UNIVERSITY OF CALIFORNIA BERKELEY
Department of Civil Engineering
Fall 2013
Structural Engineering,
Mechanics and Materials
Professor: S. Govindjee
HW 5: Solutions
1. Prove suciency in Cauchys theorem.
Tij,i + bj aj
T + bj aj dv
Rt ij,i
T n da + R bj aj d
UNIVERSITY OF CALIFORNIA BERKELEY
Department of Civil Engineering
Fall 2013
Structural Engineering,
Mechanics and Materials
Professor: S. Govindjee
HW 10: Due November 27
1. (10pts) Consider an isotropic elastic square (1 1) bar of length L in torsion.
(a
UNIVERSITY OF CALIFORNIA BERKELEY
Department of Civil Engineering
Fall 2013
Structural Engineering,
Mechanics and Materials
Professor: S. Govindjee
HW 3: Due September 27
1. WSS 3.7
2. WSS 3.8 (Sketches should be accurate and neat; use a computer if you c
UNIVERSITY OF CALIFORNIA BERKELEY
Department of Civil Engineering
Fall 2013
Structural Engineering,
Mechanics and Materials
Professor: S. Govindjee
HW 4: Solutions
1. (Area strain) The deformation gradient at the point of interest is given by:
1
00
0
1 0.
UNIVERSITY OF CALIFORNIA BERKELEY
Department of Civil Engineering
Fall 2013
Structural Engineering,
Mechanics and Materials
Professor: S. Govindjee
HW 2: Solutions
1. (Problem 2.8) To show the given relation write out in indicial notation and reduce the
r
UNIVERSITY OF CALIFORNIA BERKELEY
Department of Civil Engineering
Fall 2013
Structural Engineering,
Mechanics and Materials
Professor: S. Govindjee
HW 3: Solutions
1. Book Problem 3.7: The material description of the displacement is
u(X ) = (X ) X = X3 e2
UNIVERSITY OF CALIFORNIA BERKELEY
Department of Civil Engineering
Fall 2013
Structural Engineering,
Mechanics and Materials
Professor: S. Govindjee
HW 7: Solutions
1. Maxwell Fluid The govering equation for the Maxwell uid is E = + / . One
t/
can derive a
UNIVERSITY OF CALIFORNIA BERKELEY
Department of Civil Engineering
Fall 2013
Structural Engineering,
Mechanics and Materials
Professor: S. Govindjee
HW 8: Solutions
1. The Poisson function is computed by subjecting the material to a uniaxial step stress
=
UNIVERSITY OF CALIFORNIA BERKELEY
Department of Civil Engineering
Fall 2013
Structural Engineering,
Mechanics and Materials
Professor: S. Govindjee
HW 10 Solutions
1. (Torsion of square bar)
(a) Integrating the expresion for from lecture and multiplying b
UNIVERSITY OF CALIFORNIA BERKELEY
Department of Civil Engineering
Fall 2013
Structural Engineering,
Mechanics and Materials
Professor: S. Govindjee
HW 9: Solutions
1. 1D Nonlinear power-law hardening
From the consistency condition:
0 = f = sign( ) Hnn1 .
UNIVERSITY OF CALIFORNIA BERKELEY
Department of Civil Engineering
Fall 2013
Structural Engineering,
Mechanics and Materials
Professor: S. Govindjee
HW 11 Solutions
1. If uz = u = 0 and ur = ur (r), then the only non-zero strains are
rr = ur,r
and
=
ur
.
UNIVERSITY OF CALIFORNIA BERKELEY
Department of Civil Engineering
Fall 2013
Structural Engineering,
Mechanics and Materials
Professor: S. Govindjee
HW 6: Solutions
1. (Poissons ratio in terms of and ). The Poisson ratio is dened for a uniaxial test
as
ss
UNIVERSITY OF CALIFORNIA BERKELEY
Department of Civil Engineering
Fall 2013
Structural Engineering,
Mechanics and Materials
Professor: S. Govindjee
HW 11: Due December 9 (day of last lecture)
1. (10 pts) Consider the case of polar coordinates and the stra
UNIVERSITY OF CALIFORNIA BERKELEY
Department of Civil Engineering
Fall 2013
Structural Engineering,
Mechanics and Materials
Professor: S. Govindjee
HW 1: Due September 13
1. Consider two vectors a (0, 4, 6) and b (2, 0, 4), where the components have been
8
x 10
4
Plot of stress versus time (HW 9 Prob 2(c)
Plot of versus time (HW 9 Prob 2(c)
0.25
6
0.2
2
0.15
0
Stress,
4
0.1
-2
-4
0.05
-6
-8
0
5
10
15
Time, t
20
25
0
30
Plot of q versus time (HW 9 Prob 2(c)
400
10
15
Time, t
20
25
30
4
x 10 Plot of stress
UNIVERSITY OF CALIFORNIA BERKELEY
Department of Civil Engineering
Fall 2013
Structural Engineering,
Mechanics and Materials
Professor: S. Govindjee
HW 2: Due September 20
1. WSS 2.8.
2. WSS 2.12 (b,c)
3. WSS 2.14 (Note the A on NA is not an index/subscrip
UNIVERSITY OF CALIFORNIA BERKELEY
Department of Civil Engineering
Fall 2013
Structural Engineering,
Mechanics and Materials
Professor: S. Govindjee
Final Exam: CE231 / MSE211
Due Thursday Dec. 19
5pm
Problem
#1
#2
#3
#4
#5
#6
Total
Score
/25
/25
/15
/15
/
UNIVERSITY OF CALIFORNIA BERKELEY
Department of Civil Engineering
Fall 2013
Structural Engineering,
Mechanics and Materials
Professor: S. Govindjee
HW 5: Due October 11
1. In classes I proved suciency in Cauchys Theorem (i.e. if p, then q). Prove necessit
UNIVERSITY OF CALIFORNIA BERKELEY
Department of Civil Engineering
Fall 2013
Structural Engineering,
Mechanics and Materials
Professor: S. Govindjee
HW 7: Due November 1
1. Consider a Maxwell uid subject to an exponentially varying strain (t) = o [1
et ]H
UNIVERSITY OF CALIFORNIA BERKELEY
Department of Civil Engineering
Fall 2013
Structural Engineering,
Mechanics and Materials
Professor: S. Govindjee
HW 8: Due November 8
1. (20pts) Consider material whose volumetric behavior is elastic p = K and whose
devi
UNIVERSITY OF CALIFORNIA BERKELEY
Department of Civil Engineering
Fall 2013
Structural Engineering,
Mechanics and Materials
Professor: S. Govindjee
HW 6: Due October 25
1. Starting from Cijkl = 2Isym + ij kl nd an expression for Poissons ratio in terms of
UNIVERSITY OF CALIFORNIA BERKELEY
Department of Civil Engineering
Fall 2013
Structural Engineering,
Mechanics and Materials
Professor: S. Govindjee
HW 4: Due October 4
tt
1. Consider a thin square sheet which occupies a region [ L , L ] [ L , L ] [ 2 , 2
UNIVERSITY OF CALIFORNIA BERKELEY
Department of Civil Engineering
Fall 2013
Structural Engineering,
Mechanics and Materials
Professor: S. Govindjee
HW 9: Due November 20
1. (10 pts) Consider the following 1-D plasticity model with power-law hardening
= E
UNIVERSITY OF CALIFORNIA AT BERKELEY
Department of Civil and Environmental Engineering
CE231, Fall 2017
Instructor: F. Armero
Chapter 1
Mathematical Preliminaries
1.1. Vectors
1.1.1 Vectors are elements of a vector space V, meaning essentially that we kno