Transcript Strength of materials - Yogyakarta State University

Statically Determinate and Indeterminate System of Bars
STRENGTH OF MATERIALS
Statically Determinate
System of Bars
 Investigation to trusses and to structures
which consist of bars and rigid body
 Assumption : the elongations are small as
compared with the length of the bars
 Applying the equilibrium conditions to the
undeformed system
Statically Determinate
System of Bars
B
All bars have the axial rigidity EA
2

1
C
A
l
F
Statically Determinate
System of Bars
B
All bars have the axial rigidity EA
2
S2

1
A
l
C
F

S1
C
F
Statically Determinate
System of Bars
Calculating S1 and S2 in the bars, as follow :
𝑆2 sin 𝛼 − 𝐹 = 0
𝑆1 + 𝑆2 cos 𝛼 = 0
𝐹
𝑆1 = −
tan 𝛼
𝑆2 =
𝐹
sin 𝛼
Calculating elongation li of the bars
𝑆1 𝑙1
𝐹𝑙
𝛿1 =
=
𝐸𝐴
𝐴𝐸 tan 𝛼
𝛿2 =
𝑆2 𝑙2
𝐹𝑙
=
𝐴𝐸
𝐴𝐸 sin 𝛼 cos 𝛼
Statically Determinate
System of Bars
Statically Determinate
System of Bars
𝐹𝑙
𝑢 = ∆𝑙𝑖 =
𝐴𝐸 tan 𝛼
∆𝑙2
𝑢
𝐹𝑙 1 + 𝑐𝑜𝑠 3 𝛼
𝑣=
+
=
sin 𝛼 tan 𝛼
𝐴𝐸 𝑠𝑖𝑛2 𝛼 cos 𝛼
STATICALLY DETERMINATE SYSTEM OF BARS
ANY QUESTIONS ?
Statically Indeterminate
System of Bars
 A system of bars is statically indeterminate of
degree n if the number of unknowns exceeds
the number of the equilibrium conditions by
n.
 In order to determine the forces in the bars of
such a system, n compatibility conditions are
needed in addition to the equilibrium
conditions.
 Solving this system of equations yields the
unknown forces in the bars.
Statically Indeterminate
System of Bars
Statically Indeterminate
System of Bars
From figure (b) , we can get :
−𝑆1 sin 𝛼 + 𝑆3 sin 𝛼 = 0 → 𝑆1 = 𝑆3
(a)
𝐹 − 𝑆2
𝑆1 cos 𝛼 + 𝑆2 + 𝑆3 cos 𝛼 − 𝐹 = 0 → 𝑆1 = 𝑆3 =
2 cos 𝛼
The elongations of the bars are given by :
∆𝑙1 = ∆𝑙3 =
𝑆1 𝑙1
𝐸𝐴1
∆𝑙2 =
𝑆2 𝑙2
𝐸𝐴2
(b)
Statically Indeterminate
System of Bars
∆𝑙1 = ∆𝑙2 cos 𝛼
(c)
With (a) and (b), and l1= l / cos , we can get :
𝐹 − 𝑆2 𝑙
𝑆2 𝑙
𝐹
=
cos
𝛼
→
𝑆
=
2
𝐸𝐴
2𝐸𝐴1 𝑐𝑜𝑠 2 𝛼 𝐸𝐴2
1 + 2 𝐸𝐴1 𝑐𝑜𝑠 3 𝛼
2
𝐸𝐴1
2
𝐸𝐴2 cos 𝛼
𝑆1 = 𝑆3 =
𝐹
𝐸𝐴1
1 + 2 𝐸𝐴 cos3 𝛼
2
The vertical displacement :
𝐹𝑙
𝑆2 𝑙
𝐸𝐴2
𝑣 = ∆𝑙2 =
=
𝐸𝐴2 1 + 2 𝐸𝐴1 cos 3𝛼
𝐸𝐴2
STATICALLY INDETERMINATE SYSTEM OF BARS
ANY QUESTIONS?
Statically Determinate
System of Bars
The truss is subjected to a force F.
Given: E = 2· 102 GPa, F = 20kN.
Determine the cross-sectional area of
the three members so that
the stresses do not exceed the
allowable stress σallow = 150 MPa
and the displacement of support B is
smaller than 0.5 ‰ of the
length of bar 3.
Statically Indeterminate
System of Bars
To assemble the truss in Fig. (a), the free end of bar 3 (length l − δ, δ<<l) has to be
connected with pin C.
a) Determine the necessary force F acting at pin C (Fig. b).
b) Calculate the forces in the bars after the truss has been assembled
and force F has been removed.