7.4:
Internal Loadings in Structural Members: Problem Solving
Consider a beam OC of 5 kilonewton, inclined at an angle and supported at both ends. Determine the internal loadings at point A.
The joints supporting the beam experience reaction forces. By using the moment equation about point O and substituting the values, the reaction force at point C can be determined.
The weight of the beam acts at the center and can be resolved in its components.
Now, consider an imaginary line passing through point A, dividing the beam into two sections. Next, draw a free-body diagram of the segment with the minimum unknown forces.
By recalling the equilibrium equation and substituting the values of forces along the horizontal direction, the normal force in the section can be determined.
The negative sign indicates the opposite direction of the normal force acting on the cross-section.
Similarly, by using the equilibrium equation and substituting the values of the vertical forces, the shear force in the section can be determined.
Finally, using the moment equation, the magnitude of the moment at point A can be determined.
7.4:
Internal Loadings in Structural Members: Problem Solving
When designing or analyzing a structural member, it is important to consider the internal loadings developed within the member. These internal loadings include normal force, shear force, and bending moment. Engineers can ensure that the structural member can support the applied external forces by calculating these internal loadings.
To illustrate this, let's consider a beam OC of 5 kN, inclined at an angle of 53.13° with the horizontal and supported at both ends. Determine the internal loadings at point A.
For this, first, we need to calculate the reaction force at point C. We can do this by using the moment equation about point O and substituting the values.
This gives us the necessary information to determine the reaction force at point C in the horizontal direction, which is 1.87 kN.
Next, the weight of the beam, which is assumed to be acting at the center, is resolved into its components. Once we have calculated the reaction force at point C and resolved the weight of the beam, we can move on to determining the internal loadings at point A.
To accomplish this, consider an imaginary line passing through point A, dividing the beam into two sections. Then, a free-body diagram of the segment with the minimum unknown forces is drawn. By recalling the equilibrium equation and substituting the values of forces along the horizontal direction, the normal force in the section is obtained to be -5.12 kN. The negative sign indicates that the normal force on the cross-section is opposite to the direction assumed.
Similarly, by using the equilibrium equation and substituting the values of the vertical forces, we can determine the shear force in the section, which is 1.5 kN. Finally, using the moment equation, we can determine the magnitude of the moment at point A, which is obtained to be 8.94 kNᐧm.
By following these steps, the internal loadings at point A of the structural member can be determined. This problem-solving process is crucial for ensuring structural members can withstand the forces they are designed to support.
Suggested Reading
- Hibbeler, R.C. (2016) Engineering Mechanics: Statics and Dynamics Hoboken, New Jersey: Pearson Prentice Hall, pp 345, 355.