Kobayashi, Harutoshi

Matsuura, Mikiyoshi

Sonoda, Keiichiro

Sumino, Taichi

Osaka City University Repository

Mem. Fac. Eng., Osaka City Dniv., Vol. 39, pp. 33-38(1998)
An Experimental Study on Dynamic Response of Deep Beams by
Impact Loading Test
Mikiyoshi MATSUURA}, Taichi SUMIN02, Harutoshi KOBAYASHI3
and Keiichiro SONODA4
(Received September 30, 1998)
Synopsis:' The weights-dropping test on four different kinds of specimens was
executed to obtain stress wave propagation characteristics. Steel bar as weight and
synthetic resin mortar as specimen's material are used in the present experiment.
Applied impact load on the specimen is measured from the strain of the dropping steel
bar. Strain gauges were attached to some points of specimens, and stress responses
were recorded. From these experimental results, the velocity of stress wave is
compared with theoretical values. Furthermore, experimental dynamic response curves
are compared with results of 2D FEM analyses.
Keywords: stress wave propagation, theory ofelastodynamics, deep beam, 2D FEM,
synthetic resin mortar, steel bar
1. Introduction
It is well known that the use of 2D elastodynamic theory gives accurate stress wave phenomenon of the deep
beam than that of Bernoulli-Euler beam theory [1]. The theoretical solutions for isotropic materials consist of
longitudinal wave, transverse wave and surface wave. It is very difficult to measure the accurate stress wave
propagation by the impact test because of the problems in performance of dynamic strain measuring device.
However, they are improving in performance and cost day by day. In this study, the dynamic strain-measuring
device of 200 kHz performance (sampling time 5 J.lsec) is prepared for the dropping rod test. As the longitudinal
stress wave velocity is proportional to square root of Young's modulus, synthetic resin mortar is adopted as
tnaterial of specimens. Young's modulus of resin mortar is seemed to be about a half of the concrete's one.
2. Dropping Test
2.1 Apparatus
The test apparatus used is shown in Fig. 1. The apparatus consists of two parts. One is the part to prevent uplift
of the specimen, and another is the part of guide for dropped steel rod which is 1500 mm long. Impact load is
given by the steel rod dropping on the specimen, and then impact stress is estimated through the strain gauges,
which is attached on the rod surface. To realize the hinged support condition at near both ends, the specimen are
rested on the steel bar (~40) and fixed by the vertical supporting bar (see Fig. 1). Only four channels of dynamic
strain-measuring device were set up, and thus in order to obtain the dynamic strain data at more than four
tneasuring points, the dropping tests under the same condition were executed repeatedly. After the transformation
I Doctor Course Student, Department of Civil Engineering, also Matsuura Construction Co., Ltd.
2 Constructing Department, Nishimatsu Construction Co., Ltd.
3 Associate Professor, Department of Civil Engineering
4 Professor, Department of Civii Engineering
33
34
[rorTI analog data to digital data, the strain data are recorded in the data recorder. We used averaged data obtained
by the several experiments, hereafter for consideration of test results.
.-
o
o
L
--.-~
o
o
M
,......j
t H
-
I
II
II
II
II
(b) Side view
u
L 300 -J~I
CD Guide for steel rod
,---(D ® Loading frame
lI'n III I@
Span
1300
(a) Front view
6 - [l
----- ® Abutment
II ----.
II 0 ® Supporting S
II 0~
"-i ® Vertical suppII @ Impacter(ste®--.
0(0 ®InM
n +
I "~I I III I Specimen ::I
I III
( ) ()A
\I II
II II
II
Fig. 1 Loading apparatus (unit: mm)
2.2 Specimen and Measuring Points
Two types of specimen are prepared for the dropping test as shown in Figs. 2 and 3. The single-layer specimen
shown in Fig. 2 is made of only resin mortar. These models have three types of span whose lengths are 600, 800,
and 10001TIm. In the case of two-layered specimen shown in Fig. 3, the upper layer is made of steel and the lower
resin lTIartar. The span is 600mm long. The measurement of strain is performed on the points of Nos. 1 - 8 for
single-layer specimen, and on the middle points of both the resin mortar and steel layers for two-layered specimen.
Strain gauges except Nos. 4, 7 and 8 are attached to measure the longitudinal stress wave. Strain gauges of Nos. 4,
7 and 8 are attached for measuring the dynamic strain due to bending vibration. Strain gauges of Nos. 5 and 6 are
attached at an angle of 45-degrees to the centeriine of the specimen to obtain the longitudinal stress wave.
Material constants for mortar and steel are shown in Table 1. The longitudinal stress wave velocity in plane
stress state is given by
C=~(1_~2)p (1)
where E is Young's modulus, v is Poisson's ratio, and p is density. The longitudinal stress wave velocities for
each material calculated by Eg. (1) are given in the right column of Table 1.
Table 1 Material constants by statical loading test
Young's modulus E Poisson's ratio v Density p Wave velocity(2D)
(Pa) (g/cm3) (m/sec)
Resin JTIortar(Single-layer) 5.52XI09 0.309 1.73 1878
Resin mortar(Two-layer) 5.49 X 109 0.363 1.60 1988
Steel 2.06 X 1011 0.300 7.85 5370
351 600, 500, 400 200 :1· 200 ·11'"'4
0
·. No. 7 45° , , 45° No.8 ·. tLn
·. ·., ,
0 ·. 0.1,
·.
, 0Ln
·.
,
o. 2 ~o. 5 00 ·. ·. Nin ·. No.6 • • ~·. ·.0 ·. Resine mortar ·.Ln ·.
LJ. 1000,800,600 .u
100 100
Fig.2 Measuring points of single-layer specimen by strain gauge (unit: mm)
F
800
:1400 ·r- 400
·. a ·. r0 ·. l..!') +Strain ·.0 ·. Steel gauge ·.~ ·. ·.
·. ·. 00
+Strain gauge ·.
C'J
0 : :Resine mortar ·.
-+0 ·.~ ·. ·.·. ·.
LI. 600 .u100 100
Fig.3 Measuring points of two-layered specimen by strain gauges (unit: mm)
3. Results and Discussion
3.1 Results olTest
Figures 4 - 6 show the stress histories of single-layer specimens with 600mm, 800mm and IOOOmm spans,
respectively. Top figures in Figs. 4 - 6 show the stress of steel rod, which means the applied dynamic load on the
specimen. In these figures, recorded time texp is calibrated by Ltt that is the time duration of transmission on
longitudinal wave from the tip of steel rod to the attached strain gauge of IOOmm away from the tip. Then, the
actual loading time t is given by
1 =lexp -111 (2)
The middle figures in Figs. 4 - 6 present the strain histories at the points of Nos. 1 - 3, 5 and 6. The bottom
figures in Figs. 4 - 6 show the strain histories due to the beam vibration.
Figure 7 presents the dynamic response of two-layered specimen. The top figure shows the stress history of the
tip of steel rod. It is appear that applied impact load on the specimen is different from the case of single-layer
specilnen in spite of the same test condition. The middle and bottom figures in Fig. 7 show the strain histories at
the middle point of steel and mortar layers, respectively. Although the characteristic of stress wave propagation is
considered in these figures, high frequency vibration phenomenon which appeared in theoretical treatment [2,3] is
not seen apparently except the initial response of steel layer.
36
--Tip of the rod
-o-No.l
---No.2
-t:r- No.3
--'--No.5,6
-o-No.4
-8-No.7,8
t (msec)
4 53
--Tip of the rod
-o-No.l
--'-No.2
-tr-No.3
~N05,6
~No.4
---No.7, No.8
2o
~
~ 0
~
"-"
~ -10
CJ'.)
CJ'.)
(1)
C
rFJ -2
t (msec)
4 532o
-10
c
.~ -5
b
(/)
~40
c
.~
~ 20
Fig. 4 Response of rod and bealTI with length 600mm Fig. 5 Response of rod and beam with length 8001TIm
t (msec)
4 5
I u_ Steel I
32
I --Tip on the rod
, ~,,: : ":
L' :~:~: :.i, ~ ,o' •I.' '1' ~ ,t 11"
It I ~ II • ~: ~~
H~,~ :. {d I I
. ~': ::n: __ u MOI1a~
~.~, :::' ,
,Ii
,t
"
"
o
-10
CJ'.)
CJ'.)
~ -2
-+oJ
rFJ
t (msec)
4 53
-D-No.4
-ll-No.7,8
--The tip of the ro
-o-No.l
----No.2
-o-No.3
---A--
2o
-2
-15
~~ -50
c
.~
~-10
,-. 400
~
~200
.@
~ 0
-20
Fig. 6 Response of rod and beam with length 1000mm Fig.7 Stress response of two-layered specimen
37
3.2 Wave Velocity and Beam Vibration Period
The average time duration of stress wave propagation from the points No. 1 to No.2, which is estimated by
Figs. 4 - 6, is given in Table 2. The arriving time of stress wave is used for the purpose of estimating the time
duration. However, it is difficult to estimate the arriving time of stress wave at the points of Nos. 3, 5 and 6
because of damping phenomenon, and thus the arriving time of stress wave at No.2 is used for the case of single-
layer specimen as time duration. The right column of Table 2 shows Young's modulus, which are calculated by
Eq. (1) using these measured wave velocities. Comparing these Young's modulus with those obtained by static
test in Table 1, these dynamical Young's modulus show 2 to 3 times as much [4].
Table 2 Duration of wave propagation from point No.1 to No.2 and wave velocity
Type Span Duration Distance Wave velocity Young's modulus
(mm) Average (J-lsec) (mm) (m/sec) (Pa)
Single-layer 600 17.0 50 2941 1.35XI01o
Single-layer 800 16.7 50 2994 1.40 X 1010
Single-layer 1000 15.0 50 3333 1.74X 1010
Two-layer 600 15.7 50 3185 1.41 X 1010
The experimental natural period of single-layer specimens and theor~tical one based on Bernoulli-Euler beam
theory are given in Table 3. The former values are the estimated from the Figs. 4 - 6, and the latter is the
fundamental period based on Bernoulli-Euler beam theory. Although the boundary condition of specimen in the
present experiment gives almost both-end hinged condition, the natural period in experiment is nearer to the
theoretical period of beam with both-end fixed than both-end hinged. These results show that the assumed
experilnental boundary condition is not both-end hinged perfectly.
Table 3 Results of natural period at the points of Nos. 4,7 and 8
results of vibration period Natural period based on beam theory
Span(Single-layer) Both-end hinge Both-end fixed Both-end hinge
(mm) (JJ,sec) (j.lsec) (JJ,sec)
600 2500 1355 7052
800 3700 2410 12540
1000 4800 3764 19630
3.3 2D FEMAnalysis
Figures 8 and 9 show the results of 2D FEM (12 X 42 elements) analysis for single-layer and two-layered
specimens with 600mm-span length. Young's modulus and damping factor used in 2D FEM analyses are given in
Table 4. The damping factor of resin mortal is assumed to be higher value than steel's one, say 2 times. Applied
stress-tilne relations to the FEM model are shown in the top figures of Figs. 4 and 7. The analytical response
curves at points of Nos.l and 2 shown in Fig. 8 coincide well with experimental results in period and magnitude
of alnplitude.
Table 4 Young's modulus and damping factor for 2D FEM analysis
Type Young's modulus Damping factor
(Span; 600mm) (Pa) (%)
Resin mortar(Single-layer) 1.33x1010 12
Resin mortar(Two-layer) 1.41 X 1010 12
Steel 2.06 X 1011 6
38
--Experimen
------- 2D FEM
t (msec)
32
--Experiment
------- 2D FEM
o
2
-1
-2
-4\~----------------'
t (msec) 3
2o
Fig.8 Comparison of experiment and 2D FEM for
beam width 600 mm
Fig.9 Comparison of experiment and 20 FEM
analysis for two-layered beam
Acknowledgements
The authors would like to express our appreciation to Research Associate. H. Kitoh, Department of Civil
Engineering, Osaka City University and Mr. N. Kato, Sho-Bond Co., Ltd. for making the specimens and test
apparatus.
References
[1] Y. Sato: Theory of Elastic Wave Propagation, Iwanami, Tokyo, pp. 1-55, 1978 (in Japanese).
[2] K. Kobayashi, T. Okuda and K. Sonoda: Analyses of Stress Wave Propagation of Simply Supported Beam
under Impact Load, Journal of Structural Engineering, JSCE, Vol. 38A, pp. 1505-1514,1992 (in Japanese).
[3] K. Kobayashi, K. Sonoda and M. Kashima: Impact Response Analysis of Simply Supported, Steel-Concrete
Composite Beams Based on Theory of Two Dimensional Elasto-dynamics, Journal of Structural Engineering,
JSCE, Vol. 38A, pp. 1489-1496, 1992 (in Japanese).
[4] Impact Behavior and Design of Structures, Subcommittee of Impact Problems, Committee of Structural
Engineering, Japan Society of Civil Engineers, pp. 74-95,1993 (in Japanese).


An Experimental Study on Dynamic Response of Deep Beams by Impact Loading Test

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An Experimental Study on Dynamic Response of Deep Beams by Impact Loading Test

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