Strength of Materials Malayer University Department of Civil Engineering Taught by: Dr. Ali Reza Bagherieh In The Name of God

Subdivisions of the Mechanics of Materials

Equilibrium in the Interior of the Body Equilibrium of Rotation

Equilibrium of Translation The translation equilibrium, in terms of the forces, which act on the faces of the infinitesimal parallelepiped is verified. These forces are infinitesimal quantities of the second order: for example, the force corresponding to the stress σy is σy dx dz. The body forces acting in the parallelepiped are infinitesimal quantities of the third order: for example, the force corresponding to the body force per unit of volume in the direction x, X, is X dx dy dz.

For the case of no static balance.

Axially Loaded Members A member which has a straight axis, is said to be under purely axial loading if that axis remains a straight line after deformation, Deformation may be caused by a constant axial force or other symmetrical actions, such as a uniform temperature variation. According to this definition and to the law of conservation of plane sections, in a prismatic bar under purely axial loading, any two cross-sections remain parallel after the deformation, i.e., only the distance between them varies.

Dimensioning of Members Under Axial Loading The cross-section area ft must be given dimensions which lead to a nominal value of the acting stress σ Ed that is smaller than the nominal value of the material's resisting stress (allowable stress) σ all. Axial Deformations In the case of a material with linear elastic rheological behaviour: where l and l’ represent the length of the bar, before and after the deformation.

If, in addition to the axial force a uniform temperature variation ΔT occurs, the total elongation may be computed by the expression : where α represents the coefficient of thermal expansion of the material.

Statically Indeterminate Structures

Maximum Normal Stress in Axially Loaded Bars Maximum normal stresses develop on sections perpendicular to the bar axis. For such sections, the cross-sectional area of a bar is a minimum and the force component is a maximum, resulting in a maximum normal stress.

Stresses on Inclined Sections in Axially Loaded Bars

Consider two sections 90 degrees apart perpendicular to the bar sides:

The maximum normal stress (σ max ) in an axially loaded bar can be simply determined from the following equation: To find the maximum shear stress acting in a bar, one must differentiate the above equation with respect to ө, and set the derivative equal to zero.

Shear Stresses Some engineering materials, for example, low-carbon steel, are weaker in shear than in tension, and, at large loads, slip develops along the planes of maximum shear stress. In many routine engineering applications, large shear stresses may develop at critical locations. To determine such stresses precisely is often difficult. However, by assuming that in the plane of a section, a uniformly distributed shear stress develops, a solution can readily be found. the average shear stress σ av is determined by dividing the shear force V in the plane of the section by the corresponding area

Axial Strains and Deformations in Bars

Determine the relative displacement of point D from O for the elastic steel bar of variable cross section shown in the figure caused by the application of concentrated forces P1 = 100 kN and P3 = 200 kN acting to the left, and P2 = 250 kN and P4 = 50 kN acting to the right. The respective areas for bar segments OB, BC, and CD are 1000, 2000, and 1000 mm 2. Let E = 200 GPa.

Determine the deflection of free end B of elastic bar OB caused by its own weight w lb/in. The constant cross-sectional area is A. Assume that E is given.

For the bracket shown in the figure, determine the deflection of point B caused by the applied vertical force P = 3 kips. Also determine the vertical stiffness of the bracket at B. Assume that the members are made of T4 aluminum alloy and that they have constant cross-sectional areas, i.e., neglect the enlargements at the connections.

Poisson's Ratio