Shear Force and Bending Moment

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1 Shear Fore and Bending oent Shear Fore: is the algebrai su of the vertial fores ating to the left or right of a ut setion along the span of the bea Bending oent: is the algebrai su of the oent of the fores to the left or to the right of the setion taken about the setion

2 SFD & BD Sipl Supported Beas P P wl P P wl L L L L V +P/2 V a +P/2 V +P V a +P V -P/2 V a -P/2 a -PL a -PL/2 -wl 2 /2 a PL/4 a PL/8 wl 2 /8

3 Longitudinal strain Longitudinal stress Loation of neutral surfae oent-urvature equation

4 Bending of Beas It is iportant to distinguish between pure bending and non-unifor bending. Pure bending is the deforation of the bea under a onstant bending oent. Therefore, pure bending ours onl in regions of a bea where the shear fore is zero, beause V d/d. Non-unifor bending is deforation in the presene of shear fores, and bending oent hanges along the ais of the bea.

5 What the Bending oent does to the Bea Causes opression on one fae and tension on the other Causes the bea to deflet How uh opressive stress? How uh defletion? How uh tensile stress?

6 How to Calulate the Bending Stress It depends on the bea ross-setion We need soe partiular properties of the setion how big & what shape? is the setion we are using as a bea

7 Pure Bending Pure Bending: Prisati ebers subjeted to equal and opposite ouples ating in the sae longitudinal plane

Setri eber in Pure Bending Internal fores in an ross setion are equivalent to a ouple. The oent of the ouple is the setion bending oent. Fro statis, a ouple onsists of two equal and opposite fores.

8 Setri eber in Pure Bending Internal fores in an ross setion are equivalent to a ouple. The oent of the ouple is the setion bending oent. Fro statis, a ouple onsists of two equal and opposite fores. The su of the oponents of the fores in an diretion is zero. The oent is the sae about an ais perpendiular to the plane of the ouple and zero about an ais ontained in the plane. F z z d 0 d 0 d These requireents a be applied to the sus of the oponents and oents of the statiall indeterinate eleentar internal fores.

Bending Deforations Bea with a plane of setr in pure bending: eber reains setri bends uniforl to for a irular ar ross-setional plane passes through ar enter and reains planar length of top dereases

9 Bending Deforations Bea with a plane of setr in pure bending: eber reains setri bends uniforl to for a irular ar ross-setional plane passes through ar enter and reains planar length of top dereases and length of botto inreases a neutral surfae ust eist that is parallel to the upper and lower surfaes and for whih the length does not hange stresses and strains are negative (opressive) above the neutral plane and positive (tension) below it

10 Strain Due to Bending Consider a bea segent of length L. fter deforation, the length of the neutral surfae reains L. t other setions, L L L δ ε ε ε ( ) θ L ( ) L δ θ L θ or ε θ θ ε -κ θ (strain varies linearl) aiu strain in a ross setion e < 0 shortening opression (>0, k <0) e > 0 elongation tension (<0, k >0)

11 Curvature q q q+dq sall radius of urvature,, iplies large urvature of the bea, κ, and vie versa. In ost ases of interest, the urvature is sall, and we an approiate ds d. q+dq dq

0 d d ( E ) d ( Eκ) d d 0 First oent with respet to neutral plane

12 Stress Due to Bending For a linearl elasti aterial, Eε E Eε (stress varies linearl) aiu stress in a ross setion For stati equilibriu, F 0 E 0 d d ( E ) d ( Eκ) d d 0 First oent with respet to neutral plane (z-ais) is zero. Therefore, the neutral surfae ust pass through the setion entroid.

13 I S I I d d d Substituting 2 d I 2 is the seond oent of area The oent of the resultant of the stresses df about the N..: ( ), 2 EI EI d E E d d κ κ κ κ oent-urvature relationship

14 Deforation of a Bea Under Transverse Loading Relationship between bending oent and urvature for pure bending reains valid for general transverse loadings. 1 κ ( ) EI Cantilever bea subjeted to onentrated load at the free end, 1 P EI t the free end, 1 0, t the support B, 1 B 0, B EI PL

15 Elasti Curve tension: strethed The defletion diagra of the longitudinal ais that passes through the entroid of eah rosssetional area of the bea is alled the elasti urve, whih is haraterized b the defletion and slope along the urve. opression neutral plane elasti urve

16 oent-urvature relationship: Sign onvention aiu urvature ours where the oent agnitude is a aiu.

17 Deforations in a Transverse Cross Setion Deforation due to bending oent is quantified b the urvature of the neutral surfae 1 ε E EI 1 E I lthough ross setional planes reain planar when subjeted to bending oents, in-plane deforations are nonzero, ε νε ν ε νε z ν Epansion above the neutral surfae and ontration below it auses an in-plane urvature, 1 ν antilasti urvature

MECHANICS OF MATERIALS

MECHANICS OF MATERIALS 006 The Graw-Hill Copanies, n. ll rights reserved. Fourth E CHTER ure ECHNCS OF TERLS Ferdinand. Beer E. Russell Johnston, Jr. John T. DeWolf Bending Leture Notes: J. Walt Oler Teas Teh Universit ECHNCS