Virtual Work for Frames. Virtual Work for Frames. Virtual Work for Frames. Virtual Work for Frames. Virtual Work for Frames. Virtual Work for Frames

Size: px

Start display at page:

Download "Virtual Work for Frames. Virtual Work for Frames. Virtual Work for Frames. Virtual Work for Frames. Virtual Work for Frames. Virtual Work for Frames"

Blake Sharp
5 years ago
Views:

1 IL 32 /9 ppling the virtual work equations to a frame struture is as simple as separating the frame into a series of beams and summing the virtual work for eah setion. In addition, when evaluating the deformation of a frame struture, ou ma have to onsider both bending and aial internal fore omponents. ompute the defletion at point on the frame shown below. Inlude onl the effets of bending in our virtual work equation (no aial work). 2 k ft = 29, ksi I = 3,5 in 4 = 35 in 2 he first step is to find the equation for moment in eah setion of the frame due to the real loads. o do develop the moment epression we need the reation a points and. 2 k ft M 2 k(8 ft) Y(6 ft) F 2k Y F he net step is to find the equation for moment in eah setion of the frame. onsider setion M 2 k ft Mut M M onsider setion onsider setion 2 k Mut M 6 k( ) M 8 ft 8 ft M 6 2 k M Mut M 6 k( ) M 6 ft ft

2 IL 32 2/9 onsider setion In this problem, the virtual moments are the real moments divided b 2 (from superposition). 2 k ft M Mut M M 2 k ft Setion M m he virtual work equations are: Mm Mm Mm Mm d d I I I I ompute the slope at point on the frame shown below. Inlude onl the effets of bending in our virtual work equation (no aial work). Substituting the moment epressions into the virtual work equation and integrating ields the following: 8 8 (6 ) (6 ) 2I 2I, 24 kft I I 3 3 3, 24 kft (, 728 in / ft ) 4 (29, ksi)(3,5 in ) 2 I 8 3.7in 2 k ft = 29, psi I = 3,5 in 4 = 35 in 2 Find the moments in the frame due to a virtual ouple. First, find the reation in the frame to the virtual ouple. he net step is to find the equation for moment in eah setion of the frame. onsider setion ft M Y(6 ft) 6 F Y 6 m ft ft Mut m m F 6 6

3 IL 32 3/9 onsider setion onsider setion Mut m 6 ( ) 8 ft m ft 8 ft m 6 mft ut 6 M m ( ) m 6 ft ft onsider setion he following table lists the moment epression due to the real loading and the moment epression due to a virtual ouple at point ft ft m Mut m m 2 k ft Setion M m 6 /6 6 - /6 6 6 he virtual work equations are: Mm Mm Mm Mm d d I I I I Substituting the moment epressions into the virtual work equation and integrating ields the following: (6 ) 8 8 (6 ) 6I 6I he slope at point is zero Repeat the previous eample and inlude the effets of aial work. In order to ompute the aial work, we need the aial fore in the real and virtual loading sstems 2 k ft = 29, psi I = 3,5 in 4 = 35 in 2

4 IL 32 4/9 Find the aial fore in eah setion of the frame. onsider setion onsider setion 2 k F F F 2 k F F F F F ft ft onsider setion onsider setion 2 k F F 2 k F F F F F F ft ft In this problem, the virtual aial fores are the real aial fores divided b 2 (from superposition). 2 k ft Setion N n he virtual work equations for aial fores are: nnl Substituting the values for the aial fores into the virtual work equations ields the following: (.5)( 6 k)(2 in) (.5)( 6 k)(2 in) 72 kin 72 kin (29, ksi)(35 in ) 2.7in

5 IL 32 5/9 he displaement at point due to bending moment work and aial fore work is:.7in.7in.77in ompute the aial fores in the frame due to the virtual ouple. Reall we alread have the frame reations due to the virtual ouple from bending moment work 2 k ft from aial fore work = 29, psi I = 3,5 in 4 = 35 in 2 6 ft 6 = 29, psi I = 3,5 in 4 = 35 in 2 he net step is to find the aial fore in eah setion of the frame. onsider setion onsider setion ft n ft F n 6 n 6 F n ft n ft n onsider setion onsider setion ft F n ft 6 F n n 6 n n n ft ft

6 IL 32 6/9 he real aial fores and the virtual aial fores due to a unit virtual ouple are: 2 k ft Setion N n - -/6 - /6 he virtual work equations for aial fores are: nnl Substituting the values for the aial fores into the virtual work equations ields the following: (6 k)(2 in) (6 k)(2 in) 6 6 he ontribution to the slope at point from the aial energ is slope is zero. he total slope at point due to bending moment and aial fore work is zero. In problems involving both bending and aial deformation, be areful with the units. lso, note that the ontribution of the aial deformation is 5% of the total deformation. his is more or less tpial of the relative size of the bending and the aial effets in frame-defletion problems. herefore, it is usuall permissible to neglet the effet of aial deformation in suh ases. he primar ause of deformation in beams and frames is due to bending strain. However, in some strutures additional deformation due to aial and shear fores, torsion, and perhaps temperature ma be important. We have alread disussed deformation due to bending moments and aial fores. In this setion, we will onsider the effet of shear, torsion, and temperature on the deformation of linear elasti strutures. irtual Strain nerg From Shear onsider the following beam and a small element irtual Strain nerg From Shear he shearing deformation d aused b the real loads is d =, where is the shear strain. w = w() l d Sine we are assuming the material is linear and elasti, then Hooke s law applies he shear strain is related to the shear stress b =/G, where is the shear stress and G is the shearing modulus of elastiit. d

7 IL 32 7/9 irtual Strain nerg From Shear he shear stress ma be alulated as = K(/), where K is a form fator that depends of the shape of the beam s ross-setional area. ombining these two epressions gives d = K/(G). he internal virtual work done b the virtual shear fore v ating on the beam before it is deformed b the real loads is du i = v d d irtual Strain nerg From Shear Integrating the epression du i = v d over the entire beam gives: L v Ushear K G Remember that v is the shear due to the virtual load and is the shear due to the real loads. d irtual Strain nerg From Shear ompute the vertial defletion and rotation at point on the frame shown. k Integrating the epression du i = v d over the entire beam gives: L v Ushear K G he form fator K is based on the l ross-setional area: K =.2 for retangular setions K = /9 for irular setions K for I-beams, where is the area of the web d Inlude the effets of bending moment and both aial and shear fores in our virtual work equations. = 29, ksi G = 2, ksi I =, in 4 = 25 in 2 K =.2.5 k/ft ft 2 ft irtual Strain nerg From orsion For eample, onsider a irular ross-setion where no wrapping of the setion ours. For non-irular setions a more rigorous analsis is required. irtual Strain nerg From orsion his torque auses a shear strain: For a linear-elasti response: G J d

8 IL 32 8/9 irtual Strain nerg From orsion he angle of twist: d G GJ irtual Strain nerg From orsion If a virtual load is applied to the struture that auses an internal virtual torque t, then after appling the real loads, virtual strain energ will be: t dut t d GJ irtual Strain nerg From orsion Integrating the virtual strain over the length of the member ields: U t tl GJ irtual Strain nerg From emperature onsider a struture member is subjeted to a temperature differene aross its depth. For disussion, we will hoose the most ommon ase of a beam having a neutral ais loated at the mid-depth of the beam First ompute the amount of rotation of a differential element of the beam aused b the thermal gradient. irtual Strain nerg From emperature irtual Strain nerg From emperature m m d M m If a virtual load is applied to the struture that auses an internal virtual torque m, then after appling the real loads, virtual strain energ will be: du temp L m m 2

9 IL 32 9/9 nd of irtual Work - Frames Unless otherwise stated, in this ourse we will onsider onl beam and frame defletions due to bending. he additional defletions aused b the shear and aial fore alter the defletions b onl a few perent and are generall ignored for even small two- and three-member frames of one-stor height. n questions?

TORSION By Prof. Ahmed Amer

TORSION By Prof. Ahmed Amer ORSION By Prof. Ahmed Amer orque wisting moments or torques are fores ating through distane so as to promote rotation. Example Using a wrenh to tighten a nut in a bolt. If the bolt, wrenh and fore are