No Dokumen: SB/MMSB2/T2/BMCS2333/3

No Isu./Tarikh 1/2006

SOLID MECHANICS 2 BENDING STRESS ANALYSIS

No Semakan/Tarikh 0

Jum Mukasurat 3

KOLEJ UNIVERSITI TEKNIKAL

KEBANGSAAN MALAYSIA

OBJECTIVE To determine and analyze the strain-stress behaviour of a simply supported beam system by using strain gauges. THEORY

Beam

x

2P

2P

2L

L

P

Figure 1 A simply supported beam subjected to three points loading

Consider a beam AB , simply supported at both ends as shown in Figure 1 subjected to midspan concentrated load, . Under this condition of loading, the upper fibers (or above its neutral axis) of the beam, it is said to be subjected to compressive stress or strain and on the other hand, it is subjected to tensile stress or strain for the bottom fibers of the beam (or below its neutral axis). At a distance

P

x , from the left end , bending moment, A M , of the beam may be written generally as;

( )2PM x= ⋅ (1)

When, 2Lx = , bending moment M , will be at its maximum i.e maxM M= . Therefore;

( )( )max 2 2P L PLM = = 4 (2)

The nominal or bending stress along the beam may be determined from the general formula of bending theory. Therefore, with M is given by Equation (1);

( )( )' 2nom

P xM yI I

σh⋅⋅

= = (MPa) (3)

Equation (3), may be used to determine bending stress at any point along the beam. Now, if we substitute 2

Lx = , and maxM M= , then Equation (3) will produce the

Maximum Nominal or Bending Stress in the beam as follows;

( )( )( )' max

max. 2 2 2

8

hP LM y PLHI I

σ = = =I

(MPa) (4)

Where, I = moment of inertia of the beam cross-section given by; 3

12WhI = (m)4 (5)

By substituting Equation (5) in Equation (4), we will obtain the general equation of Maximum Bending Stress for the beam when subjected to concentrated load P;

'max 2

32

PLwh

σ = (MPa) (6)

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SB/MMSB2/T2/BMCS2333/3

In order to cause initial yielding suffered by the beam, we must increase load until

its critical value, i.e , where represents the load that cause initial yield in the beam. However, this condition may cause the beam to experience permanent deformation upon unloading and will never return to its initial shape.

PyP P= P

In general, we may use equation (3) to obtain normal stress (or normal strain) along the beam by substituting any value of x with 0 x 2

L (by exploiting the symmetrical

condition of the beam). To determine the normal strain at any distance of x , we may apply the

stress-strain relationship as follows;

' .Eσ ε= or '

Eσε = (7)

with = Young's Modulus of the beam. Equation (7) can be used to determine theoretically the strain at any location / point along the beam.

E

EQUIPMENT & SPECIMEN Equipment Universal Material Tester & Bending Device, data logger and specimen with strain gauges.

Specimen Dimension of the specimen is shown in Table 1 and Figure 2 shows the location of strain gauges on specimen.

Table 1 Dimension of specimen

Material Thickness ( h ) Length ( ) L Width ( ) w

Mild Steel 13.3 mm 320 mm 50 mm

2P 2P

P50mm 50mm 50mm 50mm 50mm 50mm

A B

1S

2S 4S

3S

5S 7S

6S

Figure 2 FBD of beam AB where 1, 2, 3,..., 7S S S S = Strain gauges mounted on the surface of the beam

PROCEDURES 1. Install the bending device to the Universal Materials Tester /Machine as provided

for this test. 2. Performing the bending test on the test beam;

a. Adjust the "support width" using mobile supports at 300mm distance. b. Load the beam gently until the load reading is shown by the "load / force display". c. Set the data logger reading (με ) to zero before starting the loading process on

the beam. d. Read and record the strain reading at 7 locations (labeled with )

as the beam is loaded with the load with an increment of 1 kN for each reading. (as shown in table 1.

1, 2, 3,..., 7S S S SP

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