Concepts of Stress and Strain

Transcription

1 Concepts of Stress and Strain One of our principal concerns in this course is material ehavior (Strength). But strength models are often intimately related to stress. Thus, we need to e ale to compute stresses. Stresses, however, cannot e directly measured, ut stain is measurale and can e directly related to stress. Stress nalysis in Perspective (t) Y(t) 1

2 Environment orces on System Dynamic Response of System Compute Nominal Loads in Memers Stress nalysis at ailure Sensitive Points Simple ormulas Stress at Point = My Or E a) Principal Stresses ) Von Mises Stress c) Etc. NORML STRESS normal stress, symolized y the Greek letter sigma, results when a memer is sujected to an axial load applied through the centroid of the cross section. The average normal stress in the memer is otained y dividing the magnitude of the resultant internal force y the cross sectional area. Normal stress is: orce VG rea 2

3 Consider the following free ody diagram of a two-force memer. Inasmuch as the stress acts in a direction perpendicular to the cut surface, it is referred to as a NORML stress. Thus, normal stressed may e either tensile or compressive. Our sign convention for normal stresses is: Tensile stresses are positive (+) Compressive stresses are negative (-) We generally assume the normal stress distriution in an axially loaded memer is uniform, except near the vicinity of the applied load know as Saint Venant s Principle. The assumption of a uniform normal stress distriution is valid if the resultant internal force is applied at the centroid of the cross section. Saint Venant, a rench mathematician, discussed the aove principle for stress distriutions in Note: centric load is one in which the resultant force passes through the centroid of the resisting section. If the resultant passes through the centroid of all resisting sections, the loading is termed XIL. SHER STRESS (Simple or Direct Shear) shear stress, symolized y the Greek letter tau τ, results when a memer is sujected to a force that is parallel or tangent to the surface. The average shear stress in the memer is otained y dividing the magnitude of the resultant shear force V y the cross sectional area. Shear stress is: Shearorce V τ VG rea 3

4 Consider the following example. a) Typical clevis joint ) ree ody diagram of olt c) ree ody of section mnqp d) Shear stresses on section mn It should e emphasized that the distriutions of shear stresses is not uniform across the cross section. Shear stress will e highest near the center of the section and ecome zero at the edge. Direct or simple shear arises in the design of olts, pins, rivets, keys, welds and glued joints. Single Shear Joint Doule Shear Joint 4

5 Punching Shear = punching force = circumference x material thickness Punchingorce τ VG rea BERING STRESS earing stress, symolized y the Greek letter sigma, is a compressive normal stress that occurs on the surface of contact etween two interacting memers. The average normal stress in the memer is otained y dividing the magnitude of the earing force y the area of interest. Bearing stress is orce rea Bolts, pins and rivets create earing stresses along the surface of contact. P P td 5

6 BERING STRESS / SHER TER OUT V τ avg cc 2 ts BERING STRESS - CONTINUED Bearing stresses are also present under the heads of olts and washers. P π 4 6 P 2 2 ( do di )

7 STRESS DISTRIBUTION UNDER XIL LODING (Saint Venant s Principle) s previously mentioned, we generally assume the normal stress distriution in an axially loaded memer is uniform, except near the vicinity of the applied load. Consider the following example. ssuming: 1. Top and ottom plates are rigid and do not rotate 2. Plates allow the memer to expand laterally 3. Centroid of each load is at the center of each plate 4. Memer is homogeneous and isotropic The distriution of stresses is uniform throughout the memer and, at any point, orce VG rea P On the other hand, if the loads are concentrated, the elements in the immediate vicinity of the points of application of the loads are sujected to very large stresses, while other elements near the ends are unaffected y the loading. 7

8 With the use of E or advanced mathematical methods, we can determine the distriution of stresses across various sections of a thin rectangular plate sujected to a concentrated load. s previously shown, stresses near the point of application of concentrated loads are much higher than the average value of stress in the memer (i.e. stress concentration). This is also true for structural memers than contain a discontinuity, such as a hole or sudden change in cross section. Lets consider two common situations, a flat ar with a circular hole and a flat ar with a reduced cross section. The stress concentration factor, K, is defined as follows: max K = avg Or as typically used y engineers: max = K avg ortunately, stress concentration factors are availale in graphs and empirical formulas from a wide variety of sources. In addition, K is dependent only on ratios of geometric parameters and is independent of memer size and material. 8

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