Statically indeterminate beams

Statically indeterminate beams Example: a beam fixed at one end while also simply supported at the other end (propped cantilever).

Redundant support(s) ( more unknown reactions than equations of statics (usually two: F = 0 and M = 0).

But the extra support(s) provide extra boundary conditions ( exploit them to solve the indeterminate problem.

Solution Procedure:

1. Write down (two) static equilibrium equations for (>two) unknown reactions.2. Use direct method to get M(x; unknown reactions) from the given loading w(x). (Ch. 4, 5)3. By two integrations, solve EIy = M(x) for y(x; unknown reactions). (Ch. 9)4. Determine integration constants (may involve unknown reactions) using the usual boundary conditions (e.g. y = y = 0 at clamped end). (Ch. 9)5. Obtain additional equation(s) involving the unknown reactions, based on boundary conditions at extra support (e.g. y = 0 at pinned end)6. Solve the full set of equations from steps (1) and (5) for all unknown reactions. If needed, substitute results back into y(x) obtained in step (3) and determine deflection, slope, bending moment, etc.Example 10-1:

Note: structure is stable even if support B is removed. 3 unknowns.1. Statics (RA + RB = qL

(Eq. 1)

qL2/2 RBL = MA

(Eq. 2)

(reaction moment at A assumed to be anti-clockwise)

2. V(x) = RA qx

( M(x) = MA + RA x qx2/2

3. EIy = M(x) (()dx (EIy = MAx + RAx2/2 qx3/6 + c1()6 (6EIy = 6MAx + 3RAx2 qx3 + C1(()dx (6EIy = 3MAx2+RAx3qx4/4+C1x+C2(...)4 (24EIy = 12MAx2+4RAx3qx4+c1x+c24. yx=0 = 0(c2 = 0; yx=0 = 0(c1 = 05. At redundant support B: yx=L = 0( -12MAL2 + 4RAL3 qL4 = 0

()L2 ( -12MA + 4RAL qL2 = 0 (Eq. 3)

6. Solve Eq.s (1),(2),(3) for the three unknown reactions RA, RB, MA( MA = qL2/8,

RA = 5qL/8, RB = 3qL/8

Read: Example 10-2** To keep you focused on the big picture (Steps 1-6) and not detour too much into peripheral yet tedious calculations, symbolic computation will be used in the following examples.

Example 10-3:

Example 10-4:

Example 10-5:(same beam as in example 10-4, but load P is replaced by uniform load q from x = 0 to x = a)

Method of Superposition for Statically Indeterminate Beams

More Examples (Ch. 10 Problems)

10.3-4

Homework 10.3-9

Check the deflection picture against

the loading on the beam:

Homework 10.3-10

Homework 10.4-7

Homework 10.4-8

Homework 10.4-22

Summary: examples covered & homework for Ch. 10:

TopicCovered in Tutorial Suggested Homework

Deflection by Direct IntegrationExamples 10-1, 3, 4, 5

10.3-4, 9, 10

10.4-7, 8, 2210.3-3, 6, 810.4-16, 21

=

+

B = 0

B)1

= EMBED Equation.3

B)2

EMBED Equation.3

EMBED Equation.3

Use Table G-2, Cases 1 & 4

Find (i) all reactions and y(x) for half beam AC; (ii) V, M for whole beam

B = ?

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