ENG202 Statics Lecture 16, Section 7.1

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1 ENG202 Statics Lecture 16, Section 7.1 Internal Forces Developed in Structural Members - Design of any structural member requires an investigation of the loading acting within the member in order to be sure the material can resist this loading. - These internal loadings can be determined by using method of sections. - Consider the simply supported beam, which is subjected to the forces F 1 and F 2 and support reactions A x, A y and B y. - If internal loadings acting on the cross section at C are to be determined, an imaginary section is passed through the beam, cutting it into two segments. Dr. Ammar T. Al-Sayegh

2 -The internal loadings at the section becomes external on the free-body diagram of each segment. C - Since both segment (AC and CB) were in equilibrium before the beam was sectioned, equilibrium of each segment is maintained provided rectangular force component NC and VC and a resultant couple moment MC are developed at the section. -These loadings must be equal in magnitude and opposite in direction on each of the segments (Newton s third law). Dr. Ammar T. Al-Sayegh 1

3 - Apply the 3 equations of equilibrium to determine magnitudes of loadings. - Σ F x = 0 N C, Σ F y = 0 V C, Σ M C = 0 M C - In mechanics, the force components: - N: acting normal to beam at the cut section = normal force, axial force. - V: acting tangent to the section = shear force. - M: couple moment = bending moment. Dr. Ammar T. Al-Sayegh 2

4 -In 3D, the internal force and couple moment resultant will act at the section. - N y : normal force. - V x, V z : shear force components. - M y : torsional or twisting moment. - M x and M z : bending moment components. - For most applications, these resultants will act at the centroid of the cross-sections. Dr. Ammar T. Al-Sayegh 3

5 Procedure for Analysis 1) Determine the members support reactions so that the equilibrium equations are only used to solve for the internal loadings when the member is sectioned. 2) If member is part of a frame or machine, use techniques learned in 6.6 to determine reactions at its connections. 3) Keep all distributed loadings, couple moments, and forces acting on the member in their exact locations then pass an imaginary section through the member where the internal loading is to be determined. 4) After sectioning draw a FBD of segment that have least number of loads on it, and indicate the x, y, z components of the force and couple moment resultants at the section. 5) If member subjected to coplanar system of forces, only N, V, and M act at the section. 6) In many cases, the proper sense of unknown loadings can be determined by inspection; however, if it seems difficult, sense can be assumed. 7) Moments should be summed about an axes passing through the centroid or geometric center of member s cross-sectional area in order to eliminate the unknown normal and shear forces to obtain direct solutions for moment components. Dr. Ammar T. Al-Sayegh 4

6 Structural Nomenclature for Beams Dr. Ammar T. Al-Sayegh 5

7 Problem 7-6 Determine the internal normal and shear forces and the bending moment in the beat at points C and D. Assume the support at B is a roller. Point C is located just to the right of the 8-kN load. Dr. Ammar T. Al-Sayegh 6