CM 197 Mechanics of Materials Chap 18: Combined Stresses Professor Joe Greene CSU, CHICO Reference: Statics and Strength of Materials, 2nd ed., Fa-Hwa Cheng, Glencoe/McGraw Hill, Westerville, OH (1997) CM 197 Copyright Joseph Greene 2003 All Rights Reserved

Copyright Joseph Greene 2003 All Rights Reserved Combined Stresses Topics Introduction Combined Axial and Bending Stresses Biaxial Bending Eccentrically Loaded Members Double Eccentricity Stresses on an inclined Plane Copyright Joseph Greene 2003 All Rights Reserved

Copyright Joseph Greene 2003 All Rights Reserved Introduction Previous chapters discussed two types of stresses that are independent. Normal stresses acting on axial portions of a member from a force that causes a pulling (Tension) or pushing action (Compression).  = P/A Bending stresses acting on internal portions of a member (or beam) from forces that are tension on one side and compression on the other side.  = M/S stress = moment / section modulus This chapter combines both of these types of stresses on the same member. Method of superposition is used to solve for the combined stresses Table 18-1 Stresses are determined separately for each type of stress. Stresses are combined in algebraic sums to give combined stresses. Normal stresses caused by Simultaneous action of axial force and bending moment Bending about two perpendicular axis. Eccentric (Off-center) loading Biaxial normal stresses Copyright Joseph Greene 2003 All Rights Reserved

Copyright Joseph Greene 2003 All Rights Reserved Introduction Table 18-1 (Condensed table) Type of Load Type of Stress Formula Eqn Axial load Direct normal stress  = P/A 9-1 Beam loading Flexural stress  = My/I 14-3  = Mc/I 14-2  = M/S 14-7 Direct shear Direct shear stress  = P/A 9-4 Beam shear Beam shear stress  = VQ/It 14-10 Max shear stress max = 1.5 V/A 14-11 in rectangular Xsec Max shear stress max = 4V/3A 14-12 in circular Xsec Copyright Joseph Greene 2003 All Rights Reserved

Combined Axial and Bending Stresses Many structural and machine members are subjected to axial forces and bending moments simultaneously. Both produce normal stresses along longitudinal directions. Normal stresses due to each load can be calculated separately and added algebraically to find the combined stresses. Example 18-1 Example 18-2 Copyright Joseph Greene 2003 All Rights Reserved

Copyright Joseph Greene 2003 All Rights Reserved Biaxial Bending Problems can arise when a load is inclined at an angle with respect to the vertical plane of symmetry. Fig 18-1 Has a beam with load P that can be resolved into two components Px and Py and the angle between them, . Results in Biaxial bending The vertical component, Py, causes bending about the horizontal axis. The horizontal component, Px, causes bending about the vertical axis. The stresses caused by wither types of bending are normal stresses along the longitudinal (length) direction. Method of superposition can be applied. Calculate the bending about each axis separately and the add the results algebraically. Example 18-3 Example 18-4 P Px Py  P Px Py Copyright Joseph Greene 2003 All Rights Reserved

Eccentrically Loaded Members Eccentric loading (off-centered) is special case of combined axial and flexural stresses. Axial load is applied not through the centroid as with normal stress loading, the stresses are NOT distributed uniformly through the cross section. If axial load at E with an eccentricity e from the centroid C (Fig 18-2a), the normal stresses are NOT distributed uniformly in cross section. An eccentric axial load can be replaced by a concentric force and a couple. Fig 18-2 (b and c) Equivalent force couple: place two equal and opposite forces F and D P B D A C E P e B D A C P E e B D A P M=Pe Copyright Joseph Greene 2003 All Rights Reserved

Eccentrically Loaded Members P B D A C E e B D A C E M=Pe Figure 18-2 a, b, c Original downward force P at E and the resulting upward force at D form a couple (moment), M = P e Now have an axial force and a bending moment as in Section 18-2. Axial force produces a uniform compressive stress throughout the section. Bending moment produces maximum compressive stress at A and maximum tensile stress at B. Method of superposition, results in Stress = Axial stress + bending stress Note: stress is + if in tension and stress is – if in compression Stress = - Axial stress (C) – bending stress (C) Stress = - Axial stress (C) + bending stress (T) Stress = Axial stress (T) + bending stress (T) Stress = Axial stress (T) – bending stress (C) Eqn 18-1 Eqn 18-2 Copyright Joseph Greene 2003 All Rights Reserved

Eccentrically Loaded Members Maximum Eccentricity Normal stress at A is always compressive (and negative) Normal stress at B could be compressive, tensile, or zero, depending on the eccentricity (how far off-centered) of the load. Some materials, Concrete, are very weak in resisting tension, they fail and crack. Important in these materials to keep the eccentricity of the compressive load to a maximum limit so that no tensile stress will develop anywhere in the member. Max eccentricity is found by setting Eqn 18-2 = 0. And solve for e. For the width, b, and height, h. The max eccentricity This is the max eccentricity for which tensile stress will not occur anywhere in the member. For e greater than h/6. T he tensile stresses will develop. In general, Compressive load applied along either centroidal axis of a rectangular section must be within the middle third of the section if no tensile stress is to occur. Important for gravitational dams where no tensile stresses are allowed at the base and the resultant force must be acting within the middle third of the base. Eqn 18-3 Copyright Joseph Greene 2003 All Rights Reserved

Copyright Joseph Greene 2003 All Rights Reserved Double Eccentricity Double Copyright Joseph Greene 2003 All Rights Reserved

Stresses on an inclined Plane Copyright Joseph Greene 2003 All Rights Reserved