Statically Indeterminate

1. B A C T LAC LCB L Statically Indeterminate Objective: To determine the internal torque and the associated shear stress for statically indeterminate problems. Find, • Reaction at A and B. • Internal torque at segment AC and CB. • Stress at AC and CB.

2. A B TA TB T TB – T + TA = 0 ---------------- A TAC TA TB TCB TCB – T + TAC = 0 ---------------- B TAC = TA TCB = TB Equation A is of better form… Statically Indeterminate Equation of equilibrium. Where TA and TB are reactions at supports. Equation of equilibrium.

3. TACLAC + TCBLCB = 0 ---------------- C Solving equations B & C for the internal torque (TAC and TCB) and then find reactions and stress. Compatibility Equations:

4. Example: The solid steel shaft as shown in figure has a diameter of 20 mm. If it is subject to the two torques, determine the reactions at the fixed support A and B.

5. Solution: Equation of equilibrium.

6. Compatibility:

7. Example: The shaft as shown in figure is made from a steel tube, which is bonded to a brass core. If a torque of T = 250 N·m is applied at its end, plot the shear-stress distribution along a radial line of its cross-sectional area. Take Gst = 80 GPa, Gbr = 36 GPa.

8. Solution: Equilibrium. Compatibility.

9. Solving Eqs. 1 and 2, we get

11. The enD