Roark's Formulas: Table 8.1-5

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. 210-211, supplied PDF excerpt. Apparent source errors are identified beside the affected cases.

5. Intermediate externally created lateral displacement

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.

	Transverse shear: $V=R_A$
	Bending moment: $M=M_A+R_Ax$
	Slope: $\theta=\theta_A+\frac{M_Ax}{EI}+\frac{R_Ax^2}{2EI}$
	Deflection: $y=y_A+\theta_Ax+\frac{M_Ax^2}{2EI}+\frac{R_Ax^3}{6EI}+\Delta_0\langle x-a\rangle^0$

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.

5a. 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, $0\leq a\leq l$
$\Delta_0 =$ 1 mm	Imposed upward displacement jump

Boundary values

Left end A	Right end B
$R_A=0 =$ 0 N	$R_B=0 =$ 0 N
$M_A=0 =$ 0 N$\cdot$mm	$M_B=0 =$ 0 N$\cdot$mm
$\theta_A=0 =$ 0 rad	$\theta_B=0 =$ 0 rad
$y_A=-\Delta_0 =$ -1 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	0 N$\cdot$mm at $x=$ 0 mm	0 N$\cdot$mm at $x=$ 0 mm
Slope	0 rad at $x=$ 0 mm	0 rad at $x=$ 0 mm	0 rad at $x=$ 0 mm
Deflection	-1 mm at $x=$ 0 mm	0 mm at $x=$ 500 mm	-1 mm at $x=$ 0 mm

• $y_{\max}=y_A=-\Delta_0$ for $x<a$; $y=0$ for $x>a$.
• Moment and slope are zero away from the imposed displacement.

5b. 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, $0\leq a\leq l$
$\Delta_0 =$ 1 mm	Imposed upward displacement jump

Boundary values

Left end A	Right end B
$R_A=0 =$ 0 N	$R_B=0 =$ 0 N
$M_A=0 =$ 0 N$\cdot$mm	$M_B=0 =$ 0 N$\cdot$mm
$\theta_A=0 =$ 0 rad	$\theta_B=0 =$ 0 rad
$y_A=-\Delta_0 =$ -1 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	0 N$\cdot$mm at $x=$ 0 mm	0 N$\cdot$mm at $x=$ 0 mm
Slope	0 rad at $x=$ 0 mm	0 rad at $x=$ 0 mm	0 rad at $x=$ 0 mm
Deflection	-1 mm at $x=$ 0 mm	0 mm at $x=$ 500 mm	-1 mm at $x=$ 0 mm

• $y_{\max}=y_A=-\Delta_0$ for $x<a$; $y=0$ for $x>a$.
• Moment and slope are zero away from the imposed displacement.

5c. 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, $0\leq a\leq l$
$\Delta_0 =$ 1 mm	Imposed upward displacement jump

Boundary values

Left end A	Right end B
$R_A=\frac{3EI\Delta_0}{l^3} =$ 600 N	$R_B=-R_A =$ -600 N
$M_A=0 =$ 0 N$\cdot$mm	$M_B=\frac{3EI\Delta_0}{l^2} =$ 600000 N$\cdot$mm
$\theta_A=-\frac{3\Delta_0}{2l} =$ -0.0015 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	0 N$\cdot$mm at $x=$ 0 mm	600000 N$\cdot$mm at $x=$ 1000 mm	600000 N$\cdot$mm at $x=$ 1000 mm
Slope	-0.0015 rad at $x=$ 0 mm	0 rad at $x=$ 1000 mm	-0.0015 rad at $x=$ 0 mm
Deflection	-0.6875 mm at $x=$ 500 mm	0.3125 mm at $x=$ 500 mm	-0.6875 mm at $x=$ 500 mm

• $M_{\max}=M_B$; maximum slope magnitude is $\theta_A$.
• Maximum positive deflection $\frac{\Delta_0}{2l^3}(2l^3+a^3-3l^2a)$ just right of $x=a$; maximum possible value $\Delta_0$ at $a=0$.
• Maximum negative deflection $-\frac{\Delta_0a}{2l^3}(3l^2-a^2)$ just left of $x=a$; maximum possible negative magnitude has signed value $-\Delta_0$ at $a=l$.

5d. 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, $0\leq a\leq l$
$\Delta_0 =$ 1 mm	Imposed upward displacement jump

Boundary values

Left end A	Right end B
$R_A=\frac{12EI\Delta_0}{l^3} =$ 2400 N	$R_B=-R_A =$ -2400 N
$M_A=-\frac{6EI\Delta_0}{l^2} =$ -1200000 N$\cdot$mm	$M_B=-M_A =$ 1200000 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	-1200000 N$\cdot$mm at $x=$ 0 mm	1200000 N$\cdot$mm at $x=$ 1000 mm	1200000 N$\cdot$mm at $x=$ 1000 mm
Slope	-0.0015 rad at $x=$ 500 mm	0 rad at $x=$ 0 mm	-0.0015 rad at $x=$ 500 mm
Deflection	-0.5 mm at $x=$ 500 mm	0.5 mm at $x=$ 500 mm	0.5 mm at $x=$ 500 mm

• Maximum positive moment $M_B$; maximum negative moment $M_A$.
• Maximum slope magnitude has signed value $-3\Delta_0/(2l)$ at $x=l/2$.
• Maximum positive deflection $\frac{\Delta_0}{l^3}(l^3+2a^3-3a^2l)$ just right of $x=a$; maximum possible value $\Delta_0$ at $a=0$.
• Maximum negative deflection $-\frac{\Delta_0a^2}{l^3}(3l-2a)$ just left of $x=a$; maximum possible negative magnitude has signed value $-\Delta_0$ at $a=l$.

5e. 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, $0\leq a\leq l$
$\Delta_0 =$ 1 mm	Imposed upward displacement jump

Boundary values

Left end A	Right end B
$R_A=0 =$ 0 N	$R_B=0 =$ 0 N
$M_A=0 =$ 0 N$\cdot$mm	$M_B=0 =$ 0 N$\cdot$mm
$\theta_A=-\frac{\Delta_0}{l} =$ -0.001 rad	$\theta_B=-\frac{\Delta_0}{l} =$ -0.001 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	0 N$\cdot$mm at $x=$ 0 mm	0 N$\cdot$mm at $x=$ 0 mm
Slope	-0.001 rad at $x=$ 0 mm	-0.001 rad at $x=$ 0 mm	-0.001 rad at $x=$ 0 mm
Deflection	-0.5 mm at $x=$ 500 mm	0.5 mm at $x=$ 500 mm	0.5 mm at $x=$ 500 mm

• Maximum positive deflection $\Delta_0(l-a)/l$ just right of $x=a$; maximum possible value $\Delta_0$ at $a=0$.
• Maximum negative deflection $-\Delta_0a/l$ just left of $x=a$; maximum possible negative magnitude has signed value $-\Delta_0$ at $a=l$.

5f. 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, $0\leq a\leq l$
$\Delta_0 =$ 1 mm	Imposed upward displacement jump

Boundary values

Left end A	Right end B
$R_A=0 =$ 0 N	$R_B=0 =$ 0 N
$M_A=0 =$ 0 N$\cdot$mm	$M_B=0 =$ 0 N$\cdot$mm
$\theta_A=0 =$ 0 rad	$\theta_B=0 =$ 0 rad
$y_A=-\Delta_0 =$ -1 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	0 N$\cdot$mm at $x=$ 0 mm	0 N$\cdot$mm at $x=$ 0 mm
Slope	0 rad at $x=$ 0 mm	0 rad at $x=$ 0 mm	0 rad at $x=$ 0 mm
Deflection	-1 mm at $x=$ 0 mm	0 mm at $x=$ 500 mm	-1 mm at $x=$ 0 mm

• $y_{\max}=y_A=-\Delta_0$ for $x<a$; $y=0$ for $x>a$.
• Moment and slope are zero away from the imposed displacement.