COMPOUND BAR

1
7. STATICALLY INDETERMINATE MEMBERS THERMAL
STRESSES
2
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
  statically indeterminate.
• The following equations are required to solve the
  problems on statically indeterminate structure.
• 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.

3
COMPOUND BAR
Material(2)
L1
L2
Material(1)
W

• A compound bar is one which is made of two or
  more than two materials rigidly fixed, so that
  they sustain together an externally applied load.
  In such cases
• Change in length in all the materials are same.
• (ii) Applied load is equal to sum of the loads
  carried by eachbar.

4

• (dL)1 (dL)2
• (s1/ E1)L1 (s2 /E2)L2

s1 s2 ( E1/E2)(L1/L2)
(1)
E1/E2 is called modular ratio Total load load
carried by material (1) load carried by

material(2)
W s1 A1 s2 A2
(2)
From Equation (1) (2) s1 and s2 can be
calculated
5
Illustrative Problems

• Q.7.1 A load of 300KN is supported by a short
  concrete column 250mm square. The column is
  strengthened by 4 steel bars in corners with
  total c/s area of 4800mm2. If Es15Ec, find
  the stress in steel and concrete.
• If the stress in concrete not to exceed
  4MPa, find the area of steel required so that the
  column can support a load of 600KN.

6
Steel
Case(i) As 4800mm2 Ac (250 250) 4800
57,700 mm2
250mm
250mm
W ss As sc Ac 300 103 15 sc
4800 sc 57,700 sc 2.31
N/mm2 ss 15sc
15 x 2.31 34.69N/mm2

• Deformation is same
• (dL)s (dL)c
• (ss / Es ) Ls (sc / Ec) Lc
• ss / 15Ec sc/Ec

(1)
ss 15sc
7

• Case (ii)
• W ss As sc Ac
• 600 103 15 sc As sc Ac
• 600 103 (15 4) As 4 (250 250 As)
• As 6250 mm2

8

• Q.7.2 A mild steel rod 5 mm diameter passes
  centrally through a copper 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

9
Steel
5mm
Since deformation are same (dL)s (dL)cu (ss /
Es)Ls ( scu / Ecu ) Lcu ss / (200 103 )
scu / (100 103) ss 2 scu W
ss As scu Acu
Copper
25mm
600mm
33mm
50KN
50 1032scu ( p/4) (5)2 scu p/4 (33)2
(25)2 scu 123.86N/mm2
ss 247.72 N/mm2 (dL)s (ss /
Es ) Ls
(dL)s 247.72/(200 103) 600
0.74mm (dL)cu
10
Solution

• Q.7.3 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
• Bar AB - L500mm, A100mm2, E200GPa
• Bar CD- L900mm, A300mm2, E100GPa
• Bar EF- L600mm, A200mm2, E200kN/mm2
• If the rigid bar remains horizontal even after
  loading, determine the stress and elongation in
  each bar.

C
F
A
600mm
900mm
500mm
D
B
E
20kN
11
Deformations are same (dL)AB (dL)CD (dL)EF
(sAB / EAB) LAB (sCD / ECD) LCD (sEF /
EEF) LEF sAB/(200 103) 500 sCD /(100
103) 900 sEF /(200 103 ) 600
sAB 3.6 sCD, sEF 3 sCD

• W (sAB AAB ) (sCD ACD) (sEF AEF)
• 20 103 (3.6 sCD 100) (sCD 300)
  (3sCD 200)
• sCD 15.87N/mm2
• sAB 3.6 15.87 57.14N/mm2
• sEF 3 15.87 47.61N/mm2
• dLAB (sAB / EAB) LAB 57.14/(200 103)
  500
• dLAB 0.14 (dL)CD (dL)EF

12

• Q.7.4 Two copper rods and one steel rod together
  supports as shown in figure. The stress in copper
  and steel not to exceed 60MPa and 120MPa
  respectively. Find the safe load that can be
  supported. Take Es 2Ecu

W
Copper (30mm30mm)
Copper (30mm30mm)
120mm
Steel (40mm40mm)
80mm
13
Solution

• Deformations are same
• (dL)s (dL)cu
• (ss / Es) Ls (scu / Ecu ) Lcu
• (ss / 2Ecu ) 200 ( scu / Ecu ) 120

s s 1.2 scu
Let scu60MPa60N/mm2,ss1.2x60 72N/mm2 lt
120N/mm2


(safe)
Safe load W ss As 2( scu Acu )
72(40 40) 2
60(30 30)
Safe load W 223.2 103 N 223.2 kN
14

• Q.7.5 A rigid bar AB 9m long is suspended by two
  vertical rods at its end 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 be applied
  if the bar remains horizontal after the load is
  applied.

Steel
Brass
5m
3m
9m
A
B
x
3000N
15
Deformations are same (dL)b (dL)s (sb / Eb )
Lb (ss / Es ) Ls (sb / 105) (3 103) ss
/ (200 103) 5 103
ss 1.2 sb

• W ssAs sbAb
• 3000 (1.2 sb 445) (sb 1000)

sb 1.95N/mm2
ss 2.34N/mm2
ve SMA 0 -(3000) (x) (2.34 445) 9000
0 x 3123.9 mm
3.12m from A If the load of 3000N is kept
at a distance of 3.12m from A, bar AB will remain
horizontal.
16

• Q.7.6 A mild steel bar of c/s 490mm2 is
  surrounded by a copper tube of 210mm2 as shown.
  When they are placed centrally over 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.

80kN
Steel bar
0.15mm
Copper tube
1000mm
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Solution

• (dL)s (dL)cu 0.15
• (ss / Es ) Ls ( scu / Ecu ) Lcu 0.15
• ss / (200 103) 1000.15 scu / (100
  103) 1000 0.15

s s 2 scu 30
W ss As scu Acu 80 103 ( 2scu 30)
490 (scu 210)
scu 54.87N/mm2
ss (254.87) 30 139.84N/mm2
18
Temperature Stress
B
A
L
B
B
A
P
L
aTL
A
B
L
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 aTL as shown in figure. Where a is the
coefficient of linear expansion, TC rise in
temperature and L original length.
19

• From the above figure it is seen that B shifts
  to B' by an amount aTL. If this expansion is
  to be prevented a compressive force is required
  at B'.

Temperature strain aTL/(L aTL) aTL/L
aT Temperature stress aTE
Hence the temperature strain is the ratio of
expansion or contraction prevented to its
original length.
If a gap d is provided for expansion then
Temperature strain (aTL d) / L Temperature
stress (aTL d)/L E
20
x

• Temperature stress in compound bars-

a1TL
P1
Material(1)
Material(2)
(dL)2
(dL)1
P2
a2TL
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)
s1A1
s2A2 (there is no external load)
s1( s2A2)/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) a1TL (dL)1 a2TL
(dL)2 a1TL (s1 / E1) L a2TL (s2
/ E2) L aT (s1 / E1) a2T s2 /
E2 --------------------------(2) From (1) and
(2) magnitude of s1 and s2 can be found out.
22
IllustrativeProblems
Q.7.7 A steel rail 30m long is at a temperature
of 24C. Estimate the elongation when
temperature increases to 44C. (1) Calculate the
thermal 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, a 18 x 10-6 /C
23

• Free expansion aTL 18 10-6 (44-24) 30
  103 10.8mm
• No expansion joints provided-
• Temperature stress aTE
• 18 10-6
  20 200 103
• 72N/mm2
• 6mm gap is provided for expansion
• temperature stress ( aTL d) / L E
• (10.8
  6)/(30 103 ) 200 103
• 32N/mm2
• when stress 60MPa
• temperature stress ( aTL d) / L E
• (10.8
  d ) / (30 103 ) 200 103

d 1.8mm
24
Q.7.8 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, as
12 x 10-6/ C, acu 17x10-6/ C
x
(dL)cu
acuTLcu
Solution
Copper
40mm
Pcu
asTLs
Steel
(dL)s
60mm
Ps
Pcu
40mm
Copper
1000mm
x
?
25
Compressive force on copper bar tensile force
on steel bar
2scu Acu ss As 2scu ( 40 5) ss ( 60
10)
scu 1.5ss
Actual expansion in copper Actual
expansion in steel
acuTLcu - (dL)cu asTLs (dL)s
acuTLcu - (scu / Ecu) Lcu asTLs (ss / Es) Ls
Since Lcu Ls
(17 10-6 80 ) 1.5ss /(100 103) (12 10-6
80) ss /(200 103)
ss 20N/mm2(T)
scu1.5 20 30N/mm2 (C)
? Change in length acu TLcu (scu / Ecu)
Lcu
1710-6 80 1000 30/(100 103) 1000
? 1.06mm
26
Q.7.9 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 40ºC. (1) Determine the load
carried by each rod and by how much the
horizontal bar descend? Given Es 200GPa.
Ecu100GPa. as 1.2 x 10-5/ºC. acu1.8x 10-5/ºC.
(2) What should be the temperature rise if the
entire load of 200kN is to be carried by steel
alone.
Copper
Copper
Steel
Pcu
Ps
Pcu
(dL)T
(dL)T
(dL)T
(dL)L
(dL)L
200kN
(dL)L
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(dL) T (dL) Lcu (dL)T (dL)Lst
acuTLcu ( PcuLcu / Acu Ecu ) asTLs (
PsLs / AsEs ) -----(1)
For equilibrium condition ?Fv 0 Ps 2Pcu
200 103 Ps 200 103 2Pcu
Substituting in (1) 1.8 10-5 40 Pcu/ (500
100 103) 1.2 10-5 40 (200 103
2Pcu ) / (500 200 103)
Pcu 44,000N
Ps 200 103 2 44,000
Ps 112 103N
Elongation acuTLcu (Pcu Lcu )/(Acu Ecu)
1.8 10-5 40 1000 44000 1000/(500 100
103)
dL1.6mm
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Case (ii)
Copper
Steel
Pcu
Ps
Pcu
Note When entire load is carried by steel,
copper will have extension due to temp. only
(dL)T
(dL)T
(dL)T
200kN
(dL)L
(dL)T cu (dL)L (dL)Ts
acuTLcu PsLs / AsEs asTLs
1.8 10-5 T (200 103) / (500 200 103)
1.2 10-5 T T 333.33ºC
29
Q.7.10 X
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 50ºC,
find stresses in each bar. Assume Ecu 100 Gpa.
Es 200GPa, as 1.2 x 10-5/ºC
acu 18 x 10-6/ºC
D
Copper
E
200mm
Steel
150mm
A
C
B
RA
C
B
A
(dL)cu
asTLs
C'
B"
(dL)s
acuTLcu
Ps
C"
B'
Pcu
30
Copper- Temp.Stress (acuTLcu) (dL)cu / 200
Ecu
Pcu (acuTLcu) (dL)cu / 200 (Ecu Acu)
(18 10-6 50 200) (dL)cu) /200 100
103 500
Pcu 45 103 - 250 103(dL)cu-----------(1)
Steel- Ps (dL)s asTLs / 150 EsAs
Ps (dL)s 12 x 10-6 50 150/150
(200 103 500)
Ps 6.66 105 (dL)s 6 104
----------------(2)
From similar ?le (dL)cu/200 (dL)s/500 (dL)s
2.5(dL)cu ----------- (3) ve S MA 0 -Ps
500 Pcu 200 0
Pcu 2.5Ps
(4)
31
Substituting (1) (2) in (4) (45 103) 250
103 (dL)cu 2.5 6.66 105 (dL)s
(6104) 45 103 250 103 (dL)cu 2.5 6.66
105 2.5(dL)cu (6104) (dL)cu
0.044mm (dL)s0.11mm
Substituting in (1) (2) Pcu (45 103)
(250 103 0.044) 34,000N Ps
34,000/2.5 13,600N ss Ps / As 13,600/500
27.2N/mm2(T) scu Pcu / Acu 34,000/500
68N/mm2(C)
32
Q.7.11 A composite bar is rigidly fixed at A and
B. Determine the reaction at the support when the
temperature is raised by 20ºC. Take EAl 70GPa,
Es 200GPa, aAl 11 x 10-6/ºC,
as 12 x 10-6/ºC.
A 600mm2
A 300mm2
A
B
Aluminium
Steel
40kN
3m
1m
33
A 600mm2
A 300mm2
RA
40kN
RB
Steel
AL
3m
1m
Steel-
Aluminium-
RB
RB
RA
RA
ve SFX 0 RA 40 103 RB 0 RA 40 103
RB (dL)T (dL)L al (dL)T (dL)L s
0 (a T L) (RA x L) /(A E )al (aTL)(
RBL )/(AE)s 0
34
(11 10-6 20 1000) (40 103
RB)1000/(600 70 103) (12 10-6 20
3000 ) (RB 3000)/(300 200 103) 0 0.22
0.95 (2.38 10-5 RB) 0.072 ( 5 10-5 RB )
0 7.38 10-5 RB 0.658 RA 8915.99 8916N (
) RB 31,084N ( )
35

• Q.7.12 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
• Supports are perfectly rigid
• Right hand support yields by 0.2mm
• Take Es 200GPa, Ecu 100GPa, Eal 70GPa, as
  12 x 10-6/ºC,
• acu 18 x 10-6/ºC, aal 24 x 10-6/ºC.

A600mm2
A400mm2
A200mm2
Copper
Aluminium
Steel
150mm
200mm
150mm
36
Case(i) Supports are perfectly rigid (dL)s
(dL)cu (dL)al asTLs acuTLcu aalTLal
(12 10-6 50 150 ) (18 10 -6 50 200)
(24 10-6 50 150) 0.45mm

(ss/Es) Ls (scu/Ecu) Lcu (sal/Eal) Lal
0.45mm
ss/(200 103)150scu/(100103)200sal/(70
103)1500.45 -(1)
From principle of compound bars ssAs scuAcu
salAal ss 200 scu 400 sal 600
sal 0.67scu
ss2scu
37
Substituting in (1) 2scu/(200
103)150scu/(100 103)2000.67scu/(70103)
1500.45 scu 91.27N/mm2 ss 2scu
182.54N/mm2, sal 0.67 scu 61.15N/mm2
Case (ii) Right hand support yield by
0.2mm (ss/Es) Ls (scu/Ecu) Lcu (sal/Eal) Lal
0.45 0.20.25 2scu/(200 103)150scu/(100
103)2000.67scu/(70103) 1500.25 scu
50.61N/mm2 ss 2scu 101.22N/mm2,sal 0.67 scu
33.91N/mm2
38
Exercise problems
Q1. A circular concrete pillar consists of
six steel rods of 22mm diameter 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 D344.3mm
39
Q2. Determine the stresses deformation
induced in Bronze steel as shown in figure.
Given As1000mm2, Ab600mm2, Es 200Gpa, Eb
83Gpa ANS ( sb55Mpa, ss93.5Mpa,
dLsdLb0.093mm)
Steel
Bronze
Bronze
160kN
40
Q3. 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 d1199.16mm
T58.330C
41
Q4. 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. Es200GPa, Eb80GPa,
as1210-6/0C, ab1910-6/0C. ANS sb8.81N/mm2
( C ) , ss47.99N/mm2( T )