EAS 209-Spring 2011
Instructors: Christine Human
EAS 209-Spring 2011
Instructors: Christine Human
Lecture 10 Chapter 3 TORSION In addition to axial loading, we are interested in stresses and strains of circular shafts subjected to twisting couples or torq

Overview of Mechanics
EAS 209: Introduction
Christine Human Department of Civil, Structural and Environmental Engineering
Statics (EAS 207) Statics Equilibrium Dynamics (EAS 208) Dynamics Newtons laws of motion Newton Mechanics of Solids (EAS 209) Mechani

Comparison of Methods: Beams with Hinge
E=29x106 psi
For the beam shown above, determine:
RA
Shear Force
(a) the slope at point A
(b) the deflection at point B
We will solve this problem using both singularity
functions and by superposition to compare the

EAS 209-Spring 2014
Instructors: Christine Human
Lecture 31
Beam Deflections
EAS 209-Spring 2014
Instructors: Christine Human
Curvature of Beams in Pure Bending
The dominant source of beam deflection is bending
We have focused on designing beams for stren

EAS 209-Spring 2014
Instructors: Christine Human
Lecture 32
Beam Deflection using Singularity Functions
For the cantilever beam with uniformly distributed
load, w can be described by a single function and
integrated 4 times to determine deflection.
EAS 20

EAS 209-Spring 2014
Instructors: Christine Human
Lecture 33
Statically Indeterminate Beams
Consider beam with fixed
support at A and roller
support at B.
From free-body diagram,
we can see that there are
four unknown reaction
components, and three
equilib

EAS 209-Spring 2014
Instructors: Christine Human
Lecture 29-Max Stress in Beams
In Ch 5 we designed beams for bending by
considering the maximum moment and calculating
EAS 209-Spring 2014
Instructors: Christine Human
Principal Stresses in Beams
Consider t

Chapter 12:
Gas Dynamics
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Introduction to Gas Dynamics
Equations of State
One-Dimensional Flow
Speed of Sound and the Mach Cone
Adiabatic, 1D Compressible Flow of a Perfect

EAS 209-Spring 2014
Instructors: Christine Human
Lecture 28
Transformation of Plane Strain
We will see that the transformation of strain is
very similar to the transformation of stress. Like
stress transformation, we can use Mohrs circle to
solve strain t

EAS 209-Spring 2014
Instructors: Christine Human
EAS 209-Spring 2014
Instructors: Christine Human
Lecture 27
Pressure Vessels
We will examine only cylindrical and spherical
thin-walled vessels.
Pressure vessels are storage containers for gases
or liquids

EAS 209-Spring 2014
Instructors: Christine Human
Lecture 30
Combined Loading
To date we have calculated the stresses due to:
EAS 209-Spring 2014
Instructors: Christine Human
We will examine a slender structural member
subjected to arbitrary loadings and f

EAS 209-Spring 2014
Instructors: Christine Human
Lecture 25
Mohrs Circle for Plane Stress
Mohrs Circle is a graphical approach to finding
the stress component at any orientation.
Procedure was developed by a German engineer
Otto Mohr (1835-1918)
EAS 209-

EAS 209-Spring 2014
Instructors: Christine Human
Lecture 26
Effect of Third Principal Stress
So far we have only determined the stresses in the
xy plane, these are called the in-plane stresses.
We have found that within this plane there are two
principal

EAS 209-Spring 2014
Instructors: Christine Human
Lecture 22
Chapter 6
Shearing Stresses in Beams
EAS 209-Spring 2014
Instructors: Christine Human
Elementary normal and shear forces
Transverse loading applied to a beam results in
both internal shear forces

EAS 209-Spring 2014
Instructors: Christine Human
Lecture 24
Transformation of Plane Stress
The stress components (normal and shear) depend
upon the orientation of the axes.
EAS 209-Spring 2014
Instructors: Christine Human
General State of Stress:
The most

EAS 209-Spring 2014
Instructors: Christine Human
Lecture 37
Design of Columns under a Centric Load
EAS 209-Spring 2014
Instructors: Christine Human
Experimental data demonstrate that the mode of
failure is highly dependent on slenderness ratio:
- for shor

EAS 209-Spring 2014
Instructors: Christine Human
EAS 209-Spring 2014
Instructors: Christine Human
Lecture 34
Superposition
We saw last lecture how we could use superposition
to solve statically indeterminate beams.
In general we can use superposition to d

EAS 209-Spring 2014
Instructors: Christine Human
Lecture 36
Stability of Columns
In preceding chapters, we have been concerned with
two main design criteria:
Strength the ability of a structure to support a
given load without exceeding the allowable
stre

p
p
V2
V2
g 2g z g 2g z H L H S
2
1
where
p2 p1 patm (free surface)
(11.37)
V2 V1 0 (large tanks)
z2 z1 H
V2 L
f K entrance K exit K elbow (where V is the mean velocity in the pipe)
2g D
Note that the mean pipe velocity can be expressed in terms of

Plot Eqn. (11.41) on the pump performance curve to determine the operating point.
system curve (Eqn. (11.41)
From the figure we observe that the operating point occurs at:
Q 1600 gpm
corresponding to a head rise and efficiency of
H 67 ft
84%
The operatin

4.
Stability issues become significant when the pump has a flat or falling (defined as a performance curve
where H as Q ) performance curve.
H
rising (H as Q )
falling (H as Q )
Q
Consider perturbations to the
system from the operating point.
Hpump > Hsys

Example:
Water is to be pumped from one large open tank to a second large open tank. The pipe diameter throughout
is 6 in. and the total length of the pipe between the pipe entrance and exit is 200 ft. Minor loss coefficients
for the entrance, exit, and t

5.
System Characteristic Curves and Pump Selection
How do we select a pump for a given system? Analyze the system to determine the shaft head required to
give a specified volumetric flow rate. Compare this result to a given pump performance curve (H-Q cur

The conditions at which the system will operate will depend on the intersection of the system head curve
with the pump performance curve as shown in the figure below.
H
system curve
pump curve
operating point
Q
Notes:
1. Ideally we would want the operatin

4.
Specific Speed, Ns
A useful dimensionless term results from the following combination of previously defined terms:
1
1
2
Q 2
Ns specific speed = 3
3
4 gH 4
(11.30)
Combining the terms in this manner eliminates the impeller diameter, D.
Notes:
1. It i

Example:
A small centrifugal pump, when tested at 2875 rpm with water, delivered a flowrate of 252 gpm and a head
of 138 ft at its best efficiency point (efficiency is 76%). Determine the specific speed of the pump at this
test condition. Sketch the impel

Example:
A centrifugal pump with a 12 in. diameter impeller requires a power input of 60 hp when the flowrate is
3200 gpm against a 60 ft head. The impeller is changed to one with a 10 in. diameter. Determine the
expected flowrate, head, and input power i

University at Buffalo
Name
EAS 209, Spring 2003
Section
Exam 1
4 questions, point distribution indicated, show all work on pages provided
Closed book, 1 page of notes allowed
1 (5 points)
2 (5 points)
3 (7 points)
4 (8 points)
_
1. Two possible configurat

University at Buffalo
Name
EAS 209, Spring 2009
Section _ A Richards
_ B Human
Exam 1
3 questions, point distribution indicated, show all work on pages provided
Closed book exam, see attached note sheet
1 (8 points)
2 (8 points)
3 (9 points)
_
1) The high

EAS209 Spring 2014
Christine Human
Lecture 38 Strain Energy
Problems can often be solved in
more than one way. During the
semester we have used direct
methods that use equilibrium and
geometric compatibility to relate force, stress, strain
and displacemen

Lec. # 18
EAS 209
Instructor: Ahmad
Flexural Stresses in Composite Beams
Many structural applications involve beams made of two (or more) materials. These
types of beams are called composite beams. Examples include wood beams
reinforced with steel plates

EAS 209
Lec. # 20
Instructor: Ahmad
Unsymmetrical Bending
If the beam cross section is unsymmetric or if the loads on the beam do not act in the
plane of bending, then the theory of bending developed in earlier sections of this
chapter is not valid. Consi

EAS 209
Lec. # 16
Instructor: Ahmad
Bending Stresses in Beams (Chapter 8)
Loads on a beam cause it to bend (or flex). The applied loads cause the initially
straight member to deform into a curved shape (Figure 8.1b), which is called
the deflection curve o

EAS 209
Lec. # 23
Instructor: Ahmad
Shear Stress and Shear Flow in Thin-Walled Members
Consider the segment of length dx of the wide-flange beam shown in fig. 9.16(a).
Next, consider the free-body diagram of a portion of the upper flange, element (1),
sho

EAS 209
Lec. # 19
Instructor: Ahmad
Bending Due to Eccentric Axial Load
An eccentric axial load is a force whose line of action does not pass through the
centroid of the cross section. When an axial force is offset from a member's
centroid, bending stress