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
Mohrscircleineachof3planes
Atangleof45toaandbaxes
(Three2DMohrscircles

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-stress theory of failure. (b) Determine the Mises
equiv

(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 increased from zero to
* Strain Energy under Axi

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 of R
b3
f3
r4
b2
r3
f2
f1
b1

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
Noyieldingfor
brittleconditionsso
cautionisrequired
Maxim

(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 . This increase in strain energy may be expressed

(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 of the deflection curve by following the same procedure

(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 is assumed to be ideal meaning that it is: 1

(60)
The unit-Load Method
:
:
There might be other external loads than P
but the term M/P is zero for them
,
.
Where M remains the same as in the standard Castiglianos metho

(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 a nodal point in the direction of real external load P

(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 (pinned, fixed, or free). We have determined the critic

(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. Under this condition, the column remains straight unti