ࡱ> OQN_}5@ 2bjbj22 (fXX*"D'rR : !;'='='='='='='$(R+a'" ""a' v'e%e%e%"v  ;'e%";'e%Fe%&& f 'kqa#&;''0'&+[$+&4"+&*!Z!@e%!4!*!*!*!a'a'DZD%^ZInternal Forces  HYPERLINK \l "Vertical" Forces from Vertical Loads  HYPERLINK \l "Horizontal" Forces from Horizontal Loads  HYPERLINK \l "Total" Total Axial Forces The internal forces in the columns are found using the reactions, the loads and the principles of equilibrium. The simplest internal forces are the axial ones, which result from the vertical loads and reactions. They reach a maximum at the base of the Tower. Forces from Vertical Loads The axial force in the columns is the inclined force, N, shown acting along the axis of one column. This force is equal to the combination of a vertical and a horizontal component. The gravity loads and reactions are vertical and create the vertical component; the horizontal component develops because the vertical forces are carried in an inclined column, i.e., for the columns to be in equilibrium there must be a horizontal component of the axial force.  These three forces create a force polygon where, drawn head to tail, the scaled components are equal to the axial force and three form a closed shape. The magnitudes of the horizontal components and the axial force can be found from the vertical force and the angle of the column axis with trigonometry. The chart shown in the geometry section gives the slope of the columns at the base as EMBED Equation.DSMT4 . The simple trigonometric relationship  EMBED Equation.DSMT4  can be rearranged to find N:  EMBED Equation.DSMT4  This is the axial force that is developed in both columns at their bases. The horizontal components at the base can be found from the trigonometric formula:  EMBED Equation.DSMT4   These horizontal forces tend to push the Tower apart but are resisted by the connection all along the Tower and the ground. This can be visualized if one imagines a ladder propped against a building. If there was no friction between the ground and ladder base, or if the base was on wheels, the ladder would slide away from the wall.  In the same manner, the columns of the Tower would slide apart if there were no connections between them. These connections, which have been idealized as continuous, experience tension force equal to the horizontal forces they are resisting.  At the base, the ground will resist with horizontal reactions of 3700 kips acting inward.  The axial force will decrease with height as the vertical load and angle of inclination decrease. As an example, the axial force at the second platform will be found. The entire top and middle parts are above this point, therefore, the vertical load is:  EMBED Equation.DSMT4   The chart given in the Geometry section shows that at this height of 380 feet, the slope of the elements is  EMBED Equation.DSMT4 . The trigonometric formula:  EMBED Equation.DSMT4  is used to find:  EMBED Equation.DSMT4  The value at the base is almost seven times this value! The compression forces from the dead load will be identical in both of the columns because they are each taking half of the vertical load. (The cross-sectional area could be decreased at the higher point because the smaller compression force would create a small compressive stress. This is indeed what Eiffel did, as shown in the earlier figure; this is neglected in the simplification.) The horizontal component of the axial force at the second platform is:  EMBED Equation.DSMT4   This force will be resisted by the connection made by the second platform itself.  Forces from Horizontal Loads The internal forces created by the overall bending moment are investigated next. The bending moment from the wind load can be found at any point if the entire structure is treated as a single unit rather than as two columns. The bending moment is found around the tower axis, whereas the axial forces were found along the columns axes. The overall bending moment from the horizontal wind load will produce tension (T) in one column and compression (C) in the other. Exaggerated action under wind load is shown:  The values of the bending moment all along the height of the Tower must be known to find the resulting tension and compression forces in the columns. A bending moment diagram illustrates the continuous values. The horizontal force creating the bending moment is the wind, simplified as a point load acting at the mid-point of the area under consideration.  At a point a distance x from the bottom as shown in the diagram below, the bending moment is found using  EMBED Equation.DSMT4 as the magnitude of the point load and  EMBED Equation.DSMT4  as the distance.  The equation for the moment at any point along the Tower uses these two values:  EMBED Equation.DSMT4  For the Eiffel Tower, l = 984 feet and p = 2.6 k/ft; the reaction calculation was made where x = 0 to find M = 1,260,000 ft-k.  The reaction formula used earlier, EMBED Equation.DSMT4 , is a simplified version of this formula where x = 0 and where  EMBED Equation.DSMT4 . The graph of the equation  EMBED Equation.DSMT4 with changing values of x is a diagram of the continuous bending moment along the Tower. The shape is also a parabola, only the magnitude is different.  The shape of the Eiffel Tower roughly imitates the shape of this diagram; the idealization imitates it exactly, but the actual tower is more sharply curved as shown earlier. At any point along the height of the Tower, the overall bending moment, M, and the width of the structure, d, are used to determine the value of both the tension (T) and compression (C) forces (which are equal) from the formula:  EMBED Equation.DSMT4  This formula was justified in the Monument analysis where it was shown that the tension or compression forces result from the bending moment acting over the depth (or width here) of the structure.  This calculation uses the bending moments found around the tower axis to find axial forces that act along the column axes.  At the base of the Tower the moment force is M = 1,260,000 ft-k (the reaction force), and the width, d, is 328 feet. Therefore,  EMBED Equation.DSMT4  where T = +3,850 k and C = -3,850 k because compression forces are designated as negative. Both of these forces are vertical reactions to the wind.  The axial forces in the inclined elements resulting from the wind load are found from the vertical reactions and the horizontal reactions to the wind found earlier.  On the leeward side, where the wind creates a vertical compression reaction, Vc and H contribute part of their forces along the axis of the column. The compression force, Vc, contributes a component of  EMBED Equation.DSMT4  to the axial force, Nc and H contributes  EMBED Equation.DSMT4  to Nc. Nc can be found from Vc, H and the angle at the base of the column,  EMBED Equation.DSMT4 . Negative signs are used because the axial force is compressive.  EMBED Equation.DSMT4  The windward side will have a positive value of N because Vt and H create axial tension:  EMBED Equation.DSMT4   The axial forces produced by the horizontal load of the wind are  EMBED Equation.DSMT4 .  To further illustrate this idea, the axial forces produced by the wind load at the second platform will be investigated. The horizontal and bending moment reactions from the wind on the top and middle parts of the Tower are found with a free-body diagram cut at the second platform.   EMBED Equation.DSMT4  where  EMBED Equation.DSMT4   EMBED Equation.DSMT4  The chart shows that d = 123 ft at the second platform, so the tension and compression are:  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Hm is found as  EMBED Equation.DSMT4  in each column. With these values Nc and Nt can be found using the formulas introduced earlier:  EMBED Equation.DSMT4   The axial forces and the vertical components are nearly equal. This illustrates that at a small angle these calculations hardly need be performed, especially considering the simplicity of this analysis. Generally at an angle less than EMBED Equation.DSMT4 , one can consider the axial force and vertical force equal. The vertical forces, Vt and Vc, at this height are roughly equal to the vertical tension and compression found at the base; they are all roughly 4,000 kips. This shows that if the shape of a structure imitates the shape of its bending moment diagram, the vertical forces created by the bending moment will be constant all along its height. The width of the base is equally capable of resisting the bending moment at the base as the width at the second platform is capable of resisting the bending moment at that point. Total Axial Forces Now that the axial forces from both vertical and horizontal loads have been calculated, the total axial forces in the column supports are found by simply adding the axial force found from the two loads. Compression forces are negative and tension forces are positive. This is shown in the first diagram, below. The second diagram shows the corresponding forces at the second platform.   At this point (the second platform) the windward column experiences tension forces rather than the compression found near the base. This creates no structural problems because wrought iron resists tension just as well as compression. Next, the total axial forces are used to find the internal stress. First, however, a quick investigation of the shear force in the columns further illustrates the efficient shape of the Tower. Since the shape of the Tower nearly imitates the shape of the moment diagram, and since the vertical compression force has been found to be constant all along the Towers height, there should, theoretically, be no shear force in the column from the horizontal wind load. This is investigated using the vertical compression and horizontal forces found earlier. (See the diagram at the top of the next page.) Where  EMBED Equation.DSMT4  and EMBED Equation.DSMT4 , the total shear force is found by the formula:  EMBED Equation.DSMT4    EMBED Equation.DSMT4  This result verifies the concept that, from the wind load, the parabolic shape of the Tower will allow only uniform vertical forces, and axial forces that decrease with the angle of the columns.  HYPERLINK "ET_Introduction.htm" Introduction  HYPERLINK "ET_Geometry.htm" Geometry  HYPERLINK "ET_Loads.htm" Loads  HYPERLINK "ET_Reactions.htm" Reactions  HYPERLINK "ET_Internal%20Forces.htm" Internal Forces  HYPERLINK "ET_Internal%20Stresses.htm" Stresses  HYPERLINK "ET_Efficiency.htm" Efficiency  HYPERLINK "ET_Explore.htm" Explore  HYPERLINK "http://www.ce.jhu.edu/perspectives/studies.htm" Structural Studies Home ./0JKUVrstŰs[sCs/jh1h1:B* CJUaJph3/jsh1h1:B* CJUaJph3'h1h10J:>*B* CJaJph3/jh1h1:B* CJUaJph3 h1h1:B* CJaJph3)jh1h1:B* CJUaJph3h1h1B* ph3h1hp&h:hp:B* CJP\]aJPph3&h:h::B* CJP\]aJPph3 0909090909Y 0909090909Y 090909090909l0909}09090909!09090909]$a$gd?c$a$gd1$a$gd:$a$gd:2 # $   F H I ` a b c ˴˘|oj9h~h~EHUjgB h~CJUVaJjJ6h~h~EHUjMB h~CJUVaJj=3h~h~EHUj+B h~CJUVaJjh~Uh~h=h?cjWh?cUh~hp6 h1h1:B* CJaJph3hph1(CDLMNOPQhijkmnopzӻӷӯӷӧӜyqj&h?cUjh1h1EHUjB h1CJUVaJ hnLh~jhnLh~Ujh?cUj)rh?cUh=jRh?cUh1h?cjAh?cUhph~jh~Uj]=h1h1EHUjB h1CJUVaJ*KLNOOPlmop23D090909090909 09090909R09090909q09090909097090909e0909$a$gd:$a$gd?c$a$gd:./01EF]^_`klôӟӟpc_W_jh?cUh?cjhG%h1EHUj-B h1CJUVaJj?h1h1EHUj#B h1CJUVaJhG%jhG%Ujh~h~EHUjB h~CJUVaJh~jh~Uh1hpjh=h=EHUj YB h=CJUVaJh=jh=U DEabjk  qrt090909090909090909090909` 09090909090909$a$gd?c$a$gd:$a$gd:2R   rstu"#:;<=>PQ󾺫󾺏zkj=B hG%CJUVaJj1ah?cUj]hG%hG%EHUjB hG%CJUVaJjZhG%hG%EHUjB hG%CJUVaJhG%jhG%UjeBh?cUj$h?cUh= h1h1:B* CJaJph3h1h?chpj& h?cU'tuOPRSABDEFbcHIef0909Q 090909090909A 09090909d090909W090909'090909090909}09$a$gd:$a$gd?c$a$gd: -.BCDFhi/0⿲枮枮sfjqhohoEHUjB hoCJUVaJjhohoEHUjԼB hoCJUVaJjhoUhohp6hojihG%hG%EHUjB hG%CJUVaJh?cjh?cUhG%hp6hG%hpjhG%Uj߁hG%hG%EHU(0noIJabcd,-./  0 1 2 I J K L M T U Z [ ^ _ k l o p ɼء{jhnLhoEHU#joB hnLhoCJUVaJ hnLhojhnLhoUj!h?cUjh?cUjhohoEHUj9B hoCJUVaJhojhoUhohp6h?cjh?cUh1hp-f+,./0 1 M N !!!!f#g#####09090909090909309090909\09090909yN090909090909 0909 090909$a$gd )8$a$gd:$a$gd?c$a$gd: !!!!!!!!!!!!<"=">"?"["\"s"t"u"v""""""""""""""""""""""ŻߤŻ߳߈{vv ho6j <hohoEHUjB hoCJUVaJj8hohoEHUjоB hoCJUVaJjhoUhoho6H*hoho6hohp6H*hohp6hoh )8j4h )8Uhph?cjh?cU-"""""# # #!#"###g#h#######################@$A$X$ƹժՇzrnh )8jKh )8UjGhkhkEHUjؿB hkCJUVaJ hk6H*jNBhkhkEHUjYB hkCJUVaJjA?hkhkEHUjCB hkCJUVaJhkjhkUhkhp6hphohk6H*hohk6%######_$`$a$c$d$%%%%%%%%%%%%?&@&\&]&09 0909'09090920909090909090909F)090909 09090909909090909}09$a$gd )8$a$gd:$a$gd:X$Y$Z$[$^$a$b$d$%%%%%%%%%%%%%%%%%%%%%%%%%%@&A&ۿ׿ۿۿۿxkۿc[jhmUhmhp6jhkhkEHUjB hkCJUVaJjhkhkEHUjB hkCJUVaJj0hkhkEHUjyB hkCJUVaJhkj1?1Y1Z1[1`1a1f1g111111111պՈmR4jh/H hDB* CJ$OJQJUaJ$ph34jh/H hDB* CJ$OJQJUaJ$ph34j(h/H hDB* CJ$OJQJUaJ$ph3,h/H hD0J>*B* CJ$OJQJaJ$ph34j`h/H hDB* CJ$OJQJUaJ$ph3%h/H hDB* CJ$OJQJaJ$ph3.jh/H hDB* CJ$OJQJUaJ$ph3111111111122 22-2.2/292:2?2@2\2]2^2e2f2h2ͶͣͣͶͣͣmͶͣͣRͶͣ4jȸh/H hDB* CJ$OJQJUaJ$ph34jh/H hDB* CJ$OJQJUaJ$ph34j2h/H hDB* CJ$OJQJUaJ$ph3%h/H hDB* CJ$OJQJaJ$ph3,h/H hD0J>*B* CJ$OJQJaJ$ph3.jh/H hDB* CJ$OJQJUaJ$ph34jdh/H hDB* CJ$OJQJUaJ$ph3h2i2222222պ%h,whDB* CJ$OJQJaJ$ph3,hU~h,w0J>*B* CJ$OJQJaJ$ph34jhU~h,wB* CJ$OJQJUaJ$ph3%hU~h,wB* CJ$OJQJaJ$ph3.jhU~h,wB* CJ$OJQJUaJ$ph3&1h:pD/ =!"#$%sDyK  VerticalwDyK  HorizontalmDyK Total1Dd`Z,>  C AbT1oR gx{*01n(1oR gx{*PNG  IHDRs4,gAMA|Q pHYs+0IDATx͏8pj4 kE͹1>:|?2A9GqZ[$!CNU*@!۰,G>%.;v?%9$So:y;:o?rOڣsE|!r8ʎ",ξv ȉFY{V#L 3g=*&˜wv z=*5$ #~o⹅(>IA߰ܘ+2s&Ҫ=9XrRcMetd︡A[[B'p<p`R-mμ=Z: O lF{ cG@aN5d7UsZUHIsZzFdz5gyN)=sȞ)B0.2h^$ǰ QȆonC,z>7=F<؃+?@'4֞̿ vz6XԸfF+3U/<y='gL㹾zoNglmr*>3)l"#q}L@w!?[J$WAP io[HStzXPH|Z1=n`V$a=*kq=znx5鹻8\(8SV)\o#{X[_N+=ף{0+vTZso=#Cm!X۽XфDAA'1ɶ.Gsc=<"EX=y=c42O#KZC5К?,UvtO^o9 yK~' u܆lO{i@uy4abK K}z}<`jHKD/"#chV[JFpLZ["#\g~` Zyq^{0U{x8>r0ax^n?n1~OA7pس`hR4=8uP'~-zL0U紐T)Xҿ>On__>Obk8|iSOYe)Qt{h4qY<zLYH.,@TTkH흵}v=; xЃfuXbB6SX|mz\ztt3=}!lZ=7Rq02xW@0$zCut,fΓ 2 =|T)򠱫Lvy.{aU<-+( tNrugj/Iq Z^g%_8FaRΈQ|$@W pW9ӯǩdIwzO#.$ ǽ'WmnK`S1z˫4c}C;<7mH@v+({=&?lCu6p)xt囝vc6qa=\^Kx-a@jG>C|Iu)z^oBY\&כhI 6,==Ga{dfgn$ L}c}#tu<1fyFpш+i{yh=P1 V<d[DW@[B|`1Cn 2U <Z:bm# 1O}Pz>$ ]_pEaI>WM`$=<4gImcEzdbwHqdgP5dȜMG<:!)ԷGBNcͬ3KzxOTB5EnClrnEG%ڬnwZ,{*Oc~@nerb==5񬚞)YHܿ==Գ5rP' i0hSi^5g)g:i|u.ýOH=pO/y6vpOsAɸ4wiΩy,w!Sz3Gtz%zTmwI[=e%;;$\n={2lߦ嗷VK[ur;$Xf{{꾧8'8j[zzz;$Xyk=ŦkGPڒ'B">ѕ? 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