Chapter 8 Deflection Structural echanics Dept of rchitecture
Outline Deflection diagras and the elastic curve Elastic-bea theory The double integration ethod oent-area theores Conjugate-bea ethod
8- Deflection diagras and the elastic curve Deflections occur due to Loads Teperature Fabrication errors Settleent Deflections and vibrations ust be liited (controlled) for the cofort of occupants Serviceability requireent
8- Deflection diagras and the elastic curve We concern only linear elastic aterial response at present This eans a structure will return to its original undefored position after the load is reoved It is useful to sketch the defored shape of the structure when it is loaded
8- Deflection diagras and the elastic curve 0 0 Deflection at soe typical supports and joints
8- Deflection diagras and the elastic curve
8- Elastic Bea theory Consider an initially straight bea Due to loading, the bea defors under shear and bending If bea L >> d, greatest deforation will be caused by bending When oent defors the eleent of a bea, the angle between the cross sections becoes d
8- Elastic Bea Theory initially straight bea strain in arc ds located at position y fro the neutral axis: ( ds ' ds) / ds Neutral axis (N..) is the iaginary axis whose length does not change before and after bending. ds ( / d and E ds' y) d d d ( y) d y N.. For linear elastic aterial, Hooke's Law holds true Fro elasticity y / I View in axial direction d Substituting, the governing equation is obtained in which d = curvature of the elastic deforation curve Side view
8- Elastic Bea theory tan tan tan tan v v v v v d v ds v dv ds ds v d ds d v d d dv dv v
8- Elastic Bea theory atheatically, the curvature is expressed as d v / [ ( dv / ) ] / d v / [ ( dv / ) ] / For sall deflection, the expression for curvature becoes dv dv 0 d v ds dv dv This iplies that points on the elastic curve will only be displaced vertically but not horizontally.
8- The Double Integration ethod v d v ( x) dv ( x) ( x) Integral constants are deterined by boundary conditions.
Exaple 8- Deterine the equation of the elastic curve x o d v dv v o ox o x C C x C at x 0, at x 0, v dv 0 C ox : slope v 0 C ox : displaceent
8-4 oent-rea theores Theore The change in slope between any points on the elastic curve equals the area of the / diagra between the points Curvature: d Change in slope between two points: B B /
8-4 oent-rea theores Theore The vertical deviation t /B of the tangent at a point on the elastic curve with respect to the tangent extended fro another point B equals the oent of the area about point under the / diagra between the points t B / B x x t d / B x B xd
8-4 oent-rea theores It is iportant to realize that the oent-area theores can only be used to deterine the angles and deviations between tangents on the bea s elastic curve In general, they do not give a direct solution for the slope or displaceent at a point However, the oent-area theores are a powerful tool giving us a significant insight and a shortcut in deterination of deforation
Exaple 8-5 Deterine the slope at points B and C of the bea where E = 00 GPa and I = 00 4
oent-rea theores Unit: N, 00 50 B 5 5 B B / B 500 5000 500 5000.50.50.750.750 0000000 0.005rad
oent-rea theores Unit: N, 00 50 B 5 5 C C / C 000 0000 50 50 0000000 0.0094 rad
Exaple 8- Deflection at points B and C N N t N t BC B C C B B B 9.0.8.5 0 4 0 00 0 0 500 0 8 0 00 0 5 0 4 0 500 500 4 4 500.5 0 8 0 00 0 4 0 500 4 4 500
Exaple 8-7 Slope at point C Use D 0 0 0 Refer to aniation in CD Ro attached in the textbook.
Exaple 8-8 Slope at point C rad rad kn kn t L t C C B C B C C 0.00 0.00078 0 8. 0.00078 0 0 0 00 0 0 0. 0 0 0 00 0 800 0 9 0 0
Exaple 8-9 Deflection at point C kn t kn t t t t C B C B B C B B C C 4. 4.7 7.5 4.7 0 4 0 00 0.75.5 7.5 0 4 0 00 0 0 5 5.5
Exaple 8-0 Deflection at point C C t C t B 9 7x x x t C 9kN 8 8 0 89 070 000 500 5. 8 8 48 x 7x x 8 x t B 9kN 8 0480 000 500 4 8 C 5. 4 4.
Exaple 8- Slope at point B 0 kn- B C D + 0 kn- B C D = 0 kn- 0 kn- B tc B BC B C D
8-5 Conjugate-Bea ethod dv w d w d d v V w w v
8-5 Conjugate-Bea ethod Here the shear V copares with the slope, the oent copares with the displaceent v and the external load w copares with the / diagra To ake use of this coparison we will now consider a bea having the sae length as the real bea but referred to as the conjugate bea
8-5 Conjugate-Bea ethod Theore : The slope at a point in the real bea is nuerically equal to the shear at the corresponding point in the conjugate bea Theore : The displaceent of a point in the real bea is nuerically equal to the oent at the corresponding point in the conjugate bea Sign convention: Upward for negative oent Downward for positive oent
8-5 Conjugate-Bea ethod pin or roller support at the end of the real bea provides zero displaceent but the bea has a non-zero slope Consequently fro Theore and, the conjugate bea ust be supported by a pin or roller since this support has zero oent but has a shear or end reaction When the real bea is fixed supported, the conjugate bea has a free end since at this end there is zero shear and oent
8-5 Conjugate-Bea ethod
Exaple 8- Deterine the slope & deflection at point B of the steel bea where E = 00 GPa and I = 4750 4
rad 0.00 0 475 00000 0.5 5000 0 00 ' V B B B B 9. 5000 5000 0.00 ' 00 50
Exaple 8- Deterine the ax deflection of the steel bea with E = 00 GPa and I = 00 4 45 x V x x 0 x 5.7 45 x x x x x 0.5kN x.7 0.50 000 00.8 ax x.7
Exaple 8-4 Deflection at center C
Exaple 8-5 Displaceent at pin B and slope of each bea segent about the pin