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208 STRENGTH OF MATERIALS
Formulas for Combined Stresses(1) Circular cantilever beam in
direct compression and bending:
(2) Circular cantilever beam in direct tension and bending:
(3) Rectangular cantilever beam in direct compression and
bending:
(4) Rectangular cantilever beam in direct tension and
bending:
(5) Circular beam or shaft in direct compression and
bending:
(6) Circular beam or shaft in direct tension and bending:
Type of Beamand Loading
Maximum NominalTens. or Comp. Stress
Maximum NominalShear Stress
Type of Beamand Loading
Maximum NominalTens. or Comp. Stress
Maximum NominalShear Stress
Type of Beamand Loading
Maximum NominalTens. or Comp. Stress
Maximum NominalShear Stress
Type of Beamand Loading
Maximum NominalTens. or Comp. Stress
Maximum NominalShear Stress
Type of Beamand Loading
Maximum NominalTens. or Comp. Stress
Maximum NominalShear Stress
Type of Beamand Loading
Maximum NominalTens. or Comp. Stress
Maximum NominalShear Stress
a 1.273
d
2-------------
8LFy
d------------- Fx
= a 0.5a
=
b 1.273
d2
-------------8LFy
d------------- Fx+
= b 0.5b
=
a 1.273
d2
------------- Fx
8LFy
d-------------+
= a 0.5a
=
b 1.273
d2------------- Fx
8LFy
d-------------
= b
0.5b
=
a 1
bh------
6LFy
h------------- Fx
= a 0.5a
=
b 1
bh------
6LFy
h------------- Fx+
= b 0.5b
=
a 1
bh------ Fx
6LFy
h-------------+
= a 0.5a
=
b 1
bh------ Fx
6LFy
h-------------
= b 0.5b
=
a 1.273d2
------------- 2LFyd
------------- Fx+ = a 0.5a=
b 1.273
d2-------------
2LFy
d------------- Fx
= b 0.5b
=
a 1.273
d2------------- Fx
2LFy
d-------------
= a 0.5a
=
b 1.273
d2------------- Fx
2LFy
d-------------+
= b 0.5b=
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STRENGTH OF MATERIALS 209
(7) Rectangular beam or shaft in direct compression and
bending:
(8) Rectangular beam or shaft in direct tension and bending:
(9) Circular shaft in direct compression and torsion:
(10) Circular shaft in direct tension and torsion:
(11) Offset link, circular cross section, in direct tension:
(12) Offset link, circular cross section, in direct
compression:
Type of Beamand Loading
Maximum NominalTens. or Comp. Stress
Maximum NominalShear Stress
Type of Beamand Loading
Maximum NominalTens. or Comp. Stress
Maximum NominalShear Stress
Type of Beamand Loading
Maximum NominalTens. or Comp. Stress
Maximum NominalShear Stress
a anywhere on surface
Type of Beam
and Loading
Maximum Nominal
Tens. or Comp. Stress
Maximum Nominal
Shear Stress
a anywhere on surface
Type of Beamand Loading
Maximum NominalTens. or Comp. Stress
Maximum NominalShear Stress
Type of Beamand Loading
Maximum NominalTens. or Comp. Stress
Maximum NominalShear Stress
a 1
bh------
3LFy
2h------------- Fx+
= a 0.5a
=
b 1bh------ 3LFy
2h------------- F x = b 0.5b=
a 1
bh------ Fx
3LFy
2h-------------
= a 0.5a
=
b 1
bh------ Fx
3LFy
2h-------------+
= b 0.5b
=
a
0.637
d2------------- F F2
8T
d------
2+
= a
0.637
d2------------- F2
8T
d------
2+
=
a
0.637
d2------------- F F2
8T
d------
2+
= a
0.637
d2------------- F2
8T
d------
2+
=
a 1.273F
d2
----------------- 18e
d
------
= a 0.5a
=
b 1.273F
d2----------------- 1
8e
d------+
= b 0.5b
=
a1.273F
d2-----------------
8e
d------ 1
= a 0.5a
=
b 1.273Fd2
----------------- 8ed
------ 1+ = b 0.5b=
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210 STRENGTH OF MATERIALS
(13) Offset link, rectangular section, in direct tension:
(14) Offset link, rectangular section, in direct
compression:
Formulas from the simple and combined stress tables, as well as
tension and shearfactors, can be applied without change in
calculations using metric SI units. Stresses
are given in newtons per meter squared (N/m2) or in N/mm2.
Three-Dimensional Stress.Three-dimensional or triaxial stress
occurs in assembliessuch as a shaft press-fitted into a gear bore
or in pipes and cylinders subjected to internal orexternal fluid
pressure. Triaxial stress also occurs in two-dimensional stress
problems ifthe loads produce normal stresses that are either both
tensile or both compressive. In eithercase the calculated maximum
shear stress, based on the corresponding two-dimensional
theory, will be less than the true maximum value because of
three-dimensional effects.Therefore, if the stress analysis is to
be based on the maximum-shear-stress theory of fail-ure, the
triaxial stress cubic equation should first be used to calculate
the three principalstresses and from these the true maximum shear
stress. The following procedure providesthe principal maximum
normal tensile and compressive stresses and the true maximumshear
stress at any point on a body subjected to any combination of
loads.
The basis for the procedure is the stress cubic equation
and Sx, Sy, etc., are as shown in Fig. 1.
The coordinate systemXYZin Fig. 1 shows the positive directions
of the normal andshear stress components on an elementary cube of
material. Only six of the nine compo-nents shown are needed for the
calculations: the normal stresses Sx, Sy, and Sz on three of
the faces of the cube; and the three shear stresses Sxy, Syz,
and Szx. The remaining three shear
stresses are known because Syx = Sxy, Szy = Syz, and Sxz = Szx.
The normal stresses Sx, Sy, and
Sz are shown as positive (tensile) stresses; the opposite
direction is negative (compressive).The first subscript of each
shear stress identifies the coordinate axis perpendicular to
theplane of the shear stress; the second subscript identifies the
axis to which the stress is par-
Type of Beamand Loading
Maximum NominalTens. or Comp. Stress
Maximum NominalShear Stress
Type of Beamand Loading
Maximum NominalTens. or Comp. Stress
Maximum NominalShear Stress
S3AS2+BSC= 0in which:
A = Sx
+Sy
+Sz
B = SxSy+SySz+SzSxSxy2Syz2Szx2
C= SxSySz+ 2SxySyzSzxSxSyz2SySzx2SzSxy2
aF
bh------ 1
6e
h------
= a 0.5a
=
bFbh------ 1
6eh
------+ = b 0.5b=
aF
bh------ 1
6e
h------
= a 0.5a
=
b
F
bh------ 1
6e
h------+
= b
0.5b
=
