Engineering Materials

posted by .

Using mechanics of materials principles (ie. equations of mechanical equilibrium applied to a free-body diagram), derive the following equations.

sigma' = sigma cos^2 θ = sigma(1+cos 2θ / 2)

and tau = sigma sin θ cos θ = sigma( sin2θ / 2)

  • Engineering Materials -

    The equations given would apply to a particular case of plane stress condition where σx and σy are the principal stresses, thus the shear stress τxy is zero. Furthermore, σx equals zero, i.e. it is a case of uniaxial tensile or compressive stress.

    The derivation of the general formula for the general case is presented in the classic work "Theory of Elasticity" by Timoshenko and Goodier (chapter 2 sec. 9) and is the basis of the Mohr's circle. Therefore, the same derivation will be found in many sources including web-pages.
    Here's one place you may look:
    http://www.efunda.com/formulae/solid_mechanics/mat_mechanics/plane_stress.cfm#transform

    The Mohr's circle is a related subject that may eventually interest you, if not for the time being.

Respond to this Question

First Name
School Subject
Your Answer

Similar Questions

  1. further mathematics

    The roots of the eqn, x^4 + px^3 + qx^2 + rx + s = 0 where p, q, r, s are constants and s does not equal to 0, are a, b, c, d. (i) a^2 + b^2 + c^2 + d^2 = p^2 -2q (in terms of p & q) (ii) 1/a + 1/b + 1/c + 1/d = -r/s (in terms of r …
  2. Math - Calculus

    The identity below is significant because it relates 3 different kinds of products: a cross product and a dot product of 2 vectors on the left side, and the product of 2 real numbers on the right side. Prove the identity below. | a …
  3. engineering

    Using mechanics of materials principles, (i.e. equations of mechanical equilibrium applied to a free-body diagram), derive the following equations: s’ = s cos2q = s[(1+ cos 2q) / 2] t' = s sinq cosq = s [sin 2q / 2] s=sigma, q=theta, …
  4. Trigonometry

    Which of the following is equivalent to 1 − cos 2 θ/cos 2 θ?
  5. Calculus AP

    hi again im really need help TextBook: James Stewart:Essential Calculus, page 311. Here the problem #27: First make a substitution and then use integration by parts to evaluate the integral. Integral from sqrt(pi/2) TO sqrt(pi)of θ^3 …
  6. math

    I just need help with this. Write the series in sigma notation. 1/4 + 1/2+ 3/4 + 1 + 5/4 + 3/2 5 A.) 1/4 sigma N n=1 6 B.) 1/4 sigma n n=1 5 C.)sigma n/n+3 n=1 6 D.)sigma n/n+3 n=1
  7. math

    I just need help with this. Write the series in sigma notation. 1/4 + 1/2+ 3/4 + 1 + 5/4 + 3/2 5 A.) 1/4 sigma N n=1 6 B.) 1/4 sigma n n=1 5 C.)sigma n/n+3 n=1 6 D.)sigma n/n+3 n=1
  8. Trigonometry

    There are four complex fourth roots to the number 4−4√3i. These can be expressed in polar form as z1=r1(cosθ1+isinθ1) z2=r2(cosθ2+isinθ2) z3=r3(cosθ3+isinθ3) z4=r4(cosθ4+isinθ4), …
  9. Calculus-antiderivative

    Find f. f ''(θ) = sin θ + cos θ, f(0) = 5, f '(0) = 3 My steps: f'(θ)=cosθ-sinθ+C When f'(0)=3, C=-2, so f'(θ)=cosθ-sinθ-2. f(θ)=-sinθ-cosθ-2x+D When f(0)=5, D=6, so f is …
  10. PRECALC

    solve the equation 1. cos(θ) − sin(θ) = 1 2.2 cos(θ) tan(θ) + tan(θ) = 1 + 2 cos(θ) 3. sin(θ) cos(3θ) + cos(θ) sin(3θ) = 0 4. sin(2θ) cos(θ) − cos(2θ) sin(θ) …

More Similar Questions