1. S T U D E N T V I N C E N Z O R O B E R T I
OPTIMIZATION OF PRESTRESSED
CONCRETE BEAMS
Civil System Analysis – CCE 6100
Instructor: Dr. Jun-Seok Oh
Final Report Presentation

2. THE PRESTRESSING CONCEPT
SIMPLY SUPPORTED BEAMS – DISTRIBUITED LOADS
Moment diagram Induced Moment that
counteract bending: F-e
Strain Stress diagram for a prestressed beam – no tensile stresses at service

3. NLP FORMULATION

4. PROBLEM VARIABLES
AND OBJECTIVE FUNCTION
Rectangular Section
𝑏1 , ℎ, 𝑒0, 𝐴 𝑝𝑠
Double Tee Section
𝑏1 , ℎ, 𝑏 𝑤, ℎ 𝑓, 𝑒0, 𝐴 𝑝𝑠
Tee Section
𝑏1 , ℎ, 𝑏 𝑤, ℎ 𝑓, 𝑒0, 𝐴 𝑝𝑠
Set of variables that minimize the final cost of the beam
𝑀𝐼𝑁: 𝐶 𝑏𝑒𝑎𝑚 = [L ∗ 𝑤𝑐 ∗ 𝐴 𝑐 − 𝐴 𝑝𝑠 ∗ 𝐶𝑐] + [𝐿 ∗ 𝑤𝑠∗ 𝐴 𝑝𝑠 ∗ 𝐶 𝑝𝑠]
Where:
w, A and C = unit weights, areas and unit costs; subscripts b, c and ps refer to beam,
concrete and prestressing steel, respectively; L = beam span
Idealized sections
Standard sections
Standard Sections
𝑒0, 𝐴 𝑝𝑠

5. CONSTRAINTS
Section limitations and practical conditions
Lower bound [in] Element Upper bound [in]
𝟏𝟐 𝑏 30
35 h 70
- h/b 3
6 𝑏 𝑤 30
6 ℎ 𝑓 10
0 𝑒0 𝑦 𝑏 − (𝑑 𝑐) 𝑚𝑖𝑛*
Integer Values Step values
[in]
Number of strands n -
height h 2
Base b 2
web width 𝑏 𝑤 1
Flange height ℎ 𝑓 1
Practical bounds
Integer values
𝑒0 ≥ 𝑘 𝑏+
1
𝐹𝑖
𝑀 𝑚𝑖𝑛 − 𝜎𝑐𝑖 ∗ 𝑍𝑡
𝑒0 ≥ 𝑘 𝑡+
1
𝐹𝑖
𝑀 𝑚𝑖𝑛 + 𝜎𝑡𝑖 ∗ 𝑍𝑏
𝑒0
≤ 𝑘 𝑏 +
1
𝜂 ∗ 𝐹𝑖
(𝑀 𝑚𝑎𝑥 − 𝜎𝑡𝑠 ∗ 𝑍𝑡)
𝑒0
≤ 𝑘 𝑡 +
1
𝜂 ∗ 𝐹𝑖
(𝑀 𝑚𝑎𝑥 + 𝜎𝑐𝑠 ∗ 𝑍𝑏)
𝑒0 ≤ (𝑒0) 𝑚𝑝 = 𝑦 𝑏 − 𝑑 𝑐,𝑚𝑖𝑛
WSD – inequality condition (ACI)
USD – ultimate strength
requirement (ACI)
Criterion for minimum
bending resistance
Criterion for minimum
reinforcement
𝑀 𝑢
𝑀 𝑛
≥ 𝜙
𝑀 𝑛
𝑀𝑐𝑟
≥
1.2
𝜙

6. FIXED PARAMETERS
Costs
Material unit cost
Prestressing steel
0,5 in - 5 strands $/lb 0.50
Concrete
5 ksi $/yd3 50.00
6 ksi $/yd3 52.00
7 ksi $/yd3 55.00
8 ksi $/yd3 58.00
Materials Properties
values
Normal weight concrete Symbol - unit
Compressive strength fc' [ksi] 6
Compressive strength at initial fci' [psi] 4.8
Concrete weight per unit volume 𝑤𝑔[pcf] 150
stress relieved tendons
Effective tensile strength of the prestressing
steel at ultimate
𝑓𝑝𝑢 [ksi] 270
Effective tensile strength of the prestressing
steel at yelding
𝑓𝑝𝑦 [ksi] 229.5
Effective tensile strength of the prestressing
steel at service, after losses
𝑓𝑝𝑒[ksi] 150
the prestressing steel losses ratio η 0.80
Area of 1 strand Aps,1 [in] 0.153
Steel weight per unit volume 𝑤𝑠 [lb/ft3] 490.00
Allowable stresses
Stress at intial load (initial) [Psi]
Extreme fiber in compression 𝜎𝑐𝑖 2880
Extreme fiber in tension 𝜎𝑡𝑖 -207.84
Stress at service loads
(service)
Estreme fiber in compression
due to prestress plus
sustained loads
𝜎𝑐𝑠𝑑
2700
Estreme fiber in compression
due to total loads
𝜎𝑐𝑠
3600
Extreme fiber in tension 𝜎𝑡𝑠 -464.75

7. IDEALIZED SECTIONS (STEP 1)
OPTIMIZATION RESULTS

8. IDEALIZED SECTIONS (STEP 1)
OPTIMIZATION RESULTS – F’C = 5-8 KSI

9. IDEALIZED SECTIONS (STEP 1)
MODULUS OF THE SECTION AND FINAL COST OF THE BEAM

10. STANDARD SECTIONS (STEP 2)
OPTIMIZATION RESULTS

11. CONCLUSION
• The idealized sections (rectangular, tee and double tee) investigated have
given for a cost minimization problem a set of section properties that
minimize the total cost of the beam for each kind;
• The minimum cost beam of the idealized sections is the double tee beam;
• Increasing the value of the concrete strength may be beneficial in term of
cost saving for certain shape of section; in this case of study for the T-beam
the 7 ksi concrete strength has revealed the optimum solution;
• The minimum Modulus of the section 𝑍 𝑏,𝑡 (eq. Guyon, sec. 3.2.1) used as a
preliminary parameter for the beam design, it is not effective to evaluate
the final cost of the beam;
• The results of the investigation of the standard sections has revealed the
same results of the idealized sections investigation;
• The investigation of the idealized sections may give an effective direction
for further investigation of standard sections to achieve the least cost
beam;

12. THANKS
Grazie per l’ascolto!
Vincenzo Roberti