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Solid Mechanics Flashcards

1
Q

How do you relate the deformation at a point given a force when the system is statically indeterminate?

A
  1. Use the strain energy equation to find U
  2. Draw the relationship that U=W
  3. W=(P*d)/2
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2
Q

What is the equation for the parallel axis theorem?

A

I=I_c + ad^2

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3
Q

What is the equation to find the neutral axis?

A

Y=summation(y_i*A_i)/Summation(A_i)

y_i*A_i=Q_i

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4
Q

How can you ensure a load will induce strictly bending with no twisting on a beam?

A

Apply the load through the shear center

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5
Q

How do you locate the shear center on a:

  1. Doubly Symmetric Shape
  2. Singly Symmetric Shape
A
  1. Centroid

2. On the axis of symmetry

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6
Q

How do you find the centroid of a composite beam?

A

E_1y_1A_1+E_2y_2A_2=0

where y are the distance from the ends to the neutral axis

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7
Q

How do you calculate the Yield moment of a beam? What is the yield moment of a beam?

A

M_y=sigma_y*I/y

The moment at which a beam no longer behaves linearly elastic and instead begins perfectly plastic deformation

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8
Q

How do you calculate the Plastic moment?

What is the plastic moment?

A

M_p=[sigma_yA(y1+y2)]/2

The maximum moment that can be sustained by a beam of elastoplastic material

To find y, break the cross-sectional area into two and y is the distance from the centroid of the divided face to the actual centroid

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9
Q

What is the difference between the stress and strain transformation equations?

A

The shear strain (gamma) takes the place of Tau except it is gamma/2. Epsilon takes the place of sigma normally

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10
Q

What is the Strain energy equation for linear deflection and angular deflection?

A
  1. U=P*d/2

2. U=M*Theta/2

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11
Q

What are the two variations of the Modified Castigliano’s Theorem?

A
  1. d_i=int((M/EI)*(dM/dP) dx)

2. theta_i=int((M/EI)*(dM/dM_0) dx)

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12
Q

What is the equation for the required section modulus of a beam under bending?

A

S=M_max/Sigma_allow

S=I/c where c is the maximum distance away

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13
Q

What is the theoretical limit for the section modulus under bending?

A

S=0.5Ah

A is the total area of both flanges
h is the distance between the center of the two flanges

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14
Q

What is an example of the superiority of a closed-section rectangular beam over an I beam?

A

Increase torsional resistance

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15
Q

How do you calculate Iyz?

A

Summation of Adydz

For each block
Where dy and dx are the distance from the individual block centroids to the complete centroid

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16
Q

How do you find the angle of the neutral axis?

A

Lambda=tantan^-1(I_yz)/I_z

The bottom MOI is perpendicular to the axis in which you find the angle

17
Q

How do you calculate the stress of a curved beam?

A

Sigma=(N/A)+(M/Ar)(A-rA_m)/(RA_m-A)

where A_m=b*ln(C/A)

b= Width of beam
A= Length from Center of the curve to the bottom portion
C= Length from Center of the curve to the top portion
18
Q

What is the governing equation for the deflection of a beam on an elastic foundation? Also, what is the equation for the slope?

A

EI d^4w/dx^4+kw=q

Theta=dw/dx

k is the modulus of rigidity of the foundation
q is the distributed load on the surface

19
Q

What is the equation for Beta for beams on elastic foundations?

A

Beta=[k/4EI]^1/4

20
Q

Draw the graph that described the relationship between a finite length beam on an elastic foundation end and the center as a function of Beta and L

A

Draw it

21
Q

What is the equation for the curvature of a beam?

A

K=M/EI

22
Q

How do you calculate the shear center of an open face with one axis of symmetry?

A

Note: You know the shear center will be on the axis of symmetry

Goal: You want to determine the eccentric distance away from the cross-section to ensure the torque caused by the shear load is equal and opposite to the torque caused by the shear flow around the cross-section.

  1. e=Vh/I

v is the shear force
h is the distance between flanges

  1. Find V as the integral of q along the entire top length. Integral ds from 0 to b

q=VQ/I where Q equal st(h/2) where s is the distance from the end of the channel

  1. Find the total MOI of both flanges and web
  2. Combine together
23
Q

What is the equation for finding the normal stress of a composite beam?

A

Sigma1=Mh1E1/(E1I1+E2I2)

Simga2=Mh2E2/(E2I2+E1I1)

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