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RELATION BETWEEN THE STATIC AND
DYNAMIC MODULUS OF ELASTICITY,
EXPERIMENTALLY MEASURED
CRISTÓBAL LARAFRANCISCO DE LA MORAGIOVANNI CASCANTEMAHESH PANDEY
OUTLINE
2
• INTRODUCTION
• BACKGROUND
• METHODOLOGY
• RESULTS
• CONCLUSION
NDT in Canada 2017 Conference (June 6-8, 2017)
INTRODUCTION
3NDT in Canada 2017 Conference (June 6-8, 2017)
BACKGROUND METHODOLOGY RESULTS CONCLUSION
ELASTIC MODULUS or YOUNG’S MODULUS
• CONSTANT STRESS/STRAIN
𝜎𝑛𝑥 = 𝐸 ∗ 𝜀𝑥
𝐸 =𝜎𝑛𝑥𝜀𝑥
• 𝜎𝑛𝑥 = Stress direction strain(x)
• 𝜀𝑥 = Strain
INTRODUCTION
4NDT in Canada 2017 Conference (June 6-8, 2017)
BACKGROUND METHODOLOGY RESULTS CONCLUSION
STATIC MODULUS (ES)
• Strain range: 10-4 to 10-3
• 5 - 50 [Gpa]
DYNAMIC MODULUS (ED)
• Strain levelsless than:~10-7
• ED > ES
0
5
10
15
0 0.001 0.002 0.003
Str
ess [
GPa]
StrainDynamic Range
Static Range
INTRODUCTION
5NDT in Canada 2017 Conference (June 6-8, 2017)
BACKGROUND METHODOLOGY RESULTS CONCLUSION
STRUCTURAL RELEVANCE
• Static modulus
• Compute Deformations
• Building Codes [GPa]:
o 80 𝑓′𝑐
o 150 𝑓′𝑐
• On site quality check
o Cost
o Time Consuming
Δx
INTRODUCTION
6NDT in Canada 2017 Conference (June 6-8, 2017)
BACKGROUND METHODOLOGY RESULTS CONCLUSION
ACI 318S-14
RC-DF (Limestone)
RC-DF (Class 2)RC-GDL
RC-DF (Basaltic)
-
10
20
30
40
5 15 25 35 45 55
Sta
tic M
odulu
s o
f Ela
sticity [
GPa]
f'c [MPa]
Samples
INTRODUCTION
7NDT in Canada 2017 Conference (June 6-8, 2017)
BACKGROUND METHODOLOGY RESULTS CONCLUSION
OBJECTIVE
• Function Relationship Es(Ed)
• Cost effective method to evaluate Es
SCOPE
• 34 concrete cylinders
• Randomly Selected
INTRODUCTION
8NDT in Canada 2017 Conference (June 6-8, 2017)
BACKGROUND METHODOLOGY RESULTS CONCLUSION
STATIC MEASURE
ASTM C 469-14
• 𝐸 =𝑆2−𝑆1
𝜖2 − 50∗10−6
S2= Stress 40% f’c
S1= Stress at ε1 (50 ∗ 10−6)
ε2 = Strain at “S2”
40% f’c
f’c
0
5
10
15
0 0.001 0.002 0.003
Str
ess [
GPa]
Strain
INTRODUCTION
9NDT in Canada 2017 Conference (June 6-8, 2017)
BACKGROUND METHODOLOGY RESULTS CONCLUSION
DYNAMIC MEASURE
ASTM C 215-14
• Force Resonance Method
• Impact Resonance Method
INTRODUCTION
10NDT in Canada 2017 Conference (June 6-8, 2017)
BACKGROUND METHODOLOGY RESULTS CONCLUSION
DYNAMIC MEASURE
ASTM C 215-97
Transverse Mode Longitudinal Mode
INTRODUCTION
11NDT in Canada 2017 Conference (June 6-8, 2017)
BACKGROUND METHODOLOGY RESULTS CONCLUSION
WAVE PROPAGATION
P-waves
• 𝑣𝑝 =𝑀
𝜌=
𝐸𝐷∗(1−𝑣)
𝜌∗(1+𝑣)∗(1−2𝑣)
• Phase Velocity 𝜐𝑝 = 𝑓 𝜆
𝜐𝑝 = 𝑓𝑛 ∗ (2 𝐿)
(Richart, Hall, & Woods, 1970).
INTRODUCTION
12NDT in Canada 2017 Conference (June 6-8, 2017)
BACKGROUND METHODOLOGY RESULTS CONCLUSION
PREVIOUS RELATIONS
• 𝐸𝑆 = 0.83 𝐸𝐷 Lydon & Balendran (1986)
• 𝐸𝑆 = 0.23 ൗ𝐸𝐷1.4
𝜌 Popovics (1975)
INTRODUCTION
13NDT in Canada 2017 Conference (June 6-8, 2017)
BACKGROUND METHODOLOGY RESULTS CONCLUSION
Dynamic Testing Static Testing Analysis
• 34 cylinder
• Dimensions
• D ≈ 15 cm
• h ≈ 30 cm
INTRODUCTION
14NDT in Canada 2017 Conference (June 6-8, 2017)
BACKGROUND METHODOLOGY RESULTS CONCLUSION
Dynamic Testing Static Testing Analysis
• Equipment
• Hammer
• Hydrophone
• 3 dynamic measures
INTRODUCTION
15NDT in Canada 2017 Conference (June 6-8, 2017)
BACKGROUND METHODOLOGY RESULTS CONCLUSION
Dynamic Testing Static Testing Analysis
• Equipment
• Hydraulic Press
• Strain Gauge
f’c
0
5
10
15
0 0.001 0.002 0.003
Str
ess [
GPa]
Strain
INTRODUCTION
16NDT in Canada 2017 Conference (June 6-8, 2017)
BACKGROUND METHODOLOGY RESULTS CONCLUSION
Dynamic Testing Static Testing Analysis
• Dynamic Analysis
• Range Sweep:
2 kHz – 10kHz
• Find peaks ratio:
𝑓𝑡 = 0.6125 𝑓𝑙
𝐶 𝑀 𝑓𝑡2 = 𝐷 𝑀 𝑓𝑙
2
𝑓𝑡 = 1.78041𝑑
𝐿 𝑇𝑓𝑙
INTRODUCTION
17NDT in Canada 2017 Conference (June 6-8, 2017)
BACKGROUND METHODOLOGY RESULTS CONCLUSION
0.00 0.03 0.05 0.08 0.10
Vib
rational
Am
plitu
de
Time [S]
n=2.60kHz
n'= 4.29kHz
2 3 4 5 6
SPECTRAL
AM
PLIT
UD
Frequency (kHz) Thousands
Fundamental Frequencies:
• Longitudinal ≈ 4.29 kHz
• Transversal ≈ 2.60 kHz
2.60
4.29= 0.606 ≈ 0.6125
INTRODUCTION
18NDT in Canada 2017 Conference (June 6-8, 2017)
BACKGROUND METHODOLOGY RESULTS CONCLUSION
-
10
20
30
40
50
10 15 20 25 30 35
Sta
tic
Modlu
sof
Ela
sticity
(GPa)
Dynamic Modulus of Elasticity (GPa)
Samples
Trendline (-10%)
Trendline (+10%)
Poly. (Samples)
𝐸𝑠 = 0.0326 𝐸𝐷2 − 0.3639 𝐸𝐷 + 8.8158
INTRODUCTION
19NDT in Canada 2017 Conference (June 6-8, 2017)
BACKGROUND METHODOLOGY RESULTS CONCLUSION
-
10
20
30
40
10 15 20 25 30 35
Sta
tic M
odlu
s o
f Ela
sticity (
GPa)
Dynamic Modulus of Elasticity (GPa)
Samples
Lydon and Balendran (1986)
-10%
10%
INTRODUCTION
20NDT in Canada 2017 Conference (June 6-8, 2017)
BACKGROUND METHODOLOGY RESULTS CONCLUSION
-
10
20
30
40
10 15 20 25 30 35
Sta
tic
Modlu
sof
Ela
sticity
(GPa)
Dynamic Modulus of Elasticity (GPa)
Samples
Popovics 1975
-10%
+10%
INTRODUCTION
21NDT in Canada 2017 Conference (June 6-8, 2017)
BACKGROUND METHODOLOGY RESULTS CONCLUSION
• New correlation ES(ED) proposed
• Further study
• New methodology – More cost effective
• No transducers
• Using transverse/longitudinal ratio
22
THANK YOU!
Questions?
NDT in Canada 2017 Conference (June 6-8, 2017)
23NDT in Canada 2017 Conference (June 6-8, 2017)
REFERENCES
ACI. (2011). Building code requirements for structural concrete (ACI 318S-11). ACI 318.
ASTM International. (1997). C 215 Standard Test Metod for Fundamental Transverse, Longitudinal and Torsional Resonant Frequencies of Concrete Specimens. Annual Book of ASTM Standards. American Society for Testing and Materials.
ASTM International. (2002). C 469 Standard Test Method for Static Modulus of Elasticity and Poisson's Ratio of Concrete in Compression. American Society for Testing and Materials.
Gobierno del Distrito Federal. (October 6th, 2004). Gaceta Oficial del Distrito Federal. Normás Técnicas Complementarias para diseño y construcción de Estructuras de Concreto, págs. 88-194.
Neville, A. (1996). Properties of Concrete (4th ed.). New York: John Wiley & Sons, Inc.
Richart, J. F., Hall, J. J., & Woods, R. D. (1970). Vibrations of Soils. Englewood Cliffs, New Jersey: Prentice Hall.
