Roark's Formulas: Table 8.1-2

Shear, Moment, Slope, and Deflection Formulas for Elastic Straight Beams.

TOC

Source: Roark's Formulas for Stress and Strain, 9th edition, Table 8.1, pp. 204-206, supplied PDF excerpt. Apparent source errors are identified beside the affected cases.

2. Partial distributed load

The beam has constant $E$ and $I$. Coordinate $x$ runs from the left end $A$ to the right end $B$, with $0\leq x\leq l$. Reactions and deflections are positive upward; slope is positive upward to the right; positive moment compresses the upper fibers. A guided end can translate vertically but cannot rotate.

Macaulay brackets satisfy ${\langle x-a\rangle}^n=0$ for $x<a$ and ${(x-a)}^n$ for $x>a$; the zeroth power is a unit step. Both one-sided limits are included when finding extrema at a jump. Cases with $a=0$ or $a=l$ are interpreted as limits from within the span.

The load varies linearly from $w_a$ to $w_l$ over $a\leq x\leq l$. The displayed general formulas assume $a<l$; at $a=l$ the loaded interval vanishes and the calculator returns zero response.

	Transverse shear: $V=R_A-w_a\langle x-a\rangle-\frac{w_l-w_a}{2(l-a)}{\langle x-a\rangle}^2$
	Bending moment: $M=M_A+R_Ax-\frac{w_a}{2}{\langle x-a\rangle}^2-\frac{w_l-w_a}{6(l-a)}{\langle x-a\rangle}^3$
	Slope: $\theta=\theta_A+\frac{M_Ax}{EI}+\frac{R_Ax^2}{2EI}-\frac{w_a}{6EI}{\langle x-a\rangle}^3-\frac{w_l-w_a}{24EI(l-a)}{\langle x-a\rangle}^4$
	Deflection: $y=y_A+\theta_Ax+\frac{M_Ax^2}{2EI}+\frac{R_Ax^3}{6EI}-\frac{w_a}{24EI}{\langle x-a\rangle}^4-\frac{w_l-w_a}{120EI(l-a)}{\langle x-a\rangle}^5$

Selected maximum formulas below assume the positive load, couple, imposed displacement, or thermal curvature shown. Signed values are retained. The current-input tables report the minimum, maximum, and greatest absolute value over the span; for negative inputs the signs reverse. A reported position is one location attaining the extremum. The formulas labeled "maximum possible" vary $a$ while holding the other inputs fixed.

2a. Left end free, right end fixed (cantilever)

Details

Inputs

$l =$ 1000 mm	Beam length, positive
$I =$ 1000000 mm$^4$	Second moment of area about the centroidal bending axis, positive
$E =$ 200000 MPa	Modulus of elasticity, positive
$a =$ 500 mm	Position from the left end, $0\leq a\leq l$
$w_a =$ 1 N/mm	Downward load intensity at $x=a$
$w_l =$ 1 N/mm	Downward load intensity at $x=l$

Boundary values

Left end A	Right end B
$R_A=0 =$ 0 N	$R_B=\frac{w_a+w_l}{2}(l-a) =$ 500 N
$M_A=0 =$ 0 N$\cdot$mm	$M_B=-\frac{w_a}{2}{(l-a)}^2-\frac{w_l-w_a}{6}{(l-a)}^2 =$ -125000 N$\cdot$mm
$\theta_A=\frac{w_a}{6EI}{(l-a)}^3+\frac{w_l-w_a}{24EI}{(l-a)}^3 =$ 0.000104167 rad	$\theta_B=0 =$ 0 rad
$y_A=-\frac{w_a}{24EI}{(l-a)}^3(3l+a)-\frac{w_l-w_a}{120EI}{(l-a)}^3(4l+a) =$ -0.0911458 mm	$y_B=0 =$ 0 mm

Maximum moments and deformations

Current inputs; positions are measured from $A$. Values at an imposed jump are one-sided limits.

Quantity	Minimum (signed)	Maximum (signed)	Greatest absolute value (signed)
Moment	-125000 N$\cdot$mm at $x=$ 1000 mm	0 N$\cdot$mm at $x=$ 0 mm	-125000 N$\cdot$mm at $x=$ 1000 mm
Slope	-2.77556e-20 rad at $x=$ 1000 mm	0.000104167 rad at $x=$ 0 mm	0.000104167 rad at $x=$ 0 mm
Deflection	-0.0911458 mm at $x=$ 0 mm	1.38778e-17 mm at $x=$ 1000 mm	-0.0911458 mm at $x=$ 0 mm

For full-span loading ($a=0$):

Loading	Maximum moment	Maximum slope	Maximum deflection
Uniform, $w_a=w_l=w$	$M_B=-wl^2/2$	$\theta_A=wl^3/(6EI)$	$y_A=-wl^4/(8EI)$
Increasing, $w_a=0$	$M_B=-w_ll^2/6$	$\theta_A=w_ll^3/(24EI)$	$y_A=-w_ll^4/(30EI)$
Decreasing, $w_l=0$	$M_B=-w_al^2/3$	$\theta_A=w_al^3/(8EI)$	$y_A=-11w_al^4/(120EI)$

Source correction: the scanned reaction for 2a appears to use $(w_a-w_l)/2$; the resultant is $(w_a+w_l)(l-a)/2$.

2b. Left end guided, right end fixed

Details

Inputs

$l =$ 1000 mm	Beam length, positive
$I =$ 1000000 mm$^4$	Second moment of area about the centroidal bending axis, positive
$E =$ 200000 MPa	Modulus of elasticity, positive
$a =$ 500 mm	Position from the left end, $0\leq a\leq l$
$w_a =$ 1 N/mm	Downward load intensity at $x=a$
$w_l =$ 1 N/mm	Downward load intensity at $x=l$

Boundary values

Left end A	Right end B
$R_A=0 =$ 0 N	$R_B=\frac{w_a+w_l}{2}(l-a) =$ 500 N
$M_A=\frac{w_a}{6l}{(l-a)}^3+\frac{w_l-w_a}{24l}{(l-a)}^3 =$ 20833.3 N$\cdot$mm	$M_B=-\frac{w_a}{6l}{(l-a)}^2(2l+a)-\frac{w_l-w_a}{24l}{(l-a)}^2(3l+a) =$ -104167 N$\cdot$mm
$\theta_A=0 =$ 0 rad	$\theta_B=0 =$ 0 rad
$y_A=-\frac{w_a}{24EI}{(l-a)}^3(l+a)-\frac{w_l-w_a}{240EI}{(l-a)}^3(3l+2a) =$ -0.0390625 mm	$y_B=0 =$ 0 mm

Maximum moments and deformations

Current inputs; positions are measured from $A$. Values at an imposed jump are one-sided limits.

Quantity	Minimum (signed)	Maximum (signed)	Greatest absolute value (signed)
Moment	-104167 N$\cdot$mm at $x=$ 1000 mm	20833.3 N$\cdot$mm at $x=$ 0 mm	-104167 N$\cdot$mm at $x=$ 1000 mm
Slope	-6.93889e-20 rad at $x=$ 1000 mm	0.0000662586 rad at $x=$ 704.124 mm	0.0000662586 rad at $x=$ 704.124 mm
Deflection	-0.0390625 mm at $x=$ 0 mm	-6.93889e-18 mm at $x=$ 1000 mm	-0.0390625 mm at $x=$ 0 mm

For full-span loading ($a=0$):

Loading	Maximum negative moment	Maximum positive moment	Maximum deflection
Uniform, $w_a=w_l=w$	$M_B=-wl^2/3$	$M_A=wl^2/6$	$y_A=-wl^4/(24EI)$
Increasing, $w_a=0$	$M_B=-w_ll^2/8$	$M_A=w_ll^2/24$	$y_A=-w_ll^4/(80EI)$
Decreasing, $w_l=0$	$M_B=-5w_al^2/24$	$M_A=w_al^2/8$	$y_A=-7w_al^4/(240EI)$

2c. Left end simply supported, right end fixed

Details

Inputs

$l =$ 1000 mm	Beam length, positive
$I =$ 1000000 mm$^4$	Second moment of area about the centroidal bending axis, positive
$E =$ 200000 MPa	Modulus of elasticity, positive
$a =$ 500 mm	Position from the left end, $0\leq a\leq l$
$w_a =$ 1 N/mm	Downward load intensity at $x=a$
$w_l =$ 1 N/mm	Downward load intensity at $x=l$

Boundary values

Left end A	Right end B
$R_A=\frac{w_a}{8l^3}{(l-a)}^3(3l+a)+\frac{w_l-w_a}{40l^3}{(l-a)}^3(4l+a) =$ 54.6875 N	$R_B=\frac{w_a+w_l}{2}(l-a)-R_A =$ 445.313 N
$M_A=0 =$ 0 N$\cdot$mm	$M_B=R_Al-\frac{w_a}{2}{(l-a)}^2-\frac{w_l-w_a}{6}{(l-a)}^2 =$ -70312.5 N$\cdot$mm
$\theta_A=-\frac{w_a}{48EIl}{(l-a)}^3(l+3a)-\frac{w_l-w_a}{240EIl}{(l-a)}^3(2l+3a) =$ -0.0000325521 rad	$\theta_B=0 =$ 0 rad
$y_A=0 =$ 0 mm	$y_B=0 =$ 0 mm

Maximum moments and deformations

Current inputs; positions are measured from $A$. Values at an imposed jump are one-sided limits.

Quantity	Minimum (signed)	Maximum (signed)	Greatest absolute value (signed)
Moment	-70312.5 N$\cdot$mm at $x=$ 1000 mm	28839.1 N$\cdot$mm at $x=$ 554.688 mm	-70312.5 N$\cdot$mm at $x=$ 1000 mm
Slope	-0.0000325521 rad at $x=$ 0 mm	0.000032464 rad at $x=$ 794.85 mm	-0.0000325521 rad at $x=$ 0 mm
Deflection	-0.0105892 mm at $x=$ 487.95 mm	7.45931e-17 mm at $x=$ 1000 mm	-0.0105892 mm at $x=$ 487.95 mm

For full-span loading ($a=0$):

Loading	Reactions and negative moment	Positive moment	Slope and deflection
Uniform, $w_a=w_l=w$	$R_A=3wl/8$, $R_B=5wl/8$, $M_B=-wl^2/8$	$9wl^2/128$ at $x=3l/8$	$\theta_A=-wl^3/(48EI)$; $y_{\max}=-0.0054wl^4/EI$ at $x=0.4215l$
Increasing, $w_a=0$	$R_A=w_ll/10$, $R_B=2w_ll/5$, $M_B=-w_ll^2/15$	$0.02981w_ll^2$ at $x=0.4472l$	$\theta_A=-w_ll^3/(120EI)$; $y_{\max}=-0.00239w_ll^4/EI$ at $x\approx0.4472l$
Decreasing, $w_l=0$	$R_A=11w_al/40$, $R_B=9w_al/40$, $M_B=-7w_al^2/120$	$0.0422w_al^2$ at $x=0.329l$	$\theta_A=-w_al^3/(80EI)$; $y_{\max}=-0.00304w_al^4/EI$ at $x=0.4025l$

Source corrections: the increasing-load positive-moment coefficient is $0.02981$ (the scan prints $0.298$); its left slope is negative. The decreasing-load moment uses $w_a$, not $w_l$.

2d. Left end fixed, right end fixed

Details

Inputs

$l =$ 1000 mm	Beam length, positive
$I =$ 1000000 mm$^4$	Second moment of area about the centroidal bending axis, positive
$E =$ 200000 MPa	Modulus of elasticity, positive
$a =$ 500 mm	Position from the left end, $0\leq a\leq l$
$w_a =$ 1 N/mm	Downward load intensity at $x=a$
$w_l =$ 1 N/mm	Downward load intensity at $x=l$

Boundary values

Left end A	Right end B
$R_A=\frac{w_a}{2l^3}{(l-a)}^3(l+a)+\frac{w_l-w_a}{20l^3}{(l-a)}^3(3l+2a) =$ 93.75 N	$R_B=\frac{w_a+w_l}{2}(l-a)-R_A =$ 406.25 N
$M_A=-\frac{w_a}{12l^2}{(l-a)}^3(l+3a)-\frac{w_l-w_a}{60l^2}{(l-a)}^3(2l+3a) =$ -26041.7 N$\cdot$mm	$M_B=R_Al+M_A-\frac{w_a}{2}{(l-a)}^2-\frac{w_l-w_a}{6}{(l-a)}^2 =$ -57291.7 N$\cdot$mm
$\theta_A=0 =$ 0 rad	$\theta_B=0 =$ 0 rad
$y_A=0 =$ 0 mm	$y_B=0 =$ 0 mm

Maximum moments and deformations

Current inputs; positions are measured from $A$. Values at an imposed jump are one-sided limits.

Quantity	Minimum (signed)	Maximum (signed)	Greatest absolute value (signed)
Moment	-57291.7 N$\cdot$mm at $x=$ 1000 mm	25227.9 N$\cdot$mm at $x=$ 593.75 mm	-57291.7 N$\cdot$mm at $x=$ 1000 mm
Slope	-0.0000180845 rad at $x=$ 277.778 mm	0.0000235177 rad at $x=$ 818.374 mm	0.0000235177 rad at $x=$ 818.374 mm
Deflection	-0.00670003 mm at $x=$ 556.721 mm	2.25514e-17 mm at $x=$ 1000 mm	-0.00670003 mm at $x=$ 556.721 mm

For full-span loading ($a=0$):

• Uniform, $w_a=w_l=w$: $M_A=M_B=-wl^2/12$; maximum positive moment $wl^2/24$ at $x=l/2$; $y_{\max}=-wl^4/(384EI)$ at $x=l/2$.
• Increasing, $w_a=0$: $R_A=3w_ll/20$, $R_B=7w_ll/20$, $M_A=-w_ll^2/30$, $M_B=-w_ll^2/20$; maximum positive moment $0.0215w_ll^2$ at $x=0.548l$; $y_{\max}=-0.001309w_ll^4/EI$ at $x=0.525l$.
• A full-span decreasing load is the reflection of the increasing-load case.

2e. Left end simply supported, right end simply supported

Details

Inputs

$l =$ 1000 mm	Beam length, positive
$I =$ 1000000 mm$^4$	Second moment of area about the centroidal bending axis, positive
$E =$ 200000 MPa	Modulus of elasticity, positive
$a =$ 500 mm	Position from the left end, $0\leq a\leq l$
$w_a =$ 1 N/mm	Downward load intensity at $x=a$
$w_l =$ 1 N/mm	Downward load intensity at $x=l$

Boundary values

Left end A	Right end B
$R_A=\frac{w_a}{2l}{(l-a)}^2+\frac{w_l-w_a}{6l}{(l-a)}^2 =$ 125 N	$R_B=\frac{w_a+w_l}{2}(l-a)-R_A =$ 375 N
$M_A=0 =$ 0 N$\cdot$mm	$M_B=0 =$ 0 N$\cdot$mm
$\theta_A=-\frac{w_a}{24EIl}{(l-a)}^2(l^2+2al-a^2)-\frac{w_l-w_a}{360EIl}{(l-a)}^2(7l^2+6al-3a^2) =$ -0.0000911458 rad	$\theta_B=\frac{w_a}{24EIl}{(l^2-a^2)}^2+\frac{w_l-w_a}{360EIl}{(l-a)}^2(8l^2+9al+3a^2) =$ 0.000117187 rad
$y_A=0 =$ 0 mm	$y_B=0 =$ 0 mm

Maximum moments and deformations

Current inputs; positions are measured from $A$. Values at an imposed jump are one-sided limits.

Quantity	Minimum (signed)	Maximum (signed)	Greatest absolute value (signed)
Moment	0 N$\cdot$mm at $x=$ 0 mm	70312.5 N$\cdot$mm at $x=$ 625 mm	70312.5 N$\cdot$mm at $x=$ 625 mm
Slope	-0.0000911458 rad at $x=$ 0 mm	0.000117187 rad at $x=$ 1000 mm	0.000117187 rad at $x=$ 1000 mm
Deflection	-0.0328168 mm at $x=$ 540.222 mm	0 mm at $x=$ 0 mm	-0.0328168 mm at $x=$ 540.222 mm

For full-span loading ($a=0$):

• Uniform, $w_a=w_l=w$: $R_A=R_B=wl/2$; $M_{\max}=wl^2/8$ at $x=l/2$; $\theta_B=-\theta_A=wl^3/(24EI)$; $y_{\max}=-5wl^4/(384EI)$ at $x=l/2$.
• Increasing, $w_a=0$: $R_A=w_ll/6$, $R_B=w_ll/3$; $M_{\max}=0.0641w_ll^2$ at $x=0.5773l$; $\theta_A=-7w_ll^3/(360EI)$, $\theta_B=w_ll^3/(45EI)$; $y_{\max}=-0.00653w_ll^4/EI$ at $x=0.5195l$.
• A full-span decreasing load is the reflection of the increasing-load case.

2f. Left end guided, right end simply supported

Details

Inputs

$l =$ 1000 mm	Beam length, positive
$I =$ 1000000 mm$^4$	Second moment of area about the centroidal bending axis, positive
$E =$ 200000 MPa	Modulus of elasticity, positive
$a =$ 500 mm	Position from the left end, $0\leq a\leq l$
$w_a =$ 1 N/mm	Downward load intensity at $x=a$
$w_l =$ 1 N/mm	Downward load intensity at $x=l$

Boundary values

Left end A	Right end B
$R_A=0 =$ 0 N	$R_B=\frac{w_a+w_l}{2}(l-a) =$ 500 N
$M_A=\frac{w_a}{2}{(l-a)}^2+\frac{w_l-w_a}{6}{(l-a)}^2 =$ 125000 N$\cdot$mm	$M_B=0 =$ 0 N$\cdot$mm
$\theta_A=0 =$ 0 rad	$\theta_B=\frac{w_a}{6EI}{(l-a)}^2(2l+a)+\frac{w_l-w_a}{24EI}{(l-a)}^2(3l+a) =$ 0.000520833 rad
$y_A=-\frac{w_a}{24EI}{(l-a)}^2(5l^2+2al-a^2)-\frac{w_l-w_a}{120EI}{(l-a)}^2(9l^2+2al-a^2) =$ -0.299479 mm	$y_B=0 =$ 0 mm

Maximum moments and deformations

Current inputs; positions are measured from $A$. Values at an imposed jump are one-sided limits.

Quantity	Minimum (signed)	Maximum (signed)	Greatest absolute value (signed)
Moment	0 N$\cdot$mm at $x=$ 1000 mm	125000 N$\cdot$mm at $x=$ 0 mm	125000 N$\cdot$mm at $x=$ 0 mm
Slope	0 rad at $x=$ 0 mm	0.000520833 rad at $x=$ 1000 mm	0.000520833 rad at $x=$ 1000 mm
Deflection	-0.299479 mm at $x=$ 0 mm	0 mm at $x=$ 1000 mm	-0.299479 mm at $x=$ 0 mm

For full-span loading ($a=0$):

Loading	Maximum moment	Maximum slope	Maximum deflection
Uniform, $w_a=w_l=w$	$M_A=wl^2/2$	$\theta_B=wl^3/(3EI)$	$y_A=-5wl^4/(24EI)$
Increasing, $w_a=0$	$M_A=w_ll^2/6$	$\theta_B=w_ll^3/(8EI)$	$y_A=-3w_ll^4/(40EI)$
Decreasing, $w_l=0$	$M_A=w_al^2/3$	$\theta_B=5w_al^3/(24EI)$	$y_A=-2w_al^4/(15EI)$

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