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Unformatted text preview: EM 319
_ 2nd MIDTERM EXAM W” NAME: SESSION: November 13, 2008, 68 pm, BEL 328 Closed books and notes exam — Formula sheet provided Exercise 1 (20 points) Derive the straincurvature (8—K) relationship for beam bending. Recail that
K =1/p z dQ/dx where p=(Os') and that the line (55) =(s’s’):dx lies on the
neutral axis, i.e. experiences no strain. Present your work clearly, and explain . 'with words evem eguation you write. Aiso, be brief, The proof should not take you more than 56 tines.
Hint: use the definition of strain, and examine the change in length of line (ef) which lies at a distance y from the neutral axis. Exercise 2 [30 points! A steel plate with modulus of elasticity E =3O><1O6 psi and Poisson’s ratio v = 0.3 is loaded in plane stress by normal stresses O”), and 01, and a shear stress rxy.
To determine these stresses, three strain gages are attached to the plate, as in
the figure (A is along the x—axis, B along the yaxis, and C at 1500 from the x—
axis). The readings from the gages are {A = 500x106 = 500m, EB :400,UE and
EC 2 100mm, respectively. Assuming the material is linearly elastic (Le. Hooke's law applies) find:
a) The stresses ox, 0‘y and TW b) The Mohr’s circle for this state of stress.
0) The thickness strain, 52 d) The principal stresses 0'1 and 02, and the principal angle 91,. Sketch the appropriate stress element.
e) The maximum in—plane shear stress, and the corresponding normal stress
GAVE and angle 85. Sketch the appropriate stress element. Note: A completely graphical answer of d) and e) based on the Mohr circle is
accepted and encouraged. Also, the shear modulus G can be found as:
G=E/[2(1+v)] ' r r u—_—m.,mwymnm,z.¢r.~é é) 7gb“ JEW Garage: '5." ' .
Q1 3; ggfi kM‘)
D _ __ Exercise 3 (50 {joints} 3) Determine the beam reactions. Then draw the Free Body Diagram of beam
ABCD onty (Le. without the vertical member AE) E 10 KN b) Draw the shear force and the bending moment distributions of the following
beam. Label the graphs completely. For the non—linear branches of the
graphs, write down whether they are quadratic or cubic etc. Also include the sign convention. (it is suggested that you use the graphical method to draw
the diagrams.) Take RA=15.34 kN and RB=3.86 kN. 1.2 kN/m c) Find the centroid ofthe crosS~section d) Find the moments of inertia about the yc and 2c axes passing through the we: centroid. fmﬁﬁ
K3} ﬁx ‘4 c:
i Ya 6 GEE}?42)(wgﬁﬂilgéﬂ P (A
l” e) Find the maximum tensile and compressive normal stresses that appear in
the following beam (magnitude and location). Noterthat you will have to
somehow calculate Mmax and Mmm. For the inverted—T cross section you may
take that the centroid is located 0.956 in. from the bottom and that'lz =1.657
In ' 29:2?00 lb q: IOlelft (gm   l ,
4 gigo hit , My: CthP‘YQb‘hshtg a. '42.,LIQJ bi E x: E H 69 OCI
(A
ﬁt.
T if 1‘) Find the maximum shear stress that appears (magnitude and location) in the
beam of the previous question. 7 ' Mm “WW MP? O'PfecziCéi} ngﬁbiai ' 23.4.15}; : V: 6043 ...
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