Solutions to Reinforced Concrete
Design: A Practical Approach
(3rd Edition) by (Brzev, 2026) | All 13
Chapters Covered newest version 2026
,3-1 What Is The Significance Of The “Critical Stress”?
(a) With Respect To The Structure Of The Concrete?
A Continuous Pattern Of Mortar Cracks Begins To Form. As A Result There Are Few Undamaged
Portions To Carry Load And The Stress-Strain Curve Is Highly Nonlinear.
(b) With Respect To Spiral Reinforcement?
At The Critical Stress The Lateral Strain Begins To Increase Rapidly. This Causes The Concrete Core
Within The Spiral To Expand, Stretching The Spiral. The Tension In The Spiral Is Equilibrated By A
Radial Compression In The Core. This In Turn, Biaxially Compresses The Core, And Thus
Strengthens It.
(c) With Respect To Strength Under Sustained Loads?
When Concrete Is Subjected To Sustained Loads Greater Than The Critical Stress, It Will Eventually
Fail.
3-2 A Group Of 45 Tests On A Given Type Of Concrete Had A Mean Strength Of 4780
Psi And A Standard Deviation Of 525 Psi. Does This Concrete Satisfy The
Requirements Of Aci Code Section 5.3.2 For 4000-Psi Concrete?
From Eq. 3-3a:
𝑓′ = 𝑓′ − 1. 34𝑠
𝑐 𝑐𝑟
Using 𝑓𝑐𝑟
′ = 4780 Psi
(For Design) 𝑐𝑓′ = 4780 − 1. 34 * 525 = 4080 Psi
From Eq. 3-3b:
𝑓′ = 𝑓′ − 2. 33𝑠 + 500
𝑐 𝑐𝑟
Using 𝑓𝑐𝑟
′ = 4780 Psi
(For Design) 𝑐𝑓′ = 4780 − 2. 33 * 525 + 500 = 4060 Psi
,Because Both Of These Exceed 4000 Psi, The Concrete Satisfies The Requirements Of Aci
Code Section 5.3.2 For 4000 Psi Concrete.
3-3 The Concrete Containing Type I Cement In A Structure Is Cured For 3 Days At
70° F Followed By 6 Days At 40° F. Use The Maturity Concept To Estimate Its
Strength As A Fraction Of The 28-Day Strength Under Standard Curing.
5
Note: C = (F − 32) , So 70° F = 21.1° C And 40° F = 4.4° C
9
From Eq. 3-6:
n
M = (Ti +10)(Ti )
I=1
= (21.1+10)(3) + (4.4 +10)(6) = 180 C Days
From Fig. 3-8 The Compressive Strength Will Be Between 0.60 And 0.70 Times The 28-Day
Strength Under Standard Curing Conditions.
3-4 Use Fig. 3-12a To Estimate The Compressive Strength 2 For Bi-Axially Loaded
Concrete Subject To:
(a) 1 = 0.0, 2 = Fc'
(b) 1 = 0.75 Ft ' In Tension, 2 = 0.5 Fc'
(c) 1 = 0.5 Fc' In Compression, 2 = 1.2 Fc'
3-5 The Concrete In The Core Of A Spiral Is Subjected To A Uniform Confining Stress
3 Of 750 Psi. What Will The Compressive Strength, 1 Be?
𝑐 Assume 𝑓′ = 4500 Psi.
From Eq. 3-16:
𝜎1 = 𝑓𝑐′ + 4.1𝜎3
𝜎1 = 4500 + 4.1 * 750 = 7580 Psi
, 3-6 What Factors Affect The Shrinkage Of Concrete?
(a) Relative Humidity. Shrinkage Increases As The Relative Humidity Decreases,
Reaching A Maximum At Rh ≤ 40%.
(b) The Fraction Of The Total Volume Made Up Of Paste. As This Fraction
Increases, Shrinkage Increases.
(c) The Modulus Of Elasticity Of The Aggregate. As This Increases, Shrinkage
Decreases.
(d) The Water/Cement Ratio. As The Water Content Increases, The Aggregate
Fraction Decreases, Causing An Increase In Shrinkage.
(e) The Fineness Of The Cement. Shrinkage Increases For Finely Ground Cement
That Has More Surface Area To Attract And Absorb Water.
(f) The Effective Thickness Or Volume To Surface Ratio. As This Ratio
Increases, The Shrinkage Occurs More Slowly And The Total Shrinkage Is
Likely Reduced.
(g) Exposure To Carbon Dioxide Tends To Increase Shrinkage.
3-7 What Factors Affect The Creep Of Concrete?
(a) The Ratio Of Sustained Stress To The Strength Of The Concrete. The Creep
Coefficient, , Is Roughly Constant Up To A Stress Of 0.5 Fc', But Increases
Above That Value.
(b) The Humidity Of The Environment. The Amount Of Creep Decreases As The Rh
Increases Above 40%.
(c) As The Effective Thickness Or Volume To Surface Ratio Increases, The Rate At
Which Creep Develops Decreases.
(d) Concretes With A High Paste Content Creep More That Concretes With A Large
Aggregate Fraction Because Only The Paste Creeps.
Design: A Practical Approach
(3rd Edition) by (Brzev, 2026) | All 13
Chapters Covered newest version 2026
,3-1 What Is The Significance Of The “Critical Stress”?
(a) With Respect To The Structure Of The Concrete?
A Continuous Pattern Of Mortar Cracks Begins To Form. As A Result There Are Few Undamaged
Portions To Carry Load And The Stress-Strain Curve Is Highly Nonlinear.
(b) With Respect To Spiral Reinforcement?
At The Critical Stress The Lateral Strain Begins To Increase Rapidly. This Causes The Concrete Core
Within The Spiral To Expand, Stretching The Spiral. The Tension In The Spiral Is Equilibrated By A
Radial Compression In The Core. This In Turn, Biaxially Compresses The Core, And Thus
Strengthens It.
(c) With Respect To Strength Under Sustained Loads?
When Concrete Is Subjected To Sustained Loads Greater Than The Critical Stress, It Will Eventually
Fail.
3-2 A Group Of 45 Tests On A Given Type Of Concrete Had A Mean Strength Of 4780
Psi And A Standard Deviation Of 525 Psi. Does This Concrete Satisfy The
Requirements Of Aci Code Section 5.3.2 For 4000-Psi Concrete?
From Eq. 3-3a:
𝑓′ = 𝑓′ − 1. 34𝑠
𝑐 𝑐𝑟
Using 𝑓𝑐𝑟
′ = 4780 Psi
(For Design) 𝑐𝑓′ = 4780 − 1. 34 * 525 = 4080 Psi
From Eq. 3-3b:
𝑓′ = 𝑓′ − 2. 33𝑠 + 500
𝑐 𝑐𝑟
Using 𝑓𝑐𝑟
′ = 4780 Psi
(For Design) 𝑐𝑓′ = 4780 − 2. 33 * 525 + 500 = 4060 Psi
,Because Both Of These Exceed 4000 Psi, The Concrete Satisfies The Requirements Of Aci
Code Section 5.3.2 For 4000 Psi Concrete.
3-3 The Concrete Containing Type I Cement In A Structure Is Cured For 3 Days At
70° F Followed By 6 Days At 40° F. Use The Maturity Concept To Estimate Its
Strength As A Fraction Of The 28-Day Strength Under Standard Curing.
5
Note: C = (F − 32) , So 70° F = 21.1° C And 40° F = 4.4° C
9
From Eq. 3-6:
n
M = (Ti +10)(Ti )
I=1
= (21.1+10)(3) + (4.4 +10)(6) = 180 C Days
From Fig. 3-8 The Compressive Strength Will Be Between 0.60 And 0.70 Times The 28-Day
Strength Under Standard Curing Conditions.
3-4 Use Fig. 3-12a To Estimate The Compressive Strength 2 For Bi-Axially Loaded
Concrete Subject To:
(a) 1 = 0.0, 2 = Fc'
(b) 1 = 0.75 Ft ' In Tension, 2 = 0.5 Fc'
(c) 1 = 0.5 Fc' In Compression, 2 = 1.2 Fc'
3-5 The Concrete In The Core Of A Spiral Is Subjected To A Uniform Confining Stress
3 Of 750 Psi. What Will The Compressive Strength, 1 Be?
𝑐 Assume 𝑓′ = 4500 Psi.
From Eq. 3-16:
𝜎1 = 𝑓𝑐′ + 4.1𝜎3
𝜎1 = 4500 + 4.1 * 750 = 7580 Psi
, 3-6 What Factors Affect The Shrinkage Of Concrete?
(a) Relative Humidity. Shrinkage Increases As The Relative Humidity Decreases,
Reaching A Maximum At Rh ≤ 40%.
(b) The Fraction Of The Total Volume Made Up Of Paste. As This Fraction
Increases, Shrinkage Increases.
(c) The Modulus Of Elasticity Of The Aggregate. As This Increases, Shrinkage
Decreases.
(d) The Water/Cement Ratio. As The Water Content Increases, The Aggregate
Fraction Decreases, Causing An Increase In Shrinkage.
(e) The Fineness Of The Cement. Shrinkage Increases For Finely Ground Cement
That Has More Surface Area To Attract And Absorb Water.
(f) The Effective Thickness Or Volume To Surface Ratio. As This Ratio
Increases, The Shrinkage Occurs More Slowly And The Total Shrinkage Is
Likely Reduced.
(g) Exposure To Carbon Dioxide Tends To Increase Shrinkage.
3-7 What Factors Affect The Creep Of Concrete?
(a) The Ratio Of Sustained Stress To The Strength Of The Concrete. The Creep
Coefficient, , Is Roughly Constant Up To A Stress Of 0.5 Fc', But Increases
Above That Value.
(b) The Humidity Of The Environment. The Amount Of Creep Decreases As The Rh
Increases Above 40%.
(c) As The Effective Thickness Or Volume To Surface Ratio Increases, The Rate At
Which Creep Develops Decreases.
(d) Concretes With A High Paste Content Creep More That Concretes With A Large
Aggregate Fraction Because Only The Paste Creeps.
