FLEXURE

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68

EXPERIMENT:FLEXURE TEST

OBJECTIVES:

(1)Develop a load-deflection diagram from a flexure

test of wood.

(2)Predict material properties based on the results

from the experiment.

(3)Explore the effects of the moment of inertia, I

INTRODUCTION:

In this experiment, beams of three types of wood are

tested in flexure to failure. The deflection, bending

stress, shear stress and failure modes associated with

this type of loading are investigated.

FIGURE 1

L

P

P/2

P/2

P/2

-P/2

PL/4

V

M

L/2

x

y

P

x

x

x

Beams loaded in flexure are among the most

common type of engineering structures in existence.

Examples range from tree branches to diving boards

to airplane wings. Although simple in nature, such

beams develop relatively complex stress

distributions and may fail in a variety of manners.

Figure 1 shows a simply supported beam loaded at

its midpoint. The free body diagram with deflected

shape as well as the corresponding shear and

moment diagrams are also presented. Four aspects of

the behavior of this beam are next reviewed; beam

deflection, bending stresses, shear stresses, and

failure modes.

BEAM DEFLECTION:

There are several methods for determining the

equations for beam deflection in the EM324

textbook. The integration method shown here for

determining the deflection starts with the differential

equation for the elastic curve:

EI (d

2

y/dx

2

) = M(x)

(1)

Using singularity functions, the moment at any

section of the beam may be expressed as:

M(x) = (P/2)x - P<x-L/2> (2)

Combining the above equations results in:

EI (d

2

y/dx

2

) = (P/2)x - P<x-L/2> (3)

The boundary conditions for this beam are:

y = 0, at x = 0 and x = L

Integrating the differential equation twice gives:

EIy

(x)

= (P/12)x

3

-(P/6)<x-L/2>

3

+ C

1

x +C

2

(4)

Applying the boundary conditions to obtain the

constants results in the following equation for the

elastic curve:

y

(x)

=(P/EI)[(x

3

/12)-(1/6)<x-L/2>

3

-(1/16)L

2

x]

(5)

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EM 327: MECHANICS OF MATERIALS LABORATORY

69

the center deflection, δ

c

, can be determined form the

above equation by substituting x=L/2.

δ

c

= y

(L/2)

= -PL

3

/48EI (6)

BENDING STRESS:

Bending stresses in the beam are determined from

the flexure formula:

σ = My/I (7)

Where:M = bending moment

y = distance form neutral axis to

point of stress

I = moment of inertia of cross-

section with respect to neutral axis.

If the beam has the rectangular cross section shown

in Figure 2, the neutral axis coincides with the

centroidal axis for symmetrical bending and is

located at the middle of the cross section. For this

section, I = bh

3

/12.

FIGURE 2

N

.A.

b

h

The maximum bending stresses occur at the top and

bottom of the beam where the value of y is largest

and at the mid-span of the beam (x=L/2) where the

bending moment, M, is largest. Substituting these

values,

σ

max

= + [(PL/4)(h/2)] / (bh

3

/12) (8)

Note that the maximum bending stress is tensile on

the bottom of the beam and compressive on the top.

SHEAR STRESS:

Shear stresses in the beam are determined from the

shearing stress formula:

τ = VQ

It (9)

Where: V is the shear force

Q is the first moment of cross

section between the location of

stress and outer location of beam.

=

c

h

tydyQ

(10)

Where:I is the moment of inertia of the

cross-section with respect to the

neutral axis.

t is the thickness of the beam

For a beam with the rectangular cross-section shown

in Figure 2, I = bh

3

/12.

The maximum shear stress occurs at the neutral axis

(y=0) where Q is maximum. For this location

Q = bh

2

/8. The maximum shear force in the beam is

+ P/2. Substituting these values

bh

P

4

3

max

±=τ

(11)

FAILURE MODES:

Three types of failure may occur in a wooden beam

loaded in flexure as shown in Figure 3. First, a

tension failure may occur at the bottom of the beam

where the bending stress is the maximum positive

value. Second, a compression failure may occur at

the top of the beam where the bending stress is the

maximum negative value. Finally, a shear failure

may occur at the neutral axis of the beam where the

shear stress is largest. These failure modes are

depicted in Figure 3.

FLEXURE

EM 327: MECHANICS OF MATERIALS LABORATORY

70

FIGURE 3

Tension Failure

Compression

Failure

Shear

Failure

Shear

Failure

A load versus deflection diagram is to be produced

for each specimen tested. The curve in Figure 4 is

typical of the load versus center deflection of a wood

beam in flexure.

FIGURE 4

Stroke,

δ

, (in.)

Load, P, (lb)

This type of plot may be used to determine material

properties. The equation for the maximum bending

stress (equation 8) at the center of the span can be

used to determine the Proportional Limit and

Modulus of Rupture. The Modulus of Elasticity can

be determined by rearranging the center deflection

equation (6).

E = (P/δ

c

)L

3

/48I (12)

Where: δ

c

is the center deflection at y=L/2.

The modulus of rupture is the maximum stress

obtained using

σ = Mc/I (13)

The average work is defined as the average force

times the distance through which it moves:

W

avg

= (F

avg

)(d) (14)

At the proportional limit this is

W

avg

= (1/2)P

pl

δ

c,pl

(15)

Where: P

pl

is the applied force at the

proportional limit

δ

c,pl

is the center deflection of the

beam at the proportional limit

The work per unit volume can be determined by

dividing this equation by the volume of the beam.

MATERIAL TO BE TESTED:

Four wood beams will be tested. The specimens are

birch, oak, and (2)pine and are approximately

14"x 1/2" x 1".

EQUIPMENT TO BE USED:

55-kip MTS testing machine

SAFETY CONSIDERATIONS:

!!USE SAFETY PLEXIGLASS SHIELD ON

MTS MACHINE AT ALL TIMES WHEN

TESTING WOOD!!

Never operate the MTS machine when someone's

hands are between the grips. Make sure all lab

participants are clear of equipment before beginning

or resuming testing.

PROCEDURE:

SPECIMEN PREPARATIONS:

TASK 1: Measure cross-sectional dimensions of the

four specimens. Visually inspect the specimens for

flaws and imperfections. Make note of any flaws and

imperfections in the lab report.

MTS SET-UP

TASK 1 and 2:

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EM 327: MECHANICS OF MATERIALS LABORATORY

71

1.) Follow Start- up Procedures

Station Manager flexure

MPT flexure.000

2.) Turn hydraulics on.

3.) Make sure MANUAL OFFSET = 0 for Stroke.

4.) Adjust 'SET POINT'' to 0.0

5.) 'AUTO OFFSET' Load.

6.) Set-up Scope to plot a/b.

Load 200 lbf -400

Stroke 0.05 in/in -0.2

Time 15 min

TESTING PROCEDURE:

TASK 1: Three specimens (oak, birch and pine)

1) Create a specimen file flex*.

2) Center specimen in flexure fixture. Be sure to

have the 1/2" side as the base.

3) Measure the length between the supports.

4) Start the Scope.

5) Close SAFETY SHIELD!

6) Lock MPT and select specimen.

7) Press `RUN' and let test proceed until rupture.

The load will drop off at this point. The test can

be continued from here until a final rupture

occurs

8) Press `STOP' button.

9) Unlock MPT.

10) Adjust 'SET POINT' to 0.0.

11) Remove specimen.

12) Repeat procedure for each specimen.

TASK 2: Investigate the effects of moment of

inertia, I

1.) Create a specimen file flex*.

2.) Center the second pine specimen in the flexure

fixture. The base should be the 1" side.

3.) Enable Manual Control on the Remote Control

Pod.

4.) Move the cross-head up until the wood is just

below the test fixture.

5.) Disable manual control on the Remote control

Pod.

6.) AUTO OFFSET Load and Stroke.

7.) Lock MPT and select specimen.

8.) Start the Scope.

9.) CLOSE SAFETY SHIELD!

10.) Press 'RUN' and let the test proceed to rupture.

11.) Press STOP.

12.) Unlock MPT.

13.) Adjust the SET POINT to 0.0.

14.) Remove specimen.

15.) Shut down hydraulics.

16.) Copy all data files (Task 1 and 2) to diskette.

c:\em327data\flex*\specimen.dat

17.) Delete specimens flex*.

REPORT:

The report outline found in Appendix A should be

used.

REPORT REQUIREMENTS:

TASK 1:

(1) Label and affix appropriate scales to graphs

(2) Determine the following properties for each

wood specimen.

a. Proportional Limit at top and bottom of

specimen

b. Maximum shear stress in the specimen at

failure

c.Maximum bending stress in the specimen

at failure

d.Modulus of Elasticity

e.Average work per unit volume at the

proportional limit.

f.Type of failure (description and sketch)

(3) Compare properties with reference values.

FLEXURE

EM 327: MECHANICS OF MATERIALS LABORATORY

72

(4) Scale your Modulus of Elasticity results to

predict the Modulus of Elasticity for the wood

sample in compression (recall E in compression

is approximately 10% greater than E for

flexure). How do your estimations for

compression compare with the values you

calculated from your compression tests?

(5) Discuss the results, possible sources of error

and other conclusions relevant to TASK 1.

(6) Answer the questions your instructor assigns.

TASK 2:

(1) Label and affix appropriate scales to graph(s).

(2) Determine the proportional limit and the

maximum bending stress in each specimen.

(3) Compare the results for the two specimen

orientations.

(4) Discuss your results and conclusions relevant to

TASK 2.

QUESTIONS:

(1) After the first failure the load once again

increases until the next fracture. Discuss what is

physically happening during this process.

(2) How does the relative importance of shearing

and bending stress vary with the span of the

beam?

(3) Both tension and compression are present in

bending. Would the wood be stronger in

compression or tension? Base your answer on

the results of your experiment.

(4) What are the assumptions made in the

derivation of the flexural formula?

(5) Is there any difference between the stresses

measured at the top of the beam and at the same

location (at the same section) on the bottom of

the beam? If so, explain.

(6) How should the Modulus of Rupture and

Modulus of Elasticity of a small clear specimen

compare with the corresponding properties

determined from testing a full-size structural

member?

(7) Is the Modulus of Rupture a true ultimate fiber

stress? Explain.

(8) Is the strength of a beam under a static load

necessarily related to its stiffness?

(9) Distinguish between primary failure and

secondary failure.

(10) Is it always possible to determine the primary

cause of failure by examining the fractured

specimen?

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