SOLID MECHANICS II (BDA 3033) CHAPTER 2: DEFLECTION OF BEAMS WEEK 2-3 FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
FAZIMAH BT MAT NOOR (UTHM) THE ELASTIC CURVE The deflection diagram of the longitudinal axis that passes through the centroid of each cross sectional area of the beam is called the elastic curve. Deflection diagram = Elastic curve Pin resist force restrict displacement Fixed wall resist moment restrict displacement and rotation FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
THE ELASTIC CURVE (CONT) A positive internal moment tends to bend the beam concave upward. Positive moment – smiley face A negative moment tends to bend the beam concave downward. Negative moment – Sad face If the moment diagram is known, it will be easy to construct the elastic curve. FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
THE ELASTIC CURVE (CONT) Force on beam in fig. (a) & moment diagram in fig. (b). displacement B & D = 0. Negative moment AC = sad curve Positive moment CD = smiley curve C = inflection point, moment change sign (sad smile) = zero moment Change moment slope = zero deflection slope, the beam’s deflection may be a maximum Whether A < E depends on the relative magnitudes of P1 and P2 and the location of the roller at B. FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
Moment-Curvature Relationship given FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
SLOPE AND DISPLACEMENT BY INTEGRATION FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
SLOPE AND DISPLACEMENT BY INTEGRATION (cont.) The elastic curve for beam can be expressed as y=f(x) To obtained this equation, the curvature (1/) must be expressed in terms of y and x. In most calculus book, this relationship is shown as: FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
SLOPE AND DISPLACEMENT BY INTEGRATION (cont.) FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
SLOPE AND DISPLACEMENT BY INTEGRATION (cont.) w = distributed load V = shear load M = moment dy/dx = slope = θ FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
SLOPE AND DISPLACEMENT BY INTEGRATION (cont.) Positive sign convention -w +w FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
Boundary and Continuity Conditions Free end no bend, no shear Unless force applied at free end of the beam (V=applied force) Pin joint wont bend Fixed end no slope FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
FAZIMAH BT MAT NOOR (UTHM) Distributed load Constant distributed load apply at 1/2 length Linear distributed load apply at 2/3 length FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
FAZIMAH BT MAT NOOR (UTHM) EXAMPLES FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
Discontinuity Functions: Macaulay If several different loadings act on a beam integration become more tedious separate loading or moment functions must be written for each region of the beam The beam shown requires four moment functions to be written. They describe the moment in regions AB, BC, CD and DE. When applying the moment-curvature relationship EI d2y/dx2=M, and integrating each moment twice, we must evaluate eight constants of integration. FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
Discontinuity Functions: Macaulay (cont.) If there is discontinuity use Discontinuity functions x = start point a = start of discontinuity Why bracket < > B’cos = function valid for x≥a FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
Discontinuity Functions: Macaulay (cont.) NOT GIVEN Sign convention positive: Forces upward Moment clockwise L@@K!!! Load towards end.. FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
Discontinuity Functions: Macaulay (cont.) FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
Discontinuity Functions: Macaulay (cont.) FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
FAZIMAH BT MAT NOOR (UTHM) EXAMPLES: FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
Statically Indeterminate Beams and Shafts In all of the problems discussed so far, it was possible to determine the forces and stresses in beams by utilizing the equations of equilibrium, that is Fx=0, Fy=0 and MA=0. When the equations of equilibrium are sufficient to determine the forces and stresses in a structural beam, we say that this beam is statically determinate. When the equilibrium equations alone are not sufficient to determine the loads or stresses in a beam, then such beam is referred to as statically indeterminate beam. (the number of unknown reactions exceeds the available number of equilibrium equations. The additional support reactions on the beam or shaft that are not needed to keep it in stable equilibrium are called redundants. The number of these redundants is referred to as the degree of indeterminacy. FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
Statically Indeterminate Beams and Shafts (cont.) Determinacy of Beams - for a coplanar (2D) beam, there are at most three equilibrium equations for each part, so that if there is a total of n parts and r reactions, we have FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
Statically Indeterminate Beams and Shafts (cont.) Example: Classify each of the beams shown as statically determinate or statically indeterminate. If statically indeterminate, report the degrees of determinacy. The beams are subjected to external loadings that are assumed to be known and can act anywhere on the beams. FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
Statically Indeterminate Beams and Shafts (cont.) Statically indeterminate to second degree (5-3=2) FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
Statically Indeterminate Beams and Shafts (cont.) r3 = r6 r4 = r5 r = 6, n = 2; 3n = 3(2) = 6 r = 3n Statically determinate FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
FAZIMAH BT MAT NOOR (UTHM) EXAMPLE FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
FAZIMAH BT MAT NOOR (UTHM) SOLUTION FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
FAZIMAH BT MAT NOOR (UTHM) SOLUTION (cont) FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
FAZIMAH BT MAT NOOR (UTHM) SOLUTION (cont) FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
FAZIMAH BT MAT NOOR (UTHM) EXAMPLE FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
FAZIMAH BT MAT NOOR (UTHM) SOLUTION: FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
FAZIMAH BT MAT NOOR (UTHM) SOLUTION (cont.) FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
FAZIMAH BT MAT NOOR (UTHM) SOLUTION (cont) FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF
FAZIMAH BT MAT NOOR (UTHM) Edited by ABD LATIF