chapt_2 SOM

8/2/2019 chapt_2 SOM

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STATICALLY INDETERMINATE MEMBERS

&

THERMAL STRESSES

Chapter II1


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STATICALLY INDETERMINATE MEMBERS

Structure for which equilibrium equations are sufficient to obtain

the solution are classified as statically determinate. But for some

combination of members subjected to axial loads, the solution

cannot be obtained by merely using equilibrium equations.The

structural problems with number of unknowns greater than the

number independent equilibrium equations are called staticallyindeterminate.

The following equations are required to solve the problems on

statically indeterminate structure.

1) Equilibrium equations based on free body diagram of the

structure or part of the structure.

2) Equations based on geometric relations regarding elastic

deformations, produced by the loads.

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COMPOUND BAR

Material(1)

Material(2)

A compound bar is one which is made of two or more than twomaterials rigidly fixed, so that they sustain together an externally

applied load. In such cases (i)Relation between deformations of

various bars can be found out . Here deformations are same.

(ii) Applied load is equal to sum of the loads carried by each bar.

W

L1 L2

3

Material(1)L1


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(dL)1 = (dL)2

(1/ E

1)L

1= (

2/E

2)L

2

1 = 2( E1/E2)(L1/L2) (1)

E1/E2 is called modular ratio

Total load = load carried by material (1) + load carried by

material(2)

W = 1 A1 + 2 A2(2)

From Equation (1) & (2) 1 and 2 can be calculated

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Problems

(1) A load of 300KN is supported by a short concrete column

250mm square. The column is strengthened by 4 steel barsin corners with total c/s area of 4800mm2. If Es=15Ec, find

the stress in steel and concrete.

If the stress in concrete not to exceed 4MPa, find the areaof steel required so that the column can support a load of

600KN.

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(2) A mild steel rod 5 mm diameter passes centrally through acopper tube of internal diameter 25mm and thickness 4mm.

The composite section is 600mm long and their ends are rigidly

connected. It is then acted upon by an axial tensile load of

50kN. Find the stresses & deformation in steel and copper. Take

Ecu = 100GPa, Es = 200GPa

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(3) Three vertical rods AB, CD, EF are hung from rigid supports and

connected at their ends by a rigid horizontal bar. Rigid bar carries a

vertical load of 20kN. Details of the bar are as follows:(i) Bar AB :- L=500mm, A=100mm2, E=200GPa

(ii) Bar CD:- L=900mm, A=300mm2, E=100GPa

(iii) Bar EF:- L=600mm, A=200mm2, E=200kN/mm2

If the rigid bar remains horizontal even after loading, determine thestress and elongation in each bar.

Solution:

600mm

900mm500mm

A

B D E

C

F

20kN

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(4) Two copper rods and one steel rod together supports as shown

in figure. The stress in copper and steel not to exceed 60MPaand 120MPa respectively. Find the safe load that can besupported. Take Es = 2Ecu

W

Copper

(30mm30mm)

Copper

(30mm30mm)

Steel

(40mm40mm)

120mm

80mm

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(5)A rigid bar AB 9m long is suspended by two vertical rods at itsend A and B and hangs in horizontal position by its own weight.

The rod at A is brass, 3m long, 1000mm2 c/s and Eb = 105N/mm2.The rod at B is steel, length 5m, 445mm2 c/s and Es = 200GPa.At what distance x from A, if a vertical load P = 3000N beapplied if the bar remains horizontal after the load is applied.

9m

5m

3m

Steel

A B

3000N

Brass

x

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(6) A mild steel bar of c/s 490mm2 is surrounded by a copper

tube of 210mm2 as shown. When they are placed centrallyover a rigid bar, it is found that steel bar is 0.15 mm longer.

Over this unit a rigid plate carrying a load of 80 kN is placed.

Find the stress in each bar, if the length of the compound bar

is 1m.Take Es = 200 GPa, Ecu = 100 GPa.

Steel bar

80kN

Copper tube

0.15mm

1000mm

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Temperature Stress

L

A B

L

AB

L

A B

B

P

TL

Any material is capable of expanding or contracting freely due to

rise or fall in temperature. If it is subjected to rise in temperature of

TC, it expands freely by an amount TL as shown in figure.

Where

is the coefficient of linear expansion,TC

= rise intemperature and L = original length.

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From the above figure it is seen that B shifts to B' by an amount

TL. If this expansion is to be prevented a compressive force isrequired at B'.

Temperature strain = TL/(L + TL) TL/L= T

Temperature stress = TE

Hence the temperature strain is the ratio of expansion or contraction

prevented to its original length.

If a gap is provided for expansion then

Temperature strain = (TL) / L

Temperature stress = [(TL)/L] E

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Temperature stress in compound bars:-

Material(2)

Material(1)

2TL

1TL

(dL)1

P1

(dL)2

P2

x

x

When a compound bar is subjected to change in temperature, both the

materials will experience stresses of opposite nature.

Compressive force on material (1) = tensile force on material (2)

1A1 = 2A2 (there is no external load)

1=( 2A2)/A1 (1)

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As the two bars are connected together, the actual position of the bars will

be at XX.

Actual expansion in material (1) = actual expansion in material (2)

1TL(dL)1 = 2TL + (dL)2

1TL (1 / E1) L =2TL + (2 / E2) L

T(1 / E1) = 2T + 2 / E2 --------------------------(2)

From (1) and (2) magnitude of1 and 2 can be found out.

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(1) A steel rail 30m long is at a temperature of 24C. Estimate the

elongation when temperature increases to 44C. (1) Calculate thethermal stress in the rail under the following two conditions :

(i) No expansion gap provided

(ii) If a 6mm gap is provided for expansion(2) If the stress developed is 60MPa , what is the gap left

between the rails?

Take E = 200GPa, = 18 x 10-6/C

PROBLEMS 22

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(2) A steel bar is placed between two copper bars. Steel bar and

copper bar has c/s 60mm 10mm and 40mm 5mm

respectively connected rigidly on each side. If the temperature is

raised by 80C, find stress in each metal and change in length.The length of bar at normal temperature is 1m. Es = 200GPa,

Ecu= 100GPa, s = 12 x 10-6/ C, cu = 17x10

-6/ C

Steel

Copper

cu

TLcu

(dL)cu

Pcu

(dL)s

Ps

Copper

40mm

60mm

40mm

1000mm

Pcu

x

s

TLs

x

Solution:

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(3) A horizontal rigid bar weighing 200 kN is hung by three vertical

rods each of 1m length and 500mm2 c/s symmetrically as shown.

Central rod is steel and the outer rods are copper. Temperature rise is

40C. (1) Determine the load carried by each rod and by how much thehorizontal bar descend? Given Es = 200GPa. Ecu=100GPa. s =1.2 x 10

-

5/C. cu=1.8x 10-5/C. (2) What should be the temperature rise if the

entire load of 200kN is to be carried by steel alone.

Copper CopperSteel

PcuPs

Pcu

(dL)T

(dL)L

(dL)

T(dL)

L

(dL)

T(dL)

L

200kN

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(4) A rigid bar AB is hinged at A and is supported by copper and steel

bars as shown each having c/s area 500mm2. If temperature is raised

by 50C, find stresses in each bar. Assume Ecu = 100 Gpa. Es=200GPa, s = 1.2 x 10

-5/C cu = 18 x 10-6/C

Copper 200mmD

Steel150mm

E

A CB

C B

B'

sTLs

Ps(dL)s

(dL)cuC'cuTLcu

P

C"

B"

RAA

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(5) A composite bar is rigidly fixed at A and B.Determine the reaction at

the support when the temperature is raised by 20C. Take EAl = 70GPa,Es= 200GPa, Al = 11 x 10

-6/C, s = 12 x 10-6/C.

A = 600mm2A = 300mm2

40kNAluminium

1m

Steel3m

BA

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(6) A bar is composed of 3 segments as shown in figure. Find the

stress developed in each material when the temperature is raised by

50C under two conditions

i)Supports are perfectly rigid

ii) Right hand support yields by 0.2mm

Take Es = 200GPa, Ecu =100GPa, Eal= 70GPa, s = 12 x 10-6/C,

cu = 18 x 10-6/C, al = 24 x 10

-6/C.

A=200mm2

A=400mm2A=600mm2

150mm

200mm 150mm

Steel Copper Aluminium

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Exercise problems 38

1) A circular concrete pillar consists of six steel rods of 22mmdiameter each reinforced into it. Determine the diameter of pillar

required when it has to carry a load of 1000kN. Take allowable

stresses for steel & concrete as 140Mpa & 8Mpa respectively. The

modular ratio is 15 ANS: D=344.3mm

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39

2) Determine the stresses & deformation induced in Bronze& steel as shown in figure. Given As=1000mm2,

Ab=600mm2, Es= 200Gpa, Eb= 83Gpa ANS: ( b=55Mpa,

s=93.5Mpa, dLs=dLb=0.093mm)

160kN

Bronze BronzeSteel


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3) A cart wheel of 1.2m diameter is to be provided with steel tyre.Assume the wheel to be rigid. If the stress in steel does not exceed

140MPa, calculate minimum diameter of steel tyre & minimum

temperature to which it should be heated before on to the wheel.

ANS: d=1199.16mm T=58.330C

4) A brass rod 20mm diameter enclosed in a steel tube of 25mm internal

diameter & 10mm thick. The bar & the tube are initially 2m long &

rigidly fastened at both the ends. The temperature is raised from 200C to

800C. Find the stresses in both the materials.

If the composite bar is then subjected to an axial pull of 50kN, find the

total stress. Es=200GPa, Eb=80GPa, s=1210-6/0C, b=1910

-6/0C.

ANS: b=8.81N/mm2 ( C ) , s=47.99N/mm2( T )


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