Application of Mohrs circle to the 3D Analysis of stress
e.g.,thinwalledpressurevessel
a b ctriaxial loadingmax int min
PlaneStressconditions(case1)
Maximum in-plane shear stress
Mohrscircleinea
A hollow structural steel flexural member is subjected to the load
shown. The yield strength of the steel is Y = 320 MPa. (a)
Determine the factors of safety predicted at point K by the
maximum-shear-
(35)
Energy Methods
External work and Elastic Strain Energy for Normal Stresses
W : External Work: dx is in the same direction as P(x)
P
Internal force of the element is
gradually incr
Bending of Curved Members
Positive direction for M
for positive M
Important: Since
generally produces a very small number it is best to calculate
with sufficient accuracy
(27)
Calculation
Failure (Fracture) Criteria for Brittle Materials under Plane Stress
For uniaxial tensile test failure occurs when = U (ultimate stress).
For plane stress conditions a criterion needs to be defined
No
(51)
CASTIGLIANOS 2nd Theorem (1847-1884, Milan)
,
,
,
Assume that after all the loads are applied, one of the loads ( ) is increased by an
infinitesimal amount d .
(79)
COLUMNS WITH OTHER SUPPORT CONDITIONS
The critical loads for columns with various kinds of support conditions (such as fixed ends or free end) can
be determined from the differential equation o
(70)
EULERS FORMULA FOR PINENDED IDEAL COLUMNS
Introduction: We aim to determine the critical buckling load (Pcr) for an ideal column that is pin
supported as shown in Figure (a). The column
(64)
Principal of Virtual Work (virtual displacement method)
virtual displacement
Limited to pinended axially
loaded bars with an external load
real loading
1: virtual unit displacement at
(94)
EMPIRICAL COLUMN FORMULAS UNDER CENTRIC LOADING
We examined the behaviour of ideal columns that are perfectly straight, made of homogeneous
materials and with exact end conditions
(87)
COLUMNS WITH ECCENTRIC AXIAL LOADING (THE SECANT FORMULA)
The Euler formula applies only to ideal columns in which the axial loads acts through the
centroid of the cross section.