structural mechanics 301be

301BE - Structural Mechanics Laboratory tests

Pre-stressed concrete beam and Concrete Flat Slab

Imran Mazumder SID: 2710356

Farah Gelle SID: 2520489

Imran MazumderFarah Gillie Structural Mechanics

Section 1: Pre-stressed Concrete beam

Introduction

In pre-tensioning the PC strands are tensioned , that are, pulled and elongated, by a calibrated tensioning apparatus (usually called stressing jack) over a jacking platform which consists a strong floor and two abutments (Neil, 2012). This method is primarily preferred in factory production of precast concrete construction. A Pre-stressed concrete beam used commonly in construction was tested to analyze its behavior under load in the lab session throughout its load history. The beam was made with a concrete mixture of 1:1.5:2.7 (cement, sand, aggregate), 10mm aggregate (3/8") (gravel uncrushed) and 0.41 liter of Cormix Accelerator. Two cubes were also casted from this mixture to determine its compressive strength. A Dial gauge was used to measure the deflection and a hydraulic jack to load the beam. The figure below roughly demonstrates the beam that was tested. It shows position of shear reinforcements in the beam and the location of the 7mm high tensile wire.

(1) State the recorded tensile force of the wire as shown in the load cell and explain the reasons why the tensions recorded is less than 40kN.

The tensile stress in the wire was 35.62kN as shown in the load cell. The reason for the tensile stress to be limited to 40kN could be to reduce the immediate drop in pre-stress force due to elastic shortening of the concrete (Mosley, Bungey, Hulse, 2007:322). It can also be to ensure that the tensile force doesn't cause the wire to be permanently deformed. If the wire is cut after tension applied, over stress might cause concrete shortening meaning when the wire is released, it contracts and tends to go back to its original length, it might cause the concrete to shorten along with it. Although this problem is overcome by using debondings, but in any unlikely event where is hasn't been applied that could be a possibility.

(2) State the recorded compressive strength of the cubes and explain the reason why this compressive strength value is not considered as fck defined in EC2. What is the theoretical value (calculated from the recorded compressive strength) of fck for the concrete considered?

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The compressive strength recorded is 55.6 MPa. The cylinder strength is considered about 20% less than the cube strength for normal concrete but with higher strength classes, the cylinder strength achieves a higher proportion of the cube strength (BS EN 206-1).

Theoretical compressive strength by considering the above explanation is equal to:

σ c=Fc

A

σ c=f ck

f ck , cube=556.6 ×103 N100 × 100 m2 =55.66 N . mm−2

f ck ,Theorical=f ck , Cylinder=0.8 × f ck , cube

f ck ,Theorical=0.8 × 55.66=44.5 N . mm−2

(EN 1992-1-1- clause 3.1.3)

(3) By using the fck estimated in (2) above and the equations provided in the EC2, calculate the value

of Ec, young modules of concrete.

Ec is calculated based on clause 3.1.3, Table 3.1Stress and deformations characteristics for normal concrete EC2- Part 1:1 (EN1992-1-1-2004)

Theoretical value of Ec = 36GPa

(4) By using the Ec calculated in (3) calculate:

(i) The Predicted elastic deflections of the beam at the mid-span

Using the equation 23 P L3

648 EI from structural engineers’ pocket book. The equation will be used to predict

the deflection of the beam. The second moment of area of the beam was calculated using the formula

b d3

12

Δmax=23 W a3

24 EI , a=L

3

3


Calculation of second moment of area

I= 112

b . h3

I= 112

60 ×1603=20480000 mm4

Substituting the value of I (Second moment of Area) in the equation for Δmax and assuming values of W we get:

W = 1 Δmax = 0.44mm

W = 2 Δmax = 0.89mm

W = 3 Δmax = 1.34mm

W = 4 Δmax = 1.78mm

W = 5 Δ max ¿2.23mm

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Testing Beam

(5) The actual “Measured” deflections for the mid-span section at all the recorded loads

Demec

LOAD DEFLECTION 1 2 3 4 5

0 0 2315 2011 2331 2210 1352

2 1.5 2284 1999 2333 2230 1389

4 3.19 2238 1982 2342 2266 1449

6 5.29 2184 1966 2363 2320 1534

8 7.49 2120 1941 2379 2367 1621

10 9.89 2053 1921 2404 2424 1720

UNLOADING

Demec


8 8.88 2084 1933 2393 2396 1673

6 7.31 2135 1957 2385 2365 1614

4 5.63 2174 1958 2361 2316 1537

2 3.69 2226 1981 2341 2270 1462

0 1.48 2275 1991 2327 2230 1391

LOADING TO FAILURE

Demec


2 3.11 2234 1972 2334 2257 1440

4 5.05 2181 1959 2355 2303 1516

6 6.82 2142 1945 2370 2344 1587

8 8.51 2091 1930 2389 2386 1657

10 10.26 2045 1918 2404 2429 1734Table 1

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(6) The actual “measured” strains for the mid-span section at all the recorded loads

Demec

LOAD DEFLECTION 1 2 3 4 50 0 0 0 0 0 0

2 1.5-

0.00012493 -0.00004836 0.00000806 0.0000806 0.00014911

4 3.19-

0.00031031 -0.00011687 0.000044330.0002256

8 0.00039091

6 5.29-

0.00052793 -0.00018135 0.00012896 0.0004433 0.00073346

8 7.49-

0.00078585 -0.0002821 0.000193440.0006327

1 0.00108407

10 9.89-

0.00105586 -0.0003627 0.000294190.0008624

2 0.00148304

UNLOADING Demec

LOAD DEFLECTION 1 2 3 4 511 8.88 0 0 0 0 010 7.31 0.00020553 0.00009672 -3.224E-05 -0.0001249 -0.00023788 5.63 0.0003627 0.00010075 -0.000129 -0.0003224 -0.00054816 3.69 0.00057226 0.00019344 -0.0002096 -0.0005078 -0.00085034 1.48 0.00076973 0.00023374 -0.000266 -0.000669 -0.0011365

LOADING TO FAILURE

DemecLOAD DEFLECTION 1 2 3 4 5

2 3.11 0 0 0 0 0

4 5.05-

0.00021359 -0.00005239 0.000084630.0001853

8 0.00030628

6 6.82-

0.00037076 -0.00010881 0.000145080.0003506

1 0.00059241

8 8.51-

0.00057629 -0.00016926 0.000221650.0005198

7 0.00087451

10 10.26-

0.00076167 -0.00021762 0.00028210.0006931

6 0.00118482Table 2

6


(7) Curves of both the measured and the predicted central deflections against load.

0 2 4 6 8 10 120

2

4

6

8

10

12Measured and Predicted Central De-

flections

Central Deflections (mm)

Load

(kN

)

Figure 1

(8) Curves for both the measured and predicted strains at the top surface of mid-span section against load.

-0.0015 -0.001 -0.0005 0 0.0005 0.0010

2

4

6

8

10

12

Measured and Predicted Strains

Deflection (mm)

Stra

ins (

Pa)

Figure 2

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(9) Curves representing the measured horizontal strains at various depths of the mid-span against the section depth for 2W = 4kN, 6kN and the maximum recorded load during the loading stage.

-0.0006 -0.0004 -0.0002 0 0.0002 0.0004 0.00060

1

2

3

4

5

6

7

Series2Series4Series6

(10) Calculate the value of 2W when the stress at the bottom of the pre-stressed concrete beam is equal to zero. You may assume a remaining pre-stress force of 30kN in the steel wire.

PA

+ PeZb

− MZb

=σ

30∗103

160∗60+ 30∗103∗20

256000− 700 w

256000=0

5.47∗256000=700 w

w=2000 N

2 W=4000 N

(11) Compare and discuss the theoretical “predicted” and the experimental “measured” results in (7) and (9) as well as the calculation in (10).

The theoretical deflection of the beam differed to the experimental deflection of the beam as shown in fig 1. This was mainly because the deflection equation did not account for the wire tendon in the concrete. The wire tendon allowed the beam to deflect more since it was the one taking the tension in the bottom of the beam.

However the theoretical strain and the experimental strain where almost similar because the equation took into account the tension in the steel wire. The discrepancies mainly came from the fact that the remaining tension left in the wire was assumed to be 30KN.

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Concrete Flat Slab

(B.1) Drawing of the crack pattern of the slab

i) When the first major crack is observed: - minor crack was first observed at 2.2KN

ii) At load = 8kN, but the slab actually failed at 7.91KN

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2.1 SLAB CALCULATIONS

(B.2) Calculate the correct value for fck according to EC2 from the measured cube strength of the

concrete.

σc= FcA

389 .48×10100×100

3

=38 .95 Nmm−2

fck = compressive strength of cylinder

fck=0 .8×fck , cube (EN 1992-1-1- clause 3.1.1 “The cylinder strength is approximately 80% of the cube strength”)

fck=0 .8×38 . 95=31 .16 Nmm−2

(B.3) By using the fck estimated in (13) above, the tensile strength of the steel from test and the

measured dimensions of the slab and steel, calculate the ultimate moment capacities of the slab in

two principle directions (no safety factor is required in the calculation). Explain why the moment

capacities are not equal in the two directions..

Tensile Test on rebar of RC slab

0

2000

4000

6000

8000

10000

12000

14000

-1 0 1 2 3 4

Extensometer (mm)

Lo

ad

kN

10

LOAD VS EXTENSOMETER


Yielding stress Calculation:

The yielding tensile load = 12900 N (according to graph)

yeildin { g stress= FA

=12900

π×62

4

=456 .24 MPa¿

Ultimate Moment Capacity Calculation:

fck=31. 16 MPa

fy=456 .24 MPa

As=62

2

π×9=254 .47 mm2

Fc=Area×σ=0. 8 x×b×fck

Ft=As×fy

Fc=Ft→0 . 8 x×b×fck=As× fy

254 .47×456. 24=0 . 8 x×1000×31 .16

x=4 .65mm

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Capacity Moment in x-direction:

d= 40 – cover – 0.5(diameter)

= 40-6- (0.5 x6) = 31mm

Mu=Fc×Z=f ×b×0. 8 x×Z

Where Z = d - 0.4x = 31 - (0.4 x 4.56) = 29.14mm

=31 . 16×1000×0 . 8(4 .65 )×29 .14=3 .38 kNm

Capacity Moment in y-direction:

d= 40 – cover – 0.5(diameter)

= 40-8- 0.5(6) = 29mm

Mu=Fc×Z=f ×b×0. 8 x×Z

Where Z = d - 0.4x = 29 - (0.4 x 4.56) = 27.18mm

=31 . 16×1000×0 . 8(4 .65 )×27 . 18=3 .15kNm

Explain why the moment capacities are not equal in two directions.

In a two way reinforced slab there are two principal directions X and Y directions. The moment capacities in this two principle directions are not equal and this because the two principles are acting at two different effective depths as show in the diagram below..

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(B.5) Carry out a yield line analysis to find the theoretical collapse load based on the ultimate moment

External Work

External work = Point Load x Area X Displacement

But displacement = 1, therefore External work = P X A (of each section)

Total External work = P X ((0.5 x 0.5) x 4) = P

Total External work = P

13


Internal Work

Internal work = (M x 0.5L x Ɵ) + (µM x 0.5L x Ɵ) but Ɵ = 1

Total internal work = 4 (M x 0.5L) + (µM x 0.5L)

Total internal work = 2M + 2µM

External work = Internal work

P = 2M + 2µM

P = (2 x 3.38) + (2 x 3.15)

P = 13.06 KN

Compare and discuss the theoretical and the measured collapse loads. State any possible reasons for the

deviations between the “Measured” and the “Predicted” results. Clearly state all possible errors in the

lab measurements and the limitations of any theory adopted in the calculations.

The theoretical collapse load is 13.06 KN as compared to the measured value 7.91 KN; this is due to lab

experimental error which has not be accounted for during the calculation of the theoretical value. Also,

the major contributing factor for the deviation between the measured value and the calculated value is

the method adopted in the calculation “Yield-line methods”. This method of analysis gives an upper

bound solution as it uses energy method in which the external work done by the point load during a

small virtual movement of the collapse mechanism is equated to the internal work (Structural Analysis

2003: 601). This means that the solution obtained is either correct or unsafe.

Also, during the calculation process no safety factors were considered and this may have led higher

value being obtained as compared to closer value

Limitations of yield-line method are: -

Yield-line method generates upper bound solutions.

Yield line method does not give the support reactions along the edges of the slab. This is a particular

issue where the slab is supported by edge beams whose design is dependent on how the slab transfers

load to them.

In Yield-line method, the pattern of fracture is assumed and the correct value may not be obtained. The

Value obtained is always within 10% of the correct value.

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At fracture, yield-line method assumes elastic deformations to be very small compared to plastic

deformations therefore ignored. This assumption means that fractured slab parts are plane and they

intersect in straight line, but this not the case in most laboratory experiment.

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References:

Amin Ghali, 2003. Structural Analysis: A Unified Classical and Matrix Approach. 5 Edition. CRC Press.

Mosley, Bungey and Hulse, 2007. Reinforced concrete design to Eurocode 2, Sixth Edition. Hampshire,

Palsgrave Macmillan.

Fiona Cobb, 2011. Structural Engineer's Pocket Book, second edition. Oxford, Elsevier Ltd.

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