Reference Problems - STRENGTH OF MATERIALS
1. There are three columns whose cross-sectional area is A0 and whose natural length attemperature T0 is L0. These three columns are bonded to the rigid plates at the top
and bottom ends, as shown in Fig. 1. The column located in the center of the rigid
plate has a longitudinal modulus of elasticity of E1 and a linear expansion coefficientof α1, while the two columns located at distance C on both sides of the central column
have a longitudinal modulus of elasticity of E2 and a linear expansion coefficient ofα2. Here, 0 < E1 < E2 and 0 < α1 < α2. Ignore the self-weight of the columns and rigidplates. Answer the following questions.
(1) At temperature T0, an object of weight W0 is placed on the center of the upper
rigid plate, as shown in Fig. 2. Find the strain in the columns. Also, find the stressacting on each column.
(2) After the object of weight W0 is removed from the state in the question (1), thetemperature is increased by ΔT from T0. Find the strain in the columns. Also, findthe stress acting on each column.
(3) The object of weight W0 is placed on the center of the upper rigid plate as in thequestion (1), and the temperature is increased by ΔT from T0. Find the stressacting on each column.
BｰBCC
Upper rigid plate

BW0B

E1 , α1Lower rigid plate

E2 , α2Fig. 1

Fig. 2

2.

Answer the following questions.
(1) The plane stresses (x＝200 MPa，y＝0 MPa，xy＝100 MPa) are given as shown inFig. 3. Draw the Mohr's stress circle corresponding to this stress state. Find the magnitude
and direction of the maximum and minimum principal stresses.
(2) The yield strength of a homogeneous isotropic material was evaluated by a uniaxial
tensile test, and the material yielded when the tensile stress reached 250 MPa. Determine
whether or not this material will yield when subjected to the plane stress as shown in Fig.
3, and the reason should also be given. Here, the yield will occur when the maximum
shear stress reaches the material-specific limit value.

xy=100 MPax=200 MPay(y=0 MPa)xFig. 3

3. Consider a stepped solid circular shaft shown in Fig. 4. The diameter and length ofsection AB are d1 and L/3, and those of section BD are d2 and 2L/3, respectively. d1
is smaller than d2. The shaft is fixed at the both ends, A and D, to vertical rigid walls.
The shear modulus of the shaft material is G. As shown in Fig. 1, the shaft is twistedby the torsional moment Mt at position C, midpoint of section BD. Answer thefollowing questions. Describe also the processes of deriving the answers. Ignore
stress concentration.
(1) Find the torsional moments developed at the ends of the shaft, position A andposition D, respectively.
(2) Find the angle of twist c at position C.
(3) Determine the ratio of torsional stress acting on section AB and torsional stressacting on section CD.

Mt

d1

A

d2

CBDFig. 4

4. A 3L-long beam is supported at two points A and B at a distance L from both ends ofthe beam, as shown in Fig. 5. A concentrated load W1 is applied at the left end of thebeam and a concentrated load W2 is applied at the right end, where 2W 2 > W１ > W２.
Answer the following questions. Neglect the weight of the beam. Describe also theprocesses of deriving the answers.
(1) Determine the reaction forces acting at position A and position B, respectively.
(2) Draw the shear force diagram.
(3) Draw the bending moment diagram.

W1

W2
ALB

L
Fig. 5L5. There is a solid circular shaft with diameter, d, and total length, 4L, both ends ofwhich are fixed to the wall. The shaft is made of a brittle material and its shear
modulus of elasticity is G. A torsional moment, Mt, is acting on the shaft at theposition L from its left end, and a torsional moment twice as large (i.e., 2Mt ) is acting
in the same direction on the shaft at the position L from its right end, Figure 6. Here,
Mt is a variable. It is known that if magnitude of Mt is gradually increased and when
Mt reaches Mt0, the shaft fails with exhibiting fracture surface characteristics of brittlematerials. Answer the following questions.

(1) Find the torsional moment of each part of the shaft and draw a graph in the formof Figure 7.
(2) Answer the direction in which the fracture surface is expected to be formed,
together with the reason.
(3) Assuming that failure occurs when the maximum principal stress of the shaft
reaches the tensile strength of the material, TS, find the relationship between TS

Mt

torsional momentand Mt0.

2Mt

dLAL

2LBC

Fig. 60DAB

PositionFig. 7CD