and modulus; concrete strength and mix; and material test methods (compression, tension, slump) and specification conformance.

4 concepts

B. Concrete

Concrete Properties, Modulus & Strength

Specified compressive strength f′c, the elastic modulus Ec = 57000√f′c (psi), modulus of rupture, the stress-strain curve, β1, mix proportioning, and reinforcement grades.

Concrete is the material the PE Civil exam returns to more than any other, and almost every concrete calculation hangs off one number you are handed in the problem statement: the specified compressive strength

f'_c

, in psi. From that single value flow the elastic modulus that controls deflection, the modulus of rupture that controls cracking, the stress-block factor

\beta_1

that controls flexural capacity, and the permissible

\sqrt{f'_c}

terms that govern shear and development. Points are lost here not on the physics but on the bookkeeping: keeping

f'_c

in psi inside every radical, not in ksi; remembering that

f'_c

is a ·specified· design strength, not a test result; and never confusing concrete's brittle compression-dominated behavior with the ductile yielding of steel. The NCEES PE Civil Reference Handbook §4.3 Concrete supplies the modulus and

\beta_1

definitions; the full design framework lives in ACI 318-14.

What f′c means and how concrete fails

The specified compressive strength

f'_c

is the 28-day cylinder strength the mix is ·designed and accepted· to reach, not the strength of any one cylinder. Concrete carries compression well and tension poorly — its tensile strength is roughly

8

to

15\%

of

f'_c

— so structural concrete is reinforced to carry every tension force with steel. Under compression the response is nonlinear from the very start, rising to a peak near a strain of about

0.002

and then descending; ACI 318 fixes the usable extreme-fiber compression strain at the crushing limit

\varepsilon_{cu}=0.003

for flexural design. Higher-strength mixes are stiffer and stronger but more brittle, with a sharper post-peak drop.

\varepsilon_{cu} = 0.003 \quad (\text{ACI 318 design crushing strain})

Elastic modulus Ec

The modulus of elasticity sets every elastic deflection, every modular ratio

n=E_s/E_c

, and the cracked-section stiffness used in service checks. For normalweight concrete the handbook gives

E_c = 57{,}000\sqrt{f'_c}

with

f'_c

and

E_c

both in psi — this is a secant modulus to about

0.45f'_c

, not a tangent modulus. The more general form

E_c = 33\,w_c^{1.5}\sqrt{f'_c}

carries the unit weight

w_c

in pcf and reduces to the

57{,}000

form when

w_c=145\,\text{pcf}

; use it for lightweight concrete, where the lower

w_c

gives a markedly lower modulus. Keep

f'_c

in psi: a

4{,}000\,\text{psi}

mix gives

E_c\approx 3{,}600\,\text{ksi}

, roughly an eighth of steel's

29{,}000\,\text{ksi}

.

E_c = 57{,}000\sqrt{f'_c}\ \text{(psi)} \;=\; 33\,w_c^{1.5}\sqrt{f'_c}\ \text{(psi, }w_c\text{ in pcf)}

Modulus of rupture and tensile behavior

Because concrete cracks in flexural tension long before it crushes, ACI characterizes its flexural tensile strength with the modulus of rupture

f_r = 7.5\,\lambda\sqrt{f'_c}

(psi), where

\lambda

is the lightweight modification factor (

1.0

for normalweight, smaller for lightweight). The modulus of rupture governs the cracking moment

M_{cr}=f_r\,I_g/y_t

used in deflection (effective moment of inertia) and in the minimum-reinforcement check. A separate

\sqrt{f'_c}

family with a

2

coefficient governs the concrete shear contribution

V_c

. In every case the radical takes

f'_c

in ·psi· — a

4{,}000\,\text{psi}

mix gives

f_r\approx 474\,\text{psi}

, not

f_r

from

\sqrt{4}

.

f_r = 7.5\,\lambda\sqrt{f'_c}\ \text{(psi)}, \qquad M_{cr} = \dfrac{f_r\,I_g}{y_t}

The equivalent stress block and β1

For strength design ACI replaces the true curved compression stress distribution with an equivalent rectangle of uniform stress

0.85f'_c

acting over a depth

a=\beta_1 c

, where

c

is the neutral-axis depth. The factor

\beta_1

relates the block depth to the true neutral axis and ·steps down· as concrete gets stronger and more brittle: it is

0.85

for

f'_c\le 4{,}000\,\text{psi}

, then drops

0.05

for each

1{,}000\,\text{psi}

above

4{,}000

, with a floor of

0.65

for

f'_c\ge 8{,}000\,\text{psi}

. Mixing up the constant

0.85

stress intensity (which never changes) with the variable

\beta_1

depth factor is one of the most common flexure errors on the exam.

\beta_1 = \begin{cases} 0.85 & f'_c \le 4{,}000 \\[2pt] 0.85 - 0.05\dfrac{f'_c-4000}{1000} & 4{,}000 < f'_c < 8{,}000 \\[2pt] 0.65 & f'_c \ge 8{,}000 \end{cases}\ \text{(psi)}

Mix proportioning concepts

Strength is driven primarily by the water-cement ratio

w/c

(or water-cementitious ratio

w/cm

when supplementary materials are used): lower

w/c

means higher strength and lower permeability, with typical structural mixes near

0.40

to

0.50

. Aggregate gradation, cement content, air entrainment for freeze-thaw durability, and admixtures round out the design, but on the PE the operative idea is the inverse strength-versus-

w/c

trend and the durability caps that a code may place on

w/c

regardless of strength. Workability is measured by slump; more water raises slump but lowers strength, which is why plasticizers are used instead of extra water to keep a mix workable.

f'_c \uparrow \;\Longleftrightarrow\; \dfrac{w}{cm} \downarrow

Reinforcement grades

Deforming bars are specified by grade, which is the minimum yield strength

f_y

in ksi: Grade 60 (

f_y=60\,\text{ksi}

) is the workhorse for flexural and column reinforcement, Grade 40 survives in some footing and slab applications, and Grade 75 or 80 appears where congestion drives the designer to higher strength. Bar sizes are designated by eighths of an inch for the imperial system — a #8 bar is

1

inch in diameter with area

0.79\,\text{in}^2

, a #4 is

\tfrac12

inch — and the handbook §4.3.1 tabulates the standard areas. The reinforcement modulus is taken as

E_s=29{,}000\,\text{ksi}

, the same as structural steel, so the modular ratio

n=E_s/E_c

falls between about

7

and

9

for ordinary mixes.

n = \dfrac{E_s}{E_c}, \qquad E_s = 29{,}000\ \text{ksi}

Exam strategy

Write

f'_c

in psi the instant you see it and keep it in psi inside every

\sqrt{f'_c}

— the

57{,}000

,

7.5\lambda

, and shear coefficients are all calibrated for psi. For modulus, default to

E_c=57{,}000\sqrt{f'_c}

unless the problem gives a unit weight or says lightweight, in which case switch to the

33\,w_c^{1.5}

form. Compute

\beta_1

from the staircase before any flexural capacity, and never let the variable

\beta_1

contaminate the fixed

0.85

stress intensity. For reinforcement, read

f_y

off the grade in ksi and the bar area off the §4.3.1 table; carry

E_s=29{,}000\,\text{ksi}

and

\varepsilon_{cu}=0.003

as constants. Cite handbook §4.3 for the modulus and

\beta_1

, and ACI 318-14 (the supplied edition, not 318-19) for the design provisions.

Key equations

Elastic modulus (normalweight)

E_c = 57{,}000\sqrt{f'_c}\ \text{(psi)}

Secant modulus of normalweight concrete. $f'_c$ and $E_c$ both in psi; e.g., $f'_c=4{,}000\,\text{psi}\Rightarrow E_c\approx 3{,}605\,\text{ksi}$

Elastic modulus (general, by unit weight)

E_c = 33\,w_c^{1.5}\sqrt{f'_c}\ \text{(psi)}

Modulus of rupture

f_r = 7.5\,\lambda\sqrt{f'_c}\ \text{(psi)}

Cracking moment

M_{cr} = \dfrac{f_r\,I_g}{y_t}

Stress-block factor β1

\beta_1 = 0.85 - 0.05\dfrac{f'_c - 4000}{1000}\quad (4000<f'_c<8000\ \text{psi})

Equivalent compression stress

C = 0.85\,f'_c\,b\,a

Resultant compression on the Whitney block; the $0.85$ stress intensity is constant (does not change with $f'_c$

Design crushing strain

\varepsilon_{cu} = 0.003

ACI extreme-fiber compression strain at flexural failure; anchors strain compatibility.

Reinforcement modulus

E_s = 29{,}000\ \text{ksi}

Same as structural steel; gives modular ratio $n=E_s/E_c \approx 7$

Modular ratio

n = \dfrac{E_s}{E_c}

Used to transform steel area to equivalent concrete in cracked-section (service) analysis.

Bar size to diameter

d_b = \dfrac{\#}{8}\ \text{in (imperial bars)}

For #3 through #8, a #n bar is n eighth-inches in diameter (#8 = 1.00 in, $A_b=0.79\,\text{in}^2$

Worked examples

Modulus and modulus of rupture of a 5,000 psi mix

Problem. A normalweight concrete is specified at $f'_c = 5{,}000\,\text{psi}$ . Find the elastic modulus $E_c$ (in ksi), the modulus of rupture $f_r$ , and the stress-block factor $\beta_1$ .

Solution. Modulus:

E_c = 57{,}000\sqrt{5{,}000} = 57{,}000(70.71) = 4{,}030{,}000\,\text{psi} = 4{,}030\,\text{ksi}

E_c = 57{,}000\sqrt{5{,}000} = 4{,}030\ \text{ksi}, \qquad \beta_1 = 0.80

Lightweight concrete modulus by unit weight

Problem. A structural lightweight concrete has $f'_c = 4{,}000\,\text{psi}$ and a unit weight $w_c = 115\,\text{pcf}$

Common pitfalls

•Putting $f'_c$ in ksi inside $\sqrt{f'_c}$ . Every ACI radical — $57{,}000\sqrt{f'_c}$ , $7.5\lambda\sqrt{f'_c}$ , the shear terms — is calibrated for $f'_c$ in psi. Using $\sqrt{4}=2$ instead of $\sqrt{4000}=63.2$ is off by a factor of $\sqrt{1000}$ .
•Confusing the constant $0.85$ stress intensity with the variable $\beta_1$ depth factor. The $0.85$ in $0.85f'_c$
•Using $E_c=57{,}000\sqrt{f'_c}$
•Treating $f'_c$ as a measured cylinder result. It is the ·specified· design strength; individual cylinders scatter above and below it, and acceptance is statistical, not pass-by-pass.
•Citing ACI 318-19 for design provisions. NCEES supplies ACI 318-14 — cite that edition; the $\beta_1$ staircase and $E_c$ form are unchanged but the clause numbers and supplied edition are 318-14.
•Reading a grade as a stress in psi. Grade 60 means $f_y = 60\,\text{ksi} = 60{,}000\,\text{psi}$ — keep $f_y$
•Forgetting $\lambda$ in $f_r$ and the shear terms for lightweight concrete, which reduces the tensile and shear contributions below the normalweight value.

References

NCEES PE Civil Reference Handbook — §4.3 Concrete (Ec, β1 definitions, §4.3.1 bar areas)
ACI 318-14, Building Code Requirements for Structural Concrete — §19.2 (Ec, fr) and §22.2 (equivalent stress block, β1)
ACI 211.1, Standard Practice for Selecting Proportions for Normal, Heavyweight, and Mass Concrete — water-cement ratio and mix proportioning background

C. Steel

Structural Steel: Stress-Strain, Grades & Testing

The steel stress-strain curve — yield Fy, ultimate Fu, E = 29000 ksi — ductility measures, the A992/A36/A572 grades, and tension and Charpy V-notch testing.

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D. Timber

Timber & Masonry Material Properties

Sawn-lumber and glulam species/grades, NDS reference design values and adjustment factors, moisture and service effects, and masonry f′m with unit, mortar, and grout types.

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F. Material test methods

Material Test Methods & Spec Conformance

Concrete cylinder compression and slump, steel tension coupons and Charpy impact, aggregate/soil tests, and the statistical acceptance criteria that decide spec conformance.

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.

$w_c$ = unit weight (pcf), $90\le w_c\le 160$ . Reduces to the $57{,}000$ form at $w_c=145\,\text{pcf}$ ; use for lightweight concrete.

Flexural tensile strength. $\lambda=1.0$ normalweight; smaller for lightweight. Drives the cracking moment and minimum-steel check.

$I_g$ = gross moment of inertia, $y_t$ = distance from centroid to tension face. Used in effective-moment-of-inertia deflection.

Depth of equivalent block $a=\beta_1 c$ . $\beta_1=0.85$ for $f'_c\le4{,}000$ , floor $0.65$ at $f'_c\ge8{,}000$ psi.

).

to

9

.

; #4 = 0.50 in); #9-#11 are larger and no longer follow the eighths rule. From handbook §4.3.1.

. Modulus of rupture:

f_r = 7.5(1.0)\sqrt{5{,}000} = 7.5(70.71) = 530\,\text{psi}

. Stress-block factor:

\beta_1 = 0.85 - 0.05\dfrac{5{,}000-4{,}000}{1{,}000} = 0.85 - 0.05 = 0.80

. **Answers:

E_c = 4{,}030\,\text{ksi}

,

f_r = 530\,\text{psi}

,

\beta_1 = 0.80

.** Sanity check:

E_c

is about

1/7

of steel's

29{,}000\,\text{ksi}

and

f_r

is about

10.6\%

of

f'_c

— both in the expected bands;

\beta_1

dropped one

0.05

step from

0.85

, consistent with one

1{,}000\,\text{psi}

above

4{,}000

.

. Compare its elastic modulus to a normalweight (

w_c=145\,\text{pcf}

) mix of the same strength.

Solution. Lightweight:

E_c = 33\,(115)^{1.5}\sqrt{4{,}000} = 33(1{,}233.6)(63.25) = 2{,}574{,}000\,\text{psi} = 2{,}570\,\text{ksi}

. Normalweight (same

f'_c

):

E_c = 57{,}000\sqrt{4{,}000} = 3{,}605{,}000\,\text{psi} = 3{,}605\,\text{ksi}

(equivalently

33(145)^{1.5}\sqrt{4{,}000}=3{,}644\,\text{ksi}

). **Answer: lightweight

E_c \approx 2{,}570\,\text{ksi}

vs normalweight

\approx 3{,}605\,\text{ksi}

— about

71\%

.** Sanity check: modulus scales with

w_c^{1.5}

, so

(115/145)^{1.5}=0.705

, matching the ratio of the two moduli; the lighter mix is correctly the more flexible one.

\dfrac{E_{c,\text{LW}}}{E_{c,\text{NW}}} = \left(\dfrac{115}{145}\right)^{1.5} = 0.705

never changes;

\beta_1

steps down from

0.85

to

0.65

as

f'_c

rises.

for lightweight concrete. That form assumes

w_c=145\,\text{pcf}

; for lightweight use

E_c=33\,w_c^{1.5}\sqrt{f'_c}

with the actual unit weight.

in the same units as the rest of the check.