Table 1. Stresses and Deflections in Beams Table 1. (Continued) Stresses and Deflections in Beams
Stresses
Deflections
General Formula for Stress at any Point
Stresses at Critical Points General Formula for Deflection at any Point a
Type of Beam
Deflections at Critical Points a
Case 4. — Supported at Both Ends, Two Symmetrical Loads
Stress at each load, and at all points between, Z Wa −
Between each support and adjacent load, y EI Wx a l a x 6 3 2 = − − ^ h 6 @ Between loads, y EI Wa v l v a 6 3 2 = − − ^ h 6 @ Between each support and adjacent end, y EIL Wu c l u u c u l 24 6 4 2 2 3 = + − − − ^ ^ h h 6 @ Between supports, y EIL Wx l x x l x l c 24 6 2 2 = − − + − ^ ^ h h 6 @
Maximum deflection at center, EI Wa l a 24 3 4 2 2 − ^ h Deflection at loads
Between each support and adjacent load, s Z Wx =− Between loads, s Z Wa =−
W
W
x
x a
v
a
W
W
l
EI Wa l a 6 3 4 2 − ^ h
Case 5. — Both Ends Overhanging Supports Symmetrically, Uniform Load Between each support and adjacent end, s Zl W c u 2 2 = − ^ h Between supports, s ZL W c x l x 2 2 = − − ^ h 6 @ Stress at each support, ZL Wc 2 2 Stress at center, ZL W c l 2 4 1 2 2 − a k If cross section is constant, the greater of these is the maximum stress. If l is greater than 2 c , the stress is zero at points l c 4 1 2 2 − on both sides of the center.
Deflection at ends, EIL Wc c c l l 24 3 2 2 3 + − ^ h 6 @ Deflection at center, EIL Wl l c 384 5 24 2 2 2 − ^ h If l is between 2 c and 2.449 c , there are maximum upward deflections at points l c 3 4 1 2 2 − a k on both sides of the center, which are, EIL W c l 96 6 2 2 2 − − ^ h
Total Load W
c u
x
u c
l L
W 2
W 2
L = 1 + 2 c
If cross section is constant and if l = 2.828 c , the stresses at supports and center are equal and opposite, and are . Z WL 4662 ±
Made with FlippingBook - Share PDF online